Performance analysis of MIMO-SESS with Alamouti scheme

University of Nebraska - LincolnDigitalCommons@University of Nebraska - LincolnFaculty Publications from the Department ofElectrical and Computer Engineering Electrical & Computer Engineering, Department of

2011

Performance analysis of MIMO-SESS withAlamouti scheme over Rayleigh fading channelsShichuan MaUniversity of Nebraska-Lincoln, [email protected]

Lim NguyenUniversity of Nebraska-Lincoln, [email protected]

Yaoqing Lamar YangUniversity of Nebraska-Lincoln, [email protected]

Won Mee JangUniversity of Nebraska-Lincoln, [email protected]

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RESEARCH Open Access

Performance analysis of MIMO-SESS withAlamouti scheme over Rayleigh fading channelsShichuan Ma*, Lim Nguyen, Yaoqing (Lamar) Yang and Won Mee Jang

Abstract

Self-encoded spread spectrum (SESS) is a novel modulation technique that acquires its spreading sequence fromthe random input data stream rather than through the use of the traditional pseudo-noise code generator. It hasbeen incorporated with multiple-input multiple-output (MIMO) systems as a means to combat fading in wirelesschannels. In this paper, we present the analytical study of the bit-error rate (BER) performance of MIMO-SESSsystems under Rayleigh fading. The BER expressions are derived in closed form, and the veracity of the analysis isconfirmed by numerical calculations that demonstrate excellent agreement with simulation results. Theperformance analysis shows that the effects of fading can be effectively mitigated by taking advantage of spatialand temporal diversities. For example, a 2 × 2 MIMO-SESS system can achieve about 7 dB performanceimprovement at 10-4 BER over a MIMO PN-coded spread spectrum system.

Keywords: self-encoded spread spectrum, iterative detection, multiple-input multiple-output, Alamouti scheme

1 IntroductionConventional direct sequence spread spectrum systememploys pseudo-noise (PN) code generators which aretypically linear feedback shift register circuits that gener-ate maximal length or related sequences. In contrast, byderiving its spreading sequences from the user datastream and obviating the need for PN code generators,self-encoded spread spectrum (SESS) provides a feasibleimplementation of random-coded spread spectrum andhas a number of unique features such as the potentialfor enhanced transmission security and anti-jammingcapability, multi-rate applications, modulation gain andinherent time diversity [1-7]. The modulation memoryassociated with self-encoding was exploited by iterativedetection to achieve a significant gain [6].Recently, SESS was incorporated with multiple-input

multiple-output (MIMO) Alamouti scheme to furtherimprove the performance [8]. The spatial diversity frommultiple antennas at the transmitter (TX) and receiver(RX) enables MIMO techniques to be effective againstchannel fading [9-11] and the technology has been stan-dardized [12-14]. In a recent report [15], we haveextended our work in [8] to orthogonal space-time

block codes with multiple iterations and have shown viasimulation that MIMO-SESS provides a novel means tocombat fading in wireless channels by exploiting diversi-ties in both space and time domains. However, the theo-retical analysis of MIMO-SESS in Rayleigh fadingchannels has not been conducted.In this paper, we thus extend our previous work in [8]

to present the analytical study of the BER performanceof SESS with MIMO-Alamouti scheme under Rayleighfading. We first derived BER of the correlation detectorin closed form and verified the performance improve-ment due to spatial diversity. Next, we derived theclosed-form BER expression of the time-diversity SESSdetector that further improves the performance fromthe combined spatial and temporal diversities. We theninvestigate the performance of the iterative detector thatcombines the outputs of the correlation and time-diver-sity detectors. By approximating the probability densityfunction (pdf) associated with the time-diversity detectorby a Dirac delta function, we obtained a lower-boundexpression for the BER of the iterative detector and veri-fied with simulations that the bound is tight even forrelatively small spreading lengths.This paper is organized as follows. MIMO-SESS sys-

tem with iterative detection is described in the next sec-tion. Section 3 presents the theoretical analysis of the

* Correspondence: [email protected] of Computer and Electronics Engineering, University ofNebraska-Lincoln, Omaha, NE 68182, USA

Ma et al. EURASIP Journal on Wireless Communications and Networking 2011, 2011:145http://jwcn.eurasipjournals.com/content/2011/1/145

© 2011 Ma et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons AttributionLicense (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium,provided the original work is properly cited.

mailto:[email protected]

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system. The BER performance of the correlation anddiversity detectors is analyzed in closed form, and alower bound for the iterative detector is derived. Section4 presents the numerical results to validate the analysis,and Section 5 concludes the paper.Notations: The superscripts * and T represent the con-

jugate and the transpose operations, respectively. |·|,E{·} , and diag(·) denote the absolute value of a scalar,the expectation operation, and the diagonal vector of amatrix, respectively. Matrix (vector) is represented bycapital (small) bold letter. The Q-function, gamma func-tion, and incomplete gamma function are denoted by Q(x), Γ(a), and Γ(a, x), respectively. Dirac delta functionis represented by δ(·).

2 System modelIn this section, the transceiver of the MIMO-SESS sys-tem with iterative detection is described.

2.1 MIMO-SESS transmitterThe block diagram of the proposed transmitter is shownin Figure 1, where the rounded corner blocks representN delay registers and Tb is the bit interval. Alamoutischeme-based space-time block coding (STBC) is uti-lized to achieve transmit diversity.

The source information b is assumed to be bi-polar

values of ±√Eb/2 , where Eb is the average transmit

energy per bit. It should be mentioned that the energyper transmit antenna is one half of the total transmitenergy in order to make the multi-antenna system com-parable to a single antenna system. The bits are firstspread by the self-encoded spreading sequence of lengthN at a chip rate of N/Tb. This sequence is constructedfrom the user’s information stored in the delay registersthat are updated every Tb. Thus, with a random inputdata stream, the sequence is also random and time-vary-ing from one bit to another. For example, the spreadingsequence for the kth bit, b(k), is given as

s(k) = [b(k − 1)b(k − 2) , . . . , b(k − N)]T , (1)

and the spreading chips are given as

c(k = b(k)s(k). (2)

To facilitate the description in the sequel, let c(k, n)denote the nth chip of the kth bit, which can beexpressed as

c(k,n) = b(k)b(k − n). (3)

Figure 1 Block diagram of the transmitter of the proposed MIMO-SESS system.


Page 2 of 13

The spreading chips are then divided into two streamsby applying Alam-outi scheme, where a block of twoconsecutive chips, c(k, 2i) and c(k, 2i + 1), are trans-mitted by sending c(k, 2i), -c*(k, 2i + 1) to the firstantenna, and c(k, 2i + 1), c*(k, 2i) to the second antenna.Here, i Î {0, 1, ... , N/2 - 1} is the block index for thechips of the kth bit. The signals are then transmittedover a MIMO fading channel.The channel between each transmit and receive

antenna pair is assumed to undergo Rayleigh fading thatremains constant over Tb but is independent from bit tobit [6,16]. The channels for different transmit/receiveantenna pair are independent.

2.2 MIMO-SESS receiverFigure 2 shows the block diagram of the receiver. Forthe mth antenna, the received signals within the ithblock are given as

xm(k, 2i) = h1,m(k)c(k, 2i) + h2,m(k)c(k, 2i + 1) + em(k, 2i),(4)

xm(k, 2i + 1) = −h1,m(k)c∗(k, 2i + 1) + h2,m(k)c∗(k, 2i) + em(k, 2i + 1), (5)

where h1,m(k) and h2,m(k) respectively denote the nor-malized complex channel impulse response coefficientsfor the kth bit between the 1st and 2nd transmit anten-nas and the mth receive antenna; em is a Gaussian noisewith zero mean and variance NN0/2. It should be notedthat the noise is broadband because it is sampled atchip level. Its variance is thus the narrow-band noisevariance N0/2 multiplied by the spreading factor N.2.2.1 Diversity combiningWe assume that the delay registers in the receiver havebeen synchronized with the transmitter [7], and that

perfect channel knowledge is available at the receiver.Under these assumptions, diversity combining from theM receiver antennas is carried out over two consecutivechip intervals according to [9,17] as

y(k, 2i) =M∑m=1

(h∗1,m(k)xm(k, 2i) + h2,m(k)x∗

m(k, 2i + 1)) + w(k, 2i)

= α(k)c(k, 2i) + w(k, 2i),

(6)

y(k, 2i + 1) =M∑m=1

(h∗2,m(k)xm(k, 2i) − h1,m(k)x∗

m(k, 2i + 1)) + w(k, 2i + 1)

= α(k)c(k, 2i + 1) + w(k, 2i + 1),

(7)

where

α(k) =M∑m=1

(|h1,m(k)|2 + |h2,m(k)|2) (8)

and

w(k,n) =

⎧⎪⎪⎨⎪⎪⎩M∑m=1

(h∗1,m(k)em(k,n) + h2,m(k)e∗

m(k,n + 1)), if n is even

M∑m=1

(h∗2,m(k)em(k,n − 1) − h1,m(k)e∗

m(k,n)), if n is odd(9)

Because the channel coefficients and the white noiseare independent, the noise term w is a random variablewith zero means and variance MNN0.The combined signals given in (6) and (7) can be writ-

ten in one equation as

y(k,n) = α(k)c(k,n) + w(k,n), (10)

and further in vector form as

y(k) = α(k)c(k) +w(k), (11)

Figure 2 Block diagram of the receiver of the proposed MIMO-SESS system with iterative detection.


Page 3 of 13

where

y(k) = [y(k, 1)y(k, 2), . . . , y(k,N)]T , (12)

and

w(k) = [w(k, 1)w(k, 2), . . . ,w(k,N)]T . (13)

2.2.2 Correlation detectionAfter diversity combining, the chips are despread anddetected. The correlation estimate of the kth bit is givenas

b(k) =1NyT(k)s(k)

= α(k)b(k) + v1(k)(14)

where

v1(k) =1N

N∑n=1

w(k,n)b(k − n) (15)

is a random variable with zero mean and varianceMN0. The conditional SNR of the correlation detectiongiven the channel coefficients is

γ1 =|α(k)b(k)|2E{|v1(k)|2} .

(16)

Because the channel fading coefficients, the informa-tion bits, and the thermal noise are independent of eachother, it is easy to see that Equation (16) is reduced to

γ1 =EbN0

α(k). (17)

A hard decision is then performed on the correlationoutput, which in turn is fed back to the receiver delayregisters in order to update the dispreading sequencefor the next bit.

rClb(k) = sgn[b(k)] =

{1, b(k) > 0

−1, b(k) < 0(18)

This also provides an estimate of the correspondingbit if no iterative detection is employed.2.2.3 Time-diversity detectionSESS provides a unique encoding scheme that can beutilized to achieve temporal diversity. To better illustratethis mechanism, let us write the following N2 chips in asquare matrix as

P(k) =

⎡⎢⎢⎢⎣α(k + 1)b(k + 1)b(k) α(k + 1)b(k + 1)b(k − 1) ... α(k + 1)b(k + 1)b(k − N + 1)

α(k + 2)b(k + 2)b(k + 1) α(k + 2)b(k + 2)b(k) ... α(k + 2)b(k + 2)b(k − N)...

.... . .

...α(k +N)b(k +N)b(k +N − 1) α(k +N)b(k +N)b(k +N − 2) ... α(k +N)b(k +N)b(k)

⎤⎥⎥⎥⎦ (19)

where each element represents a chip as given in (10)and each row includes the chips of one bit. For simpli-city, the noise term is omitted.We observe that the kth bit, b(k), is present in the

diagonal elements of the matrix P(k). This means thatthe information of the kth bit is effectively transmittedin the next N bits, which is a unique characteristic ofthe SESS modulation scheme. By defining d(k) = diag (P(k)), the diversity estimate of b(k) can be obtained fromthe hard decisions of the next N bits,

[b(k + 1)b(k + 2), . . . , b(k +N)] , as

b(k) =1Nd(k)[b(k + 1)b(k + 2), . . . , b(k +N)]T + v2(k)

=1N

N∑n=1

α(k + n)b(k) + v2(k),(20)

where

v2(k) =1N

N∑n=1

w(k + n,n)b(k + n) (21)

is a random variable with zero mean and varianceMN0. Notice that the effect of the possible bit errorsfrom the correlation detection has been ignored - this isjustified with sufficient SNR, as shown in Section 4.The conditional SNR, given the channel coefficients

for the diversity detection, is

γ2 =

∣∣∣∣ 1N ∑Nn=1 α(k + n)b(k)

∣∣∣∣2E{|v2(k)|2} .

(22)

Because a(k +n), n Î {1, 2, ... , N}, b(k), and v2(k) areindependent of each other, Equation (22) reduces to

γ2 =Eb

NN0

N∑n=1

α(k + n). (23)

It should be noted that the diversity detection intro-duces a time delay of NTb and that the structure is simi-lar to a Rake receiver employing N fingers to exploit thetemporal diversity of SESS signals.2.2.4 Iterative detectionThe estimate of b(k) can be improved iteratively with

the summation of the correlation estimate, b(k) , and

the diversity estimate, b(k) . Thus, we define

�

b(k) = b(k) +b(k)=

(α(k) +

1N

N∑n=1

α(k + n)

)b(k) + v(k),

(24)


Page 4 of 13

where v(k) = v1(k) + v2(k).The conditional SNR of the iterative detection, condi-

tioned on the channel coefficients, is the summation ofthe SNR of the two independent signals, as follows:

γ =

∣∣∣∣(α(k) +1N

∑Nn=1 α(k + n)

)b(k)

∣∣∣∣2E{|v(k)|2}

=EbN0

[α(k) +

1N

N∑n=1

α(k + n)

]= γ1 + γ2.

(25)

The final hard decision is then obtained by

rClb(k) = sgn[�

b(k)] =

⎧⎨⎩ 1,�

b(k) > 0

−1,�

b(k) < 0(26)

3 Performance analysisIn this section, we analyze the distribution of the SNRsgiven in (17), (23) and (25), and the corresponding BERexpressions. We proceed with the distribution of the nor-malized channel gain, |h(k)|2. It is well known that thecoefficients of the Rayleigh fading channel is generated as

h(k) = (hr + jhi)/√2 , where hr and hi are real values and

normally distributed as N (0, 1) . The denominator√2 is

for power normalization and the channel gain |h(k)|2 = (|hr|

2 + |hi|2)/2, is then gamma-distributed as Γ(1, 1) [18].

Because the summation of independent and identicallydistributed (i.i.d.) gamma random variables still followsa gamma distribution [18], the spatial diversity gain a(k)given in (8) has a Γ(2M, 1) distribution. Furthermore,the conditional SNR of the correlation detector given in(17) follows Γ(2M, Eb/N0) distribution, i.e.,

pγ1 (γ ) =γ 2M−1e

− γ

Eb/N0

(2M − 1)!(Eb/N0)2M , γ ≥ 0. (27)

Now the bit-error probability for binary phase shiftkeying (BPSK) in AWGN with an SNR of Eb/N0 is givenin [16] as

Pb = Q

(√2EbN0

)(28)

Given the conditional SNR of pγ1 (γ ) , the BER expres-sion for the correlation detection is given by [[16], Eq.14.4-15]:

Pe,corr =∫ ∞

0Q

(√2γ

)pγ1 (γ )dγ

=[12(1 − μ1)

]2M 2M−1∑i=0

(2M − 1 + i

i

)[12(1 + μ1)

]i (29)

where

μ1 �√

Eb/N0

1 + Eb/N0. (30)

A similar analysis shows that the conditional SNR ofthe time-diversity detector given in (23) follows aΓ(2NM, Eb/N0/N) distribution, i.e.,

pγ2 (γ ) =γ 2NM−1e

− γ

Eb/N0/N

(2NM − 1)!(Eb/N0

/N)

2NM , γ ≥ 0. (31)

and the BER expression for the diversity detection isthen obtained as

Pe,div =

∞∫0

Q(√

2γ)pγ2 (γ )dγ

=[12(1 − μ2)

]2NM 2NM−1∑i=0

(2NM − 1 + i

i

)[12(1 + μ2)

]i (32)

where

μ2 �√

Eb/N0

/N

1 + Eb/N0

/N. (33)

It should be noted that the correlative detector has anM-fold (spatial) diversity, whereas the time-diversitydetector has an MN-fold (spatial and temporal) diversity.For the iterative detection, since the conditional SNR g

is the summation of the independent conditional SNRsg1 and g2, the pdf of g is the convolution of the gammapdfs of g1 and g2. This is written as follows:

pγ (γ ) = pγ1 (γ ) ⊗ pγ2 (γ )

=

(γM−1e

− γ

Eb/N0

)⊗

(γNM−1e

− γ

Eb/N0/N)

(M − 1)!(NM − 1)!(Eb/N0)M(Eb

/N0

/N)NM , γ ≥ 0.

(34)

The BER expression can then be obtained by aver-aging Q

(√2γ

)over the conditional pdf of g.

The complexity of such a calculation can be obviatedwith the observation that the pdf pγ2 (γ ) approaches aDirac delta function as N becomes large.This stems from the fact that the gamma distribution

approaches a Gaussian distribution if the degree of free-dom, here 2NM, is large [18]. Furthermore, the Gaus-sian distribution tends toward a Dirac delta function asits variance, here 2NM(Eb/N0/N)2 = 2M(Eb/N0)

2/N,becomes very small [18]. It follows that if the spreadingfactor N is sufficiently large, the distribution of pγ2 (γ )can be approximated by an impulse located at the meanvalue of g2, which is given as 2NM(Eb/N0/N ) = 2M (Eb/N0). In other words, pγ2 (γ ) is approximatively equal toδ(g - 2MEb/N0).


Page 5 of 13

So as N becomes large, the pdf of g2 approaches animpulse and the pdf of g can be obtained as the shiftedpdf of g1:

pγ (γ ) ≈ pγ1 (γ ) ⊗ δ(γ − 2MEb/N0)

=

⎧⎪⎪⎨⎪⎪⎩ (γ−2MEb/N0)2M−1e

−γ − 2MEb

/N0

Eb/N0

(2M−1)!(Eb/N0)2M , γ ≥ 2M Eb

N0

0 else

.(35)

The BER of the iterative detector can be calculated as

Pe =

∞∫0

Q(√

2γ)pγ (γ )dγ

≈∞∫

2MEb/N0

Q(√

2γ) (

γ − 2MEb/N0

)2M−1e− γ−2MEb/N0

Eb/N0

(2M − 1)!(Eb/N0)

2M dγ

= Q

(√4M

EbN0

)−

√μ1

2πe2M

2M−1∑k=0

Nk0

Ekbk!

k∑l=0

(k

l

)(μ1)

k−l(−2MEbN0

)l�(k − l +

12, 2M

(1 +

EbN0

))(36)

Appendix A presents the derivation of this result.We note that the impulse approximation of the pdf

yields a lower bound for the BER, because it can only beapproached when the spreading factor N is sufficientlylarge.

4 Analytical and simulation results4.1 Probabilistic modelsIn this section, we present the distribution of the condi-tional SNRs and discuss the delta approximation relatedto the iterative detection. Figure 3 shows the probabilitydensity functions of the conditional SNRs of the correla-tion detection g1, diversity detection g2, and iterativedetection g, for different scenarios. Each sub-figureincludes an extra curve that shows the distribution of g1that has been right-shifted by 2M Eb

N0. The coordinates

for different sub-figures have been adjusted in order toillustrate the distributions in detail. As analyzed in

0 10 20 30 40 50 60 70 800

0.02

0.04

0.06

0.08

0.1

0.12

Conditional SNR

Pro

babi

lity

dens

ity

γ1γ2γshifted γ1

(a) 2×2, N=8, SNR=10 dB

0 10 20 30 40 50 60 70 800

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Conditional SNR

Pro

babi

lity

dens

ity

γ1γ2γshifted γ1

(b) 2×1, N=8, SNR=10 dB

0 10 20 30 40 50 60 70 800

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Conditional SNR

Pro

babi

lity

dens

ity

γ1γ2γshifted γ1

(c) 2×2, N=64, SNR=10 dB

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

3.5

Conditional SNR

Pro

babi

lity

dens

ity

γ1γ2γshifted γ1

(d) 2×2, N=64, SNR=0 dB

Figure 3 Distribution of the conditional SNRs.


Page 6 of 13

Section 3, the mean and variance of the conditional SNR

of the diversity detection g2 are 2M EbN0

and 2MN ( Eb

N0)2 ,

respectively. This means that while the number ofreceive antennas and the SNR affect both position andshape of the “pulse,” the spreading length only influ-ences its shape. Figure 3a and 3b show that the positionand the shape of the pulse vary with the number ofantennas. In both cases, the pulse is not very narrowdue to the small value of N = 8, resulting in a discern-able difference between the shifted version of the distri-bution of g1 (dotted curve) and the distribution of g(dashed curve). The “pulse” looks sharper in Figure 3csince N = 64 is much larger. Because the area under thepdf is fixed, this sharper pulse approximates a deltafunction such that there is negligible difference betweenthe shifted distribution of g1 and the distribution of g.Also, Figure 3c and 3d shows that the delta

approximation is not very sensitive to SNR, although itis more accurate with lower SNR.

4.2 Performance comparisonsNext, we present the BER performance of MIMO-SESSand compared with a conventional MIMO, PN-codedspread spectrum (MIMO-PNSS) system. The channelbetween each transmit and receive antenna pair isassumed to be Rayleigh fading that remains constantover Tb but is independent from bit to bit [6,16]. Inaddition, the channels for different transmit/receiveantenna pair are independent. The length of the spread-ing sequence is set to N = 64 unless noted otherwise.For each scenario and bit signal-to-noise ratio (SNR),100 runs of 100,000 bits are simulated to evaluate theaverage bit error rate (BER).Figure 4 compares the BER of MIMO-SESS with cor-

relation detection to a MIMO-PNSS system. It is

0 1 2 3 4 5 6 7 8 9 1010

−4

10−3

10−2

10−1

100

SNR (dB)

BE

R

MIMO−PNSS 2x1MIMO−PNSS 2x2MIMO−SESS 2x1 Corr SimMIMO−SESS 2x2 Corr SimMIMO−SESS 2x1 Corr AnaMIMO−SESS 2x2 Corr Ana

Figure 4 Comparison between MIMO-PNSS and MIMO-SESS with correlation detection, N = 64.


Page 7 of 13

observed that without iterative detection, there is notime diversity gain, and thus, the performance with orwithout self-encoding is the same as expected underhigh SNR. The performance degradation of MIMO-SESS at low SNR is due to error propagation [1]. In par-ticular, Figure 4 shows that the effect of error propaga-tion is quite severe for a BER greater than 10%. Theeffect of error propagation can be efficiently alleviatedwith more antennas, as shown by the example 2 × 2scenario, since larger spatial diversity reduces BER tobelow 10% even at low SNR. The plots in Figure 4 showexcellent agreement between the analytical and simula-tion results at high SNR, where error propagationbecomes insignificant.The performance of MIMO-SESS with time-diversity

detection is shown in Figure 5. The plots show that theproposed system significantly outperforms conventionalMIMO-PNSS systems. As an example, there is about 4.5

dB gain for the 2 × 2 configuration at 10-4 BER. Thisperformance improvement can be attributed to the N-fold time diversity introduced by self encoding. Due tothe significantly improved BER, error propagation iseffectively mitigated as shown by the 2 × 2 scenario.Moveover, the agreement with the simulation resultsverifies that the analysis has been justified in ignoringerror propagation.Figure 6 compares the BER with the iterative detector.

Again, there is excellent agreement between the analyti-cal and simulation results at sufficiently high SNR. Theperformance gain over conventional MIMO-PNSS sys-tems is almost 7 dB at 10-4 BER for the 2 × 2 configura-tion. The excess gain beyond the expected 3 dB SNRimprovement can be attributed to the diversity gainfrom the time-diversity detection. In fact, the plot showsthat a 2 × 2 MIMO-SESS would require an SNR of only3 dB to achieve a 10-4 BER. The results verify the

0 1 2 3 4 5 6 7 8 9 1010

−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

BE

R

MIMO−PNSS 2x1MIMO−PNSS 2x2MIMO−SESS 2x1 T−D SimMIMO−SESS 2x2 T−D SimMIMO−SESS 2x1 T−D AnaMIMO−SESS 2x2 T−D Ana

Figure 5 Comparison between MIMO-PNSS and MIMO-SESS with diversity detection, N = 64.


Page 8 of 13

veracity of the performance analysis and demonstratethat MIMO-SESS can completely mitigate the effect ofRayleigh fading. The difference between the numericaland simulation results at low SNR is due to error propa-gation. This can be easily demonstrated in simulation byfeeding the original (transmitted) bits into the delay reg-isters at the receiver. The results of this simulation arecompared with the numerical calculations in Figure 7.The excellent agreement between the simulation andnumerical results demonstrates the validity of the statis-tical models and approximations.Figure 8 shows the numerical results of MIMO-SESS

with the iterative detection for different spreadinglengths. For both 2 × 1 and 2 × 2 cases, the perfor-mance approaches the lower bound when the spreadinglength increases. It should be noted that the lowerbound is very tight for N ≥ 64. Also, the lower bound istighter for the 2 × 1 case and at lower SNR. These

results are consistent with the discussion in Section 3. Itcan be seen by comparing the lower bound and the per-formance of BPSK in AWGN that 2 × 2 MIMO-SESScan provide about 5.2 dB gain over BPSK at 10-4 BER.This significant performance improvement is clearly dueto the combined temporal and spatial diversities asso-ciated with MIMO-SESS design.

4.3 RemarksThe effect of error propagation is largely confined tohigh BERs (>10-2) as the bit errors are fed back to thede-spreading chips and propagate through the receiverdelay registers. Thus, the benefit of MIMO reducing theeffect of error propagation occurs in the low SNRregions (primarily through the 3 dB array gain) asshown by the simulation results. However, the perfor-mance gaps between error propagation and “no” errorpropagation (true bits) remain for low BER. The

0 1 2 3 4 5 6 7 8 9 1010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

BE

R

MIMO−PNSS 2x1MIMO−PNSS 2x2MIMO−SESS 2x1 Iter SimMIMO−SESS 2x2 Iter SimMIMO−SESS 2x1 Iter NumMIMO−SESS 2x2 Iter Num

Figure 6 Comparison between MIMO-PNSS and MIMO-SESS with iterative detection, N = 64.


Page 9 of 13

comparison with the analytical results also showed thatthe discrepancies are primarily in high BER, low SNRregions. This suggests that in practical systems wherethe BER is sufficiently low, the effect of error propaga-tion and hence its mitigation with MIMO scheme arerather minimal and relatively insignificant. For exampleat 10-3 BER, the performance gap is negligible for corre-lation and time diversity detectors, and amounts to lessthan 0.5 dB for the iterative detector.The results showed that MIMO scheme provides spa-

tial diversity under fading that is most beneficial to cor-relation detection (which has no diversity). Timediversity detection on the other hand can achieve N-folddiversity that completely mitigates the effect of fading;so there is very little spatial diversity benefit fromMIMO scheme especially under high SNR. Since itera-tive detection combines time diversity and correlationdetection, it also benefit from spatial diversity but to a

lesser extent. The advantage of the combination ofMIMO and SESS therefore is to exploit the array gainto mitigate error propagation at low SNR and the spatialdiversity to improve the overall BER.

5 ConclusionIn this paper, we presented the analytical study of theBER performance of MIMO-SESS over Rayleigh fadingchannels. We derived the closed-form BER expressionsfor the correlation detector and time diversity detector,and obtained a lower bound for the iterative detector.The veracity of the analysis has been confirmed with thenumerical calculations based on the analytical expres-sions, which have demonstrated excellent agreementwith the simulation results. The performance analysishas shown that MIMO-SESS offers almost 7 dB of gainat a 10-4 BER compared with a 2 × 2 MIMO PN-codedspread spectrum system. Furthermore, the system

0 1 2 3 4 5 6 7 8 9 1010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

BE

R

MIMO−SESS 2x1 Iter Sim (TrueBits)MIMO−SESS 2x2 Iter Sim (TrueBits)MIMO−SESS 2x1 Iter NumMIMO−SESS 2x2 Iter Num

Figure 7 Comparison between numerical results and simulation results without error propagation, N = 64.


Page 10 of 13

requires only about 3 dB SNR to achieve a BER of 10-4.This demonstrates that MIMO-SESS can provide a veryeffective means to exploit both spatial and temporaldiversities in order to achieve robust performance inwireless environments.Although we have examined an Alamouti scheme-

based MIMO system with two transmitting antennas,the MIMO-SESS analysis that has been developed herecan be easily applied to other orthogonal space-timeblock coded MIMO systems with any number of anten-nas at transmitter side. Additional study is being under-taken to extend this work toward multi-code MIMO-SESS that can improve the transmission rate, systemthroughput, and spectral efficiency.

AppendixA Derivation of Equation 36This appendix provides a detailed derivation of Equation(36). Define a = 2MEb/N0, b = Eb/N0, and L = 2M, thenthe BER of the iterative detector can be written as

Pe ≈∞∫a

Q(√

2γ) (γ − a)L−1e− γ−a

b

(L − 1)!bLdγ

=

∞∫0

Q(√

2(x + a)) xL−1e− x

b

(L − 1)!bLdx

=

∞∫0

1√2π

∫ ∞√

2(x+a)e− u2

2 duxL−1e− x

b

(L − 1)!bLdx

(37)

By changing the order of the integrals, (37) becomes

Pe ≈ 1√2π

1(L − 1)!bL

∞∫√2a

e− u2

2 P1du (38)

where

P1 =

u22 −a∫0

xL−1e− xb dx (39)

0 1 2 3 4 5 6 7 8 9 1010

−14

10−12

10−10

10−8

10−6

10−4

10−2

100

SNR (dB)

BE

R

BPSK (AWGN)MIMO−SESS Iter Num N=16MIMO−SESS Iter Num N=32MIMO−SESS Iter Num N=64MIMO−SESS Iter Num N=128MIMO−SESS Iter Num N=256MIMO−SESS Iter LowerBound

2×1

2×2

Figure 8 Performance of the proposed MIMO-SESS system with various N values.


Page 11 of 13

Let z = xb , then

P1 = bL

u22b− a

b∫0

zL−1e−zdx (40)

Using the indefinite integral equation given in [[19],Eq. 7.4.322], P1 can evaluated as

P1 = −bLe− u22b +

ab

L−1∑k=1

(L − 1)!k!

(u2

2b− a

b

)k

− bL(L − 1)!(1 − e− u2

2b +ab

)(41)

Substituting (41) into (38) to determine Pe:

Pe ≈∞∫

√2a

1√2π

e− u2

2 du − 1√2π

eab

∞∫√2a

e− u2

2

(1+bb

)du

− 1√2π

eab

L−1∑k=1

1k!

(12b

)k ∞∫√2a

e−u2

2

⎛⎝1 + b

b

⎞⎠(u2 − 2a

)kdu

= Q(√

2a)

− 1√2π

eab

L−1∑k=0

1k!

(12b

)k

P2

(42)

where

P2 =

∞∫√2a

e− u2

2

(1+bb

)(u2 − 2a

)kdu (43)

This can be further transformed with the substitution

P2 = 2k−1

√2b

1 + b

∞∫a(1+b)

b

e−tt−12

(b

1 + bt − a

)k

dt

= 2k−1

√2b

1 + b

∞∫a(1+b)

b

e−tt−12

k∑l=0

(k

l

)(b

1 + bt)k−l

(−a)ldt

= 2k−1

√2b

1 + b

k∑l=0

(k

l

)(b

1 + b

)k−l

(−a)l�(k − l +

12,a(1 + b)

b

):

P2 = 2k−1

√2b

1 + b

∞∫a(1+b)

b

e−tt−12

(b

1 + bt − a

)k

dt

= 2k−1

√2b

1 + b

∞∫a(1+b)

b

e−tt−12

k∑l=0

(k

l

)(b

1 + bt)k−l

(−a)ldt

= 2k−1

√2b

1 + b

k∑l=0

(k

l

)(b

1 + b

)k−l

(−a)l�(k − l +

12,a(1 + b)

b

)(44)

where Γ(a, x) denotes the incomplete gamma functionwith shape a and scale x [[20], Eq. 6.5.3]. Substituting(44) into (42), we have:

Pe ≈ Q(√

2a)

− 1√2π

eab

√b

1 + b

×L−1∑k=0

1bkk!

k∑l=0

(k

l

)(b

1 + b

)k−l

(−a)l�(k − l +

12,a(1 + b)

b

) (45)

The final expression for Pe in (36) can be obtainedwith the defined values for a, b, and L:

Pe ≈ Q

(√4M

EbN0

)−

√μ1

2πe2M

2M−1∑k=0

Nk0

Ekbk!

k∑l=0

(k

l

)(μ1)

k−l(−2MEbN0

)l�(k − l +

12, 2M

(1 +

EbN0

)) (46)

where μ1 is given by (30).

AcknowledgementsThis work was funded in part by contract award FA9550-08-1-0393 from theUS Air Force Office of Scientific Research. The authors wish to thank Dr. J.Sjogren for his support of this study. This paper was presented in part atIEEE International Conference on Communications (ICC), Cape Town, SouthAfrica, May 2010, and at ISITA/ISSSTA2010, Taichung, Taiwan, Oct. 2010.

Competing interestsThe authors declare that they have no competing interests.

Received: 28 April 2011 Accepted: 27 October 2011Published: 27 October 2011

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doi:10.1186/1687-1499-2011-145Cite this article as: Ma et al.: Performance analysis of MIMO-SESS withAlamouti scheme over Rayleigh fading channels. EURASIP Journal onWireless Communications and Networking 2011 2011:145.

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