Mabruk Jour Adaptive Full Paper

8/8/2019 Mabruk Jour Adaptive Full Paper

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Design of adaptive MIMO system using linear

dispersion code

Mabruk Gheryani, Zhiyuan Wu, and Yousef R. Shayan

Concordia University, Department of Electrical Engineering

Montreal, Quebec, Canada

email: (m gherya, zy wu, yshayan)@ece.concordia.ca

Abstract

In this paper, first we studied the statistics of signal-to-interference-noise for a MIMO transceiver using linear

dispersion code and linear minimum-mean-square-error (MMSE) receiver over a Rayleigh fading channel. The

associated probability density function of the signal-to-interference-noise is derived and verified. The average

BER over MIMO fading channel for a given constellation using the MMSE receiver is calculated numerically.

The numerical and simulation results match very well. With the statistics as a guideline, we study new design of

selection-mode adaptation using a linear dispersion code. A new adaptive parameter, called space-time symbol rate,

can be applied due to the use of linear dispersion code. An adaptive algorithm for the selection-mode adaptation

is proposed. Based on the proposed algorithm, two adaptive techniques using constellation and space-time symbol

rate are studied, respectively. If constellation and space-time symbol rate are considered jointly, more selection

modes can be available. Theoretical analysis demonstrates that the average transmission rate of selection-mode

adaptation can be improved in this case. Simulation results are provided to show the benefits of our new design.

Index Terms

MIMO, LDC, FRFD

I. INTRODUCTION

In the future, the demand for bandwidth efficiency in wireless communications has experienced an

unprecedented growth. One significant advance to improve radio spectrum efficiency is the so-called

multiple-input-multiple-output (MIMO) technology [1] [2] and space-time (ST) codes are the most promis-

ing technique for MIMO systems [3] [4]. However, due to battery life and device size, the power available

for radio communications is limited. Under this power constraint, MIMO technology shall cooperate with

adaptive technique to further exploit radio spectrum [5] [6].



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In an adaptive system, a feedback channel is utilized to provide channel state information (CSI) from

the receiver to the transmitter. According to the feedback of CSI, the transmitter will adjust transmission

parameters, such as power allocation, modulation, coding rate, etc. This is conditioned by the fact that

the channel keeps relatively constant before the transmitter receives the CSI and then transmits next

data block accordingly. That is, the channel is “slow”. A lot of adaptive MIMO schemes have been

proposed, such as water-filling-based schemes [1] [7]- [10] and various beamforming schemes [6] [11]-

[14]. The above schemes often need near-perfect CSI feedback for adaptation calculation and consume

large feedback bandwidth. In practice, the channel estimation will exhibit some inaccuracy depending on

the estimation method. The receive will need time to process the channel estimate, the feedback is subject

to some transmission delay, the transmitter needs some time to choose a more proper code, and there

are possible errors in the feedback channel. All these factors make the CSI at the transmitter inaccurate.

Additionally, the feedback bandwidth is often limited. In these cases, adaptive schemes with a set of

discrete transmission modes are often more preferable. We can call them “selection-mode” adaptation. At

the receiver, the channel is measured and then one transmission mode with the highest transmission rate

is chosen, which meanwhile meets the BER requirement. The optimal mode is fed back to the transmitter.

For selection-mode MIMO adaptation, the most convenient adaptive parameter is constellation size

for uncoded systems. For example, constellation adaptation, such as 2η-QAM, is applied to space-time

block code (STBC) [15] and to space-time trellis code (STTC) [16]. The disadvantage of these schemes

is not flexible for different rates, which is the key requirement in the future wireless communications.

Additionally, the gap between the available transmission rates are often very large due to the use of

discrete constellations [12].

In this study, we propose to apply the linear dispersion code (LDC) for adaptation. This is because it

subsumes many existing block codes as its special cases, allows suboptimal linear receivers with greatly

reduced complexity, and provides flexible rate-versus-performance tradeoff [17]- [19]. The LDC breaks

the data stream into sub-streams that are dispersed over space and time and then combined linearly at the

transmitter [17].

With the application of LDC, a linear MMSE detector is more attractive due to its simplicity and good

performance [20] [21]. However, the performance analysis in this case is still deficient. Most of the related

works address only the V-BLAST [22]- [24] scheme, a special case of the full-rate LDC. For example,

the case of two transmit antennas was analyzed in [25] and the distribution of the angle between two

complex Gaussian vectors was presented. The layer-wise signal-to-interference-plus-noise ratio (SINR)

distribution for V-BLAST with successive interference cancellation at the receiver was provided in [26].

Hereby, the error-rate probability and statistics of SINR for LDCs with a MMSE receiver is studied in



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this paper and the results will be a guideline for adaptation. Since the LDC is applied, it makes ST

symbol rate available for adaptation. By adjusting this new parameter together with constellation size,

more available transmission modes can be provided. Hence, the throughput under a power constraint can

be further improved while the target bit error rate (BER) is satisfied.

This paper will be organized as follows. Our system model is presented in Section II. In Section III,

the statistics of SINR using MMSE receiver and the associated average BER are studied. In Section IV,

selection-mode adaptation using constellation or ST symbol rate is designed. In Section V, joint adaptive

technique is proposed to improve throughput further. Finally, in Section VI, conclusions are drawn.

II. SYSTEM MODEL

In this study, during one ST modulation block, the channel is assumed to be the same as estimated at

the receiver. Furthermore, the channel is assumed to be a Rayleigh flat fading channel with N t transmit

and N r receive antennas. Let’s denote the complex gain from transmit antenna n to receiver antenna m

by hmn and collect them to form an N r × N t channel matrix H = [hmn], known perfectly to the receiver.

The entries in H are assumed to be independently identically distributed (i.i.d.) symmetrical complex

Gaussian random variables with zero mean and unit variance.

The selection-mode adaptive system is depicted in Fig. 1. In this system, the information bits are first

mapped into symbols. After that, the symbol stream is parsed into blocks of length L. The symbol vector

associated with one modulation block is denoted by x = [x1, x2, . . . , xL]T with xi ∈ Ω ≡ Ωm|m =

0, 1, . . . , 2η

− 1, η ≥ 1, i.e., a complex constellation of size 2η

, such as 2η

-QAM). The average symbolenergy is assumed to be 1, i.e., 1

2η

2η−1m=0

|Ωm|2 = 1. Each block of symbols will be mapped by the ST

modulator to a dispersion matrix of size N t × T and then transmitted over the N t transmit antennas over

T channel uses. The following model will be considered in this study, i.e.,

X =L

i=1

Mixi (1)

where Mi is defined by its L N t × T dispersion matrices Mi = [mi1,mi2, . . . ,miT ]. The so-obtained

results can be extended to the model in [17]. With a constellation of size 2η, the data rate of the space-time

modulator in bits per channel use is

Rm = η · L/T (2)

Hence, one can adjust ST symbol rate L/T and constellation size η according to the feedback from the

receiver.

At the receiver, the received signals associated with one modulation block can be written as

Y =

P

N tHX+ Z =

P

N tH

Li=1

Mixi + Z (3)



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where Y is a complex matrix of size N r × T whose (m, n)-th entry is the received signal at receive

antenna m and time instant n, Z is the additive white Gaussian noise matrix with i.i.d. symmetrical

complex Gaussian elements of zero mean and variance σ2z , and P is the average energy per channel use

at each receive antenna. It is often desirable to write the matrix input-output relationship in (3) in an

equivalent vector notation. Letvec()

be the operator that forms a column vector by stacking the columns

of a matrix and define y = vec(Y), z = vec(Z), and mi = vec(Mi), then (3) can be rewritten as

y =

P

N tHGx+ z =

P

N tHx+ z (4)

where H = IT ⊗H with ⊗ as the Kronecker product operator and G = [m1,m2, . . . ,mL] will be referred

to as the modulation matrix. Since the average energy of the signal per channel use at a receive antenna

is assume to be P , we have tr(GGH ) = N tT .

III . THE STATISTICS OF SINR WITH THE MMSE RECEIVER

Since the LDC is linear, an MMSE detector can be applied as suboptimal receiver due to its simplicity

and good performance [20]. The main goal of this section is to study the error-rate probability and the

statistics of SINR for LDCs [17]- [19] using linear MMSE receiver over a Rayleigh fading channel.

We consider a general system model as shown in Section II. In our study, N r ≥ N t is assumed. For

simplicity, we choose T equal to N t and L equal to N tT .

Equation (4) can also be written as

y = P N thixi + P

N t j=i

h jx j + z (5)

In the sequel, the i-th column of H, denoted as hi, will be referred to as the signature signal of symbol

xi.

Without loss of generality, we consider the estimation of one symbol, say xi. Collect the rest of the

symbols into a column vector xI and denote HI = [h1,.., hi−1, hi+1, ..., hL] as the matrix obtained by

removing the i-th column from H.

A linear MMSE receiver is applied and the corresponding output is given by

xi = wH i y = xi + zi. (6)

where zi is the noise term of zero mean. The corresponding wi can be found as

wi =

hih

H i +RI

−1hi

P N thH

i

hih

H i +RI

−1hi

(7)



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where RI = HI HH

I + N tσ2zP I. Note that the scaling factor 1

P N thHi (hihHi +RI)

−1hi

in the coefficient vector

of the MMSE receiver wi is added to ensure an unbiased detection as indicated by (6). The variance of

the noise term zi can be found from (6) and (7) as

σ2i = wH

i RI wi (8)

Substituting the coefficient vector for the MMSE receiver in (7) into (8), the variance can be written as

σ2i =

1P

N thH

i R−1I hi

(9)

Then, the SINR of MMSE associated with xi is 1/σ2i .

γ i =1

σ2i

=

P

N t

hH

i R−1I hi (10)

Closed-form BER for a channel mode such as (6) can be found in [29]. The average BER over MIMO

fading channel for a given constellation can be found as follows.

BE Rav = E γ i

1

L

i

BER(γ i(H, γ ))

(11)

where γ = P N tσ2z

.

In our LDC design, all the symbols has the same SINR, i.e., γ 1 = γ 2 = · · · = γ L = γ . Equation (11)

can be written as

BERav =

BER(γ, γ )P Γ(γ )dγ (12)

By using singular value decomposition (SVD), (10) can be written as

γ =

P

N t

hH

i UΛ−1UH hi (13)

where UH is an N 2t −1×N 2t −1 unitary matrix and the matrix Λ is (N 2t −1)×(N 2t −1) with nonnegative

numbers on the diagonal and zeros off the diagonal. Let’s define

h′ = UH hi

which is the transformed propagation vector with components h′l, l = 1, ...........N rN t.

Equation (13) can be written as

γ =

P

N t

N rN tl=1

|h′l|2λl

(14)

with

λl =

λl = 1σ2z

(γλl + 1) l = 1,.......,r ≤ L − 1

σ2z l = r + 1, ........., N rN t

(15)

where r is the rank of HI HH

I which is less than N 2t − 1.



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The vector h′ has the same statistics as the original vector hi [1]. For analytical purpose, we can replace

|h′l|2 by |hil|2. Now, we can write (14) as

γ =r

i=1

γ |hil|2

(γλl + 1)+

N tN rr+1

γ |hil|2 (16)

The probability density function (PDF) of γ can be found using the moment generating function (MGF)

as follows [28]. First, we find the conditional (on the eigenvalues) MGF of γ as

M γ/λ (s) = [M γ (γs)]N rN t−rr

l=1

M γ

γs

(γλl + 1)

(17)

the f λi(λi) denotes the PDF of the ith nonzero eigenvalue of the HI HH

I . If we let f λ(λ) denote the PDF

of any unordered λi for i = 1, ....., m,, then (17) can be written as

M γ/λ (s) = [M γ (γs)]N rN t−r

M γ

γs

(γλ + 1)

r

(18)

Further,

M γ/λ (s) =1

(1 − γs)N rN t−r × 11 − γs

(1+γλ)

r (19)

we can find the probability density function (PDF) of γ conditionally on λ by using inverse Laplace

transform for (19) as [27]

P Γ/λ(γ ) =(γ )−N rN t

Γ(N rN t)(γ )N rN t−1 exp(−γ

γ ) ×

(1 + γλ)r1 F 1(N rN t − r, N rN t, λγ )exp(−λγ ) (20)

where 1F 1(.,.,.) is Kummer’s confluent hypergeometric function [30] and defined as

1F 1(a,b,x) =∞n

(a)n

(c)n

xn

n!

where (∗)n = Γ(∗+n)Γ(∗)

.

Let’s define

K γ =(γ )−N rN t

Γ(N rN t)(γ )N rN t−1 exp(−γ

γ )

Then equation (20) can be written as

P Γ/λ(γ ) = K γ

∞n

(N rN t − r)n

(r)n×

γ n

n!λn (1 + γλ)r exp(−λγ ) (21)

Now, we can find the PDF of γ as follows.

P Γ(γ ) = ∞0

P γ/λ (γ )f λ(λ) dλ (22)



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f λ(λ) was given in [1] and can be written as

f λ(λ) =1

r

ri=1

Φi(λ)2λN rN t−r exp(−λ) (23)

where

Φk+1(λ) = k!

(k + N rN t − r)!12

LN rN t−rk (λ)

k = 0,...r − 1

where LN rN t−rk (λ) is the associated Laguere polynomial of order k [30]. Equation (23) can be written as

f λ(λ) =1

r

r−1k=0

k!

(k + N rN t − r)![LN rN t−r

k (λ)]2 (24)

Let’s define

K 1(k) =k!

(k + N rN t−

r)!

Γ(k + n′

)

22kk!

K 2(i) =(2i)!(2k − 2i)!

i![(k − i)!]2Γ(k + n′)

K 3(d) =(−2)d

d!

2k + 2N rN t − 2r

2k − d

where n

′

= N rN t − r + 1. Then we can write (22) as

P Γ(γ ) =K γ

r

r−1k=0

K 1(k)k

i=0

K 2(i)2k

d=0

K 3(d) ∞0

(1 + γλ)r

λN r

N t−

r+d

1F 1(N rN t − r, N rN t, λγ ) exp(−γλ)dλ (25)

The term (1 + γλ)rcan be written as

(1 + γλ)r = γ rr

v=0

r

v

γ v−rλv

Then equation (25) can be written as

P Γ(γ ) =

K γ γ r

r ×∞n

γ nK (n)r

v=0

K (v)K 1(k)k

i=0

K 2(i)2k

d=0

K 3(d)×

r−1k=0

∞0

λN rN t−r+d+v+n exp(−γλ) dλ (26)

with

K (n) =(N rN t − r)n

(r)nn!



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and

K (v) =

r

v

γ v−r

The general form of the integration of (26) can be found in [30]

∞0 xΘ

exp(−µx)dx = Θ!µ−Θ−1

where

Θ = N rN t − r + d + v + n

Then (26) can be written as

P Γ(γ ) =K γ γ r

r

∞n

γ nK (n)r

v=0K (v)

r−1

k=0K 1(k)

k

i=0K 2(i)×

2kd=0

K 3(d)γ −N rN t+r−d−v−n−1(N rN t − r + d + v + n)! (27)

Further,

P Γ(γ ) =K γ γ rγ r

rγ N rN t+1

r−1k=0

K 1(k)k

i=0

K 2(i)2k

d=0

K 3(d)×

γ −dr

v=0

K (v)γ −v∞n

K (n)(N rN t − r + d + v + n)! (28)

Let’s define

K (v, d) = (N rN t − r + d + v)!×Γ(N rN t − r + d + v + 1)Γ(d + v − r + 1)

Γ(d + v + 1)Γ(N rN t + d + v + 1)`K (v)

and

`K (v) =

r

v

Then (28) can be written as

P Γ(γ ) = K γ γ r−N rN t−1

r

r−1k=0

K 1(k)k

i=0K 2(i)×

2kd=0

K 3(d)γ −dr

v=0

K (v, d) (29)

This is the PDF of SINR for our system over Rayleigh fading channels.

We verify our derivation by simulation. In the simulation, the dispersion matrices are given by

M(k−1)N t+i = diag[f k]P−(i−1) (30)



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for k = 1, 2, . . . , N t and i = 1, 2, . . . , N t, P is the permutation matrix of size N t and given by

P =

01×(N t−1) 1

IN t−1 0(N t−1)×1

(31)

where f k denotes the k-th column vector of F. F = [f mn] is a Fast Fourier Transform (FFT) matrix and

f mn is calculated by

f mn =1√N t

exp(−2πj(m − 1)(n − 1)/N t) (32)

In the simulation, N t = N r = T = 2 and N t = N r = T = 4 were assumed. In Fig. 2, the theoretical

PDFs of the SINR in (29) and results by Monte Carlo simulation were compared for 2 × 2 and 4 × 4

channels, respectively at P/σ2z = 20dB. Simulation results match to the analytical result very well.

The closed-form formula for the average BER in (12) is difficult to find. For example, the BERav for

2η-PSK can be written as

BERav = 2η

Q

2η γ sin( π2η

)

P Γ(γ ) dγ (33)

and for rectangular 2η-QAM can be written as

BERav =4

η

Q

3η γ

2η − 1

P Γ(γ ) dγ (34)

where Q(·) denotes the Gaussian-Q function. Here, the above average BER is calculated numerically. In

Fig. 3, numerical and simulation results are compared for 8PSK over 3 × 3 and 4PSK over 4 × 4 fading

channels, respectively. As can be seen, the numerical and simulation results match very well.

IV. DESIGN OF SELECTION-M OD E ADAPTATION

The general idea of selection-mode adaptation is to maximize the average transmission rate by choosing

a proper transmission mode from a set of available modes. Based on some certain strategy, the transmitter is

informed by the receiver to increase or decrease the transmission rate depending on the channel condition,

i.e., CSI. For selection-mode adaptation, the signal-to-noise ratio (SNR) will be considered as a proper

metric. The corresponding adaptive algorithm is proposed as follows.

1) Find the SNR, saying γ o, at the receiver;

2) Find the BERs of each mode at the obtained SNR γ o from BER curves by experiment;

3) Select a proper transmission mode with the maximum rate while satisfying the target BER;

4) Feed back the selected mode to the transmitter.

We can formulate the selection of transmission modes as follows.

Θopt = arg maxΘn,∀n=1,2,...,N

RΘn(35)



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subject to

BERΘn(γ o) ≤ BERtarget (36)

where Θn, ∀n = 1, 2, . . . , N is the set of transmission modes, RΘn is the rate of transmission mode Θn,

BERΘn(γ o) is the BER of transmission mode Θn at SNR γ o and BERtarget is the target BER. Without

loss of generality, we assume RΘ1 < RΘ2 < . . . < RΘN . Θopt is the optimal transmission mode at SNR

γ o.

Below, we consider the average transmission rate using the proposed adaptive algorithm. Let γ Θndenote

the minimum SNR satisfying the following condition.

γ Θn= arg min

γ [BERΘn

(γ ) ≤ BERtarget)] (37)

That is, for the SNR region γ Θn ≤ γ ≤ γ Θn+1, the transmission rate RΘn (i.e., the transmission mode Θn)

should be selected while the target BER is satisfied.

Then, the average transmission rate is

R =N

n=1

RΘn

γ Θn+1

γ Θn

pΓ(γ )dγ (38)

where pΓ(γ ) is the probability density function (PDF) of the SNR γ and γ ΘN +1= ∞. Maximization of

the average transmission rate R can be solved using Lagrange multipliers. However, due to the structure

of both the objective function and the inequality constraint, an analytical solution is extremely difficult to

find. Therefore, we will find the SNR region corresponding to each transmission mode by measurement.

In our simulations, we assume N t = N r = 4 using the dispersion matrices defined in (30) and the

MMSE receiver is applied. First, we perform constellation adaptation alone with a fixed ST symbol rate.

Secondly, we perform the ST symbol rate adaptation alone with a fixed constellation. Finally, we will

consider these two parameter jointly to maximize the average transmission rate meanwhile maintaining

the target BER, which is equal to 10−3 in our design examples.

A. Adaptation Using Variable Constellations

Although the system design for continuous-rate scenario provide intuitive and useful guidelines [12],the associated constellation mapper requires high implementation complexity. In practice, discrete con-

stellations are preferable. That is, η only takes integer number, such as η = 1, 2, 3,..... For a given

adaptive system, we can adjust the constellation to maximize the transmission rate meanwhile keeping

the target BER satisfied. The proposed adaptive algorithm is applied to the case. Here, we only consider

BPSK (η = 1), QPSK (η = 2), 8PSK (η = 3) and 16QAM (η = 4) as examples. That is, Θn ∈BPSK,QPSK, 8PSK, 16QAM with a fixed ST symbol rate. The optimal transmission mode is



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selected by the proposed adaptive algorithm, i.e., by equation (35) and (36). Simulation results are shown

in Fig. 4, where each subfigure has its own ST symbol rate. We summarize our simulation results in

Table I. In the following context, γ LT

η denotes the SNR associated with the transmission mode with 2η

constellation and LT ST symbol rate.

B. Adaptation Using Variable ST Symbol Rate

In other existing schemes, only the orthogonal designs, such as Alamouti scheme, are applied as the ST

modulation. In this case, the most convenient adaptive parameter is the constellation size. For our adaptive

scheme, the application of LDC makes another adaptive parameter available, i.e., ST symbol rate. In this

subsection, we fix the constellation size but adjust the ST symbol rate for adaptation. Additionally, one

advantage of using ST symbol rate is that it is easier to change ST symbol rate than constellation size

for adaptation as can be seen in Fig. 1. The proposed adaptive algorithm described by (35) and (36) can

be applied to ST symbol rate adaptation.

Note that, this system with 4 transmit antennas can have 16 choices of ST symbol rates, i.e., (14

←· · · → 16

4). For convenience and less complexity, we use 4 choices, i.e., L

T = 1, 2, 3, 4. That is, Θn ∈ L

T =

1, LT = 2, L

T = 3, LT = 4 with a fixed constellation. In the following context, the integer of L

T is referred

as “layer”. The simulation results are shown in Fig. 5, where each subfigure has its own constellation.

We summarize these results in Table II.

V. JOINT ADAPTIVE TECHNIQUE

As shown in the previous two subsections, either constellation adaptation or ST symbol rate adaptation

can increase the average transmission rate while the given BER is satisfied as compared to non-adaptive

schemes. However, we can further improve the average transmission rate by applying a joint adaptation.

The joint adaptation is performed by choosing the best pair of constellation size and ST symbol rate. The

available transmission modes are increased. That is,

Θn ∈ (BPSK, LT

= 1), . . . , (BPSK, LT

= 4),

(QPSK,LT = 1), . . . , (QPSK,

LT = 4),

(8PSK, LT = 1), . . . , (8PSK, L

T = 4),

(16QAM, LT = 1), . . . , (16QAM, L

T = 4)We can reduce the gap between the selection modes further by adding more choices of the transmission

rates. For the target BER, a scheme with the joint adaptation can improve the average transmission rate

significantly as compared to the two techniques in the previous section as shown in Table III.

From the simulation results, we have the following observations:



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• If the ST symbol rate is reduced, the slope of the associated BER curve becomes steeper, which

suggests a larger diversity;

• If the constellation size is reduced, the BER curve will shift to left with the similar slope, which

suggests the diversity keeps the same but the coding gain is improved.

There exists a tradeoff between diversity gain and multiplexing gain [31]. However, this tradeoff can notprovide insight for the adaptive system with discrete constellations. From the above observations, we find

that we can improve data rate by using the two adaptive parameters jointly. Specifically, in some cases,

we can adjust constellation size to improve rate and performance; which in the other cases, we will adjust

ST symbol rate, i.e., multiplexing gain, for adaptation. To proceed, we have the following proposition.

Proposition 1: The average transmission rate in the adaptive selection-mode system can be improved

by adding more possible transmission modes providing higher data rate than the corresponding original

mode at the same SNR region.

Proof: Let us define the SNR regions of our adaptive system using one set of selection modes as

follows.

ℜ1 −→ γ 1 < γ < γ 2 associated with −→ R1

...

ℜi −→ γ i < γ < γ i+1 associated with −→ Ri

If we add more possible selection modes, the SNR regions will be changed as follows.

ℜ1 −→ γ 1 < γ < γ ′

1 associated with −→ R1

ℜ′

1 −→ γ ′

1 < γ < γ 2 associated with −→ R′

1

...

ℜi −→ γ i < γ < γ ′

i associated with −→ Ri

ℜ′

i −→ γ ′

i < γ < γ i+1 associated with −→ R′

i

We assume R′

i > Ri for any i. The total average rate for original scheme can be written as

R =

i

Ri

γ i+1

γ i pΓ(γ )dγ

The total average rate when for the scheme with more transmission modes can be written as

A =

i

(Ri

γ ′

i

γ i pΓ(γ )dγ + R

′

i

γ i+1

γ ′

i

pΓ(γ )dγ )



13

It is obvious that

A > R

In Fig. 6, we compare the average spectral efficiency (ASE) for the three adaptive techniques. As can be

seen from Fig. 6, The ASE of the joint adaptive scheme outperforms the other two schemes significantly

from 0dB to 25dB. At high SNR (larger than 25 dB), three schemes have the same performance. As

predicted by Proposition 1, if there are more available modes, the ASE can be improved further.

V I. CONCLUSIONS

In this paper, statistics of signal-to-interference-noise ratio has been studied for linear dispersion code

with linear minimum-mean-square-error receiver. The associated probability density function of the signal-

to-interference-noise is derived. The average bit-error rate for linear dispersion code with linear minimum-

mean-square-error is found numerically. The simulation and numerical results are provided to verify our

analysis. With these results as guidelines, we proposed a novel adaptive design with discrete selection

modes, in which the linear dispersion code is applied. Since the linear dispersion code is applied, it makes

space-time symbol rate available for adaptation. An adaptive algorithm is proposed for selection-mode

adaptation. Based on the proposed algorithm, two adaptation techniques using constellation and space-time

symbol rate are studied, respectively. With joint adaptation of space-time symbol rate and constellation

size, more transmission modes can be provided to reduce rate gap among transmission modes. Theoretical

analysis shows that the average transmission rate can be improved with more available transmission

modes. Additionally, with space-time symbol rate of the linear dispersion code, the adaptive design can

be simplified and various levels of diversity and multiplexing gain can be provided. Simulation results

were provided to demonstrate merits of the joint adaptation of constellation and space-time symbol rate.

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16

Fig. 1. Selection-mode adaptive system block diagram.

(a) N r = N t = 2 at P/σ2z = 20dB (b) N r = N t = 4 at P/σ2

z = 20dB

Fig. 2. Comparison between the theoretical PDF of SINR and Monte Carlo simulation

(a) 3× 3, 8PSK (b) 4× 4, 4PSK

Fig. 3. Numerical and simulation results for LDC with MMSE reciver



17

(a) L/T = 1 (b) L/T = 2 (c) L/T = 3 (d) L/T = 4

Fig. 4. Adaptive Constellation.

TABLE I

ADAPTIVE CONSTELLATION WITH ST SYMBOL RATE L/T = 1, 2, 3, 4

MODE Constellation L/T Rm γ L

T η

0 - - - γ < −0.6309

1 BPSK 1 1 −0.6309 ≤ γ 11 < −0.1893

2 QPSK 1 2 −0.1893 ≤ γ 12 < 3.384

3 8PSK 1 3 3.384 ≤ γ 13 < 11.7479

4 16QAM 1 4 γ 14 ≥ 11.7479

MODE Constellation L/T Rm γ 2η

0 - - - γ < 0.8385

1 BPSK 2 2 0.8385 ≤ γ 21 < 1.4058

2 QPSK 2 4 1.4058 ≤ γ 22 < 5.3886

3 8PSK 2 6 5.3886 ≤ γ 23 < 15.4452

4 16QAM 2 8 γ 24 ≥ 15.4452


0 - - - γ < 3.1014

1 BPSK 3 3 3.1014 ≤ γ 31 < 4.4833

2 QPSK 3 6 4.4833 ≤ γ 32 < 8.9696

3 8PSK 3 9 8.9696 ≤ γ 33

< 26.5898

4 16QAM 3 12 γ 34 ≥ 26.5898


0 - - - γ < 8.1509

1 BPSK 4 4 8.1509 ≤ γ 41 < 14.2812

2 QPSK 4 8 14.2812 ≤ γ 42 < 24.2533

3 8PSK 4 12 24.2533 ≤ γ 43 < 30.8208

4 16QAM 4 16 γ 44 ≥ 30.8208



18

(a) BPSK (η = 1) (b) QPSK (η = 2) (c) 8PSK (η = 3) (d) 16QAM (η = 4)

Fig. 5. Adaptive ST symbol rate.

TABLE II

ADAPTIVE ST SYMBOL RATE WHEN CONSTELLATION IS BPSK, QPSK, 8PSK AN D 16QAM, RESPECTIVELY.


T η

0 - - - γ < −0.6309

1 BPSK 1 1 −0.6309 ≤ γ 11 < 0.8385

2 BPSK 2 2 0.8385 ≤ γ 21 < 3.1014

3 BPSK 3 3 3.1014 ≤ γ 31 < 8.1509

4 BPSK 4 4 γ 41 ≥ 8.1509

MODE Constellation L/T Rm γ i2

0 - - - γ < −0.1893

1 QPSK 1 2 −0.1893 ≤ γ 12 < 1.4058

2 QPSK 2 4 1.4058 ≤ γ 22 < 4.4833

3 QPSK 3 6 4.4833 ≤ γ 32 < 14.2812

4 QPSK 4 8 γ 42 ≥ 14.2812


0 - - - γ < 3.384

1 8PSK 1 3 3.384 ≤ γ 13 < 5.3886

2 8PSK 2 6 5.3886 ≤ γ 23 < 8.9696

3 8PSK 3 9 8.9696 ≤ γ 33 < 24.2533

4 8PSK 4 12 γ 43 ≥ 24.2533


0 - - - γ < 11.7479

1 16QAM 1 4 11.7479 ≤ γ 14 < 15.4452

2 16QAM 2 8 15.4452 ≤ γ 24 < 26.5898

3 16QAM 3 12 26.5898 ≤ γ 34 < 30.8208

4 16QAM 4 16 γ 44 ≥ 30.8208



19

TABLE III

JOINT ADAPTATION OF ST SYMBOL RATE AND CONSTELLATION SIZE


T η

0 - - - γ < −0.6309

1 BPSK 1 1 −0.6309 ≤ γ 11 < −0.1893

2 QPSK 1 2 −0.1893 ≤ γ 21 < 1.4058

3 QPSK 2 4 1.4058 ≤ γ 22 < 4.4833

4 QPSK 3 6 4.4833 ≤ γ 23 < 8.9696

5 8PSK 3 9 8.9696 ≤ γ 33 < 24.2533

6 8PSK 4 12 24.2533 ≤ γ 34 < 30.8208

7 16QAM 4 16 γ 44 ≥ 30.8208

Fig. 6. Average spectral efficiency comparison for the three adaptive schemes.

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