8/14/2019 Performance Analysis of Convolutionally Coded DS-CDMA Systems

1/19

Performance Analysis of Convolutionally Coded

DS-CDMA Systems with Spatial and Temporal

Channel Correlations

Jie Lai and Narayan B. Mandayam

Wireless Information Network Laboratory (WINLAB)

Department of Electrical & Computer EngineeringRutgers University

73 Brett Road, Piscataway, NJ 08854-8060

Abstract

Combined spatial and temporal processing has been shown to increase the potential link

capacity enormously for wireless communication systems, especially when the channels

between different transmit and receive antenna pairs are uncorrelated. In this paper,

we consider both spatial and temporal channel correlations that may be encountered in

space-time processing and present the performance analysis of convolutionally coded

DS-CDMA systems. An upper b ound for the average bit error probability (Pb) is

derived for the case of perfect channel estimation and an analytical approximation for

Pb is derived in the case of erroneous channel estimates. The analytical approach is

general enough to be applicable to various space and time diversity situations, such as

wideband multipath channels and antenna arrays.

Keywords: Space-time processing, Convolutional codes, Correlated channels, CDMA

This work was supported in part by the NSF under grant number NCR 97-06036.

8/14/2019 Performance Analysis of Convolutionally Coded DS-CDMA Systems

2/19

1 Introduction

Space-time processing has been shown to yield tremendous capacity gains in multiple-input

and multiple-output (MIMO) wireless communication systems, especially when the transferfunctions between different transmit and receive antenna pairs are uncorrelated [13]. The

coherent rake receiver with pilot-aided channel estimation will be used to improve the re-

verse link performance versus the non-coherent rake receiver in wideband DS-CDMA based

third-generation cellular systems. Most of the analytical studies in the literature on the per-

formance of convolutional codes in DS-CDMA systems do not consider channel correlation

and channel estimation errors. The usual assumptions are spatially uncorrelated multi-

paths/antenna signals, temporally independent symbol errors by perfect interleaving andperfect channel estimation. In the absence of the above idealistic assumptions, performance

of such systems will be degraded. In practice, the temporal correlation in each fading path

could be caused by imperfect interleaving due to application delay constraints. The spatial

correlation between different rake fingers could exist when multipath components and their

associated fingers are close to each other or the fingers come from different antennas with

limited spacing in the case of antenna diversity. It has been shown in [4, 5] that spatial

correlation between signals at different antenna elements may limit potential capacity gains.Thus it is of interest to analyze the performance of convolutional codes in DS-CDMA sys-

tems with spatial and temporal channel correlations. Further, since the success of space-time

processing relies on accurate channel estimates, it is also of interest to study the performance

under channel estimation errors.

In previous work, the performance analysis of CDMA systems has been conducted exten-

sively [68]. The union bound has been used to analyze the coding performance in DS-CDMA

systems under perfect channel state information and memoryless fading [7, 8]. The coding

performance over non-independent Rician fading channels has been studied with perfect

channel information [9]. Pairwise error probability of non-interleaved codes over Rayleigh

channel with channel estimation was analyzed in [10] for a general maximum likelihood (ML)

decoder. In this paper, considering the effect of temporal and spatial channel correlations, we

2

8/14/2019 Performance Analysis of Convolutionally Coded DS-CDMA Systems

3/19

first derive an upper bound for the average bit error probability for a convolutionally coded

DS-CDMA system under perfect channel estimation. We then present an approximation for

the average bit error probability by taking channel estimation errors into account. Simula-

tions using SPW (Signal Processing WorkSystem, Cadence) are performed to compare withthe analytical results.

2 System Model with Spatial and Temporal Channel

Correlations

The system model considered here is that of a convolutionally coded DS-CDMA system

employing a coherent rake receiver. It is assumed that every user is received with equal

average signal power at the base station, i.e., power control is implicitly considered to have

eliminated the near-far effect due to the pathloss and also large-scale fading. However, the

short-scale (such as Rayleigh) fading is assumed to remain uncompensated.

The wideband frequency-selective channel is modeled as a random delay, discrete resolv-

able path model [11]. The number of resolvable paths of the channel for the k-th user is

assumed to be L(k). The received signal over each path is temporally correlated due to imper-

fect interleaving and channel correlation. In most analytical studies, the different paths are

generally assumed to be mutually independent. However, when the resolvable delay between

two consecutive paths becomes smaller (especially in the case of systems with relatively high

chip rates as in WCDMA), the fading signals in two consecutive paths could be correlated.

Such correlation may come from either the spatial correlation between two paths with small

delay spread or the unresolvable paths due to pulse shaping filtering. Also, in the situation

of antenna diversity, the correlation between different rake fingers may come from the spatial

correlation between different antennas due to limited antenna spacing. In this paper, we do

not distinguish the correlation between paths due to different reasons but simply characterize

such correlation as spatial correlation. Thus, both the temporal correlation in each fading

path and the spatial correlation between different fading paths could exist and affect the

3

8/14/2019 Performance Analysis of Convolutionally Coded DS-CDMA Systems

4/19

coding performance.

2.1 Temporal Channel Correlation

For the k-th user, the envelope of each path in the wideband channel is assumed to be

Rayleigh distributed. Thus, the sample value of each path at time iT (time index i) is a

complex Gaussian random variable given as

h(k)l (i) = h

(k)xl (i) +jh

(k)yl (i), 1 l L(k), (1)

where T is the sample period, h(k)xl (i) and h

(k)yl (i) represent respectively the I and Q compo-

nents of the l-th path at time index i of the k-th user. The temporal autocorrelation function

of h(k)l (i) is given as [12]

(k)ll (m) =

1

2E[h

(k)l (i)(h

(k)l (i + m))

] = ((k)hl )

2J0(2fDmT), 1 l L(k), (2)

where fD is the maximum Doppler frequency and J0() is the zero-order Bessel function.

2.2 Spatial Channel Correlations

The spatial (cross) correlation coefficient between two Rayleigh faded envelopes r(k)l (i) =

|h(k)

l (i)| and r(k)

n (i) = |h(k)

n (i)| is expressed as (page 51 in [12])

(k)r =(1 +

(k)ln )Ei(

2q

(k)ln

1+(k)ln

) 22 2

(3)

where (k)ln denotes the correlation coefficient between the complex amplitudes h

(k)l (i) and

h(k)n (i), and Ei() denotes the complete elliptical integral of the second kind. The spatial and

temporal correlation function between two complex amplitudes with time index difference

m can be expressed as

(k)ln (m) =

1

2E[h

(k)l (i)(h

(k)n (i + m))

] = (k)ln

(k)hl

(k)hn J0(2fDmT), 1 l L(k). (4)

Readers are referred to [12] for further details and [13] for the algorithm for the generation

of Rayleigh fading signals with spatial correlation.

4

8/14/2019 Performance Analysis of Convolutionally Coded DS-CDMA Systems

5/19

3 Bit Error Probability Analysis

In this section, we analyze the bit error probability of a convolutional code with maximum-

ratio combining (MRC) rake receiver in DS-CDMA systems. We first consider the case ofperfect channel estimation and derive an upper bound for the average probability of bit error

under both spatial and temporal channel correlations.

3.1 Upper bound for average bit error probability under perfect

channel estimation

It is well known that the output of the maximum-ratio combining (MRC) rake receiver with

channel estimation hl (i) can be expressed as [68]

U(i) = Re{Ll=1

(

Eshl(i)b(i) + wl(i))h

l (i)} (5)

in the case of BPSK data modulation with b(i) {1, +1} being the modulated data symboland Es being the symbol energy. In the above equation and henceforth, we drop the super-

index notation (k) and refer to the reference user only. Using the Gaussian assumption for

the interference from (K1) users, the total interference wl(i) = wxl(i)+jwyl(i) is a complex

white Gaussian variable with zero mean and variance I0/2 [7,8]

2w =1

2E[wl(i)w

l (i)] =I02

= (K 1) Ec2

+N02

(6)

where Ec = Es/g = REb/g is the chip energy with g being the processing gain, Eb being the

bit energy, and R being the code rate.

When the transmitted codeword is all-zero codeword C0, a Viterbi decoder will select the

erroneous codeword Cj if the received signals Euclidean distance to Cj is less than that to

C0 [9,10], i.e.,Ni=1

[U(i) bj(i)]2