Research Article Energy-Efficiency Analysis of Per ...downloads.hindawi.com/journals/ijap/2014/313195.pdfIn this paper, per-subcarrier antenna selection MIMO-OFDM systems with linear

Research ArticleEnergy-Efficiency Analysis of Per-Subcarrier Antenna Selectionwith Peak-Power Reduction in MIMO-OFDM Wireless Systems

Ngoc Phuc Le Le Chung Tran and Farzad Safaei

School of Electrical Computer and Telecommunications Engineering University of Wollongong Wollongong NSW 2522 Australia

Correspondence should be addressed to Ngoc Phuc Le pnl750uoweduau

Received 8 January 2014 Accepted 14 May 2014 Published 11 June 2014

Academic Editor Ahmad Safaai-Jazi

Copyright copy 2014 Ngoc Phuc Le et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The use of per-subcarrier antenna subset selection in OFDM wireless systems offers higher system capacity andor improved linkreliabilityHowever the implementation of the conventional per-subcarrier selection schememay result in significant fluctuations ofthe average power and peak power across antennas which affects the potential benefits of the system In this paper power efficiencyof high-power amplifiers and energy efficiency in per-subcarrier antenna selection MIMO-OFDM systems are investigated Todeliver the maximum overall power efficiency we propose a two-step strategy for data-subcarrier allocation This strategy consistsof an equal allocation of data subcarriers based on linear optimization and peak-power reduction via cross-antenna permutationsFor analysis we derive the CCDF (complementary cumulative distribution function) of the power efficiency as well as the analyticalexpressions of the average power efficiency It is proved from the power-efficiency perspective that the proposed allocation schemeoutperforms the conventional scheme We also show that the improvement in the power efficiency translates into an improvedcapacity and in turn increases energy efficiency of the proposed system Simulation results are provided to validate our analyses

1 Introduction

The last few years have seen an increasing demand for veryfast data speeds in wireless multimedia applications Mean-while reducing energy consumption in wireless networkshas become a problem of concern among academic andindustrial communities As a result high-speed systems withhigh-energy efficiency has emerged as a main stream fordesigning future wireless networks [1 2] The potential forimproving energy efficiency in wireless systems could be incomponent level link level or network level [1] From aphysical layer viewpoint it is well known that high-poweramplifier (HPAs) is a major source of RF (radio frequency)power consumption For example in mobile networks HPAsconsume up to 50ndash80 of overall power at a base station [13]Thus increasing power efficiency ofHPAs is of importanceto achieve high energy-efficient wireless networks

To date MIMO-OFDM (multi-input multi-outputorthogonal frequency division multiplexing) has beenconsidered as a key technique for high-speed wirelesstransmission [4] This is mainly because OFDM could

offer high spectral efficiency and robustness againstintersymbol interference in multipath fading channelsAlso an increase in capacity andor diversity gains could beachieved with MIMO [5 6] In fact MIMO-OFDM has beenadopted in current and future standards such as WiMAXWLAN 80211n or LTE-Advanced Among various MIMOschemes antenna selection appears to be promising forOFDM systems This MIMO scheme requires a low-costimplementation and small amount of feedback informationcompared to other precoding or beamforming methods [7]

Many research works have considered the applicationof antenna selection in OFDM systems for example see[8ndash12] To achieve a large coding gain resulting from thefrequency-selective nature of the channels a per-subcarrierantenna selection (ie selecting antennas on each subcarrierbasis) is applied [9] However as thismethod selects antennasindependently on each subcarrier a large number of datasymbols may be allocated to some antennas depending onthe channel condition As a result the peak power andaverage power of the signals on these antennas might bevery large whereas those on the other antennas might be

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 313195 13 pageshttpdxdoiorg1011552014313195

2 International Journal of Antennas and Propagation

small The fluctuation of the powers clearly affects the powerefficiency of HPAs or distorts signals which in turn reducesthe potential benefits of the antenna selectionOFDM systems[13]

One possible approach to deal with the problem ofimbalance allocation of data subcarriers is selecting antennasunder a constraint that the number of data subcarriersallocated to each antenna is equal Some research workshave studied such a constrained selection approach in theliterature for example see [10ndash12] In [10] an ad hoc algo-rithm was developed to realize the constrained selectionscheme Meanwhile the authors in [11] considered linearoptimization to devise the constrained selection scheme Itwas shown that the selection scheme based on optimizationcould offer a better performance than the suboptimal solutionin [10] In [12] we generalized the approach in [11] to thesystem with an arbitrary number of multiplexed data streamsand analyzed the performance directly in nonlinear fadingchannels However all of these works only study the efficacyof the constrained selection scheme from error-performanceperspective Moreover even though the same number of datasubcarriers is allocated to each transmit antenna (ie allantennas have an equal average power) an occurrence of highpeak-to-average power ratio (PAPR) still affects the systemDespite an increasing concern about energy consumption inwireless networks to the best of our knowledge a research onenergy efficiency in the context of antenna selection MIMO-OFDM systems is still missing

It is also essential to emphasize the need of research onimproving energy efficiency in antenna selection MIMO-OFDM systems First per-subcarrier antenna selectionMIMO configurations need multiple active RF (radio fre-quency) chains which immediately raises a concern aboutenergy consumption compared to single RF-chain OFDMsystems Second OFDM inherently suffers from a high PAPRproblem which leads to a poor power efficiency of HPAs[13 14] In per-subcarrier antenna selection OFDM systemsthis effect is intuitively more serious due to a combinationof the high PAPR and power fluctuation resulting from animbalance allocation of data subcarriers This problem alsobecomes crucial in the OFDM systems where linear scaling(ie scale the peak power of the time-domain OFDM signalsto the saturation level of the HPAs [15ndash17]) is implemented torealize OFDM transmissions with no nonlinear distortions

In this paper per-subcarrier antenna selection MIMO-OFDM systems with linear scaling are investigated froman energy perspective for the first time The importantcontributions of this work include (a) the analysis of powerefficiency ofHPAs and energy efficiency and (b) the proposedstrategy to improve these useful performance metrics Themain results are summarized as follows

(1) A two-step data allocation strategy is proposed todeliver a maximum overall power efficiency of HPAsThis strategy consists of an equal allocation of datasubcarriers among transmit antennas based on linearoptimization and a peak-power reduction algorithmvia cross-antenna permutations

(2) Analytical expressions characterizing the achievedpower efficiency of HPAs including the CCDFs(complementary cumulative distribution function)and the average power efficiency are derived It isproved that from the power-efficiency viewpoint theproposed allocation scheme outperforms the conven-tional scheme

(3) The improvements in capacity and energy efficiencyresulting from the improved power efficiency ofHPAsare analyzed

Numerical results are provided to verify the analyses as wellas demonstrate the benefits in terms of the power efficiencyof HPAs and capacity as well as energy efficiency

The remainder of the paper is organized as follows InSection 2 a per-subcarrier antenna subset selection MIMO-OFDM system with linear scaling is described In Section 3a data allocation strategy that could allocate evenly data sub-carriers across antennas with a low peak power is proposedAnalysis of power efficiency is carried out in Section 4 Theachievable capacity and energy efficiency are considered inSection 5Numerical results are provided in Section 6 FinallySection 7 concludes the paper

Notation A bold letter denotes a vector or a matrix whereasan italic letter denotes a variable (sdot)119879 (sdot)119867 119864sdot and det(sdot)denote transpose Hermitian transpose expectation anddeterminant of a matrix respectively otimes denotes the Kro-necker product I

119899indicates the 119899times119899 identity matrix and 1

119870

is a119870times 1 vector of onesR indicates the set of real numbers

2 Antenna Selection for MIMO-OFDMSystems with Linear Scaling

21 SystemModel Weconsider aMIMO-OFDMsystemwith119870 subcarriers 119899

119879transmit antennas and 119899

119877receive antennas

as shown in Figure 1 At the transmitter the input data aredemultiplexed into 119899

119863independent data streams Each data

stream is then mapped onto 119872-QAM (119872-ary quadratureamplitudemodulation) constellations For the 119896th subcarrierwe denote 119906

119896

119897and 119909119896

119894 1 le 119897 le 119899

119863 1 le 119894 le 119899

119879

0 le 119896 le 119870 minus 1 to be the symbols that the subcarrierblock takes at its 119897th input and outputs at its 119894th outputrespectively The allocation block assigns the elements ofu119896

= [1199061198961 1199061198962 119906119896

119899119863]119879to 119899119863selected antennas at the 119896th

subcarrier based on feedback information As a result only119899119863elements in a vector x

119896= [1199091198961 1199091198962 119909119896

119899119879]119879 are assigned

values from u119896 whereas the others are zeros It is assumed

that 119864u119896u119867119896 = 1205902I

119899119863 The output sequences from the

subcarrier allocation block are then fed into 119870-point IFFT(inverse fast Fourier transform) blocks In this paper theNyquist sampling signal is considered Thus the discrete-time baseband OFDM signals can be expressed as

119904119894 (119899) =

1

radic119870

119870minus1

sum119896=0

119909119896

1198941198901198952120587119899119896119870

0 le 119899 le 119870 minus 1 1 le 119894 le 119899119879

(1)

International Journal of Antennas and Propagation 3

Inputdata

Symbolmapping

Subcarrierallocation

IFFTadd GI

Linearscaling

NonlinearHPA

1

nD nT

Dem

ultip

lexe

r

Symbolmapping

IFFTadd GI

NonlinearHPA

Antenna selection index from receiver

1 1

nD

Tx 1

Tx nT

(a)

middot middot middot

ChannelH

Remove GIFFT

Rece

iver

proc

essin

g Estimateddata

Channel estimation andantenna selection decision

Remove GIFFT

To transmitter

Rx 1

Rx nR

(b)

Figure 1 A simplified block diagram of a MIMO-OFDM system with per-subcarrier transmit antenna selection

For simplicity we consider ideal predistortion HPAs(ie soft envelope limiters) with a unity gain and class-Aoperation To deliver the maximum power efficiency withno nonlinear distortions in the system with nonlinear HPAsthe peak power across transmit antennas is linearly scaledto the saturation level 119875sat of the HPAs In addition asfeedback information (ie the selected antenna indiceswhichare calculated based on the channel state information) isdeployed by the transmitter all transmit branches are scaledwith the same scaling factor [16] Thus the signal after linearscaling can be expressed as

119904119894 (119899) = radic120572119904

119894 (119899) (2)

where the scaling factor 120572 = 119875sat119875 and the peak poweracross antennas 119875 = max119904

119894(119899)|119899 = 0 119870 minus 1 119894 =

1 119899119879 Each time-domain OFDM signal is then amplified

by the HPA before being transmitted via its correspondingtransmit antenna

At the receiver the received signal at each antenna is fedinto the FFT block after the GI (guard interval) is removedThe systemmodel in the frequency domain corresponding tothe 119896th subcarrier can be expressed as

y119896= radic120572H

119896x119896+ n119896

= radic120572H119896u119896+ n119896

(3)

where

x119896= [119909119896

1119909119896

2sdot sdot sdot 119909119896

119899119879]119879

H119896=

[[[[

[

ℎ119896

11ℎ11989612

sdot sdot sdot ℎ1198961119899119879

ℎ11989621

ℎ11989622

sdot sdot sdot ℎ1198962119899119879

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot

ℎ1198961198991198771

ℎ1198961198991198772

sdot sdot sdot ℎ119896119899119877119899119879

]]]]

]

y119896= [119910119896

11199101198962

sdot sdot sdot 119910119896119899119877]119879

n119896= [119899119896

11198991198962

sdot sdot sdot 119899119896119899119877]119879

(4)

In the above equations ℎ119896119895119894indicates the channel coefficient

between the 119894th transmit antenna and the 119895th receive antennaThe effective channel matrix H

119896is obtained by eliminating

the columns of H119896corresponding to the unselected transmit

Table 1 Antenna subsets (119899119879= 4 119899

119863= 2 and Γ = 6)

120574 Γ120574

1 1 2

2 1 3

3 1 4

4 2 3

5 2 4

6 3 4

antennas 119910119896119895and 119899119896119895denote the received signal and the noise

at the 119895th receive antenna respectively Here the noise ismodeled as a Gaussian random variable with zero mean and119864n119896n119867119896 = 1205902119899I119899119877 We assume that the receiver can perfectly

estimate the channel coefficients for example using a block-type pilot arrangement for channel estimation [18]

22 Per-Subcarrier Antenna Subset Selection In a MIMO-OFDM system with the conventional per-subcarrier antennasubset selection antenna subsets are selected independentlyfor each subcarrier [9] On each subcarrier only 119899

119863antennas

out of 119899119879available transmit antennas are active Denote

Γ120574 120574 = 1 2 Γ to be the 120574th subset consisting of 119899

119863

selected transmit antennas where Γ = (119899119863119899119879

) = 119899119879119899119863(119899119879minus

119899119863) is the number of all possible 119899

119863-element subsets Each

subset consists of 119899119863transmit antenna indices that are chosen

based on the feedback information from the receiver Forexample when 119899

119879= 4 and 119899

119863= 2 then Γ = 6 and all possible

subsets Γ120574 120574 = 1 2 6 are defined in Table 1 The choice

of the best antenna subset depends on a particular selectioncriterion

Several antenna selection criteria such as maximumcapacity or maximum SNR (signal-to-noise ratio) [19] canbe extended to this system In this paper we consider thecapacity criterion Accordingly the optimal subset at the119896th subcarrier is determined by maximizing the mutualinformation of the 119896th subchannel that is

Γ120574lowast (119896) = arg max

Γ120574 120574=1Γ

119868119896

120574 (5)

where

119868119896

120574= log2(det(I

119899119877+

120588

119899119863

H119896H119867119896)) (6)


is the instantaneous mutual information associated with the119896th subchannel [5] Here 120588 = 119875

1199051205902119899

= 12057211989911986312059021205902119899 and

119875119905= 1205721198991198631205902 is the total transmit power per subchannel Also

we have assumed in (6) that the transmit power is allocateduniformly across antennas This is due to the fact that thefeedback information in our system is only the selectedantenna indices (ie not sufficient enough to perform powerallocation algorithms across subcarriers as well as antennas)The average mutual information across subcarriers can nowbe expressed as

119868 (120588H) =1

119870

119870minus1

sum119896=0

119868119896

120574lowast (7)

3 A Proposed Strategy forPeak-Power Reduction

In Section 2 we have described the MIMO-OFDM systemwith the conventional selection scheme It can be noted thatthe number of data subcarriers assigned to each transmitantenna might be significantly different depending on thechannel condition In the systemwith identical linear scalingto achieve the maximal overall power efficiency of HPAs thepeak power across antennas should be as small as possibleWe note that the peak power of the signal on each branchdepends on both the transmitted constellation symbols andthe number of data subcarriers allocated in each OFDMsymbol (cf (1)) Thus it is not sufficient to reduce the peakpower by solely implementing PAPR reduction techniquesMore specifically PAPR reduction techniques themselvescannot solve the problem of imbalance allocation of datasubcarriers across antennas To reduce the peak power acrossantennas we propose a two-step strategy consisting of thefollowing

Step 1 Allocate the same number of data subcarriers to alltransmit antennas (ie selecting antennas under a constraintthat all antennas have the same number of data subcarriersas illustrated in Figure 2(b)) Once this is achieved thetime-domain signals on all transmit branches have the sameaverage power Moreover as we will mathematically prove inSection 41 the peak power across antennas is reduced

Step 2 Reallocate data symbols across antennasThis processwill alter the statistical distribution of signals and thus furtherreduce the peak power

We note that this paper does not focus on developingoriginal techniques for either equal allocation of data subcar-riers or peak-power reduction Instead we are interested inanalyzing power efficiency of HPAs and energy efficiency inper-subcarrier antenna selection OFDM systems which hasnot been considered so far The two steps in our proposedstrategy described in Sections 31 and 32 are accomplishedby extending the suitable approaches available in the litera-ture to the context of the considered system

31 Optimal Equal Allocation of Data Subcarriers The opti-mal constrained antenna selection scheme based on linearoptimization was considered in [11 12] to improve error per-formance of OFDM systems suffering nonlinear distortionsdue to HPAs We now consider this method for the first stepof our strategy to achieve a better power delivery in a linearlyscaled MIMO-OFDM system Specifically we define a vari-able 119911

119896

120574 where 119911119896

120574= 1 if Γ

120574is chosen for the 119896th subcarrier

and 119911119896120574= 0 otherwise Also denote 119888119896

120574to be the cost associated

with the chosen subset Γ120574 Here 119888119896

120574= 119868119896120574as the maximum

capacity criterion is considered By denoting vectors z =

(11991101 1199110

Γ 11991111 1199111

Γ 119911119870minus1

1 119911119870minus1

Γ)119879

isin 0 1119870Γtimes1 and

c = (11988801 1198880

Γ 11988811 1198881

Γ 119888119870minus1

1 119888119870minus1

Γ)119879

isin R119870Γtimes1 anoptimal solution for an equal allocation of data subcarriersis obtained by solving the following linear optimizationproblem [11 12]

maxzisin01119870Γtimes1

c119879z

subject to Az = a(8)

where A1= I119870otimes 1119879Γisin 0 1

119870times119870Γ A2= 1119879119870otimes IΓisin 0 1

Γtimes119870ΓA = (A119879

1A1198792)119879

isin 0 1(119870+Γ)times119870Γ a = (1119879

119870120582119879)119879 and 120582 =

(1205821 1205822 120582

Γ)119879 where 120582

120574is the number of times that the

subset Γ120574is selected The constraint in (8) means that only

119899119863antennas are allowed to transmit data symbols on each

subcarrier and all transmit antennas have the same numberof allocated data subcarriers It is important to note that thisbinary linear optimization problem can be relaxed to linearprogramming with integral solutions [12] Hence (8) can besolved efficiently by the known linear programmingmethodssuch as the simplex methods or interior point methods [20]

32 Data Allocation with Peak-Power Reduction To furtherreduce the peak power of the whole system various availablePAPR reduction techniques (eg see [14] and the referencestherein) can be now adopted In this paper we are interestedin a selected mapping (SLM) technique [21] as SLM is adistortionless PAPR technique that could achieve a goodPAPR reduction [14 21] One of the most important steps inSLM is creating a set of candidates that represents the samedata information To exploit the available degrees of freedomin multiple-antenna systems for peak-power reduction wepropose to create a set of candidates using cross-antennapermutations In the literature a SLM-based scheme thatcould exploit the available degrees of freedom was firstdeveloped in [22] The scheme in [22] creates candidatesby performing cross-antenna rotation and inversion (CARI)based on a defined random matrix However that scheme isproposed for an Alamouti code basedMIMO-OFDM systemonly In a per-subcarrier antenna selection OFDM systemCARI cannot be implemented directly as only 119899

119863out of 119899

119879

antennas are active on each subcarrier To create candidates inour scheme we perform cross-antenna permutations insteadof CARI In addition we utilize an antenna allocation patternthat is already known by the transmitter and receiver insteadof storing a defined random matrix at both transmitter and


Antennas

Subcarriers

Tx 1

Tx 2

Tx 3

Tx 4

Data

Not used

(a) Unbalanced-power selection

Data

Not used

Antennas

Subcarriers

Tx 1

Tx 2

Tx 3

Tx 4

(b) Balanced-power selection

Figure 2 Illustration of per-subcarrier antenna subset selection (119899119879= 4 119899

119863= 2 and 119870 = 12)

Ant

enna

s

Subcarriers

1st candidate(original)

x11 x

21

x12

x32

x33 x

43

x24 x

44

0 0

0

00

0 0

0

(a)

2nd candidate (permutedata on the 1st antenna)

Subcarriers

x12 x

24 0 0

00

00

0 0

x11 x

32

x33 x

43

x44

x21

(b)

3rd candidate (permutedata on the 2nd antenna)

Subcarriers

Ant

enna

s

x12

x21 0 0

00

00

0 0

x11 x

33

x32 x

43

x44

x24

(c)

4th candidate (permutedata on the 3rd antenna)

Subcarriers

0 0

00

00

0 0

x11 x

21

x12 x

33

x32 x

44

x43x

24

(d)

Figure 3 Illustration of cross-antenna permutations (119899119879= 4 119899

119863= 2 and 119870 = 4)

receiver as in [22] The proposed algorithm is described asfollows

(1) Create 119882 candidates by performing cross-antennaspermutations An illustration of this process in thesystem with 119899

119863= 2 119899

119879= 4 and 119870 = 4 is

shown in Figure 3 Accordingly the first candidateis the original data allocation The second candidateis obtained by permuting all symbols on the firstantennas with their associated symbols on the otherantennasThe third and fourth candidates are createdin a similar manner To obtain a larger number ofcandidates all symbols on the antenna that are goingbe permutated need first have their phase rotated (iebeing multiplied with an element of a phase set ega 4-phase set is 0 1205872 120587 31205872)

(2) Calculate the peak powers of all available candidates(3) Select the candidate with the minimum peak power

for transmission

To recover the transmitted data the transmitter needs toinform the receiver which candidates have been selectedThus the number of side information bits in this scheme islog2119882 which is similar to that in [22]

33 Complexity Considerations In this subsection the com-plexity of the proposed allocation scheme is compared tothat of the conventional (imbalance) allocation scheme Inthe first step of the proposed scheme to realize an equalallocation of data subcarriers the optimization problem in (8)

needs to be solved at the receiverWenote that this linear opti-mization problem can be solved in polynomial time Morespecifically the complexity to solve this problem using inte-rior point methods can be reduced to 119874([(119870Γ)

3 ln(119870Γ)]120585)

where119874(sdot) denotes an order of complexity and 120585 is the bit sizeof the optimization problem [23] In addition it is noted thatthis step is transparent to the transmitter (ie no additionalcomplexity is required at the transmitter) In the secondstep a major additional complexity lies in the required IFFToperations due to additional 119882 candidates As an 119870 point-IFFT requires 119870log

2119870 complex additions and (1198702)log

2119870

complex multiplications the numbers of complex additionsand complex multiplications in the conventional scheme are119899119879119870log2119870 and 119899

119879(1198702)log

2119870 respectivelyMeanwhile in the

proposed scheme with 119882 candidates 119882119899119879119870log2119870 complex

additions and 119882119899119879(1198702)log

2119870 complex multiplications are

required However as we will show analytically in Section 4and numerically in Section 6 an improvement in the peak-power reduction reduces when 119882 becomes large Thus asmall value of 119882 is generally chosen which does not incurmuch additional complexity Finally the amount of feedbackinformation in the proposed scheme is similar to that in theconventional scheme

4 Analysis of Power Efficiency

41 Statistical Distribution of Peak Power Before proceedingto analyze the power efficiency of HPAs we need to investi-gate the distribution of the peak power of the MIMO-OFDMsignals (ie the peak power across transmit antennas) We


consider the complementary cumulative distribution func-tion (CCDF) of the peak power defined as the probability thatthe peak power 119875 exceeds a given threshold 119875

0 that is

CCDF119875 (1198750) = Pr (119875 gt 119875

0) (9)

Note that although a procedure for calculating CCDF ofPAPR in OFDM systems is known to the best of ourknowledge all the CCDF expressions with respect toMIMO-OFDM signals available in the literature assume that all datasubcarriers are active which can be considered as a specialcase in the considered system when all the transmit antennashave the same number of allocated data symbols In thefollowing we calculate the CCDF of the peak power in oursystem

Let us begin with the discrete-time OFDM signal 119904119894(119899)

119899 = 0 1 119870minus1 corresponding to the 119894th transmit antennaThe peak power of this signal is defined as

119875119894= max0le119899le119870minus1

|119904119894 (119899) |2 (10)

For analytical tractability we assume that both the real partand imaginary part of the signal 119904

119894(119899) are asymptotically

independent and identically distributed Gaussian randomvariables Note that this assumption which is based on thecentral limit theorem [24] only holds when the number ofassigned data subcarriers on the 119894th antenna denoted as119870119894 is large enough As a result |119904

119894(119899)| follows the Rayleigh

distribution and |119904119894(119899)|2 has a chi-square distribution with

two degrees of freedom The probability density function ofthe signal |119904

119894(119899)|2 can be expressed as [24]

119901|119904|2 (

100381610038161003816100381611990411989410038161003816100381610038162) =

1

1205902119870119894

119890minus|119904119894|21205902

119870119894 (11)

where 1205902119870119894

= 1205902119870119894119870 is the variance of the signal |119904

119894(119899)| Note

that sum119899119879

119894=1119870119894

= 119899119863119870 thus we have sum

119899119879

119894=11205902119870119894

= 1198991198631205902 The

CDF (cumulative distribution function) of the signal |119904119894(119899)|2

is given as

Pr (100381610038161003816100381611990411989410038161003816100381610038162le 120579) = 1 minus 119890

minus1205791205902

119870119894 120579 ge 0 (12)

Suppose that 119870 samples of |119904119894(119899)| 119899 = 0 1 119870 minus 1 are

independent the CDF of the peak power 119875119894can be expressed

as

CDF119875119894 = Pr (119875119894le 1198750)

= Pr (1003816100381610038161003816119904119894 (0)10038161003816100381610038162le 1198750)Pr (1003816100381610038161003816119904119894 (1)

10038161003816100381610038162le 1198750) sdot sdot sdot

Pr (1003816100381610038161003816119904119894 (119870 minus 1)10038161003816100381610038162le 1198750)

= (1 minus 119890minus1198750120590

2

119870119894 )119870

(13)

In MIMO-OFDM systems with linear scaling the peakpower across transmit antennas 119875 can be defined as

119875 = max1le119894le119899119879

119875119894 (14)

Given the statistical independence of data among transmitantennas which is the case in the considered spatial mul-tiplexed OFDM system the CDF of the peak power 119875 iscalculated as

CDF119875 = Pr (119875 le 1198750)

= Pr (1198751le 1198750)Pr (119875

2le 1198750) sdot sdot sdotPr (119875

119899119879le 1198750)

=

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870

(15)

Therefore the CCDF of the peak power of the antennaselection MIMO-OFDM signals can be expressed as

CCDF119875imbalance (1198750) = 1 minus CDF119875

= 1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870

(16)

In the MIMO-OFDM system with a power balancingconstraint the number of allocated data subcarriers pertransmit antenna is equal to one another (ie 119870

119894=

119899119863119870119899119879

= 119870 forall119894 = 1 2 119899119879) Thus the variances of the

signals are 1205902119870119894

= 1198991198631205902119899119879= 1205902119870 forall119894 = 1 2 119899

119879 As a result

the CCDF expression could be simplified to as

CCDF119875balance (1198750) = 1 minus (1 minus 119890minus1198750120590

2

119870)119899119879119870

(17)

A comparison of the CCDF of the peak powers in the twosystems is presented in the following theorem

Theorem 1 In MIMO-OFDM transmission schemes that con-sist of inactive data subcarriers (eg per-subcarrier antennaselection) the probability of occurrences of high peak power isthe smallest when the same number of data symbols is allocatedto all transmit antennas that is

CCDF119875balance (1198750) le CCDF119875imbalance (1198750) (18)

Proof The proof is given in Appendix AWhen the peak-power reduction algorithm proposed in

Section 32 is implemented in theMIMO-OFDMsystemwitha power-balancing constraint the CCDF of the peak powercan be expressed as

CCDF119875balance+reduced (1198750) = (CCDF119875balance (1198750))119882

= (1 minus (1 minus 119890minus1198750120590

2

119870)119899119879119870)119882

(19)

where 119882 is the number of candidates that are assumed tobe independent Recall that by definition the CCDF value isalways smaller than one (cf (9)) Therefore the CCDF valuein (19) is smaller than that in (17) that is

CCDF119875balance+reduced (1198750) le CCDF119875balance (1198750) (20)


42 Power Efficiency of HPAs We now analyze the powerefficiency (PE) of high-power amplifiers (HPAs) The drainefficiency of HPAs which is defined as a ratio between thepower drawn from the DC source 119875dc and the average outputpower 119875out is considered in this paper Denote 119875119894in and 119875119894out tobe the average input and output powers of the HPA for the119894th antenna respectively Recall that all HPAs are assumedto have a unity gain that is 119875119894out = 119875119894in forall119894 = 1 2 119899

119879

Hence the instantaneous overall power efficiency of HPAs inthe MIMO-OFDM system can be expressed as [16]

120578PE

=sum119899119879

119894=1119875119894out

119899119879119875dc

=sum119899119879

119894=1119875119894in

119899119879119875dc

= 1205721198991198631205902

119899119879119875dc

=1198991198631205902119875sat

119899119879119875dc

1

119875=

1198991198631205902

2119899119879

1

119875

(21)

In the above manipulations we have used the fact thatsum119899119879

119894=1119875119894in = 120572119899

1198631205902 and119875dc = 2119875sat for class-AHPAs regardless

of the average powers of the input time-domain signalsDenoting CCDFPE (120578PE

0) = Pr(120578PE gt 120578PE

0) to be the CCDF

of the power efficiency we obtain the following result withrespect to 120578

PE

Theorem 2 In per-subcarrier antenna selection MIMO-OFDM systems with linear scaling the probability of achievinghigh instantaneous overall power efficiency of HPAs is thelargest when all transmit antennas have the same number ofallocated data symbols that is

CCDFPEbalance (120578PE0) ge CCDFPEimbalance (120578

PE0) (22)

where

CCDFPEbalance (120578PE0) = (1 minus 119890

minus12120578PE0 )119899119879119870

CCDFPEimbalance (120578PE0) =

119899119879

prod119894=1

(1 minus 119890minus1198991198631198702119899119879119870119894120578

PE0 )119870

(23)

Proof From (18) and (21) it is readily to obtain (22) and (23)With respect to the average value of the overall power

efficiency from (21) we can express

120578PE

= 119864 120578PE =

1198991198631205902

2119899119879

1198641

119875 =

1198991198631205902

2119899119879

int1

119909119901 (119909) 119889119909 (24)

where 119901(119909) is the pdf (probability distribution function) ofthe peak power In the system with a balance constraint (ieonly use Step 1) the pdf of the peak power can be calculatedas

119901balance (119909) =119889

119889119909CDF119875balance (119909) =

119889

119889119909(1 minus 119890

minus1199091205902

119870)119899119879119870

=119899119879119870

1205902119870

119890minus1199091205902

119870(1 minus 119890minus1199091205902

119870)119899119879119870minus1

(25)

Hence

120578PEbalance =

1198991198631205902

2119899119879

int1

119909119901balance (119909) 119889119909

=119899119879119870

2int1198701205902

119870

1205902

119870

1

119909119890minus1199091205902

119870(1 minus 119890minus1199091205902

119870)119899119879119870minus1

119889119909

=119899119879119870

2int119870

1

1

119909119890minus119909

(1 minus 119890minus119909

)119899119879119870minus1

119889119909

(26)

Similarly the power efficiency of HPAs in the conventionalsystem is

120578PEimbalance =

1198991198631205902

2119899119879

int1

119909119901imbalance (119909) 119889119909

=1198991198631198701205902

2119899119879

int119870max120590

2

max

1205902

max

1

119909

119899119879

sum119895=1

119890minus1199091205902

119870119895

(1 minus 119890minus1199091205902

119870119895 ) 1205902119870119895

times

119899119879

prod119894=1

(1 minus 119890minus1199091205902

119870119894 )119870

119889119909

(27)

where 1205902max = max12059021198701

1205902119870119899119879

and 119870max = max1198701

119870119899119879When the peak-power reduction algorithm is also imple-

mented (ie the system employs both Steps 1 and 2) it isreadily from (20) and (21) that

CCDFPEbalance+reduced (120578PE0

) ge CCDFPEbalance (120578PE0

) (28)

where


) = 1 minus (1 minus (1 minus 119890minus12120578

PE0 )119899119879119870

)119882

(29)

Also the average power efficiency can now be calculated as

120578PEbalance+reduced =

1198991198631205902

2119899119879

int1

119909119901balance+reduced (119909) 119889119909

=119899119879119870119882

2int119870

1

1

119909119890minus119909

(1 minus 119890minus119909

)119899119879119870minus1

times (1 minus (1 minus 119890minus119909

)119899119879119870

)119882minus1

119889119909

(30)

It can be seen from (30) that the smaller the peak poweris reduced (ie the larger the number of candidates 119882 isused) the higher the power efficiency could be achievedNote that although the integrals in (26) (27) and (30) haveno closed-form solutions they can be evaluated numericallyAlso the average power efficiencies in (26) (27) and (30) aretaken with respect to the input data In other words they areconsidered as instantaneous power efficiencieswith respect tothe channel distribution Consequently the power efficiencyin the systems is obtained by averaging these values over thefading channel distribution


5 Analyses of Capacity and Energy Efficiency

It has been shown in (22) and (28) that the proposed systemcould achieve a better power efficiency of HPAs than itscounterpart Thus when the power 119875dc is fixed it is intuitivethat an increased average power efficiency results in anincreased average transmit power and in turn leads to anincrease in the achievable rate Moreover an increase in thedata rate under a constant consumption power will translateinto an improvement in energy efficiency The achievedcapacity and energy efficiency are now investigated in thissection

51 System Capacity We begin by rewriting the mutualinformation in (7) with respect to the average SNR valueof 120588 = 119875

1199051205902119899 where 119875

119905= 120572119899

1198631205902 = 120578

PE119899119879119875dc (cf (21))

The ergodic capacity is then calculated by averaging themutual information over the fading channel distributionthat is 119862(120588) = 119864H119868(120588H) From (6) and (7) the mutualinformation in the proposed and conventional systems canbe expressed respectively as

119868 (120588proposedH)

=1

119870

119870minus1

sum119896=0

log2(det(I

119899119877+

120588proposed

119899119863

H119896H119867119896))

(31)

119868 (120588imbalanceH)

=1

119870

119870minus1

sum119896=0

log2(det(I

119899119877+

120588imbalance119899119863

H119896H119867119896))

(32)

where 120588proposed = 120578PEproposed119899119879119875dc120590

2

119899and 120588imbalance =

120578PEimbalance119899119879119875dc120590

2

119899 Here 120578

PEproposed = 120578

PEbalance+reduced if the

peak-power reduction algorithm is implemented otherwise120578PEproposed = 120578

PEbalance Also H119896 in (31) denotes the effective

channel matrix on the 119896th subcarrier in the proposed systemwhich is obtainedwhen solving the problem in (8)This chan-nel matrix is generally different from the effective channelmatrix H

119896in the conventional system because the selected

antenna subset may be differentThe difference in the mutualinformation between the two systems can be now calculatedas

Δ119868 = 119868 (120588proposedH) minus 119868 (120588imbalanceH) =1

119870

119870minus1

sum119896=0

Δ119868119896 (33)

where

Δ119868119896= log2(det(I

119899119877+

120588proposed

119899119863

H119896H119867119896))

minus log2(det(I

119899119877+

120588imbalance119899119863

H119896H119867119896))

(34)

For analytical simplicity we focus on the high-SNRregime At the high SNR the mutual information at the 119896thsubcarrier can be approximated as [25]

119868119896= log2(det(

120588

119899119863

Ω119896)) (35)

where

Ω119896= Ω (H

119896) =

H119896H119867119896 119899119877le 119899119863

H119867119896H119896 119899119877gt 119899119863

(36)

Thus the difference in the mutual information can berewritten as

Δ119868 =1

119870

119870minus1

sum119896=0

(log2(det(

120588proposed

119899119863

Ω119896))

minuslog2(det(

120588imbalance119899119863

Ω119896)))

= 119901log2

120578PEproposed

120578PEimbalance

minus1

119870

119870minus1

sum119896=0

Δ119896= 1198791+ 1198792

(37)

where 119901 = min(119899119863 119899119877)Ω119896= Ω(H

119896) and

Δ119896= log2(det (Ω

119896)) minus log

2(det (Ω

119896)) (38)

is the loss in the mutual information associated with the 119896thsubcarrier due to the constrained allocation Note that if bothsystems have the same selected antenna subset at the 119896thsubcarrier then Δ

119896= 0 otherwise Δ

119896gt 0Thus the total loss

in the mutual information Δ = (1119870)sum119870minus1

119896=0Δ119896gt 0

We have some important observations with respect to thevalue of Δ119868 in (37)

(i)The change in mutual information Δ119868 comes fromtwo sources The first source 119879

1is a benefit in mutual

information due to the improvement in the powerefficiency of HPAs The second source 119879

2is a penalty

that incurs because the chosen effective channelmatrices in the proposed system are different from theones in the conventional system(ii) For each channel realization thematrixH

119896is fixed

and the first term 1198791in (37) is a constant Thus the

value of Δ119868 depends on how the effective channelmatrix H

119896is selected in the constrained selection

scheme From this observation it is clear that tomake the value Δ119868 become as positive as possiblethe constrained selection method should result inthe cost penalty Δ as small as possible We note thatthe formulated optimization in (8) could achieve theminimum possible value of the total cost Hence it isexpected that the constrained selection scheme basedon linear optimization will guarantee the maximumachievable value of Δ119868 In addition to have an insightinto the cost penalty we derive the upper bound ofthe expected value of the cost penalty in Appendix BBased on the obtained bound it is observed that forfixed values of 119899

119879and 119899

119863 the cost penalty becomes

smaller with an increasing value of 119902 = max(119899119863 119899119877)


(iii) As Δ gt 0 the upper bound of the capacityimprovement can be given as

Δ119862 = 119864H Δ119868 le 119901log2119864H

120578PEproposed

120578PEimbalance

(39)

In (39) we have used Jensenrsquos inequality of119864log(119909) le log(119864119909) as log(119909) is a concavefunction Based on this bound we could estimatethe maximum improvement in capacity that couldbe realized in the proposed system compared to itscounterpart

It is now necessary to evaluate the change in capacitythat is Δ119862 We note that although the distribution of themutual information at high SNRs can be well approximatedby a Gaussian distribution [25] it is still challenging toperform a mathematical evaluation of Δ119862 from a statisticalviewpoint This is mainly due to the fact that the two termsin (37) are complicated dependent random variables Thuswe perform a numerical evaluation of Δ119862 instead Figure 4plots the empirical CCDF of 119879

1 CDF of 119879

2 and CCDF

of Δ119868 In the figure ldquo119882 = 1rdquo stands for the case in whichonly Step 1 in Section 31 is implemented The results areobtained in the systems with 119899

119879= 4 119899

119863= 2 119899

119877= 2 119870= 128

and are averaged over 103 channel realizations Details aboutother simulation parameters are described in Section 6 Thenumerical results confirm that 119879

1gt 0 and 119879

2lt 0 Moreover

as shown in Figure 4(c) Δ119868 is always positive when the peak-power reduction algorithm is implemented (ie 119882= 4 and119882 = 8) For the case of 119882 = 1 the probability of ΔI beingpositive is significant Therefore the proposed system attainsa better ergodic capacity than that in the conventional systemThe achieved capacities in the considered systems will beprovided in Section 6

52 Energy Efficiency In this subsection we further exam-ine the efficacy of the proposed system from an energy-efficiency (EE) perspective Normalized energy efficiency(bitsHzJoule) in MIMO-OFDM systems can be defined as[3 26]

120578EE

=119862 (120588)

119875total (40)

where 119862(120588) is the achievable rate and the total powerconsumption per-subchannel is 119875total = 119899

119879119875dc + 119899

119879119875RF + 119875sp

where 119875RF is the RF power consumption in each transmitbranch excluding the associatedHPA and119875sp is the basebandprocessing power consumption It can be seen from (40)that given a fixed value of 119875total a comparison of energyefficiency achieved in the two systems is based on the capacitycomparison that has been analyzed in Section 51We are nowinterested in evaluating a useful metric of energy efficiency-spectral efficiency performance Recall that the average SNR120588 is given as 120588 = 119875

1199051205902119899= 120578

PE119899119879119875dc1205902

119899 Thus we can rewrite

the energy efficiency in (40) as a function of 119875dc as

120578EE

(119875dc) =119862 (120578

PE119899119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

(41)

Table 2 Simulation parameters

Parameter ValueBandwidth 528MHzFFT size 128Number of samples in zero-padded suffix 37Modulation scheme 4-QAMIEEE 802153a channel model CM1

The energy efficiency of the proposed and conventionalsystems can now be respectively expressed as

120578EEproposed (119875dc) =

119862proposed (120578PEproposed 119899

119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEproposed 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(42)

120578EEimbalance (119875dc) =

119862imbalance (120578PEimbalance 119899

119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEimbalance 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(43)

Similarly to the case of capacity we compare the energyefficiency achieved in the two systems by means of numericalresults in the next section Note that the calculation of energyefficiency in the proposed system (ie (42)) has assumed thata reduction in spectral efficiency due to the side informationas well as additional processing power required for the peak-power reduction algorithm is negligible In fact a reductionin spectral efficiency is very small For example in a systemwith 16-QAM FFT size of 128 119899

119863= 2 and 119882= 4 (ie 2 bits

are needed for side information) a spectral efficiency lossis 2 bits(128 times 4 times 2 + 2) bits = 019 Also it was shownin [27] that the additional power cost when implementingSLM schemes is minuscule Thus the proposed peak-powerreduction algorithm in Section 32 which is a SLM-basedscheme requires a small additional power cost

6 Numerical Results

In this section we provide numerical results to validatethe analyses mentioned in the previous sections as wellas demonstrate the effectiveness of the proposed allocationscheme over its counterpart A MIMO-OFDM system with119899119879= 4 119899

119863= 2 and 119899

119877= 2 is considered in our simulations

The system parameters are listed in Table 2These parametersare chosen based on the legacy WiMedia MB-OFDM UWB(Multiband-OFDM UWB) standard [28] Also the channelCM1 defined in the IEEE 802153a channel model [29] isbased on ameasurement of a line-of-sight scenario where thedistance between the transmitter and the receiver is up to 4mMoreover the multipath gains are modeled as independentlog normally distributed random variables We assume thatperfect channel state information is available at the receiverAlso the feedback link has no delay and is error-free


Empirical CCDF of T110

0

10minus1

10minus2

minus2 0 2

W = 1

W = 4

W = 8

(a)

100

10minus1

10minus2

minus04 minus02 0

Empirical CDF of T2

(b)

Empirical CCDF of (T1 + T2)10

0

10minus1

10minus2

minus1 minus05 0 05 1 15 2

W = 1

W = 4

W = 8

(c)

Figure 4 Statistical distributions (note 1198792is independent of119882)

61 Evaluations of Peak-Power Distribution In Figure 5 weplot the CCDFs of the peak power of time-domain signalsThe analytical curves are based on (16) (17) and (19) Mean-while the simulation curves are empirical CCDF values Thesimulation result confirms that a system with the proposedallocation scheme offers a better CCDF performance than itscounterpart As expected the occurrence of high peak poweris significantly reduced when the peak-power reductionalgorithm is implemented Also it can be seen that theimprovement associated with this algorithm is reduced withincreasing 119882 In other words a very large value of 119882 whilerequiring higher complexity in terms on the number of IFFToperations results in a marginal improvement Thus it isreasonable to choose a relatively small value for 119882 (eg119882 = 4) It is also worth noting that the analytical curvesare relatively close to the simulation curves The small gapsexist due to the fact that the assumption of independentsamples |119904

119894(119899)| to obtain (13) does not strictly hold as we have

sum119870minus1

119899=0|119904119894(119899)|2 = 1205902119870

119894by Parsevalrsquos relation [24]

62 Evaluations of Power Efficiency of HPAs Figure 6 com-pares the CCDFs of the power efficiency achieved in theproposed and conventional systems It can be seen that theprobability of power efficiency being large highly likely occursin the proposed system compared to its counterpart Also

the simulation results agree well with the analytical resultsderived in (23) and (29) In Table 3 we compare the averagepower efficiencies Here the analytical values are obtainedaccording to (26) (27) and (30) Meanwhile the simulationvalues are empirical values based on the original definitionof the drain efficiency in (21) Also these values are averagedover the fading channel realizations It can be seen that thederived expressions approximate well the achieved powerefficiencies Table 3 also provides relative improvements ofthe power efficiencies achieved in the proposed system overthe conventional system Here only 120578

PE (Simulation) valuesare used for calculating these improvements It is clear thatthe proposed system could achieve a significant improvementin terms of average power efficiency

63 Evaluations of Capacity and Energy Efficiency Figure 7shows the system capacity inMbps (ie the normalized valuein (6) is scaled upwith the system bandwidth) versus the SNRvalue of 12059021205902

119899 It is clear that a systemwith the proposed allo-

cation scheme achieves a higher capacity than its counterpartThis agrees with the analysis in Section 51 that the changein capacity Δ119862 is positive In Figure 8 we plot the energyefficiency (MbitsJoule) versus spectral efficiency (MbpsHz)This figure is obtained based on (42) and (43) when varying119875dc Other parameters are 119875RF = 100mW 119875sp = 10mW and


Table 3 A comparison of average power efficiencies

Imbalance Proposed (119882 = 1) Proposed (119882 = 4) Proposed (119882 = 8)120578PE (Simulation) 00697 00768 00894 00942

120578PE (Analysis) 00691 00757 00892 00943

Improvement120578PEproposed minus 120578

PEimbalance

120578PEimbalance

mdash 1019 2826 3515

1 2 3 4 5 6 7 8 9

100

10minus1

10minus2

10minus3

Peak power P0

Imbalance (analysis)Imbalance (simulation)Proposed W = 1 (analysis)Proposed W = 1 (simulation)

Proposed W = 4 (analysis)Proposed W = 4 (simulation)Proposed W = 8 (analysis)Proposed W = 8 (simulation)

Pr (P

gtP0)

Figure 5 Comparison of CCDFs of the peak powers

002 004 006 008 01 012 014Power efficiency

CCD

F

100

10minus1

10minus2

10minus3



Figure 6 Comparison of CCDFs of the power efficiencies

4500

4000

3500

3000

2500

2000

1500

1000

Ergo

dic c

apac

ity (M

bps)

2 4 6 8 10 12 14

SNR (dB)

3300

3200

3100

3000

2900

2800

97 98 99 10 101

ImbalanceProposed W = 1

Proposed W = 4

Proposed W = 8

Figure 7 Comparison of the ergodic capacities

12059021205902119899= 15 dB As expected the improvement in the power

efficiency of HPAs results in an improved energy efficiencyIn addition it can be observed that there exists an energyefficiency-spectral efficiency tradeoff in the systems Thistradeoff clearly needs to be taken into consideration whendesigning energy-efficient per-subcarrier antenna selectionbased OFDM wireless systems

7 Conclusions

In this paper a per-subcarrier antenna subset selectionMIMO-OFDM system with linear scaling has been investi-gated from an energy-efficiency perspective We have shownthat an imbalance allocation of data subcarriers associatedwith the conventional selection scheme affects the power effi-ciency of HPAs as well as the energy-efficiency of the wholesystem To deliver the maximum overall power efficiencywe have proposed the two-step strategy consisting of equalallocation data subcarriers across antennas and peak-powerreduction It has been proved from the power-efficiency view-point that the proposed allocation scheme outperforms theconventional scheme We have also derived the expressionsfor measuring the average power efficiency Moreover theimprovements in terms of capacity and energy efficiency


0 2 4 6 8 10400

600

800

1000

1200

1400

1600

Spectral efficiency (MbpsHz)

Ener

gy effi

cien

cy (M

bits

J)


Proposed W = 4

Proposed W = 8

Figure 8 Energy efficiency versus spectral efficiency

resulting from the improved power efficiency have beenanalyzed The analytical results are validated by simulationresults The simulation result also shows that the systemwith the proposed allocation scheme could achieve a betterefficiency efficiency-spectral energy tradeoff compared to itscounterpart

Appendices

A Proof of Theorem 1

Let us consider a function 119891(120592) = 1 minus 119890minus1198750120592 0 lt 120592 le 1205902maxwhere 1205902max = max1205902

1198701 12059021198702

1205902119870119899119879

The second derivativeof this function is

11989110158401015840(120592) =

21198750120592 minus 11987520

1205924119890minus1198750120592 (A1)

From a HPAsrsquo viewpoint it is of interest to consider thesituation when the peak power across antennas 119875

0is large

Thus we consider the scenarios of 1198750

gt 21205902max where 1205902maxis the maximum average power across antennas Under thesesituations it is clear that 11989110158401015840(120592) lt 0 forall120592 isin (0 1205902max] Hencethe function 119891(120592) is concave By applying Jensenrsquos inequalitywe obtain the following inequality

1

119899119879

119899119879

sum119894=1

119891 (1205902

119870119894) le 119891(

1

119899119879

119899119879

sum119894=1

1205902

119870119894)

= 119891(1

119899119879

119899119879

sum119894=1

1205902

119870) = 119891 (120590

2

119870)

(A2)

where the equality comes from the fact that sum119899119879

119894=11205902119870119894

=

sum119899119879

119894=11205902119870

= 1198991198631205902 Expression (A2) can be rewritten as119899119879

sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le 119899119879(1 minus 119890

minus11987501205902

119870) (A3)

with equality if and only if 1205902119870119894

= 1205902119870 forall119894 = 1 119899

119879

On the other hand by applying the arithmetic-geometricmean inequality and note that (1 minus 119890

minus119909) gt 0 forall119909 gt 0 we have

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1

119899119879

119899119879

sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ))

119899119879

(A4)

Note that the equality in (A4) holds if and only if 1205902119870119894

= 1205902119870

forall119894 = 1 119899119879

Combining (A3) and (A4) results in

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1 minus 119890minus1198750120590

2

119870)119899119879

(A5)


= 1205902119870 forall119894 = 1 119899

119879 Thus we

get the following desired inequality

1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870

ge 1 minus (1 minus 119890minus1198750120590

2

119870)119899119879119870

(A6)

or

CCDF119875imbalance (1198750) ge CCDF119875balance (1198750) (A7)

This completes the proof

B Upper Bound of an ExpectedValue of Cost Penalty

In this appendix we derive an upper bound of the expectedvalue of Δ

119896 It can be seen from (5) that among all pos-

sible matrices H119896 the matrix H

119896with the highest value of

log2(det(Ω

119896))Ω119896

= Ω(H119896) will be selected as the effective

channel matrix for the 119896th subcarrier in the conventionalscheme Meanwhile in the proposed scheme due to thebalance constraint the effective channel matrix associatedwith the kth subcarrier is not necessarily the one with thehighest log

2(det(Ω

119896)) that is log

2(det(Ω

119896)) le log

2(det(Ω

119896))

Thus the expected value of Δ119896can be computed by using

order statistics In particular an upper bound on the expecteddifference of two-order statistics the Γth and 120574th 1 le 120574 lt Γis given by [30]

119864 Δ119896 = 119864 log

2(det (Ω

119896)) minus 119864 log

2(det (Ω

119896))

le 120590119862radic

Γ (120574 + 1)

120574

(B1)

where 1205902119862is the variance of log

2(det(Ω

119896)) that is assumed to

be the same for all possible matricesH119896

On the other hand suppose that the entries of the 119899119877times119899119879

matrixH119896are iid complex Gaussian random variables with

zero mean and unit variance then for any effective channelmatrix H

119896 Ω119896is a complex Wishart matrix It follows from

[25] that the variance of log2(det(Ω

119896)) can be expressed as

1205902

119862= [log

2(119890)]2

119901

sum119898=1

1205951015840(119902 minus 119898 + 1) (B2)


where 119901 = min(119899119863 119899119877) 119902 = max(119899

119863 119899119877) and 1205951015840(119909) =

suminfin

120593=11(120593 + 119909 minus 1)

2 is the first derivative of the digammafunction By approximating 1205951015840(119909) asymp 1119909 [25] the simplerexpression for 1205902

119862in (B2) is given as

1205902

119862= [log

2(119890)]2

119901

sum119898=1

1

119902 minus 119898 + 1 (B3)

Substitute (B3) into (B1) we finally arrive at

119864 Δ119896 le log

2(119890)radic(

119901

sum119898=1

1

119902 minus 119898 + 1) times

Γ (120574 + 1)

120574 (B4)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L M Correia D Zeller O Blume et al ldquoChallenges andenabling technologies for energy aware mobile radio networksrdquoIEEECommunicationsMagazine vol 48 no 11 pp 66ndash72 2010

[2] Z Hasan H Boostanimehr and V K Bhargava ldquoGreen cellularnetworks a survey some research issues and challengesrdquo IEEECommunications Surveys and Tutorials vol 13 no 4 pp 524ndash540 2011

[3] J Joung C K Ho and S Sun ldquoGreenwireless communicationsa power amplifier perspectiverdquo in Proceedings of the 4th Asia-Pacific Signal and Information Processing Association AnnualSummit and Conference (APSIPA ASC rsquo12) pp 1ndash8 December2012

[4] G L Stuber J R Barry S W Mclaughlin Y E Li M AIngram and T G Pratt ldquoBroadband MIMO-OFDM wirelesscommunicationsrdquoProceedings of the IEEE vol 92 no 2 pp 271ndash294 2004

[5] H Bolcskei D Gesbert and A J Paulraj ldquoOn the capacity ofOFDM-based spatial multiplexing systemsrdquo IEEE Transactionson Communications vol 50 no 2 pp 225ndash234 2002

[6] V Tarokh N Seshadri and A R Calderbank ldquoSpace-timecodes for high data rate wireless communication performancecriterion and code constructionrdquo IEEE Transactions on Infor-mation Theory vol 44 no 2 pp 744ndash765 1998

[7] C M Vithanage M Sandell J P Coon and Y Wang ldquoPrecod-ing in OFDM-based multi-antenna ultra-wideband systemsrdquoIEEE Communications Magazine vol 47 no 1 pp 41ndash47 2009

[8] Z Tang H Suzuki and I B Collings ldquoPerformance of antennaselection forMIMO-OFDMsystems based onmeasured indoorcorrelated frequency selective channelsrdquo in Proceedings of theAustralian Telecommunication Networks and Applications Con-ference pp 435ndash439 December 2006

[9] H Zhang and R U Nabar ldquoTransmit antenna selection inMIMO-OFDM systems bulk versus per-tone selectionrdquo inProceedings of the IEEE International Conference on Communi-cations (ICC rsquo08) pp 4371ndash4375 May 2008

[10] H Shi M Katayama T Yamazato H Okada and A OgawaldquoAn adaptive antenna selection scheme for transmit diversityin OFDM systemsrdquo in Proceedings of the IEEE 54th VehicularTechnology Conference (VTC rsquo01) pp 2168ndash2172 October 2001

[11] M Sandell and J P Coon ldquoPer-subcarrier antenna selectionwith power constraints in OFDM systemsrdquo IEEE Transactionson Wireless Communications vol 8 no 2 pp 673ndash677 2009

[12] N P Le F Safaei and L C Tran ldquoTransmit antenna subsetselection for high-rate MIMO-OFDM systems in the presenceof nonlinear power amplifiersrdquo EURASIP Journal on WirelessCommunications and Networking vol 2014 article 27 2014

[13] R PrasadOFDM for Wireless Communications Systems ArtechHouse 1st edition 2004

[14] T Jiang and Y Wu ldquoAn overview peak-to-average power ratioreduction techniques for OFDM signalsrdquo IEEE Transactions onBroadcasting vol 54 no 2 pp 257ndash268 2008

[15] H Ochiai ldquoPerformance analysis of peak power and band-limited OFDM system with linear scalingrdquo IEEE Transactionson Wireless Communications vol 2 no 5 pp 1055ndash1065 2003

[16] C Zhao R J Baxley and G T Zhou ldquoPeak-to-average powerratio and power efficiency considerations in MIMO-OFDMsystemsrdquo IEEE Communications Letters vol 12 no 4 pp 268ndash270 2008

[17] D Wulich and I Gutman ldquoImpact of linear power amplifier onpower loading in OFDM part I principlesrdquo in Proceedings ofthe IEEE 26th Convention of Electrical and Electronics Engineersin Israel (IEEEI rsquo10) pp 360ndash364 November 2010

[18] S Coleri M Ergen A Puri and A Bahai ldquoChannel estimationtechniques based on pilot arrangement in OFDM systemsrdquoIEEE Transactions on Broadcasting vol 48 no 3 pp 223ndash2292002

[19] R W Heath S Sandhu and A Paulraj ldquoAntenna selectionfor spatial multiplexing systems with linear receiversrdquo IEEECommunications Letters vol 5 no 4 pp 142ndash144 2001

[20] R Fletcher Practical Methods of Optimization John Willey ampSons 2nd edition 1987

[21] R W Bauml R F H Fischer and J B Huber ldquoReducingthe peak-to-average power ratio of multicarrier modulation byselected mappingrdquo Electronics Letters vol 32 no 22 pp 2056ndash2057 1996

[22] M Tan Z Latinovic and Y Bar-Ness ldquoSTBC MIMO-OFDMpeak-to-average power ratio reduction by cross-antenna rota-tion and inversionrdquo IEEE Communications Letters vol 9 no 7pp 592ndash594 2005

[23] K M Anstreicher ldquoLinear programming in O(n3lnnL) opera-tionsrdquo SIAM Journal on Optimization vol 9 no 4 pp 803ndash8121999

[24] J G Proakis Digital Communications McGraw Hill 4th edi-tion 2001

[25] O Oyman R U Nabar H Bolcskei and A J Paulraj ldquoChar-acterizing the statistical properties of mutual information inMIMO channelsrdquo IEEE Transactions on Signal Processing vol51 no 11 pp 2784ndash2795 2003

[26] S Cui A J Goldsmith and A Bahai ldquoEnergy-constrainedmodulation optimizationrdquo IEEE Transactions onWireless Com-munications vol 4 no 5 pp 2349ndash2360 2005

[27] R J Baxley and G T Zhou ldquoPower savings analysis of peak-to-average power ratio reduction in OFDMrdquo IEEE Transactions onConsumer Electronics vol 50 no 3 pp 792ndash797 2004

[28] A Batra et al Multiband OFDM Physical Layer SpecificationRelease 1 5 WiMedia Alliance 2009

[29] J Foerster et al ldquoChannel modeling sub-committee reportfinalrdquo IEEE P802 15-02490r1-SG3a 2003

[30] B C Arnold and R A Groeneveld ldquoBounds on expectationsof linear systematic statistics based on dependent samplesrdquoTheAnnals of Statistics vol 7 no 1 pp 220ndash223 1979

International Journal of

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Control Scienceand Engineering

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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



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Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



small The fluctuation of the powers clearly affects the powerefficiency of HPAs or distorts signals which in turn reducesthe potential benefits of the antenna selectionOFDM systems[13]

One possible approach to deal with the problem ofimbalance allocation of data subcarriers is selecting antennasunder a constraint that the number of data subcarriersallocated to each antenna is equal Some research workshave studied such a constrained selection approach in theliterature for example see [10ndash12] In [10] an ad hoc algo-rithm was developed to realize the constrained selectionscheme Meanwhile the authors in [11] considered linearoptimization to devise the constrained selection scheme Itwas shown that the selection scheme based on optimizationcould offer a better performance than the suboptimal solutionin [10] In [12] we generalized the approach in [11] to thesystem with an arbitrary number of multiplexed data streamsand analyzed the performance directly in nonlinear fadingchannels However all of these works only study the efficacyof the constrained selection scheme from error-performanceperspective Moreover even though the same number of datasubcarriers is allocated to each transmit antenna (ie allantennas have an equal average power) an occurrence of highpeak-to-average power ratio (PAPR) still affects the systemDespite an increasing concern about energy consumption inwireless networks to the best of our knowledge a research onenergy efficiency in the context of antenna selection MIMO-OFDM systems is still missing

It is also essential to emphasize the need of research onimproving energy efficiency in antenna selection MIMO-OFDM systems First per-subcarrier antenna selectionMIMO configurations need multiple active RF (radio fre-quency) chains which immediately raises a concern aboutenergy consumption compared to single RF-chain OFDMsystems Second OFDM inherently suffers from a high PAPRproblem which leads to a poor power efficiency of HPAs[13 14] In per-subcarrier antenna selection OFDM systemsthis effect is intuitively more serious due to a combinationof the high PAPR and power fluctuation resulting from animbalance allocation of data subcarriers This problem alsobecomes crucial in the OFDM systems where linear scaling(ie scale the peak power of the time-domain OFDM signalsto the saturation level of the HPAs [15ndash17]) is implemented torealize OFDM transmissions with no nonlinear distortions

In this paper per-subcarrier antenna selection MIMO-OFDM systems with linear scaling are investigated froman energy perspective for the first time The importantcontributions of this work include (a) the analysis of powerefficiency ofHPAs and energy efficiency and (b) the proposedstrategy to improve these useful performance metrics Themain results are summarized as follows

(1) A two-step data allocation strategy is proposed todeliver a maximum overall power efficiency of HPAsThis strategy consists of an equal allocation of datasubcarriers among transmit antennas based on linearoptimization and a peak-power reduction algorithmvia cross-antenna permutations

(2) Analytical expressions characterizing the achievedpower efficiency of HPAs including the CCDFs(complementary cumulative distribution function)and the average power efficiency are derived It isproved that from the power-efficiency viewpoint theproposed allocation scheme outperforms the conven-tional scheme

(3) The improvements in capacity and energy efficiencyresulting from the improved power efficiency ofHPAsare analyzed

Numerical results are provided to verify the analyses as wellas demonstrate the benefits in terms of the power efficiencyof HPAs and capacity as well as energy efficiency

The remainder of the paper is organized as follows InSection 2 a per-subcarrier antenna subset selection MIMO-OFDM system with linear scaling is described In Section 3a data allocation strategy that could allocate evenly data sub-carriers across antennas with a low peak power is proposedAnalysis of power efficiency is carried out in Section 4 Theachievable capacity and energy efficiency are considered inSection 5Numerical results are provided in Section 6 FinallySection 7 concludes the paper

Notation A bold letter denotes a vector or a matrix whereasan italic letter denotes a variable (sdot)119879 (sdot)119867 119864sdot and det(sdot)denote transpose Hermitian transpose expectation anddeterminant of a matrix respectively otimes denotes the Kro-necker product I

119899indicates the 119899times119899 identity matrix and 1

119870

is a119870times 1 vector of onesR indicates the set of real numbers

2 Antenna Selection for MIMO-OFDMSystems with Linear Scaling

21 SystemModel Weconsider aMIMO-OFDMsystemwith119870 subcarriers 119899

119879transmit antennas and 119899

119877receive antennas

as shown in Figure 1 At the transmitter the input data aredemultiplexed into 119899

119863independent data streams Each data

stream is then mapped onto 119872-QAM (119872-ary quadratureamplitudemodulation) constellations For the 119896th subcarrierwe denote 119906

119896

119897and 119909119896

119894 1 le 119897 le 119899

119863 1 le 119894 le 119899

119879

0 le 119896 le 119870 minus 1 to be the symbols that the subcarrierblock takes at its 119897th input and outputs at its 119894th outputrespectively The allocation block assigns the elements ofu119896

= [1199061198961 1199061198962 119906119896

119899119863]119879to 119899119863selected antennas at the 119896th

subcarrier based on feedback information As a result only119899119863elements in a vector x

119896= [1199091198961 1199091198962 119909119896

119899119879]119879 are assigned

values from u119896 whereas the others are zeros It is assumed

that 119864u119896u119867119896 = 1205902I

119899119863 The output sequences from the

subcarrier allocation block are then fed into 119870-point IFFT(inverse fast Fourier transform) blocks In this paper theNyquist sampling signal is considered Thus the discrete-time baseband OFDM signals can be expressed as

119904119894 (119899) =

1

radic119870

119870minus1

sum119896=0

119909119896

1198941198901198952120587119899119896119870

0 le 119899 le 119870 minus 1 1 le 119894 le 119899119879

(1)


Inputdata

Symbolmapping


IFFTadd GI

Linearscaling

NonlinearHPA

1

nD nT

Dem

ultip

lexe

r

Symbolmapping

IFFTadd GI

NonlinearHPA


1 1

nD

Tx 1

Tx nT

(a)


ChannelH

Remove GIFFT

Rece

iver

proc

essin

g Estimateddata


Remove GIFFT

To transmitter

Rx 1

Rx nR

(b)



119904119894 (119899) = radic120572119904

119894 (119899) (2)


119894(119899)|119899 = 0 119870 minus 1 119894 =




y119896= radic120572H

119896x119896+ n119896

= radic120572H119896u119896+ n119896

(3)

where

x119896= [119909119896

1119909119896


119899119879]119879

H119896=

[[[[

[

ℎ119896

11ℎ11989612

sdot sdot sdot ℎ1198961119899119879

ℎ11989621

ℎ11989622

sdot sdot sdot ℎ1198962119899119879


ℎ1198961198991198771

ℎ1198961198991198772

sdot sdot sdot ℎ119896119899119877119899119879

]]]]

]

y119896= [119910119896

11199101198962

sdot sdot sdot 119910119896119899119877]119879

n119896= [119899119896

11198991198962

sdot sdot sdot 119899119896119899119877]119879

(4)






119863= 2 and Γ = 6)

120574 Γ120574

1 1 2

2 1 3

3 1 4

4 2 3

5 2 4

6 3 4





119863antennas



119863


) = 119899119879119899119863(119899119879minus





119879= 4 and 119899






Γ120574 120574=1Γ

119868119896

120574 (5)

where

119868119896

120574= log2(det(I

119899119877+

120588

119899119863

H119896H119867119896)) (6)



1199051205902119899

= 12057211989911986312059021205902119899 and



119868 (120588H) =1

119870

119870minus1

sum119896=0

119868119896

120574lowast (7)







119896

120574 where 119911119896

120574= 1 if Γ





120574= 119868119896120574as the maximum


(11991101 1199110

Γ 11991111 1199111

Γ 119911119870minus1

1 119911119870minus1

Γ)119879


c = (11988801 1198880

Γ 11988811 1198881

Γ 119888119870minus1

1 119888119870minus1

Γ)119879



c119879z




Γtimes119870ΓA = (A119879

1A1198792)119879

isin 0 1(119870+Γ)times119870Γ a = (1119879

119870120582119879)119879 and 120582 =

(1205821 1205822 120582

Γ)119879 where 120582






119863out of 119899

119879



Antennas

Subcarriers

Tx 1

Tx 2

Tx 3

Tx 4

Data

Not used


Data

Not used

Antennas

Subcarriers

Tx 1

Tx 2

Tx 3

Tx 4



119863= 2 and 119870 = 12)

Ant

enna

s

Subcarriers


x11 x

21

x12

x32

x33 x

43

x24 x

44

0 0

0

00

0 0

0

(a)


Subcarriers

x12 x

24 0 0

00

00

0 0

x11 x

32

x33 x

43

x44

x21

(b)


Subcarriers

Ant

enna

s

x12

x21 0 0

00

00

0 0

x11 x

33

x32 x

43

x44

x24

(c)


Subcarriers

0 0

00

00

0 0

x11 x

21

x12 x

33

x32 x

44

x43x

24

(d)


119863= 2 and 119870 = 4)



119863= 2 119899

119879= 4 and 119870 = 4 is



for transmission




3 ln(119870Γ)]120585)



2119870


119879(1198702)log










0 that is

CCDF119875 (1198750) = Pr (119875 gt 119875

0) (9)




119875119894= max0le119899le119870minus1

|119904119894 (119899) |2 (10)








119901|119904|2 (

100381610038161003816100381611990411989410038161003816100381610038162) =

1

1205902119870119894

119890minus|119904119894|21205902

119870119894 (11)

where 1205902119870119894


119894(119899)| Note

that sum119899119879

119894=1119870119894

= 119899119863119870 thus we have sum

119899119879

119894=11205902119870119894

= 1198991198631205902 The


is given as

Pr (100381610038161003816100381611990411989410038161003816100381610038162le 120579) = 1 minus 119890

minus1205791205902

119870119894 120579 ge 0 (12)



as

CDF119875119894 = Pr (119875119894le 1198750)

= Pr (1003816100381610038161003816119904119894 (0)10038161003816100381610038162le 1198750)Pr (1003816100381610038161003816119904119894 (1)

10038161003816100381610038162le 1198750) sdot sdot sdot

Pr (1003816100381610038161003816119904119894 (119870 minus 1)10038161003816100381610038162le 1198750)

= (1 minus 119890minus1198750120590

2

119870119894 )119870

(13)


119875 = max1le119894le119899119879

119875119894 (14)


CDF119875 = Pr (119875 le 1198750)

= Pr (1198751le 1198750)Pr (119875


119899119879le 1198750)

=

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870

(15)



= 1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870

(16)


119894=

119899119863119870119899119879


signals are 1205902119870119894

= 1198991198631205902119899119879= 1205902119870 forall119894 = 1 2 119899

119879 As a result



2

119870)119899119879119870

(17)








2

119870)119899119879119870)119882

(19)





119879


120578PE

=sum119899119879

119894=1119875119894out

119899119879119875dc

=sum119899119879

119894=1119875119894in

119899119879119875dc

= 1205721198991198631205902

119899119879119875dc

=1198991198631205902119875sat

119899119879119875dc

1

119875=

1198991198631205902

2119899119879

1

119875

(21)


119894=1119875119894in = 120572119899



0) = Pr(120578PE gt 120578PE

0) to be the CCDF


PE



PE0) (22)

where


minus12120578PE0 )119899119879119870


119899119879

prod119894=1

(1 minus 119890minus1198991198631198702119899119879119870119894120578

PE0 )119870

(23)



120578PE

= 119864 120578PE =

1198991198631205902

2119899119879

1198641

119875 =

1198991198631205902

2119899119879

int1

119909119901 (119909) 119889119909 (24)


119901balance (119909) =119889

119889119909CDF119875balance (119909) =

119889

119889119909(1 minus 119890

minus1199091205902

119870)119899119879119870

=119899119879119870

1205902119870

119890minus1199091205902

119870(1 minus 119890minus1199091205902

119870)119899119879119870minus1

(25)

Hence

120578PEbalance =

1198991198631205902

2119899119879

int1

119909119901balance (119909) 119889119909

=119899119879119870

2int1198701205902

119870

1205902

119870

1

119909119890minus1199091205902

119870(1 minus 119890minus1199091205902

119870)119899119879119870minus1

119889119909

=119899119879119870

2int119870

1

1

119909119890minus119909

(1 minus 119890minus119909

)119899119879119870minus1

119889119909

(26)


120578PEimbalance =

1198991198631205902

2119899119879

int1

119909119901imbalance (119909) 119889119909

=1198991198631198701205902

2119899119879

int119870max120590

2

max

1205902

max

1

119909

119899119879

sum119895=1

119890minus1199091205902

119870119895

(1 minus 119890minus1199091205902

119870119895 ) 1205902119870119895

times

119899119879

prod119894=1

(1 minus 119890minus1199091205902

119870119894 )119870

119889119909

(27)

where 1205902max = max12059021198701

1205902119870119899119879

and 119870max = max1198701





) (28)

where



PE0 )119899119879119870

)119882

(29)



1198991198631205902

2119899119879

int1

119909119901balance+reduced (119909) 119889119909

=119899119879119870119882

2int119870

1

1

119909119890minus119909

(1 minus 119890minus119909

)119899119879119870minus1


)119899119879119870

)119882minus1

119889119909

(30)






1199051205902119899 where 119875

119905= 120572119899

1198631205902 = 120578

PE119899119879119875dc (cf (21))


119868 (120588proposedH)

=1

119870

119870minus1

sum119896=0

log2(det(I

119899119877+

120588proposed

119899119863

H119896H119867119896))

(31)


=1

119870

119870minus1

sum119896=0

log2(det(I

119899119877+

120588imbalance119899119863

H119896H119867119896))

(32)


2


120578PEimbalance119899119879119875dc120590

2

119899 Here 120578

PEproposed = 120578








119870

119870minus1

sum119896=0

Δ119868119896 (33)

where

Δ119868119896= log2(det(I

119899119877+

120588proposed

119899119863

H119896H119867119896))

minus log2(det(I

119899119877+

120588imbalance119899119863

H119896H119867119896))

(34)


119868119896= log2(det(

120588

119899119863

Ω119896)) (35)

where

Ω119896= Ω (H

119896) =

H119896H119867119896 119899119877le 119899119863

H119867119896H119896 119899119877gt 119899119863

(36)


Δ119868 =1

119870

119870minus1

sum119896=0

(log2(det(

120588proposed

119899119863

Ω119896))

minuslog2(det(

120588imbalance119899119863

Ω119896)))

= 119901log2

120578PEproposed

120578PEimbalance

minus1

119870

119870minus1

sum119896=0

Δ119896= 1198791+ 1198792

(37)

where 119901 = min(119899119863 119899119877)Ω119896= Ω(H

119896) and


119896)) minus log

2(det (Ω

119896)) (38)





119896=0Δ119896gt 0





2is a penalty


119896is fixed





119879and 119899





Δ119862 = 119864H Δ119868 le 119901log2119864H

120578PEproposed

120578PEimbalance

(39)



1 CDF of 119879

2 and CCDF


119879= 4 119899

119863= 2 119899

119877= 2 119870= 128


1gt 0 and 119879

2lt 0 Moreover



120578EE

=119862 (120588)

119875total (40)


119879119875dc + 119899

119879119875RF + 119875sp


1199051205902119899= 120578

PE119899119879119875dc1205902



120578EE

(119875dc) =119862 (120578

PE119899119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

(41)






119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEproposed 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(42)



119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEimbalance 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(43)


119863= 2 and 119882= 4 (ie 2 bits


6 Numerical Results


119863= 2 and 119899





0

10minus1

10minus2

minus2 0 2

W = 1

W = 4

W = 8

(a)

100

10minus1

10minus2

minus04 minus02 0

Empirical CDF of T2

(b)


0

10minus1

10minus2

minus1 minus05 0 05 1 15 2

W = 1

W = 4

W = 8

(c)




sum119870minus1

119899=0|119904119894(119899)|2 = 1205902119870











120578PE (Analysis) 00691 00757 00892 00943


PEimbalance

120578PEimbalance

mdash 1019 2826 3515

1 2 3 4 5 6 7 8 9

100

10minus1

10minus2

10minus3

Peak power P0



Pr (P

gtP0)



CCD

F

100

10minus1

10minus2

10minus3




4500

4000

3500

3000

2500

2000

1500

1000

Ergo

dic c

apac

ity (M

bps)

2 4 6 8 10 12 14

SNR (dB)

3300

3200

3100

3000

2900

2800

97 98 99 10 101


Proposed W = 4

Proposed W = 8




7 Conclusions



0 2 4 6 8 10400

600

800

1000

1200

1400

1600


Ener

gy effi

cien

cy (M

bits

J)


Proposed W = 4

Proposed W = 8



Appendices



1198701 12059021198702

1205902119870119899119879


11989110158401015840(120592) =

21198750120592 minus 11987520

1205924119890minus1198750120592 (A1)


0is large



1

119899119879

119899119879

sum119894=1

119891 (1205902

119870119894) le 119891(

1

119899119879

119899119879

sum119894=1

1205902

119870119894)

= 119891(1

119899119879

119899119879

sum119894=1

1205902

119870) = 119891 (120590

2

119870)

(A2)


119894=11205902119870119894

=

sum119899119879

119894=11205902119870


sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le 119899119879(1 minus 119890

minus11987501205902

119870) (A3)


= 1205902119870 forall119894 = 1 119899

119879



119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1

119899119879

119899119879

sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ))

119899119879

(A4)


= 1205902119870

forall119894 = 1 119899119879


119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1 minus 119890minus1198750120590

2

119870)119899119879

(A5)


= 1205902119870 forall119894 = 1 119899

119879 Thus we


1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870


2

119870)119899119879119870

(A6)

or








log2(det(Ω

119896))Ω119896



2(det(Ω


2(det(Ω

119896)) le log

2(det(Ω

119896))



119864 Δ119896 = 119864 log

2(det (Ω

119896)) minus 119864 log

2(det (Ω

119896))

le 120590119862radic

Γ (120574 + 1)

120574

(B1)


2(det(Ω









1205902

119862= [log

2(119890)]2

119901

sum119898=1

1205951015840(119902 minus 119898 + 1) (B2)


where 119901 = min(119899119863 119899119877) 119902 = max(119899

119863 119899119877) and 1205951015840(119909) =

suminfin

120593=11(120593 + 119909 minus 1)



1205902

119862= [log

2(119890)]2

119901

sum119898=1

1

119902 minus 119898 + 1 (B3)


119864 Δ119896 le log

2(119890)radic(

119901

sum119898=1

1

119902 minus 119898 + 1) times

Γ (120574 + 1)

120574 (B4)



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Inputdata

Symbolmapping


IFFTadd GI

Linearscaling

NonlinearHPA

1

nD nT

Dem

ultip

lexe

r

Symbolmapping

IFFTadd GI

NonlinearHPA


1 1

nD

Tx 1

Tx nT

(a)


ChannelH

Remove GIFFT

Rece

iver

proc

essin

g Estimateddata


Remove GIFFT

To transmitter

Rx 1

Rx nR

(b)



119904119894 (119899) = radic120572119904

119894 (119899) (2)


119894(119899)|119899 = 0 119870 minus 1 119894 =




y119896= radic120572H

119896x119896+ n119896

= radic120572H119896u119896+ n119896

(3)

where

x119896= [119909119896

1119909119896


119899119879]119879

H119896=

[[[[

[

ℎ119896

11ℎ11989612

sdot sdot sdot ℎ1198961119899119879

ℎ11989621

ℎ11989622

sdot sdot sdot ℎ1198962119899119879


ℎ1198961198991198771

ℎ1198961198991198772

sdot sdot sdot ℎ119896119899119877119899119879

]]]]

]

y119896= [119910119896

11199101198962

sdot sdot sdot 119910119896119899119877]119879

n119896= [119899119896

11198991198962

sdot sdot sdot 119899119896119899119877]119879

(4)






119863= 2 and Γ = 6)

120574 Γ120574

1 1 2

2 1 3

3 1 4

4 2 3

5 2 4

6 3 4





119863antennas



119863


) = 119899119879119899119863(119899119879minus





119879= 4 and 119899






Γ120574 120574=1Γ

119868119896

120574 (5)

where

119868119896

120574= log2(det(I

119899119877+

120588

119899119863

H119896H119867119896)) (6)



1199051205902119899

= 12057211989911986312059021205902119899 and



119868 (120588H) =1

119870

119870minus1

sum119896=0

119868119896

120574lowast (7)







119896

120574 where 119911119896

120574= 1 if Γ





120574= 119868119896120574as the maximum


(11991101 1199110

Γ 11991111 1199111

Γ 119911119870minus1

1 119911119870minus1

Γ)119879


c = (11988801 1198880

Γ 11988811 1198881

Γ 119888119870minus1

1 119888119870minus1

Γ)119879



c119879z




Γtimes119870ΓA = (A119879

1A1198792)119879

isin 0 1(119870+Γ)times119870Γ a = (1119879

119870120582119879)119879 and 120582 =

(1205821 1205822 120582

Γ)119879 where 120582






119863out of 119899

119879



Antennas

Subcarriers

Tx 1

Tx 2

Tx 3

Tx 4

Data

Not used


Data

Not used

Antennas

Subcarriers

Tx 1

Tx 2

Tx 3

Tx 4



119863= 2 and 119870 = 12)

Ant

enna

s

Subcarriers


x11 x

21

x12

x32

x33 x

43

x24 x

44

0 0

0

00

0 0

0

(a)


Subcarriers

x12 x

24 0 0

00

00

0 0

x11 x

32

x33 x

43

x44

x21

(b)


Subcarriers

Ant

enna

s

x12

x21 0 0

00

00

0 0

x11 x

33

x32 x

43

x44

x24

(c)


Subcarriers

0 0

00

00

0 0

x11 x

21

x12 x

33

x32 x

44

x43x

24

(d)


119863= 2 and 119870 = 4)



119863= 2 119899

119879= 4 and 119870 = 4 is



for transmission




3 ln(119870Γ)]120585)



2119870


119879(1198702)log










0 that is

CCDF119875 (1198750) = Pr (119875 gt 119875

0) (9)




119875119894= max0le119899le119870minus1

|119904119894 (119899) |2 (10)








119901|119904|2 (

100381610038161003816100381611990411989410038161003816100381610038162) =

1

1205902119870119894

119890minus|119904119894|21205902

119870119894 (11)

where 1205902119870119894


119894(119899)| Note

that sum119899119879

119894=1119870119894

= 119899119863119870 thus we have sum

119899119879

119894=11205902119870119894

= 1198991198631205902 The


is given as

Pr (100381610038161003816100381611990411989410038161003816100381610038162le 120579) = 1 minus 119890

minus1205791205902

119870119894 120579 ge 0 (12)



as

CDF119875119894 = Pr (119875119894le 1198750)

= Pr (1003816100381610038161003816119904119894 (0)10038161003816100381610038162le 1198750)Pr (1003816100381610038161003816119904119894 (1)

10038161003816100381610038162le 1198750) sdot sdot sdot

Pr (1003816100381610038161003816119904119894 (119870 minus 1)10038161003816100381610038162le 1198750)

= (1 minus 119890minus1198750120590

2

119870119894 )119870

(13)


119875 = max1le119894le119899119879

119875119894 (14)


CDF119875 = Pr (119875 le 1198750)

= Pr (1198751le 1198750)Pr (119875


119899119879le 1198750)

=

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870

(15)



= 1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870

(16)


119894=

119899119863119870119899119879


signals are 1205902119870119894

= 1198991198631205902119899119879= 1205902119870 forall119894 = 1 2 119899

119879 As a result



2

119870)119899119879119870

(17)








2

119870)119899119879119870)119882

(19)





119879


120578PE

=sum119899119879

119894=1119875119894out

119899119879119875dc

=sum119899119879

119894=1119875119894in

119899119879119875dc

= 1205721198991198631205902

119899119879119875dc

=1198991198631205902119875sat

119899119879119875dc

1

119875=

1198991198631205902

2119899119879

1

119875

(21)


119894=1119875119894in = 120572119899



0) = Pr(120578PE gt 120578PE

0) to be the CCDF


PE



PE0) (22)

where


minus12120578PE0 )119899119879119870


119899119879

prod119894=1

(1 minus 119890minus1198991198631198702119899119879119870119894120578

PE0 )119870

(23)



120578PE

= 119864 120578PE =

1198991198631205902

2119899119879

1198641

119875 =

1198991198631205902

2119899119879

int1

119909119901 (119909) 119889119909 (24)


119901balance (119909) =119889

119889119909CDF119875balance (119909) =

119889

119889119909(1 minus 119890

minus1199091205902

119870)119899119879119870

=119899119879119870

1205902119870

119890minus1199091205902

119870(1 minus 119890minus1199091205902

119870)119899119879119870minus1

(25)

Hence

120578PEbalance =

1198991198631205902

2119899119879

int1

119909119901balance (119909) 119889119909

=119899119879119870

2int1198701205902

119870

1205902

119870

1

119909119890minus1199091205902

119870(1 minus 119890minus1199091205902

119870)119899119879119870minus1

119889119909

=119899119879119870

2int119870

1

1

119909119890minus119909

(1 minus 119890minus119909

)119899119879119870minus1

119889119909

(26)


120578PEimbalance =

1198991198631205902

2119899119879

int1

119909119901imbalance (119909) 119889119909

=1198991198631198701205902

2119899119879

int119870max120590

2

max

1205902

max

1

119909

119899119879

sum119895=1

119890minus1199091205902

119870119895

(1 minus 119890minus1199091205902

119870119895 ) 1205902119870119895

times

119899119879

prod119894=1

(1 minus 119890minus1199091205902

119870119894 )119870

119889119909

(27)

where 1205902max = max12059021198701

1205902119870119899119879

and 119870max = max1198701





) (28)

where



PE0 )119899119879119870

)119882

(29)



1198991198631205902

2119899119879

int1

119909119901balance+reduced (119909) 119889119909

=119899119879119870119882

2int119870

1

1

119909119890minus119909

(1 minus 119890minus119909

)119899119879119870minus1


)119899119879119870

)119882minus1

119889119909

(30)






1199051205902119899 where 119875

119905= 120572119899

1198631205902 = 120578

PE119899119879119875dc (cf (21))


119868 (120588proposedH)

=1

119870

119870minus1

sum119896=0

log2(det(I

119899119877+

120588proposed

119899119863

H119896H119867119896))

(31)


=1

119870

119870minus1

sum119896=0

log2(det(I

119899119877+

120588imbalance119899119863

H119896H119867119896))

(32)


2


120578PEimbalance119899119879119875dc120590

2

119899 Here 120578

PEproposed = 120578








119870

119870minus1

sum119896=0

Δ119868119896 (33)

where

Δ119868119896= log2(det(I

119899119877+

120588proposed

119899119863

H119896H119867119896))

minus log2(det(I

119899119877+

120588imbalance119899119863

H119896H119867119896))

(34)


119868119896= log2(det(

120588

119899119863

Ω119896)) (35)

where

Ω119896= Ω (H

119896) =

H119896H119867119896 119899119877le 119899119863

H119867119896H119896 119899119877gt 119899119863

(36)


Δ119868 =1

119870

119870minus1

sum119896=0

(log2(det(

120588proposed

119899119863

Ω119896))

minuslog2(det(

120588imbalance119899119863

Ω119896)))

= 119901log2

120578PEproposed

120578PEimbalance

minus1

119870

119870minus1

sum119896=0

Δ119896= 1198791+ 1198792

(37)

where 119901 = min(119899119863 119899119877)Ω119896= Ω(H

119896) and


119896)) minus log

2(det (Ω

119896)) (38)





119896=0Δ119896gt 0





2is a penalty


119896is fixed





119879and 119899





Δ119862 = 119864H Δ119868 le 119901log2119864H

120578PEproposed

120578PEimbalance

(39)



1 CDF of 119879

2 and CCDF


119879= 4 119899

119863= 2 119899

119877= 2 119870= 128


1gt 0 and 119879

2lt 0 Moreover



120578EE

=119862 (120588)

119875total (40)


119879119875dc + 119899

119879119875RF + 119875sp


1199051205902119899= 120578

PE119899119879119875dc1205902



120578EE

(119875dc) =119862 (120578

PE119899119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

(41)






119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEproposed 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(42)



119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEimbalance 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(43)


119863= 2 and 119882= 4 (ie 2 bits


6 Numerical Results


119863= 2 and 119899





0

10minus1

10minus2

minus2 0 2

W = 1

W = 4

W = 8

(a)

100

10minus1

10minus2

minus04 minus02 0

Empirical CDF of T2

(b)


0

10minus1

10minus2

minus1 minus05 0 05 1 15 2

W = 1

W = 4

W = 8

(c)




sum119870minus1

119899=0|119904119894(119899)|2 = 1205902119870











120578PE (Analysis) 00691 00757 00892 00943


PEimbalance

120578PEimbalance

mdash 1019 2826 3515

1 2 3 4 5 6 7 8 9

100

10minus1

10minus2

10minus3

Peak power P0



Pr (P

gtP0)



CCD

F

100

10minus1

10minus2

10minus3




4500

4000

3500

3000

2500

2000

1500

1000

Ergo

dic c

apac

ity (M

bps)

2 4 6 8 10 12 14

SNR (dB)

3300

3200

3100

3000

2900

2800

97 98 99 10 101


Proposed W = 4

Proposed W = 8




7 Conclusions



0 2 4 6 8 10400

600

800

1000

1200

1400

1600


Ener

gy effi

cien

cy (M

bits

J)


Proposed W = 4

Proposed W = 8



Appendices



1198701 12059021198702

1205902119870119899119879


11989110158401015840(120592) =

21198750120592 minus 11987520

1205924119890minus1198750120592 (A1)


0is large



1

119899119879

119899119879

sum119894=1

119891 (1205902

119870119894) le 119891(

1

119899119879

119899119879

sum119894=1

1205902

119870119894)

= 119891(1

119899119879

119899119879

sum119894=1

1205902

119870) = 119891 (120590

2

119870)

(A2)


119894=11205902119870119894

=

sum119899119879

119894=11205902119870


sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le 119899119879(1 minus 119890

minus11987501205902

119870) (A3)


= 1205902119870 forall119894 = 1 119899

119879



119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1

119899119879

119899119879

sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ))

119899119879

(A4)


= 1205902119870

forall119894 = 1 119899119879


119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1 minus 119890minus1198750120590

2

119870)119899119879

(A5)


= 1205902119870 forall119894 = 1 119899

119879 Thus we


1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870


2

119870)119899119879119870

(A6)

or








log2(det(Ω

119896))Ω119896



2(det(Ω


2(det(Ω

119896)) le log

2(det(Ω

119896))



119864 Δ119896 = 119864 log

2(det (Ω

119896)) minus 119864 log

2(det (Ω

119896))

le 120590119862radic

Γ (120574 + 1)

120574

(B1)


2(det(Ω









1205902

119862= [log

2(119890)]2

119901

sum119898=1

1205951015840(119902 minus 119898 + 1) (B2)


where 119901 = min(119899119863 119899119877) 119902 = max(119899

119863 119899119877) and 1205951015840(119909) =

suminfin

120593=11(120593 + 119909 minus 1)



1205902

119862= [log

2(119890)]2

119901

sum119898=1

1

119902 minus 119898 + 1 (B3)


119864 Δ119896 le log

2(119890)radic(

119901

sum119898=1

1

119902 minus 119898 + 1) times

Γ (120574 + 1)

120574 (B4)



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1199051205902119899

= 12057211989911986312059021205902119899 and



119868 (120588H) =1

119870

119870minus1

sum119896=0

119868119896

120574lowast (7)







119896

120574 where 119911119896

120574= 1 if Γ





120574= 119868119896120574as the maximum


(11991101 1199110

Γ 11991111 1199111

Γ 119911119870minus1

1 119911119870minus1

Γ)119879


c = (11988801 1198880

Γ 11988811 1198881

Γ 119888119870minus1

1 119888119870minus1

Γ)119879



c119879z




Γtimes119870ΓA = (A119879

1A1198792)119879

isin 0 1(119870+Γ)times119870Γ a = (1119879

119870120582119879)119879 and 120582 =

(1205821 1205822 120582

Γ)119879 where 120582






119863out of 119899

119879



Antennas

Subcarriers

Tx 1

Tx 2

Tx 3

Tx 4

Data

Not used


Data

Not used

Antennas

Subcarriers

Tx 1

Tx 2

Tx 3

Tx 4



119863= 2 and 119870 = 12)

Ant

enna

s

Subcarriers


x11 x

21

x12

x32

x33 x

43

x24 x

44

0 0

0

00

0 0

0

(a)


Subcarriers

x12 x

24 0 0

00

00

0 0

x11 x

32

x33 x

43

x44

x21

(b)


Subcarriers

Ant

enna

s

x12

x21 0 0

00

00

0 0

x11 x

33

x32 x

43

x44

x24

(c)


Subcarriers

0 0

00

00

0 0

x11 x

21

x12 x

33

x32 x

44

x43x

24

(d)


119863= 2 and 119870 = 4)



119863= 2 119899

119879= 4 and 119870 = 4 is



for transmission




3 ln(119870Γ)]120585)



2119870


119879(1198702)log










0 that is

CCDF119875 (1198750) = Pr (119875 gt 119875

0) (9)




119875119894= max0le119899le119870minus1

|119904119894 (119899) |2 (10)








119901|119904|2 (

100381610038161003816100381611990411989410038161003816100381610038162) =

1

1205902119870119894

119890minus|119904119894|21205902

119870119894 (11)

where 1205902119870119894


119894(119899)| Note

that sum119899119879

119894=1119870119894

= 119899119863119870 thus we have sum

119899119879

119894=11205902119870119894

= 1198991198631205902 The


is given as

Pr (100381610038161003816100381611990411989410038161003816100381610038162le 120579) = 1 minus 119890

minus1205791205902

119870119894 120579 ge 0 (12)



as

CDF119875119894 = Pr (119875119894le 1198750)

= Pr (1003816100381610038161003816119904119894 (0)10038161003816100381610038162le 1198750)Pr (1003816100381610038161003816119904119894 (1)

10038161003816100381610038162le 1198750) sdot sdot sdot

Pr (1003816100381610038161003816119904119894 (119870 minus 1)10038161003816100381610038162le 1198750)

= (1 minus 119890minus1198750120590

2

119870119894 )119870

(13)


119875 = max1le119894le119899119879

119875119894 (14)


CDF119875 = Pr (119875 le 1198750)

= Pr (1198751le 1198750)Pr (119875


119899119879le 1198750)

=

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870

(15)



= 1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870

(16)


119894=

119899119863119870119899119879


signals are 1205902119870119894

= 1198991198631205902119899119879= 1205902119870 forall119894 = 1 2 119899

119879 As a result



2

119870)119899119879119870

(17)








2

119870)119899119879119870)119882

(19)





119879


120578PE

=sum119899119879

119894=1119875119894out

119899119879119875dc

=sum119899119879

119894=1119875119894in

119899119879119875dc

= 1205721198991198631205902

119899119879119875dc

=1198991198631205902119875sat

119899119879119875dc

1

119875=

1198991198631205902

2119899119879

1

119875

(21)


119894=1119875119894in = 120572119899



0) = Pr(120578PE gt 120578PE

0) to be the CCDF


PE



PE0) (22)

where


minus12120578PE0 )119899119879119870


119899119879

prod119894=1

(1 minus 119890minus1198991198631198702119899119879119870119894120578

PE0 )119870

(23)



120578PE

= 119864 120578PE =

1198991198631205902

2119899119879

1198641

119875 =

1198991198631205902

2119899119879

int1

119909119901 (119909) 119889119909 (24)


119901balance (119909) =119889

119889119909CDF119875balance (119909) =

119889

119889119909(1 minus 119890

minus1199091205902

119870)119899119879119870

=119899119879119870

1205902119870

119890minus1199091205902

119870(1 minus 119890minus1199091205902

119870)119899119879119870minus1

(25)

Hence

120578PEbalance =

1198991198631205902

2119899119879

int1

119909119901balance (119909) 119889119909

=119899119879119870

2int1198701205902

119870

1205902

119870

1

119909119890minus1199091205902

119870(1 minus 119890minus1199091205902

119870)119899119879119870minus1

119889119909

=119899119879119870

2int119870

1

1

119909119890minus119909

(1 minus 119890minus119909

)119899119879119870minus1

119889119909

(26)


120578PEimbalance =

1198991198631205902

2119899119879

int1

119909119901imbalance (119909) 119889119909

=1198991198631198701205902

2119899119879

int119870max120590

2

max

1205902

max

1

119909

119899119879

sum119895=1

119890minus1199091205902

119870119895

(1 minus 119890minus1199091205902

119870119895 ) 1205902119870119895

times

119899119879

prod119894=1

(1 minus 119890minus1199091205902

119870119894 )119870

119889119909

(27)

where 1205902max = max12059021198701

1205902119870119899119879

and 119870max = max1198701





) (28)

where



PE0 )119899119879119870

)119882

(29)



1198991198631205902

2119899119879

int1

119909119901balance+reduced (119909) 119889119909

=119899119879119870119882

2int119870

1

1

119909119890minus119909

(1 minus 119890minus119909

)119899119879119870minus1


)119899119879119870

)119882minus1

119889119909

(30)






1199051205902119899 where 119875

119905= 120572119899

1198631205902 = 120578

PE119899119879119875dc (cf (21))


119868 (120588proposedH)

=1

119870

119870minus1

sum119896=0

log2(det(I

119899119877+

120588proposed

119899119863

H119896H119867119896))

(31)


=1

119870

119870minus1

sum119896=0

log2(det(I

119899119877+

120588imbalance119899119863

H119896H119867119896))

(32)


2


120578PEimbalance119899119879119875dc120590

2

119899 Here 120578

PEproposed = 120578








119870

119870minus1

sum119896=0

Δ119868119896 (33)

where

Δ119868119896= log2(det(I

119899119877+

120588proposed

119899119863

H119896H119867119896))

minus log2(det(I

119899119877+

120588imbalance119899119863

H119896H119867119896))

(34)


119868119896= log2(det(

120588

119899119863

Ω119896)) (35)

where

Ω119896= Ω (H

119896) =

H119896H119867119896 119899119877le 119899119863

H119867119896H119896 119899119877gt 119899119863

(36)


Δ119868 =1

119870

119870minus1

sum119896=0

(log2(det(

120588proposed

119899119863

Ω119896))

minuslog2(det(

120588imbalance119899119863

Ω119896)))

= 119901log2

120578PEproposed

120578PEimbalance

minus1

119870

119870minus1

sum119896=0

Δ119896= 1198791+ 1198792

(37)

where 119901 = min(119899119863 119899119877)Ω119896= Ω(H

119896) and


119896)) minus log

2(det (Ω

119896)) (38)





119896=0Δ119896gt 0





2is a penalty


119896is fixed





119879and 119899





Δ119862 = 119864H Δ119868 le 119901log2119864H

120578PEproposed

120578PEimbalance

(39)



1 CDF of 119879

2 and CCDF


119879= 4 119899

119863= 2 119899

119877= 2 119870= 128


1gt 0 and 119879

2lt 0 Moreover



120578EE

=119862 (120588)

119875total (40)


119879119875dc + 119899

119879119875RF + 119875sp


1199051205902119899= 120578

PE119899119879119875dc1205902



120578EE

(119875dc) =119862 (120578

PE119899119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

(41)






119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEproposed 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(42)



119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEimbalance 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(43)


119863= 2 and 119882= 4 (ie 2 bits


6 Numerical Results


119863= 2 and 119899





0

10minus1

10minus2

minus2 0 2

W = 1

W = 4

W = 8

(a)

100

10minus1

10minus2

minus04 minus02 0

Empirical CDF of T2

(b)


0

10minus1

10minus2

minus1 minus05 0 05 1 15 2

W = 1

W = 4

W = 8

(c)




sum119870minus1

119899=0|119904119894(119899)|2 = 1205902119870











120578PE (Analysis) 00691 00757 00892 00943


PEimbalance

120578PEimbalance

mdash 1019 2826 3515

1 2 3 4 5 6 7 8 9

100

10minus1

10minus2

10minus3

Peak power P0



Pr (P

gtP0)



CCD

F

100

10minus1

10minus2

10minus3




4500

4000

3500

3000

2500

2000

1500

1000

Ergo

dic c

apac

ity (M

bps)

2 4 6 8 10 12 14

SNR (dB)

3300

3200

3100

3000

2900

2800

97 98 99 10 101


Proposed W = 4

Proposed W = 8




7 Conclusions



0 2 4 6 8 10400

600

800

1000

1200

1400

1600


Ener

gy effi

cien

cy (M

bits

J)


Proposed W = 4

Proposed W = 8



Appendices



1198701 12059021198702

1205902119870119899119879


11989110158401015840(120592) =

21198750120592 minus 11987520

1205924119890minus1198750120592 (A1)


0is large



1

119899119879

119899119879

sum119894=1

119891 (1205902

119870119894) le 119891(

1

119899119879

119899119879

sum119894=1

1205902

119870119894)

= 119891(1

119899119879

119899119879

sum119894=1

1205902

119870) = 119891 (120590

2

119870)

(A2)


119894=11205902119870119894

=

sum119899119879

119894=11205902119870


sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le 119899119879(1 minus 119890

minus11987501205902

119870) (A3)


= 1205902119870 forall119894 = 1 119899

119879



119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1

119899119879

119899119879

sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ))

119899119879

(A4)


= 1205902119870

forall119894 = 1 119899119879


119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1 minus 119890minus1198750120590

2

119870)119899119879

(A5)


= 1205902119870 forall119894 = 1 119899

119879 Thus we


1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870


2

119870)119899119879119870

(A6)

or








log2(det(Ω

119896))Ω119896



2(det(Ω


2(det(Ω

119896)) le log

2(det(Ω

119896))



119864 Δ119896 = 119864 log

2(det (Ω

119896)) minus 119864 log

2(det (Ω

119896))

le 120590119862radic

Γ (120574 + 1)

120574

(B1)


2(det(Ω









1205902

119862= [log

2(119890)]2

119901

sum119898=1

1205951015840(119902 minus 119898 + 1) (B2)


where 119901 = min(119899119863 119899119877) 119902 = max(119899

119863 119899119877) and 1205951015840(119909) =

suminfin

120593=11(120593 + 119909 minus 1)



1205902

119862= [log

2(119890)]2

119901

sum119898=1

1

119902 minus 119898 + 1 (B3)


119864 Δ119896 le log

2(119890)radic(

119901

sum119898=1

1

119902 minus 119898 + 1) times

Γ (120574 + 1)

120574 (B4)



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Antennas

Subcarriers

Tx 1

Tx 2

Tx 3

Tx 4

Data

Not used


Data

Not used

Antennas

Subcarriers

Tx 1

Tx 2

Tx 3

Tx 4



119863= 2 and 119870 = 12)

Ant

enna

s

Subcarriers


x11 x

21

x12

x32

x33 x

43

x24 x

44

0 0

0

00

0 0

0

(a)


Subcarriers

x12 x

24 0 0

00

00

0 0

x11 x

32

x33 x

43

x44

x21

(b)


Subcarriers

Ant

enna

s

x12

x21 0 0

00

00

0 0

x11 x

33

x32 x

43

x44

x24

(c)


Subcarriers

0 0

00

00

0 0

x11 x

21

x12 x

33

x32 x

44

x43x

24

(d)


119863= 2 and 119870 = 4)



119863= 2 119899

119879= 4 and 119870 = 4 is



for transmission




3 ln(119870Γ)]120585)



2119870


119879(1198702)log










0 that is

CCDF119875 (1198750) = Pr (119875 gt 119875

0) (9)




119875119894= max0le119899le119870minus1

|119904119894 (119899) |2 (10)








119901|119904|2 (

100381610038161003816100381611990411989410038161003816100381610038162) =

1

1205902119870119894

119890minus|119904119894|21205902

119870119894 (11)

where 1205902119870119894


119894(119899)| Note

that sum119899119879

119894=1119870119894

= 119899119863119870 thus we have sum

119899119879

119894=11205902119870119894

= 1198991198631205902 The


is given as

Pr (100381610038161003816100381611990411989410038161003816100381610038162le 120579) = 1 minus 119890

minus1205791205902

119870119894 120579 ge 0 (12)



as

CDF119875119894 = Pr (119875119894le 1198750)

= Pr (1003816100381610038161003816119904119894 (0)10038161003816100381610038162le 1198750)Pr (1003816100381610038161003816119904119894 (1)

10038161003816100381610038162le 1198750) sdot sdot sdot

Pr (1003816100381610038161003816119904119894 (119870 minus 1)10038161003816100381610038162le 1198750)

= (1 minus 119890minus1198750120590

2

119870119894 )119870

(13)


119875 = max1le119894le119899119879

119875119894 (14)


CDF119875 = Pr (119875 le 1198750)

= Pr (1198751le 1198750)Pr (119875


119899119879le 1198750)

=

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870

(15)



= 1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870

(16)


119894=

119899119863119870119899119879


signals are 1205902119870119894

= 1198991198631205902119899119879= 1205902119870 forall119894 = 1 2 119899

119879 As a result



2

119870)119899119879119870

(17)








2

119870)119899119879119870)119882

(19)





119879


120578PE

=sum119899119879

119894=1119875119894out

119899119879119875dc

=sum119899119879

119894=1119875119894in

119899119879119875dc

= 1205721198991198631205902

119899119879119875dc

=1198991198631205902119875sat

119899119879119875dc

1

119875=

1198991198631205902

2119899119879

1

119875

(21)


119894=1119875119894in = 120572119899



0) = Pr(120578PE gt 120578PE

0) to be the CCDF


PE



PE0) (22)

where


minus12120578PE0 )119899119879119870


119899119879

prod119894=1

(1 minus 119890minus1198991198631198702119899119879119870119894120578

PE0 )119870

(23)



120578PE

= 119864 120578PE =

1198991198631205902

2119899119879

1198641

119875 =

1198991198631205902

2119899119879

int1

119909119901 (119909) 119889119909 (24)


119901balance (119909) =119889

119889119909CDF119875balance (119909) =

119889

119889119909(1 minus 119890

minus1199091205902

119870)119899119879119870

=119899119879119870

1205902119870

119890minus1199091205902

119870(1 minus 119890minus1199091205902

119870)119899119879119870minus1

(25)

Hence

120578PEbalance =

1198991198631205902

2119899119879

int1

119909119901balance (119909) 119889119909

=119899119879119870

2int1198701205902

119870

1205902

119870

1

119909119890minus1199091205902

119870(1 minus 119890minus1199091205902

119870)119899119879119870minus1

119889119909

=119899119879119870

2int119870

1

1

119909119890minus119909

(1 minus 119890minus119909

)119899119879119870minus1

119889119909

(26)


120578PEimbalance =

1198991198631205902

2119899119879

int1

119909119901imbalance (119909) 119889119909

=1198991198631198701205902

2119899119879

int119870max120590

2

max

1205902

max

1

119909

119899119879

sum119895=1

119890minus1199091205902

119870119895

(1 minus 119890minus1199091205902

119870119895 ) 1205902119870119895

times

119899119879

prod119894=1

(1 minus 119890minus1199091205902

119870119894 )119870

119889119909

(27)

where 1205902max = max12059021198701

1205902119870119899119879

and 119870max = max1198701





) (28)

where



PE0 )119899119879119870

)119882

(29)



1198991198631205902

2119899119879

int1

119909119901balance+reduced (119909) 119889119909

=119899119879119870119882

2int119870

1

1

119909119890minus119909

(1 minus 119890minus119909

)119899119879119870minus1


)119899119879119870

)119882minus1

119889119909

(30)






1199051205902119899 where 119875

119905= 120572119899

1198631205902 = 120578

PE119899119879119875dc (cf (21))


119868 (120588proposedH)

=1

119870

119870minus1

sum119896=0

log2(det(I

119899119877+

120588proposed

119899119863

H119896H119867119896))

(31)


=1

119870

119870minus1

sum119896=0

log2(det(I

119899119877+

120588imbalance119899119863

H119896H119867119896))

(32)


2


120578PEimbalance119899119879119875dc120590

2

119899 Here 120578

PEproposed = 120578








119870

119870minus1

sum119896=0

Δ119868119896 (33)

where

Δ119868119896= log2(det(I

119899119877+

120588proposed

119899119863

H119896H119867119896))

minus log2(det(I

119899119877+

120588imbalance119899119863

H119896H119867119896))

(34)


119868119896= log2(det(

120588

119899119863

Ω119896)) (35)

where

Ω119896= Ω (H

119896) =

H119896H119867119896 119899119877le 119899119863

H119867119896H119896 119899119877gt 119899119863

(36)


Δ119868 =1

119870

119870minus1

sum119896=0

(log2(det(

120588proposed

119899119863

Ω119896))

minuslog2(det(

120588imbalance119899119863

Ω119896)))

= 119901log2

120578PEproposed

120578PEimbalance

minus1

119870

119870minus1

sum119896=0

Δ119896= 1198791+ 1198792

(37)

where 119901 = min(119899119863 119899119877)Ω119896= Ω(H

119896) and


119896)) minus log

2(det (Ω

119896)) (38)





119896=0Δ119896gt 0





2is a penalty


119896is fixed





119879and 119899





Δ119862 = 119864H Δ119868 le 119901log2119864H

120578PEproposed

120578PEimbalance

(39)



1 CDF of 119879

2 and CCDF


119879= 4 119899

119863= 2 119899

119877= 2 119870= 128


1gt 0 and 119879

2lt 0 Moreover



120578EE

=119862 (120588)

119875total (40)


119879119875dc + 119899

119879119875RF + 119875sp


1199051205902119899= 120578

PE119899119879119875dc1205902



120578EE

(119875dc) =119862 (120578

PE119899119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

(41)






119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEproposed 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(42)



119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEimbalance 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(43)


119863= 2 and 119882= 4 (ie 2 bits


6 Numerical Results


119863= 2 and 119899





0

10minus1

10minus2

minus2 0 2

W = 1

W = 4

W = 8

(a)

100

10minus1

10minus2

minus04 minus02 0

Empirical CDF of T2

(b)


0

10minus1

10minus2

minus1 minus05 0 05 1 15 2

W = 1

W = 4

W = 8

(c)




sum119870minus1

119899=0|119904119894(119899)|2 = 1205902119870











120578PE (Analysis) 00691 00757 00892 00943


PEimbalance

120578PEimbalance

mdash 1019 2826 3515

1 2 3 4 5 6 7 8 9

100

10minus1

10minus2

10minus3

Peak power P0



Pr (P

gtP0)



CCD

F

100

10minus1

10minus2

10minus3




4500

4000

3500

3000

2500

2000

1500

1000

Ergo

dic c

apac

ity (M

bps)

2 4 6 8 10 12 14

SNR (dB)

3300

3200

3100

3000

2900

2800

97 98 99 10 101


Proposed W = 4

Proposed W = 8




7 Conclusions



0 2 4 6 8 10400

600

800

1000

1200

1400

1600


Ener

gy effi

cien

cy (M

bits

J)


Proposed W = 4

Proposed W = 8



Appendices



1198701 12059021198702

1205902119870119899119879


11989110158401015840(120592) =

21198750120592 minus 11987520

1205924119890minus1198750120592 (A1)


0is large



1

119899119879

119899119879

sum119894=1

119891 (1205902

119870119894) le 119891(

1

119899119879

119899119879

sum119894=1

1205902

119870119894)

= 119891(1

119899119879

119899119879

sum119894=1

1205902

119870) = 119891 (120590

2

119870)

(A2)


119894=11205902119870119894

=

sum119899119879

119894=11205902119870


sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le 119899119879(1 minus 119890

minus11987501205902

119870) (A3)


= 1205902119870 forall119894 = 1 119899

119879



119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1

119899119879

119899119879

sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ))

119899119879

(A4)


= 1205902119870

forall119894 = 1 119899119879


119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1 minus 119890minus1198750120590

2

119870)119899119879

(A5)


= 1205902119870 forall119894 = 1 119899

119879 Thus we


1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870


2

119870)119899119879119870

(A6)

or








log2(det(Ω

119896))Ω119896



2(det(Ω


2(det(Ω

119896)) le log

2(det(Ω

119896))



119864 Δ119896 = 119864 log

2(det (Ω

119896)) minus 119864 log

2(det (Ω

119896))

le 120590119862radic

Γ (120574 + 1)

120574

(B1)


2(det(Ω









1205902

119862= [log

2(119890)]2

119901

sum119898=1

1205951015840(119902 minus 119898 + 1) (B2)


where 119901 = min(119899119863 119899119877) 119902 = max(119899

119863 119899119877) and 1205951015840(119909) =

suminfin

120593=11(120593 + 119909 minus 1)



1205902

119862= [log

2(119890)]2

119901

sum119898=1

1

119902 minus 119898 + 1 (B3)


119864 Δ119896 le log

2(119890)radic(

119901

sum119898=1

1

119902 minus 119898 + 1) times

Γ (120574 + 1)

120574 (B4)



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0 that is

CCDF119875 (1198750) = Pr (119875 gt 119875

0) (9)




119875119894= max0le119899le119870minus1

|119904119894 (119899) |2 (10)








119901|119904|2 (

100381610038161003816100381611990411989410038161003816100381610038162) =

1

1205902119870119894

119890minus|119904119894|21205902

119870119894 (11)

where 1205902119870119894


119894(119899)| Note

that sum119899119879

119894=1119870119894

= 119899119863119870 thus we have sum

119899119879

119894=11205902119870119894

= 1198991198631205902 The


is given as

Pr (100381610038161003816100381611990411989410038161003816100381610038162le 120579) = 1 minus 119890

minus1205791205902

119870119894 120579 ge 0 (12)



as

CDF119875119894 = Pr (119875119894le 1198750)

= Pr (1003816100381610038161003816119904119894 (0)10038161003816100381610038162le 1198750)Pr (1003816100381610038161003816119904119894 (1)

10038161003816100381610038162le 1198750) sdot sdot sdot

Pr (1003816100381610038161003816119904119894 (119870 minus 1)10038161003816100381610038162le 1198750)

= (1 minus 119890minus1198750120590

2

119870119894 )119870

(13)


119875 = max1le119894le119899119879

119875119894 (14)


CDF119875 = Pr (119875 le 1198750)

= Pr (1198751le 1198750)Pr (119875


119899119879le 1198750)

=

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870

(15)



= 1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870

(16)


119894=

119899119863119870119899119879


signals are 1205902119870119894

= 1198991198631205902119899119879= 1205902119870 forall119894 = 1 2 119899

119879 As a result



2

119870)119899119879119870

(17)








2

119870)119899119879119870)119882

(19)





119879


120578PE

=sum119899119879

119894=1119875119894out

119899119879119875dc

=sum119899119879

119894=1119875119894in

119899119879119875dc

= 1205721198991198631205902

119899119879119875dc

=1198991198631205902119875sat

119899119879119875dc

1

119875=

1198991198631205902

2119899119879

1

119875

(21)


119894=1119875119894in = 120572119899



0) = Pr(120578PE gt 120578PE

0) to be the CCDF


PE



PE0) (22)

where


minus12120578PE0 )119899119879119870


119899119879

prod119894=1

(1 minus 119890minus1198991198631198702119899119879119870119894120578

PE0 )119870

(23)



120578PE

= 119864 120578PE =

1198991198631205902

2119899119879

1198641

119875 =

1198991198631205902

2119899119879

int1

119909119901 (119909) 119889119909 (24)


119901balance (119909) =119889

119889119909CDF119875balance (119909) =

119889

119889119909(1 minus 119890

minus1199091205902

119870)119899119879119870

=119899119879119870

1205902119870

119890minus1199091205902

119870(1 minus 119890minus1199091205902

119870)119899119879119870minus1

(25)

Hence

120578PEbalance =

1198991198631205902

2119899119879

int1

119909119901balance (119909) 119889119909

=119899119879119870

2int1198701205902

119870

1205902

119870

1

119909119890minus1199091205902

119870(1 minus 119890minus1199091205902

119870)119899119879119870minus1

119889119909

=119899119879119870

2int119870

1

1

119909119890minus119909

(1 minus 119890minus119909

)119899119879119870minus1

119889119909

(26)


120578PEimbalance =

1198991198631205902

2119899119879

int1

119909119901imbalance (119909) 119889119909

=1198991198631198701205902

2119899119879

int119870max120590

2

max

1205902

max

1

119909

119899119879

sum119895=1

119890minus1199091205902

119870119895

(1 minus 119890minus1199091205902

119870119895 ) 1205902119870119895

times

119899119879

prod119894=1

(1 minus 119890minus1199091205902

119870119894 )119870

119889119909

(27)

where 1205902max = max12059021198701

1205902119870119899119879

and 119870max = max1198701





) (28)

where



PE0 )119899119879119870

)119882

(29)



1198991198631205902

2119899119879

int1

119909119901balance+reduced (119909) 119889119909

=119899119879119870119882

2int119870

1

1

119909119890minus119909

(1 minus 119890minus119909

)119899119879119870minus1


)119899119879119870

)119882minus1

119889119909

(30)






1199051205902119899 where 119875

119905= 120572119899

1198631205902 = 120578

PE119899119879119875dc (cf (21))


119868 (120588proposedH)

=1

119870

119870minus1

sum119896=0

log2(det(I

119899119877+

120588proposed

119899119863

H119896H119867119896))

(31)


=1

119870

119870minus1

sum119896=0

log2(det(I

119899119877+

120588imbalance119899119863

H119896H119867119896))

(32)


2


120578PEimbalance119899119879119875dc120590

2

119899 Here 120578

PEproposed = 120578








119870

119870minus1

sum119896=0

Δ119868119896 (33)

where

Δ119868119896= log2(det(I

119899119877+

120588proposed

119899119863

H119896H119867119896))

minus log2(det(I

119899119877+

120588imbalance119899119863

H119896H119867119896))

(34)


119868119896= log2(det(

120588

119899119863

Ω119896)) (35)

where

Ω119896= Ω (H

119896) =

H119896H119867119896 119899119877le 119899119863

H119867119896H119896 119899119877gt 119899119863

(36)


Δ119868 =1

119870

119870minus1

sum119896=0

(log2(det(

120588proposed

119899119863

Ω119896))

minuslog2(det(

120588imbalance119899119863

Ω119896)))

= 119901log2

120578PEproposed

120578PEimbalance

minus1

119870

119870minus1

sum119896=0

Δ119896= 1198791+ 1198792

(37)

where 119901 = min(119899119863 119899119877)Ω119896= Ω(H

119896) and


119896)) minus log

2(det (Ω

119896)) (38)





119896=0Δ119896gt 0





2is a penalty


119896is fixed





119879and 119899





Δ119862 = 119864H Δ119868 le 119901log2119864H

120578PEproposed

120578PEimbalance

(39)



1 CDF of 119879

2 and CCDF


119879= 4 119899

119863= 2 119899

119877= 2 119870= 128


1gt 0 and 119879

2lt 0 Moreover



120578EE

=119862 (120588)

119875total (40)


119879119875dc + 119899

119879119875RF + 119875sp


1199051205902119899= 120578

PE119899119879119875dc1205902



120578EE

(119875dc) =119862 (120578

PE119899119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

(41)






119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEproposed 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(42)



119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEimbalance 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(43)


119863= 2 and 119882= 4 (ie 2 bits


6 Numerical Results


119863= 2 and 119899





0

10minus1

10minus2

minus2 0 2

W = 1

W = 4

W = 8

(a)

100

10minus1

10minus2

minus04 minus02 0

Empirical CDF of T2

(b)


0

10minus1

10minus2

minus1 minus05 0 05 1 15 2

W = 1

W = 4

W = 8

(c)




sum119870minus1

119899=0|119904119894(119899)|2 = 1205902119870











120578PE (Analysis) 00691 00757 00892 00943


PEimbalance

120578PEimbalance

mdash 1019 2826 3515

1 2 3 4 5 6 7 8 9

100

10minus1

10minus2

10minus3

Peak power P0



Pr (P

gtP0)



CCD

F

100

10minus1

10minus2

10minus3




4500

4000

3500

3000

2500

2000

1500

1000

Ergo

dic c

apac

ity (M

bps)

2 4 6 8 10 12 14

SNR (dB)

3300

3200

3100

3000

2900

2800

97 98 99 10 101


Proposed W = 4

Proposed W = 8




7 Conclusions



0 2 4 6 8 10400

600

800

1000

1200

1400

1600


Ener

gy effi

cien

cy (M

bits

J)


Proposed W = 4

Proposed W = 8



Appendices



1198701 12059021198702

1205902119870119899119879


11989110158401015840(120592) =

21198750120592 minus 11987520

1205924119890minus1198750120592 (A1)


0is large



1

119899119879

119899119879

sum119894=1

119891 (1205902

119870119894) le 119891(

1

119899119879

119899119879

sum119894=1

1205902

119870119894)

= 119891(1

119899119879

119899119879

sum119894=1

1205902

119870) = 119891 (120590

2

119870)

(A2)


119894=11205902119870119894

=

sum119899119879

119894=11205902119870


sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le 119899119879(1 minus 119890

minus11987501205902

119870) (A3)


= 1205902119870 forall119894 = 1 119899

119879



119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1

119899119879

119899119879

sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ))

119899119879

(A4)


= 1205902119870

forall119894 = 1 119899119879


119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1 minus 119890minus1198750120590

2

119870)119899119879

(A5)


= 1205902119870 forall119894 = 1 119899

119879 Thus we


1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870


2

119870)119899119879119870

(A6)

or








log2(det(Ω

119896))Ω119896



2(det(Ω


2(det(Ω

119896)) le log

2(det(Ω

119896))



119864 Δ119896 = 119864 log

2(det (Ω

119896)) minus 119864 log

2(det (Ω

119896))

le 120590119862radic

Γ (120574 + 1)

120574

(B1)


2(det(Ω









1205902

119862= [log

2(119890)]2

119901

sum119898=1

1205951015840(119902 minus 119898 + 1) (B2)


where 119901 = min(119899119863 119899119877) 119902 = max(119899

119863 119899119877) and 1205951015840(119909) =

suminfin

120593=11(120593 + 119909 minus 1)



1205902

119862= [log

2(119890)]2

119901

sum119898=1

1

119902 minus 119898 + 1 (B3)


119864 Δ119896 le log

2(119890)radic(

119901

sum119898=1

1

119902 minus 119898 + 1) times

Γ (120574 + 1)

120574 (B4)



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119879


120578PE

=sum119899119879

119894=1119875119894out

119899119879119875dc

=sum119899119879

119894=1119875119894in

119899119879119875dc

= 1205721198991198631205902

119899119879119875dc

=1198991198631205902119875sat

119899119879119875dc

1

119875=

1198991198631205902

2119899119879

1

119875

(21)


119894=1119875119894in = 120572119899



0) = Pr(120578PE gt 120578PE

0) to be the CCDF


PE



PE0) (22)

where


minus12120578PE0 )119899119879119870


119899119879

prod119894=1

(1 minus 119890minus1198991198631198702119899119879119870119894120578

PE0 )119870

(23)



120578PE

= 119864 120578PE =

1198991198631205902

2119899119879

1198641

119875 =

1198991198631205902

2119899119879

int1

119909119901 (119909) 119889119909 (24)


119901balance (119909) =119889

119889119909CDF119875balance (119909) =

119889

119889119909(1 minus 119890

minus1199091205902

119870)119899119879119870

=119899119879119870

1205902119870

119890minus1199091205902

119870(1 minus 119890minus1199091205902

119870)119899119879119870minus1

(25)

Hence

120578PEbalance =

1198991198631205902

2119899119879

int1

119909119901balance (119909) 119889119909

=119899119879119870

2int1198701205902

119870

1205902

119870

1

119909119890minus1199091205902

119870(1 minus 119890minus1199091205902

119870)119899119879119870minus1

119889119909

=119899119879119870

2int119870

1

1

119909119890minus119909

(1 minus 119890minus119909

)119899119879119870minus1

119889119909

(26)


120578PEimbalance =

1198991198631205902

2119899119879

int1

119909119901imbalance (119909) 119889119909

=1198991198631198701205902

2119899119879

int119870max120590

2

max

1205902

max

1

119909

119899119879

sum119895=1

119890minus1199091205902

119870119895

(1 minus 119890minus1199091205902

119870119895 ) 1205902119870119895

times

119899119879

prod119894=1

(1 minus 119890minus1199091205902

119870119894 )119870

119889119909

(27)

where 1205902max = max12059021198701

1205902119870119899119879

and 119870max = max1198701





) (28)

where



PE0 )119899119879119870

)119882

(29)



1198991198631205902

2119899119879

int1

119909119901balance+reduced (119909) 119889119909

=119899119879119870119882

2int119870

1

1

119909119890minus119909

(1 minus 119890minus119909

)119899119879119870minus1


)119899119879119870

)119882minus1

119889119909

(30)






1199051205902119899 where 119875

119905= 120572119899

1198631205902 = 120578

PE119899119879119875dc (cf (21))


119868 (120588proposedH)

=1

119870

119870minus1

sum119896=0

log2(det(I

119899119877+

120588proposed

119899119863

H119896H119867119896))

(31)


=1

119870

119870minus1

sum119896=0

log2(det(I

119899119877+

120588imbalance119899119863

H119896H119867119896))

(32)


2


120578PEimbalance119899119879119875dc120590

2

119899 Here 120578

PEproposed = 120578








119870

119870minus1

sum119896=0

Δ119868119896 (33)

where

Δ119868119896= log2(det(I

119899119877+

120588proposed

119899119863

H119896H119867119896))

minus log2(det(I

119899119877+

120588imbalance119899119863

H119896H119867119896))

(34)


119868119896= log2(det(

120588

119899119863

Ω119896)) (35)

where

Ω119896= Ω (H

119896) =

H119896H119867119896 119899119877le 119899119863

H119867119896H119896 119899119877gt 119899119863

(36)


Δ119868 =1

119870

119870minus1

sum119896=0

(log2(det(

120588proposed

119899119863

Ω119896))

minuslog2(det(

120588imbalance119899119863

Ω119896)))

= 119901log2

120578PEproposed

120578PEimbalance

minus1

119870

119870minus1

sum119896=0

Δ119896= 1198791+ 1198792

(37)

where 119901 = min(119899119863 119899119877)Ω119896= Ω(H

119896) and


119896)) minus log

2(det (Ω

119896)) (38)





119896=0Δ119896gt 0





2is a penalty


119896is fixed





119879and 119899





Δ119862 = 119864H Δ119868 le 119901log2119864H

120578PEproposed

120578PEimbalance

(39)



1 CDF of 119879

2 and CCDF


119879= 4 119899

119863= 2 119899

119877= 2 119870= 128


1gt 0 and 119879

2lt 0 Moreover



120578EE

=119862 (120588)

119875total (40)


119879119875dc + 119899

119879119875RF + 119875sp


1199051205902119899= 120578

PE119899119879119875dc1205902



120578EE

(119875dc) =119862 (120578

PE119899119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

(41)






119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEproposed 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(42)



119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEimbalance 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(43)


119863= 2 and 119882= 4 (ie 2 bits


6 Numerical Results


119863= 2 and 119899





0

10minus1

10minus2

minus2 0 2

W = 1

W = 4

W = 8

(a)

100

10minus1

10minus2

minus04 minus02 0

Empirical CDF of T2

(b)


0

10minus1

10minus2

minus1 minus05 0 05 1 15 2

W = 1

W = 4

W = 8

(c)




sum119870minus1

119899=0|119904119894(119899)|2 = 1205902119870











120578PE (Analysis) 00691 00757 00892 00943


PEimbalance

120578PEimbalance

mdash 1019 2826 3515

1 2 3 4 5 6 7 8 9

100

10minus1

10minus2

10minus3

Peak power P0



Pr (P

gtP0)



CCD

F

100

10minus1

10minus2

10minus3




4500

4000

3500

3000

2500

2000

1500

1000

Ergo

dic c

apac

ity (M

bps)

2 4 6 8 10 12 14

SNR (dB)

3300

3200

3100

3000

2900

2800

97 98 99 10 101


Proposed W = 4

Proposed W = 8




7 Conclusions



0 2 4 6 8 10400

600

800

1000

1200

1400

1600


Ener

gy effi

cien

cy (M

bits

J)


Proposed W = 4

Proposed W = 8



Appendices



1198701 12059021198702

1205902119870119899119879


11989110158401015840(120592) =

21198750120592 minus 11987520

1205924119890minus1198750120592 (A1)


0is large



1

119899119879

119899119879

sum119894=1

119891 (1205902

119870119894) le 119891(

1

119899119879

119899119879

sum119894=1

1205902

119870119894)

= 119891(1

119899119879

119899119879

sum119894=1

1205902

119870) = 119891 (120590

2

119870)

(A2)


119894=11205902119870119894

=

sum119899119879

119894=11205902119870


sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le 119899119879(1 minus 119890

minus11987501205902

119870) (A3)


= 1205902119870 forall119894 = 1 119899

119879



119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1

119899119879

119899119879

sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ))

119899119879

(A4)


= 1205902119870

forall119894 = 1 119899119879


119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1 minus 119890minus1198750120590

2

119870)119899119879

(A5)


= 1205902119870 forall119894 = 1 119899

119879 Thus we


1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870


2

119870)119899119879119870

(A6)

or








log2(det(Ω

119896))Ω119896



2(det(Ω


2(det(Ω

119896)) le log

2(det(Ω

119896))



119864 Δ119896 = 119864 log

2(det (Ω

119896)) minus 119864 log

2(det (Ω

119896))

le 120590119862radic

Γ (120574 + 1)

120574

(B1)


2(det(Ω









1205902

119862= [log

2(119890)]2

119901

sum119898=1

1205951015840(119902 minus 119898 + 1) (B2)


where 119901 = min(119899119863 119899119877) 119902 = max(119899

119863 119899119877) and 1205951015840(119909) =

suminfin

120593=11(120593 + 119909 minus 1)



1205902

119862= [log

2(119890)]2

119901

sum119898=1

1

119902 minus 119898 + 1 (B3)


119864 Δ119896 le log

2(119890)radic(

119901

sum119898=1

1

119902 minus 119898 + 1) times

Γ (120574 + 1)

120574 (B4)



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













1199051205902119899 where 119875

119905= 120572119899

1198631205902 = 120578

PE119899119879119875dc (cf (21))


119868 (120588proposedH)

=1

119870

119870minus1

sum119896=0

log2(det(I

119899119877+

120588proposed

119899119863

H119896H119867119896))

(31)


=1

119870

119870minus1

sum119896=0

log2(det(I

119899119877+

120588imbalance119899119863

H119896H119867119896))

(32)


2


120578PEimbalance119899119879119875dc120590

2

119899 Here 120578

PEproposed = 120578








119870

119870minus1

sum119896=0

Δ119868119896 (33)

where

Δ119868119896= log2(det(I

119899119877+

120588proposed

119899119863

H119896H119867119896))

minus log2(det(I

119899119877+

120588imbalance119899119863

H119896H119867119896))

(34)


119868119896= log2(det(

120588

119899119863

Ω119896)) (35)

where

Ω119896= Ω (H

119896) =

H119896H119867119896 119899119877le 119899119863

H119867119896H119896 119899119877gt 119899119863

(36)


Δ119868 =1

119870

119870minus1

sum119896=0

(log2(det(

120588proposed

119899119863

Ω119896))

minuslog2(det(

120588imbalance119899119863

Ω119896)))

= 119901log2

120578PEproposed

120578PEimbalance

minus1

119870

119870minus1

sum119896=0

Δ119896= 1198791+ 1198792

(37)

where 119901 = min(119899119863 119899119877)Ω119896= Ω(H

119896) and


119896)) minus log

2(det (Ω

119896)) (38)





119896=0Δ119896gt 0





2is a penalty


119896is fixed





119879and 119899





Δ119862 = 119864H Δ119868 le 119901log2119864H

120578PEproposed

120578PEimbalance

(39)



1 CDF of 119879

2 and CCDF


119879= 4 119899

119863= 2 119899

119877= 2 119870= 128


1gt 0 and 119879

2lt 0 Moreover



120578EE

=119862 (120588)

119875total (40)


119879119875dc + 119899

119879119875RF + 119875sp


1199051205902119899= 120578

PE119899119879119875dc1205902



120578EE

(119875dc) =119862 (120578

PE119899119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

(41)






119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEproposed 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(42)



119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEimbalance 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(43)


119863= 2 and 119882= 4 (ie 2 bits


6 Numerical Results


119863= 2 and 119899





0

10minus1

10minus2

minus2 0 2

W = 1

W = 4

W = 8

(a)

100

10minus1

10minus2

minus04 minus02 0

Empirical CDF of T2

(b)


0

10minus1

10minus2

minus1 minus05 0 05 1 15 2

W = 1

W = 4

W = 8

(c)




sum119870minus1

119899=0|119904119894(119899)|2 = 1205902119870











120578PE (Analysis) 00691 00757 00892 00943


PEimbalance

120578PEimbalance

mdash 1019 2826 3515

1 2 3 4 5 6 7 8 9

100

10minus1

10minus2

10minus3

Peak power P0



Pr (P

gtP0)



CCD

F

100

10minus1

10minus2

10minus3




4500

4000

3500

3000

2500

2000

1500

1000

Ergo

dic c

apac

ity (M

bps)

2 4 6 8 10 12 14

SNR (dB)

3300

3200

3100

3000

2900

2800

97 98 99 10 101


Proposed W = 4

Proposed W = 8




7 Conclusions



0 2 4 6 8 10400

600

800

1000

1200

1400

1600


Ener

gy effi

cien

cy (M

bits

J)


Proposed W = 4

Proposed W = 8



Appendices



1198701 12059021198702

1205902119870119899119879


11989110158401015840(120592) =

21198750120592 minus 11987520

1205924119890minus1198750120592 (A1)


0is large



1

119899119879

119899119879

sum119894=1

119891 (1205902

119870119894) le 119891(

1

119899119879

119899119879

sum119894=1

1205902

119870119894)

= 119891(1

119899119879

119899119879

sum119894=1

1205902

119870) = 119891 (120590

2

119870)

(A2)


119894=11205902119870119894

=

sum119899119879

119894=11205902119870


sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le 119899119879(1 minus 119890

minus11987501205902

119870) (A3)


= 1205902119870 forall119894 = 1 119899

119879



119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1

119899119879

119899119879

sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ))

119899119879

(A4)


= 1205902119870

forall119894 = 1 119899119879


119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1 minus 119890minus1198750120590

2

119870)119899119879

(A5)


= 1205902119870 forall119894 = 1 119899

119879 Thus we


1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870


2

119870)119899119879119870

(A6)

or








log2(det(Ω

119896))Ω119896



2(det(Ω


2(det(Ω

119896)) le log

2(det(Ω

119896))



119864 Δ119896 = 119864 log

2(det (Ω

119896)) minus 119864 log

2(det (Ω

119896))

le 120590119862radic

Γ (120574 + 1)

120574

(B1)


2(det(Ω









1205902

119862= [log

2(119890)]2

119901

sum119898=1

1205951015840(119902 minus 119898 + 1) (B2)


where 119901 = min(119899119863 119899119877) 119902 = max(119899

119863 119899119877) and 1205951015840(119909) =

suminfin

120593=11(120593 + 119909 minus 1)



1205902

119862= [log

2(119890)]2

119901

sum119898=1

1

119902 minus 119898 + 1 (B3)


119864 Δ119896 le log

2(119890)radic(

119901

sum119898=1

1

119902 minus 119898 + 1) times

Γ (120574 + 1)

120574 (B4)



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Δ119862 = 119864H Δ119868 le 119901log2119864H

120578PEproposed

120578PEimbalance

(39)



1 CDF of 119879

2 and CCDF


119879= 4 119899

119863= 2 119899

119877= 2 119870= 128


1gt 0 and 119879

2lt 0 Moreover



120578EE

=119862 (120588)

119875total (40)


119879119875dc + 119899

119879119875RF + 119875sp


1199051205902119899= 120578

PE119899119879119875dc1205902



120578EE

(119875dc) =119862 (120578

PE119899119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

(41)






119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEproposed 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(42)



119879119875dc1205902

119899)

119899119879119875dc + 119899

119879119875RF + 119875sp

=119864H 119868 (120578

PEimbalance 119899

119879119875dc1205902

119899H)

119899119879119875dc + 119899

119879119875RF + 119875sp

(43)


119863= 2 and 119882= 4 (ie 2 bits


6 Numerical Results


119863= 2 and 119899





0

10minus1

10minus2

minus2 0 2

W = 1

W = 4

W = 8

(a)

100

10minus1

10minus2

minus04 minus02 0

Empirical CDF of T2

(b)


0

10minus1

10minus2

minus1 minus05 0 05 1 15 2

W = 1

W = 4

W = 8

(c)




sum119870minus1

119899=0|119904119894(119899)|2 = 1205902119870











120578PE (Analysis) 00691 00757 00892 00943


PEimbalance

120578PEimbalance

mdash 1019 2826 3515

1 2 3 4 5 6 7 8 9

100

10minus1

10minus2

10minus3

Peak power P0



Pr (P

gtP0)



CCD

F

100

10minus1

10minus2

10minus3




4500

4000

3500

3000

2500

2000

1500

1000

Ergo

dic c

apac

ity (M

bps)

2 4 6 8 10 12 14

SNR (dB)

3300

3200

3100

3000

2900

2800

97 98 99 10 101


Proposed W = 4

Proposed W = 8




7 Conclusions



0 2 4 6 8 10400

600

800

1000

1200

1400

1600


Ener

gy effi

cien

cy (M

bits

J)


Proposed W = 4

Proposed W = 8



Appendices



1198701 12059021198702

1205902119870119899119879


11989110158401015840(120592) =

21198750120592 minus 11987520

1205924119890minus1198750120592 (A1)


0is large



1

119899119879

119899119879

sum119894=1

119891 (1205902

119870119894) le 119891(

1

119899119879

119899119879

sum119894=1

1205902

119870119894)

= 119891(1

119899119879

119899119879

sum119894=1

1205902

119870) = 119891 (120590

2

119870)

(A2)


119894=11205902119870119894

=

sum119899119879

119894=11205902119870


sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le 119899119879(1 minus 119890

minus11987501205902

119870) (A3)


= 1205902119870 forall119894 = 1 119899

119879



119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1

119899119879

119899119879

sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ))

119899119879

(A4)


= 1205902119870

forall119894 = 1 119899119879


119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1 minus 119890minus1198750120590

2

119870)119899119879

(A5)


= 1205902119870 forall119894 = 1 119899

119879 Thus we


1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870


2

119870)119899119879119870

(A6)

or








log2(det(Ω

119896))Ω119896



2(det(Ω


2(det(Ω

119896)) le log

2(det(Ω

119896))



119864 Δ119896 = 119864 log

2(det (Ω

119896)) minus 119864 log

2(det (Ω

119896))

le 120590119862radic

Γ (120574 + 1)

120574

(B1)


2(det(Ω









1205902

119862= [log

2(119890)]2

119901

sum119898=1

1205951015840(119902 minus 119898 + 1) (B2)


where 119901 = min(119899119863 119899119877) 119902 = max(119899

119863 119899119877) and 1205951015840(119909) =

suminfin

120593=11(120593 + 119909 minus 1)



1205902

119862= [log

2(119890)]2

119901

sum119898=1

1

119902 minus 119898 + 1 (B3)


119864 Δ119896 le log

2(119890)radic(

119901

sum119898=1

1

119902 minus 119898 + 1) times

Γ (120574 + 1)

120574 (B4)



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

10minus1

10minus2

minus2 0 2

W = 1

W = 4

W = 8

(a)

100

10minus1

10minus2

minus04 minus02 0

Empirical CDF of T2

(b)


0

10minus1

10minus2

minus1 minus05 0 05 1 15 2

W = 1

W = 4

W = 8

(c)




sum119870minus1

119899=0|119904119894(119899)|2 = 1205902119870











120578PE (Analysis) 00691 00757 00892 00943


PEimbalance

120578PEimbalance

mdash 1019 2826 3515

1 2 3 4 5 6 7 8 9

100

10minus1

10minus2

10minus3

Peak power P0



Pr (P

gtP0)



CCD

F

100

10minus1

10minus2

10minus3




4500

4000

3500

3000

2500

2000

1500

1000

Ergo

dic c

apac

ity (M

bps)

2 4 6 8 10 12 14

SNR (dB)

3300

3200

3100

3000

2900

2800

97 98 99 10 101


Proposed W = 4

Proposed W = 8




7 Conclusions



0 2 4 6 8 10400

600

800

1000

1200

1400

1600


Ener

gy effi

cien

cy (M

bits

J)


Proposed W = 4

Proposed W = 8



Appendices



1198701 12059021198702

1205902119870119899119879


11989110158401015840(120592) =

21198750120592 minus 11987520

1205924119890minus1198750120592 (A1)


0is large



1

119899119879

119899119879

sum119894=1

119891 (1205902

119870119894) le 119891(

1

119899119879

119899119879

sum119894=1

1205902

119870119894)

= 119891(1

119899119879

119899119879

sum119894=1

1205902

119870) = 119891 (120590

2

119870)

(A2)


119894=11205902119870119894

=

sum119899119879

119894=11205902119870


sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le 119899119879(1 minus 119890

minus11987501205902

119870) (A3)


= 1205902119870 forall119894 = 1 119899

119879



119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1

119899119879

119899119879

sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ))

119899119879

(A4)


= 1205902119870

forall119894 = 1 119899119879


119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1 minus 119890minus1198750120590

2

119870)119899119879

(A5)


= 1205902119870 forall119894 = 1 119899

119879 Thus we


1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870


2

119870)119899119879119870

(A6)

or








log2(det(Ω

119896))Ω119896



2(det(Ω


2(det(Ω

119896)) le log

2(det(Ω

119896))



119864 Δ119896 = 119864 log

2(det (Ω

119896)) minus 119864 log

2(det (Ω

119896))

le 120590119862radic

Γ (120574 + 1)

120574

(B1)


2(det(Ω









1205902

119862= [log

2(119890)]2

119901

sum119898=1

1205951015840(119902 minus 119898 + 1) (B2)


where 119901 = min(119899119863 119899119877) 119902 = max(119899

119863 119899119877) and 1205951015840(119909) =

suminfin

120593=11(120593 + 119909 minus 1)



1205902

119862= [log

2(119890)]2

119901

sum119898=1

1

119902 minus 119898 + 1 (B3)


119864 Δ119896 le log

2(119890)radic(

119901

sum119898=1

1

119902 minus 119898 + 1) times

Γ (120574 + 1)

120574 (B4)



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












120578PE (Analysis) 00691 00757 00892 00943


PEimbalance

120578PEimbalance

mdash 1019 2826 3515

1 2 3 4 5 6 7 8 9

100

10minus1

10minus2

10minus3

Peak power P0



Pr (P

gtP0)



CCD

F

100

10minus1

10minus2

10minus3




4500

4000

3500

3000

2500

2000

1500

1000

Ergo

dic c

apac

ity (M

bps)

2 4 6 8 10 12 14

SNR (dB)

3300

3200

3100

3000

2900

2800

97 98 99 10 101


Proposed W = 4

Proposed W = 8




7 Conclusions



0 2 4 6 8 10400

600

800

1000

1200

1400

1600


Ener

gy effi

cien

cy (M

bits

J)


Proposed W = 4

Proposed W = 8



Appendices



1198701 12059021198702

1205902119870119899119879


11989110158401015840(120592) =

21198750120592 minus 11987520

1205924119890minus1198750120592 (A1)


0is large



1

119899119879

119899119879

sum119894=1

119891 (1205902

119870119894) le 119891(

1

119899119879

119899119879

sum119894=1

1205902

119870119894)

= 119891(1

119899119879

119899119879

sum119894=1

1205902

119870) = 119891 (120590

2

119870)

(A2)


119894=11205902119870119894

=

sum119899119879

119894=11205902119870


sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le 119899119879(1 minus 119890

minus11987501205902

119870) (A3)


= 1205902119870 forall119894 = 1 119899

119879



119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1

119899119879

119899119879

sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ))

119899119879

(A4)


= 1205902119870

forall119894 = 1 119899119879


119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1 minus 119890minus1198750120590

2

119870)119899119879

(A5)


= 1205902119870 forall119894 = 1 119899

119879 Thus we


1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870


2

119870)119899119879119870

(A6)

or








log2(det(Ω

119896))Ω119896



2(det(Ω


2(det(Ω

119896)) le log

2(det(Ω

119896))



119864 Δ119896 = 119864 log

2(det (Ω

119896)) minus 119864 log

2(det (Ω

119896))

le 120590119862radic

Γ (120574 + 1)

120574

(B1)


2(det(Ω









1205902

119862= [log

2(119890)]2

119901

sum119898=1

1205951015840(119902 minus 119898 + 1) (B2)


where 119901 = min(119899119863 119899119877) 119902 = max(119899

119863 119899119877) and 1205951015840(119909) =

suminfin

120593=11(120593 + 119909 minus 1)



1205902

119862= [log

2(119890)]2

119901

sum119898=1

1

119902 minus 119898 + 1 (B3)


119864 Δ119896 le log

2(119890)radic(

119901

sum119898=1

1

119902 minus 119898 + 1) times

Γ (120574 + 1)

120574 (B4)



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 10400

600

800

1000

1200

1400

1600


Ener

gy effi

cien

cy (M

bits

J)


Proposed W = 4

Proposed W = 8



Appendices



1198701 12059021198702

1205902119870119899119879


11989110158401015840(120592) =

21198750120592 minus 11987520

1205924119890minus1198750120592 (A1)


0is large



1

119899119879

119899119879

sum119894=1

119891 (1205902

119870119894) le 119891(

1

119899119879

119899119879

sum119894=1

1205902

119870119894)

= 119891(1

119899119879

119899119879

sum119894=1

1205902

119870) = 119891 (120590

2

119870)

(A2)


119894=11205902119870119894

=

sum119899119879

119894=11205902119870


sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le 119899119879(1 minus 119890

minus11987501205902

119870) (A3)


= 1205902119870 forall119894 = 1 119899

119879



119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1

119899119879

119899119879

sum119894=1

(1 minus 119890minus1198750120590

2

119870119894 ))

119899119879

(A4)


= 1205902119870

forall119894 = 1 119899119879


119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 ) le (1 minus 119890minus1198750120590

2

119870)119899119879

(A5)


= 1205902119870 forall119894 = 1 119899

119879 Thus we


1 minus

119899119879

prod119894=1

(1 minus 119890minus1198750120590

2

119870119894 )119870


2

119870)119899119879119870

(A6)

or








log2(det(Ω

119896))Ω119896



2(det(Ω


2(det(Ω

119896)) le log

2(det(Ω

119896))



119864 Δ119896 = 119864 log

2(det (Ω

119896)) minus 119864 log

2(det (Ω

119896))

le 120590119862radic

Γ (120574 + 1)

120574

(B1)


2(det(Ω









1205902

119862= [log

2(119890)]2

119901

sum119898=1

1205951015840(119902 minus 119898 + 1) (B2)


where 119901 = min(119899119863 119899119877) 119902 = max(119899

119863 119899119877) and 1205951015840(119909) =

suminfin

120593=11(120593 + 119909 minus 1)



1205902

119862= [log

2(119890)]2

119901

sum119898=1

1

119902 minus 119898 + 1 (B3)


119864 Δ119896 le log

2(119890)radic(

119901

sum119898=1

1

119902 minus 119898 + 1) times

Γ (120574 + 1)

120574 (B4)



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










where 119901 = min(119899119863 119899119877) 119902 = max(119899

119863 119899119877) and 1205951015840(119909) =

suminfin

120593=11(120593 + 119909 minus 1)



1205902

119862= [log

2(119890)]2

119901

sum119898=1

1

119902 minus 119898 + 1 (B3)


119864 Δ119896 le log

2(119890)radic(

119901

sum119898=1

1

119902 minus 119898 + 1) times

Γ (120574 + 1)

120574 (B4)



References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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