SINR_IEEE_ISWCS

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SINR Analysis for Full-Rate Linear Dispersion

Code Using Linear MMSE

Mabruk Gheryani, Zhiyuan Wu, Yousef R. Shayan

Department of Electrical Engineering, Concordia University Montreal, Quebec Canada

{m gherya, zy wu, yshayan}@ece.concordia.ca

Abstract— We have studied the statistics of signal-to-interference-noise for a MIMO transceiver using full-rate lin-ear dispersion code and linear minimum-mean-square-error(MMSE) receiver over a Rayleigh fading channel. The associatedprobability density function of the signal-to-interference-noiseis derived, which will benefit the future study, such as error-rate probability. Simulation results are provided to verify thetheoretical analysis.

I. INTRODUCTION

The use of multiple antennas at the transmitter and receiver

can lead to significant spectral efficiency on a scattering-rich

wireless channel [1][2]. This has spurred a remarkable thrust

into the so-called Multiple-Input Multiple-Output (MIMO)

technologies. One of the most promising technologies is space-

time (ST) coding, e.g., [3]-[6].

Among the existing space-time codes, the linear disper-

sion code (LDC) [4]-[6] is often preferable for full rate.

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 [4][6]. The LDC breaks the data streaminto sub-streams that are dispersed over space and time and

then combined linearly at the transmitter [4]. As a subopti-

mal receiver for the LDC, linear minimum-mean-square-error

detector is more attractive due to its simplicity and good

performance [7][8].

However, the performance analysis in this case is still

deficient. Most of related works were addressed only for the

V-BLAST [9][10] scheme, a special case of the full-rate LDC.

For example, the case of two transmit antennas was analyzed

in [11] 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 wasprovided in [12].

The main goal of this paper is to study the statistics of SINR

for full-rate LDCs [4]-[6] using linear minimum-mean-square-

error(MMSE) receiver over a Rayleigh fading channel. In this

paper, the associated probability density function of the SINR

is derived, which will benefit future studies, such as error-

rate probability. This paper will be organized as follows. Our

system model is presented in section II. The statistics of SINR

with the MMSE receiver is derived in section III. Simulation

ML

Ant-N t

Symbols

S/P

M1

1 Ant-1

MMSE

1

Nt

Nt

Ant-1

Ant-N r

Fig. 1. System Block Diagram.

results are provided in section IV to verify the theoretical

analysis. Finally, in section V, conclusions are drawn.

I I . SYSTEM MODEL

In this study, a block fading channel model is assumed

where the channel keeps constant in one modulation block

but may change from block to block. That is, the channel

is not necessarily constant within a coding frame which often

consists of a large number of modulation blocks. 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 but

unknown to the transmitter. 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 considered MIMO 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 . We assume the symbols xi

are Gaussian with zero mean and unit variance and indepen-dent to each other. Each block of symbols will be mapped to

a dispersion matrix of size N t × T and then transmitted over

the N t transmit antennas over T channel uses. An N t × T codeword matrix is constructed as [4].

X =Li=1

Mixi +Li=1

Nix∗i (1)

where Mi, Ni are the dispersion matrices associated with

the i-th symbol. For simplicity, the following model will be



considered in this study, i.e.,

X =Li=1

Mixi (2)

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 (1). Received signals associated with

one modulation block can be written as

Y =

P

N tHX+Z =

P

N tH

Li=1

Mixi +Z (3)

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. Let vec() 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 . For full-rate LDC, we have GH G = I [4]

[5] [6]. Denoting hi = Hmi as the i-th column vector of H,

the above equation can also be written as

y =

P

N t

L

i=1hixi + z (5)

III. SINR ANALYSIS WITH THE LINEAR MMSE

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

hjxj + z (6)

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 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 detector is applied and the corresponding

output is given by

xi = wH i y = xi + zi. (7)

where zi is the noise term of zero mean. The corresponding

wi can be found as

wi =

hih

H i +RI

−1hi

hH i

hih

H i +RI

−1hi

(8)

where RI = HI HH I + σ2zI. Note that the scaling factor

1

hHi (hihHi +RI)

−1

hi

in the coefficient vector of the MMSE

estimator wi is added to ensure an unbiased estimation as

indicated by (7). The variance of the noise term zi can be

found from (7) and (8) as

σ2i = wH i RI wi (9)

Substituting the coefficient vector for the MMSE estimator in(8) into (9), the variance can be written as

σ2i =1

hH i R

−1I hi

(10)

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

γ i =1

σ2i=

P

N t

hH i R

−1I hi (11)

In our system model, all the symbols has the same SINR, i.e.,

γ 1 = γ 2 = .........γ L = γ By using singular value decomposition (SVD), (11) can be

written asγ =

P

N t

hH i UΛ

−1UH hi (12)

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

hl, l = 1, ...........N rN t.

Equation (12) can be written as

γ = P N tN rN t

l=1

|˜hl|

2

λl(13)

with

λl =

λl = 1

σ2z

(γλl + 1) l = 1, ......., r ≤ L − 1

σ2z l = r + 1, ........., N rN t(14)

where γ = P N tσ2z

and r is the rank of HI HH I which is less

than N 2t − 1.

The vector h has the same statistics as the original vector

hi [1]. For analytical purpose, we can replace |hl|2 by |hil|

2.

Now, we can write (13) as

γ =r

i=1

γ |hil|2

(γλl + 1)+

N tN rr+1

γ hil|2 (15)

The probability density function (PDF) of γ can be found

using the moment generating function (MGF) as follows [13].

First, we find the conditional (on the eigenvalues) MGF of γ as

M γ/λ (s) = [M γ (γs)]N rN t−r

rl=1

M γ

γs

(γλl + 1)

(16)



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, ....., r,, then (16) can be written as

M γ/λ (s) = [M γ (γs)]N rN t−r

M γ

γs

(γλ + 1)

r(17)

Further,

M γ/λ (s) = 1(1 − γs)

N rN t−r× 1

1 − γs(1+γλ)

r (18)

we can find the probability density function (PDF) of γ conditionally on λ by using inverse laplace transform for (18)

as [15]

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 ) (19)

where 1F 1(. , . , .) is Kummer’s confluent hypergeometric func-

tion [14] 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 (19) 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 ) (20)

Now, we can find the probability density function (PDF) of γ as follows.

P Γ(γ ) =

∞0

P γ/λ(γ )f λ(λ) dλ (21)

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) (22)

where

Φk+1(λ) =

k!

(k + N rN t − r)!

1

2

LN rN t−rk (λ)

k = 0,...r − 1

where LN rN t−rk (λ) is the associated Laguere Polynomial of

order k [14]. Equation (22) 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 (23)

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 (21) as

P Γ(γ ) =K γ

r

r−1k=0

K 1(k)ki=0

K 2(i)2kd=0

K 3(d)

∞0

(1 + γλ)r λN rN t−r+d

1F 1(N rN t − r, N rN t, λγ )exp(−γλ)dλ (24)

The term (1 + γλ)r

can be written as

(1 + γλ)r

= γ rr

v=0rv γ v−rλv

Then equation (24) will be as

P Γ(γ ) =K γ γ r

r×

∞n

γ nK (n)r

v=0

K (v)K 1(k)ki=0

K 2(i)

2kd=0

K 3(d)×

r−1k=0

∞0

λN rN t−r+d+v+n exp(−γλ) dλ (25)

with

K (n) =(N rN t − r)n

(r)nn!

and

K (v) =

rv

γ v−r

The general form of the integration of (25) can be found in

[14] ∞0

xΘ exp(−µx)dx = Θ!µ−Θ−1

where

Θ = N rN t − r + d + v + n

Then (25) can be written as

P Γ(γ ) =K γ γ r

r

∞n

γ nK (n)r

v=0

K (v)r−1k=0

K 1(k)k

i=0

K 2(i)×

2kd=0

K 3(d)γ −N rN t+r−d−v−n−1(N rN t−r+d+v+n)! (26)



0 10 20 30 40 50 60 70 80 900

5

10

15x 10

4

Monte Carlo Simulation

Theoretical PDF

−2

10

Fig. 2. Comparison between the theoretical PDF of SINR and Monte CarloSimulation when Nt=Nr=2 at P/σ2

z=20dB

Further,

P Γ(γ ) =K γ γ rγ r

rγ N rN t+1

r−1k=0

K 1(k)ki=0

K 2(i)

2kd=0

K 3(d)×

γ −dr

v=0

K (v)γ −v∞n

K (n)(N rN t − r + d + v + n)! (27)

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) =

rv

Then (27) can be written as

P Γ(γ ) =K γγ r−N rN t−1

r

r−1k=0

K 1(k)ki=0

K 2(i)×

2kd=0

K 3(d)γ −dr

v=0

K (v, d) (28)

This is the PDF of SINR for our system over Rayleigh

fading channels.

0 10 20 30 40 50 600

1

2

3

4

5

6

7x 10

4

Monte Carlo Simulation

Theoretical PDF

−2

10

Fig. 3. Comparison between the theoretical PDF of SINR and Monte CarloSimulation when Nt=Nr=4 at P/σ2

z=20dB

IV. SIMULATION RESULTS

In this section, we verify our derivation by simulation. In

the simulation, N t = N r = T = 2 and N t = N r = T = 4were assumed. In Fig. 2 and Fig. 3, the theoretical results

of the SINR in (28) and Monte Carlo simulation results were

compared for 2×2 and 4×4 channels, respectively at P/σ2z =20dB. Simulation results match to the analytical result very

well.

V. CONCLUSIONS

In this paper, over a Rayleigh fading channel, the probability

density function of signal-to-interference-plus-noise ratio of

a MIMO transceiver using full-rate linear dispersion code

and linear minimum-mean-square-error receiver is derived.

Monte Carlo results are presented to verify our derivation.

The analytical results will shed light on the future study of

linear dispersion codes.

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SINR_IEEE_ISWCS