Comparison and analysis of combining techniques for spatial multiplexing space time block coded systems in rayleigh fading channel

International Journal of Electronics and Communication
International Journal of Electronics and Communication Engineering & Technology (IJECET),
Engineering & Technology (IJECET) IJECET
ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - Feb (2011), © IAEME
ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online)
Volume 2, Number 1, Jan - Feb (2011), pp. 01-10 ©IAEME
© IAEME, http://www.iaeme.com/ijecet.html

COMPARISON AND ANALYSIS OF COMBINING
TECHNIQUES FOR SPATIAL MULTIPLEXING/SPACE
TIME BLOCK CODED SYSTEMS IN RAYLEIGH FADING
CHANNEL
Prof. Vijay K. Patel
Assistant Professor & HOD, Electronics & Communication Department
U. V. Patel College of Engineering
Ganpat vudyanagar, Mehsana, Gujarat
E-Mail: vijayk_patel@yahoo.com

Dr D. J. Shah
Principal
L. C. Institute of Technonogy, Bhandu, Mehsana, Gujarat
E-Mail: research@dharmeshshah.org

ABSTRACT
Space divisional multiplexing (SDM) and space-time block coding (STBC) have
begun to appear in the latest wireless communication systems. In recent years different
detection strategies for these schemes have been proposed which can be broadly
categorized as group based or direct-detection techniques. This paper presents the
detection technique that gives superior bit error rate compared to conventional
techniques. STBC coded signals are detected using different combining techniques like
equal gain combining, selection combining. No diversity scheme (i.e. single input single
output (SISO), Alamouti and Maximum Ratio Combining (MRC) schemes are compared.
But Maximum Ratio Combining (MRC) outperforms all the other techniques for any type
of STBC code configuration. This is true even in Rayleigh faded environment.
Keywords: space division multiplexing, space-time block code, Rayleigh fading.
1. INTRODUCTION
The capability of the wireless systems have been greatly improved by the
introduction of spatial diversity mechanisms at the transmitter side and at the receiver
side (i.e. multiple transmit and receive antennas). The two common forms of spatial

1


transmit processing are space division multiplexing [1] and space-time block coding.
[2][3] While SDM increases the data rate by transmitting independent data streams
through the different transmit antennas, in STBC, the rate is kept constant or even
reduced but the performance (bit error rate (BER)) is improved by means of a space-time
code that indicates the transmission pattern of a block of symbols over the available
transmit antennas and several time periods. The Alamouti coding scheme [2] has
achieved great popularity due to its simple optimal decoding and large performance
improvement it offers. There is a problem in combining the two approaches, SDM and
STBC. [4]-[8], with the aim of achieving high transmission rates at low BERs. Also
varieties of combining techniques are developed at the receiver side. [9][10]
The receiver with multiple receive antennas can combine signals from different
antennas by various techniques. These combining techniques include selection combining
(SC), maxilla ratio combining (MRC) and equal gain combining (EGC). In SC, the
received signal with the highest SNR among NR receive antennas is selected for
decoding. In MRC all NR branches are combined by weighted sum of all the branches.
Weight factor of each branch must be matched to the corresponding channel and then
added. EGC is a special case of MRC in the sense that all signals from multiple branches
are combined with equal weights.
The paper is organized as follows: Section II describes the basics of STBC.
Section III describes the Alamouti Space Time Code (STC) with decoding. The
combining techniques are described in IV. Simulation results and discussions are
presented in section V and conclusion is reflected in Section VI.
2. SPACE TIME BLOCK CODES: OVERVIEW
This section describes the mathematical description of STBC. Based on the
mathematical model, a pairwise error probability is derived. Finally a space-time code
design criterion is described by using a pairwise error probability.
Consider NT transmit antennas and NR receive antennas. In space-time coded
Multiple Input Multiple Output (MIMO) systems, bit streams is mapped into symbol
N
{ }
stream xi
i =1
. As shown in figure 1, a symbol stream of size N is space-time-encoded

NT
into { xi(t ) } , t = 1, 2, K , T , where i is the antenna index and t is the symbol time index.
i =1

2


Note that the number of symbols in a space-time codeword is NT •T (i.e., N= NT •T )
resulting in symbol rate of N/T.

Figure 1 Scenario of Different MIMO Channels.
As shown in Figure 1, there are three different systems like SISO with single
input and single output. Other system is Single Input Multiple Output (SIMO) and
Multiple input Single Output (MISO). Third system is Multiple Input, Multiple Output
(MIMO) system.
N
At the receiver side, the symbol stream {x }
i
i =1
is estimated by using the received

NR
signals { y (jt ) } , t = 1, 2, K , T . Let h(jit ) denote the Rayleigh-distributed channel gain from
j =1

the ith transmit antenna to the jth receive antenna over the tth symbol period (i=1,2,…,NT,
j=1,2,…,NR, and t=1,2,…,T). If we assume that the channel gains do not change during T
symbol periods, the symbol time index can be omtted. Furthermore, as long as the
transmit antennas and receive antennas are spaced sufficiently apart, NR × NT fading gains

{h( ) } can be assumed to be statistically independent. If x( ) is the transmitted signal from
t
ji i
t

the ith transmit antenna during tth symbol period, the received signal at the jth receive
antenna during tth symbol period is

3


 x1( t ) 
 (t ) 
Ex ( t )   x2 
 h j1 , h j 2 , K , h jN   + z (jt ) --------------------------------------- (1)
y (j ) = (t ) ( t )
t

N 0 NT  T 

 M
 x(t ) 
 NT 

where z (j ) is the noise process at the jth receive antenna during tth symbol period,
t

which is modeled as the zero mean circularly symmetric complex Gaussian (ZMCSCG)
noise of unit variance, and Ex is the average energy of each transmitted signal.
Meanwhile, the total transmitted power is considered as
NT

∑E
i =1
{ x } = N , t = 1, 2,K ,T
(t )
i
2
T ----------------------------------- (2)

Note that when variance is assumed to be 0.5 for real and imaginary parts of hji, the
probability density function (PDF) of each channel gain is given as

(
f H ji ( h ji ) = f H ji Re {h ji } , Im {h ji } )
------------------------------------- (3)
=
1
π
exp − h ji( 2
)
In a similar manner, the PDF of the additive noise can be expressed as

j
( )
f Z (t ) z (j ) =
t 1
π (
exp − z (j )
t
2
). ------------------------------------- (4)
Pairwise Error Probability:
Assuming that Channel State Information (CSI) is exactly known at the receiver
side and the noise components are independent, the conditional PDF of the received
signal is given as

fY (Y H , X ) = f ( Z ) Z

NR

( )
T
1 2
= ∏∏ exp − z (jt ) --------------------------------------- (5)
j =1 t =1 π
1
=
πN T R
(
exp −tr ( ZZ H ) )
Using the above conditional PDF, the ML codeword XML can be found by
maximizing (5), that is

4


X ML = arg max f y (Y H , X )
X
--------------------------- (6)
    
H
1 EX EX
= arg max exp  −tr  Y −
 HX  Y −
 HX   
 
X πN T
R
  N 0 NT  N 0 NT  
 
 EX  EX  
H

= arg max tr  Y − HX  Y − HX  
 N 0 NT  N 0 NT  
   
X

Note that the detected symbol X is erroneous (i.e. XML≠X ) when the following
condition is satisfied:
 EX  EX  
H

tr  Y − HX  Y − HX   ≥
 N 0 NT  N 0 NT  
   
------------------------------- (7)
 EX  EX  
H

tr  Y − HX ML  Y − HX ML  
 N 0 NT  N 0 NT  
   
Probability that X is transmitted but XML ≠ X is given as
 EX 
Pr ( X → X ML ) = Q 
 2N N H ( X − X ML ) F 

 0 T 
This is upper-bounded as
1  E H ( X − X ML ) F
2

Pr ( X → X ML ) ≤ exp  − X 
2  N 0 NT 4 
 
3. ALAMOUTI SPACE TIME CODE DESIGN
The very first and well-known STBC is the Alamouti code, which is a complex
orthogonal space-time code specialized for the case of two transmit antennas [2]. In this
section, we first consider the Alamouti STBC and it can be generalized to the case of
three antennas or more [11].
In the Alamouti encoder, two consecutive symbols x1 and x2 are encoded with the
following space-time code word matrix:
 x1 − x2 
*
X = * 
----------------------------------- (8)
 x2 x1 
Alamouti encoded signal is transmitted from the two transmit antennas over two
symbol periods. During the first symbol period, two symbols x1 and x2 are simultaneously
transmitted from the two transmit antennas. During the second symbol period, these
*
symbols are transmitted again, where − x2 is transmitted from the first transmit antenna
*
and x1 transmitted from the second antenna. The Alamouti code has been shown to have a

5


diversity gain of 2. We now discuss ML signal detection for Alamouti space-time coding
scheme. Here, we assume that two channel gains h1(t) and h2(t) are time-invariant over
two consecutive symbol periods, that is,
h1 ( t ) = h1 ( t + Ts ) = h1 = h1 e jθ1
------------------------------------- (9)
h2 ( t ) = h2 ( t + Ts ) = h2 = h2 e jθ2
Where │h1│and θi denote the amplitude gain and phase rotation over the two
symbol periods, i=1,2. Let y1 and y2 denote the received signals at time t and t + Ts,
respectively, then
y1 = h1 x1 + h2 x2 + z1
* *
y2 = −h1 x2 + h2 x1 + z2
where z1 and z2 are the additive noise at time t and t + Ts, respectively. ML
receiver structure for this scheme can be given by:
 y 
xi ,ML = Q  2 i 2  , i = 1, 2. ---------------------------------------- (10)
h +h 
 1 2 
Where Q (.) denotes a slicing function that determines a transmit symbol for the
given constellation set. The above equation implies that x1 and x2 can be decided
separately, which reduces the decoding complexity of original ML-decoding algorithm
from │C│2 to 2│C│ where C represents a constellation for the modulation symbols x1
and x2.
4. RECEIVE DIVERSITY
Consider a receive diversity system with NR receiver antennas. Assuming a single
transmit antenna as in the single input multiple output (SIMO) channel, the channel is
expressed as
T
h =  h1h2 L hN R 
 
For NR independent Rayleigh fading channels. Let x denote the transmitted signal
with the unit variance in the SIMO channel. The received signal y ∈ C N R ×1 is expressed as

Ex
y= hx + z ------------------------------------------------ (11)
N0
Where z is ZMCSCG noise with E { zz H } = I N R . The received signals in the

different antennas can be combined by various techniques. These combining techniques
include selection combining (SC), maximal ratio combining (MRC), and equal gain

6


combining (EGC). In SC, the received signal with the highest SNR among NR branches is
selected for decoding. Let γi be the instantaneous SNR for the ith branch, which is given
as
Ex 2
yi = hi , i = 1, 2, L , N R -------------------------------------------------- (12)
N0
Then the average SNR for SC is given as

ρ SC = E max hi{ ( )} ⋅ N , i = 1, 2,L , N
i
E 2 x

0
R -------------------------------------------------- (13)

In MRC, all NR branches are combined by the following weighted sum:
NR
yMRC =  w1( MRC ) w2MRC ) L wNMRC )  y = ∑ wi( MRC ) yi ------------------------------- (14)

( (
R  i =1

where y is the received signal in (11) and wMRC is the weight vector. As
Ex
yi = hi x + zi from (11), the combined signal can be decomposed into the signal
N0

and noise parts, that is,

 Ex 
yMRC = w T 
MRC  hx + z 

 N0 
---------------------------------------------------- (15)
Ex T
= w MRC hx + w T z
MRC
N0
Average power of the instantaneous signal part and that of the noise part in equation
(15) are respectively given as
 E

2
 E
 E
{ }
2
Ps = E  x
w MRC hx  = x E w T hx = x w T h and
T
MRC MRC
 N0
  N0

N0

{
Pz = E w T z
MRC
2
}= w T 2
MRC 2 -----------------------------------------------(16)
From equation (16), the average SNR for the MRC is given as
T 2
Ps Ex w MRC h
ρ MRC = =
Pz N 0 w T 2
MRC 2
Which is upper bounded as
T 2 2
Ex w MRC 2 h 2 Ex 2
ρ MRC ≤ 2
= h 2
---------------------------------------------------- (17)
N0 wT N0
MRC 2
Here the SNR in above equation is maximized at wMRC=h*, which
2
yields ρ MRC = E x h 2 / N 0 . In other words, the weight factor of each branch in equation

7


(14) must be matched to the corresponding channel for MRC. Equal gain combining
(EGC) is a special case of MRC in the sense that all signals from multiple branches are
combined with equal weights.
5. SIMULATION RESULTS AND DISCUSSIONS
Simulation is carried out by considering the Rayleigh faded channel with STBC in
MIMO environment. It is assumed that the channel response remains constant during the
period of observation. Symbols are transmitted by multiple transmit antennas and
received by multiple receive antennas. QPSK modulation scheme is used in all the
results. Perfect channel state information is assumed to be available at the receiver.
Comparison is done among SISO, MRC with one transmit and two receive antennas and
MRC with one transmit and four receive antennas. This result is shown in figure 3.
Performance of MRC for Rayleigh faded Channel

SISO
-1 MRC(Tx=1,Rx=2)
10 MRC(Tx=1,Rx=4)

-2
10
B E R -->

-3
10

-4
10

0 2 4 6 8 10 12 14 16 18 20
SNR dB -->

Figure 2 The Performance of MRC for Rayleigh fading Channels
As shown in Figure 2, MRC with multiple receive antennas give better
performance compared to SISO channel. Also as the number of receive antennas
increases, BER improves by large amount as SNR increases.
Figure 3 shows the comparison of SISO, Alamouti with two transmit and one
receive (2 × 1), Alamouti with two transmit and two receive antennas (2 × 2), MRC (1 ×
2) and MRC (1 × 4). Note that the Alamouti coding achieves the same diversity order
as 1×2 MRC technique (implied by the same slope of the BER curves).

8


0
BER Performance of Alamouti scheme and MRC with equal diversity
10
SISO
Alamouti (2x1)
-1
10 Alamouti (2x2)
MRC(1x2)
MRC(1x4)
-2
10

BER -->
-3
10

-4
10

-5
10

-6
10
0 2 4 6 8 10 12 14 16 18 20
SNR dB -->

Figure 3 Performance Comparison of Alamouti and MRC schemes
Also shown is the 2×2 Alamouti technique which achieves the same diversity
order as 1×4 MRC technique.
CONCLUSION
From these results it is found that Alamouti scheme achieves the same diversity
order as 1 × 2 MRC technique. Due to a total transmit power constraint (i.e., total
transmit power split into each antenna by one half in the Alamouti coding), however,
MRC technique gives better performance than Alamouti technique in providing a power
combining gain in the receiver. Also with the same diversity order of 2×2 Alamouti and
1×4 MRC technique, MRC outperforms Alamouti scheme.
REFERENCES
1. G. Foshini, ’Layered space-time architecture for Wireless communication in a fading
environment when using multi-element antennas,’ Bell Labs Technical J., vol. 1,
pp.41-59, 1996.
2. A. Alamouti, ‘A simplet transmit diversity technique for wireless communications,’
IEEE J. Sel. Areas Comm. vol 16, pp. 1451-1458, 1998.
3. V. Tarokh, H. Jafarkhani and A. Calderbank, ‘Space-time block codes from orthogonal
designs,’ IEEE trans. Inform. Theory, vol. 45, pp. 1456-1467, July 1999.
4. A. F. Naguib, N. Seshadri, and A. R. Calderbank, “Applications of space-time block
codes and interference supression for high capacity and high data rate wireless

9


systems," in Proc. Asilomar Conf. Signals, Systems Computers, Asilomar, USA, Nov.
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coded scheme: performance and improvement in correlated fading channels," in Proc.
IEEE Wireless Commun. Networking Conf., Orlando, USA, Mar. 2002, pp. 194-199.
7. S. N. Diggavi, N. Al-Dhahir, and A. R. Calderbank, “Algebraic properties of space-
time block codes in intersymbol interference multiple access channels," IEEE Trans.
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8. W. F. Jr., F. Cavalcanti, and R. Lopes, “Hybrid transceiver schemes for spatial
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10. Todd K. Moon, “Error Correction Coding Mathematical Methods and Algorithms,”
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