Performance of General Order Selection Diversity in Spatially Correlated Rayleigh Fading Channel

Keon-Wook Lee, Hee-Nam Cho, and Yong-Hwan Lee

School of Electrical Engineering and INMC, Seoul National University Kwanak P. O. Box 34, Seoul, 151-600 Korea

Email: {kwlee, hncho}@ttl.snu.ac.kr and ylee@snu.ac.kr

Abstract - In this paper, we investigate the performance of

general order selection diversity in multi-antenna Rayleigh fading channel. Most of previous works considered the selection of the maximum of independently and identically distributed random variables. We consider the use of the n -th order selection, and analyze its performance in terms of the expected signal-to-noise power ratio (SNR) and capacity in independently but not necessarily identically distributed Rayleigh fading channel where the fading process between the transmit antennas is correlated. The performance can be represented in a recursive-form and verified by computer simulation. Simulation results show that expected SNR and capacity of the n -th order selection in spatially correlated channel become higher than those of spatially uncorrelated channel for small values of n , and opposite for large values of n .

I. INTRODUCTION

The use of selection diversity has been considered as one of efficient and simple techniques for wireless communications in the presence of multipath fading. Recently, generalizations of selection diversity that selects multiple diversity links or users have been studied. For example, generalized selection combining (GSC) combines the signals received through a number of strong paths instead of all the paths [1], [2]. Multi-user opportunistic scheduling selects multiple users for simultaneous services at the same time, termed parallel multi-user scheduling (PMUS) [3], [4]. Thus, in order to design and analyze such systems, it is necessary to examine the performance corresponding to the n -th order selection.

A few previous works investigated diversity schemes with general order selection in multi-antenna wireless systems [5]–[7]. The asymptotic performance of generalized transmit antenna selection was analyzed in Rayleigh fading channel [5]. However, it considered independently and identically distributed (i.i.d.) fading channel. The performance of general order selection combining was analyzed in independently but not necessarily identically distributed (i.n.d.) Weibull and Nakagami fading channels [6]. The signal-to-noise power ratio (SNR) of the n -th order selection was analyzed in correlated

This work was supported by the IT R&D program of MKE/KEIT

[KI002139, development of cooperative operation profiles in multicell wireless systems].

Nakagami fading channel [7]. However, these works did not fully consider the joint effect of selection diversity and multi-antenna transmission techniques on the system performance. Moreover, it is required to consider the effect of spatial correlation which is an unavoidable in real multi-antenna channel environments [8].

In this paper, we investigate the performance of general order selection diversity with transmit beamforming in multi-antenna Rayleigh fading channel. We analyze the performance in terms of the expected SNR and the capacity, assuming that the multi-antenna channel of each user (or link) is spatially correlated and the fading processes on different users are i.n.d.. The analytic results are given in a recurrence relation for efficient computation.

This paper is organized as follows. Section II introduces the system model in consideration. Section III analyzes the expected SNR and the capacity of the n -th order selection. Section IV verifies the analytic results by computer simulation. Finally, Section V summarizes conclusions.

II. SYSTEM MODEL

Consider the downlink of a wireless system where the base station (BS) serves K users. We assume that the BS has TN transmit antennas and each user has a single receive antenna. Fig. 1 illustrates the system model when 2TN = and 5K = . Assuming the use of transmit beamforming, the received signal of user k can be represented as

*k k k k ky s n= +h w (1)

where kh denotes the ( )1TN × channel vector of user k , whose elements are zero-mean complex Gaussian random variables, kw denotes the ( )1TN × normalized beam weight vector for user k , ks denotes the data symbol transmitted to user k with power constraint 2{ }kE s P= , kn denotes additive white Gaussian noise (AWGN) with variance 2

Nσ , and the superscript * denotes conjugate transpose. In a spatially correlated Rayleigh fading channel, the channel covariance matrix of user k defined by

{ }*k k kER h h (2)

is a positive semi-definite Hermitian matrix and thus can be decomposed as [8]

*k k k k=R V Σ V (3)

where ( )1, , [ ]Tk k N k=V v v is an ( )T TN N× unitary

matrix whose columns are the eigenvectors of kR and kΣ is an ( )T TN N× diagonal matrix whose diagonal elements { }1, ,, ,

Tk N kλ λ are descending ordered distinct and non-negative real eigenvalues (i.e., 1, , 0

Tk N kλ λ≥ ≥ ≥ ). Then, kh can be represented as [9]

1/ 2

1/ 2

k k k

k k k

=

=

h R h

V Σ h (4)

where ( )1, ,[ ]T

Tk k N kh h=h is the ( )1TN × virtual channel

vector of user k , whose elements are i.i.d. zero-mean complex Gaussian random variables with variance 2

kσ . Here, the superscript T denotes the transpose operation. Note that the second equality in (4) comes from the fact that the unitary transformation does not change the isotropicity of an i.i.d. channel. We consider two transmit beamforming schemes: One is coherent beamforming (CBF) with the use of instantaneous channel state information (CSI) and the other is eigen-beamforming (EBF) with the use of statistical CSI [10], [11]. When the CBF is employed with optimum beam weight vector

2/ || ||k k k=w h h , the instantaneous SNR of user k can be represented as [10]

2

, , 21

.TN

k m k m km N

Phγ λσ=

= ∑ (5)

Since the instantaneous SNR of EBF with beam weight ,Tk N k=w v can easily be calculated from (5) with 1TN = , we

consider only the CBF case. For ease of description, let :n Kγ denote the n -th largest instantaneous SNR among K i.n.d. instantaneous SNR vlaues { }1, , Kγ γ . Then, the complementary cumulative density function (cdf) of :n Kγ can be represented as [12]

( ) { }

( ):

1 2

:

, , , 1 1

Pr

( ) 1 ( )satisfying

n K

k kj jK

i

n K

i KK

i n k k k j j iA

G z z

G z G z

γ

γ γ

γ

= = = +

= >

= −∑ ∑ ∏ ∏ (6)

where ( )XG z

denotes the complementary cdf of random variable X and the condition iA is given by [6]

{ }1 2

1 2

1 2

, , , 1, 2, ,

; ,

,.

K

i j

i

i i K

k k k K

k k i j

k k kk k k

∀

+ +

∈

≠ ≠

< < << < <

(7)

Note that the condition iA can have Ki

⎛ ⎞⎜ ⎟⎝ ⎠

cases. Then, (6) can be rewritten in a recursive-form as

( ) ( ) ( )1: :1 2

1

1

, , , 1

( ) 1 ( ) .satisfying

n K n K k kj jK

n

n K

k k k j j nA

G z G z G z G zγ γ γ γ−

−

−

= =

= + −∑ ∏ ∏ (8)

Thus, expected SNR and the capacity of n -th order user can

be calculated as

( ):

:

: 0

0

1 ( )

( ) ,

n K

n K

n K

d G zz dz

dz

G z dz

γ

γ

γ∞

∞

−=

=

∫

∫ (9)

( )

( )

( )

:

:

: 20

0

1 ( )log 1

1 ( ) .1

n K

n K

n K

G zC z dz

dz

G z dzz

γ

γ

∞

∞

−= +

=+

∫

∫ (10)

III. EXPECTED SNR AND CAPACITY OF n -th ORDER USER

It can be seen from (5) that kγ is a sum of TN independent exponential random variables with scale parameter , ,m k m k kθ λ γ= . It can be shown that the complementary cdf of

kγ can be represented as [13]

,

,

,

1 , ,

,1

( )

.

Tm k

k

Tm k

zNm k

m i m m k i k

zN

m km

G z e

e

θγ

θ

θθ θ

π

−

= ≠

−

=

⎛ ⎞= ⎜ ⎟⎜ ⎟−⎝ ⎠

=

∑ ∏

∑ (11)

It can be shown from (9) that

,1: ,0

11

1 1T

m k

zNK

K m kmk

e dzθγ π−∞

==

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟= − −

⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠∑∏∫ (x)

Fig. 1. An example of the system model when 2TN = and 5K = .

( )

( )

,1 1

1 11 1

,1

1

,01 1

1,

1 1 1

1,

1

1:

1

1

Tm k

K

T Tm ii i

iK

zNK

m kk m

zN N KK

m im m i

Kj

j Kj

K e

e dz

s

θ

θ

π

π

γ

=

−∞

= =

−−

= = =

−

=

⎛⎜= +⎜⎝

⎞∑⎛ ⎞ ⎟+ − ⎜ ⎟ ⎟⎝ ⎠ ⎠

= −

∑ ∑∫

∑ ∑ ∏

∑

(12)

where

1 2 1

,1

,1 1 1

1 ,

.1

i iT T

j j

i i

j

m kN Ni

j K jk k k K m m

i m k

sπ

θ

=

≤ < < < ≤ = =

=

=∏

∑∑ ∑ ∑∑

(13)

Similarly, it can also be shown from (6) and (8) that

,

: ,011

,

Tm i

zNK

K K m imi

K K

e dz

s

θγ π−∞

==

⎛ ⎞⎜ ⎟=⎜ ⎟⎝ ⎠

=

∑∏∫ (14)

and

( )

( )

( )

1 21

1

1: : 0, , , 1

satisfying

1

: 1,0

1: ,

1

( ) 1 ( )

11

1

1

1 .1

k ki jK

n

n K

n K n Kk k k j j n

A

K nj

n K n j Kj

Kj n

n K j Kj n

G z G z dz

K K nn j

sK

n j

js

j n

γ γγ γ

γ

γ

−

−∞

−= =

− +

+ −=

− +

= −

= + −

− +⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠= + −

⎛ ⎞⎜ ⎟+ −⎝ ⎠

⎛ ⎞= + − ⎜ ⎟− +⎝ ⎠

∑ ∏ ∏∫

∑

∑

(15)

When the channels are i.i.d. (i.e., , ; ,i k i i kθ θ ∀= ), (15) can be rewritten as

( )

( )

( )

1 2

:, , , 1 1

satisfying

01 1

,0

( ) 1 ( )

1

1

11

k kj jK

i

T Tm m

i KK

n Ki n k k k j j i

A

i K iz zN NK

m mi n m m

K K ip

i p i pi n p

K n p

p

G z G z

Ke e dz

i

K K is

i p

KK

K n

γ γ

θ θ

γ

π π

= = = +

−− −∞

= = =

−

+ += =

− −

= −

⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟= −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠−⎛ ⎞ ⎛ ⎞

= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

−⎛ ⎞= −⎜ ⎟−⎝ ⎠

∑ ∑ ∏ ∏

∑ ∑ ∑∫

∑ ∑

( ),

0

K nK p K psK n

p K p

−− −

=

−⎛ ⎞⎜ ⎟ −⎝ ⎠

∑

(16)

where

( )

1

1,

1 2

1

., , ,

Ti

TTNT

Nm

ii

K K NNm m K i

i i

Ks

m m m m

π

θ

=

+ + =

=

⎛ ⎞= ⎜ ⎟

⎝ ⎠

∏∑

∑ (17)

Similarly, it can be shown from [14]

( ) ( )10

11 ln 2

z ee dz Ez

αα α

∞ − =+∫ (18)

that the capacity of n -th order user can be represented as

( ) 11: ,

11 ,

Kj

K j Kj

C q−

=

= −∑ (19)

: , ,K K K KC q= (20)

( ) 11: : ,

1

11

Kj n

n K n K j Kj n

jC C q

j n− +

−= −

⎛ ⎞= + − ⎜ ⎟− +⎝ ⎠

∑ (21)

where

,1

1 1

1

, 1 ,1 1 1 1 1,

1ln 2

j

m ki i iT T

i i

j j i i

N N jj

j K m kk k K m m i im k

eq Eθ

πθ

=

≤ < < ≤ = = = =

∑⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

∑∑ ∑ ∑ ∑ ∏ (22)

and ( )1t

xE x e t dt

∞ −= ∫ denotes exponential integral function.

When the channels are i.i.d., (21) can be rewritten as

( ) ( ),

:0

11

K nK n p K p K p

n Kp

qK K nC K

K n p K p

−− − − −

=

− −⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠

∑ (23)

where

1

1

, 11 1 1

., , ln 2

ji

i Tii

TNT

mNj

miK K i

Nm m K i ii

K meq Em m

θ

πθ

=

+ + = = =

∑⎛ ⎞ ⎛ ⎞

= ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

∑ ∑ ∏ (24)

It was shown that multi-user diversity is diminished in antenna-dominant environments (i.e., TN K ) [15]. However, this is not valid when the spatial channel has full correlation (i.e.,

Tk N=R J ), where TNJ is an ( )T TN N×

matrix whose elements are all equal to one. To measure the gain of multi-user diversity, we define the opportunistic scheduling gain oppΔ by the difference of the average capacity between the opportunistic scheduling and the round-robin scheduling among K

users, i.e.,

( ) 1: 1:1.opp KK C CΔ = − (25)

The following theorem states asymptotic behavior of ( )opp KΔ as TN goes to infinity in fully correlated spatial channel.

Theorem: In fully correlated spatial channel, as TN goes to infinity, we have

( ) ( )( )

,

a contant when is a constant lim log log when =O

log when =Ok NT

T T

opp T T

N NT

KK N K N

N K e∀ →

→∞

⎧⎪⎪Δ = ⎨⎪⎪⎩

R J∼

∼

(26)

Proof: It can be shown from (19) that

( ) ( ) 1

11

1

1

1lim 1ln 2

1 1 .ln 2

T

k NT

T

kKk N

oppk T

N

T

K kK e Ek N

e EN

γ

γ

γ

γ

∀

−

→ =

⎛ ⎞⎛ ⎞Δ = − ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞− ⎜ ⎟

⎝ ⎠

∑R J

(27)

Since (27) does not give any intuition, we can approximate it using Jensen’s inequality as

( ) ( )2 2

1

21

1lim log 1 log

1 1log .

k NT

K

opp T Tk

K

kT

K N Nk

N k

γ γ

γ

∀ → =

=

⎛ ⎞Δ + −⎜ ⎟

⎝ ⎠⎛ ⎞

= +⎜ ⎟⎝ ⎠

∑

∑

R J (28)

Since 011/ logK

kk K ε

=+∑ , we have

( ) ( )2 0,

lim log logk NT

T

opp

N

K K ε∀ →

→∞

Δ ≈ +R J

(29)

where 0ε is the Euler constant.

It can be seen from (29) that even if TN goes to infinite, ( )opp KΔ is an increasing function of K in fully correlated

spatial channel, yielding a multi-user diversity gain. This result is opposite to the channel hardening property (i.e.,

( )lim 0T

oppNK

→∞Δ =

regardless of K ) [15]. It can be also seen from (26) that the difference growth rate in ( )opp KΔ according to the relative growth rate between the TN and K . Thus, it can be said that multi-user scheduling is still effective in antenna-dominant environments when the spatial channel is fully correlated.

IV. SIMULATION RESULTS

To verify the analytic results, we consider the multi-user downlink with CBF in Rayleigh fading spatial channel with and without the presence of spatial correlation. As in [16], we consider a channel with spatial correlation given by

{ }*, ,

i ji k j k kE h h ρ −= , where kρ is the transmit correlation

coefficient between two adjacent antennas of user k . For modeling of i.n.d. channel environments, we assume that the average SNR and the amplitude of the transmit channel correlation coefficient are linearly increased at a rate of

5%/user from the initial value 0 dB and 0.7, respectively. Fig. 2 and 3 respectively depict the expected SNR and capacity of the n -th order user with the use of CBF in i.n.d. Rayleigh fading channel when 4TN = and 8K = . It can be seen that the analytic results agree well with the simulation results. It can also be seen that the expected SNR and capacity of the n -th order user in spatially correlated channel become higher than those of spatially uncorrelated channel for small values of n , and opposite for large values of n . As the spatial correlation increases, the dominant channel component associated with the largest eigenvalue plays a major role, which may decrease the overall channel diversity or increase the entire channel fluctuation. This high channel fluctuation not only increases the fading peaks but also increases the deep fades, thus performance of higher order users become better while that of lower order users become worse. Fig. 4 and 5 depicts the oppΔ in fully correlated Rayleigh

n

:8nγ

1 2 3 4 5 6 7 80

2

4

6

8

10

12

Uncorrelated channel - SimulationSpatially correlated channel - SimulationSpatially uncorrelated channel - AnalysisSpatially correlated channel - Analysis

Fig. 2. The expected SNR of the n -th order user when 4TN = and 8K = .

n

:8 (

bps/

Hz)

nC

1 2 3 4 5 6 7 81

1.5

2

2.5

3

3.5

4

Uncorrelated channel - SimulationSpatially correlated channel - SimulationSpatially uncorrelated channel - AnalysisSpatially correlated channel - Analysis

Fig. 3. The capacity of the n -th order user when 4TN = and 8K = .

fading channel where all users have the same 10 dB average SNR. Due to the computation constraint, since it’s difficult to simulate the case of TNK e= when TN is large, the case of

TNK e= is only evaluated when TN is small in Fig. 4. Despite of the use of Jensen’s inequality, it can be seen that the analytic results in (26) agree well with the simulation results for the three relative growth rate between the TN and K .

V. CONCLUSIONS

In this paper, general order selection diversity in spatially correlated i.n.d. Rayleigh fading channels is analyzed. The expected SNR and capacity of n -th order user with transmit beamforming are analyzed with a recurrence relation for efficient computation. It can be seen that expected SNR and capacity of the n -th order user in spatially correlated channel become higher than those of spatially uncorrelated channel for small values of n , and opposite for large values of n . It can

be also shown that selection diversity is still effective in antenna-dominant environments when the spatial channel is fully correlated.

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1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Number of Tx antenna (NT)

Opp

ortu

nist

ic s

ched

ulin

g ga

in (b

ps/H

z)

K=4K=Nt

K=eNt

( )scales as log TN

( )( )scales as log log TN

Fig. 4. oppΔ according to K when TN is small.

10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Number of Tx antenna (NT)

Opp

ortu

nist

ic s

ched

ulin

g ga

in (b

ps/H

z)

K=4K=16K=NT

K=5NT

( )( )scales as log log TN

Fig. 5. oppΔ according to K when TN is large.