New Generalized Selection Combining for BPSK Signals in Rayleigh Fading
Channels
Young Gil Kim and Sang Wu Kim
Department of Electrical Engineering and Computer Science
Korea Advanced Institute of Science and Technology
373-1 Kusong-dong, Yusong-gu, Taejon 305-701, Korea
Abstract-We propose a new generalized selection combining
(GSC) scheme that selects diversity branches out of (
)
diversity branches based on the magnitude of log-likelihood ratio (LLR), which, for BPSK signals, is proportional to the product of the fading amplitude and the matched filter output. The
proposed GSC is shown to provide a significant power gain over
the conventional GSC which selects diversity branches based on
signal-to-noise ratio. The proposed GSC scheme for
is
found to be only
dB inferior to the maximal ratio combining
scheme (
), when
, in a Rayleigh fading channel.
1 Introduction
Diversity combining is an efficient method to combat multipath fading because the combined signal-to-noise ratio
(SNR) is increased compared with the SNR of each diversity
branch. The optimum combiner is the maximal ratio combiner (MRC) [1]. But the receiver complexity for the MRC
is directly proportional to the number of resolvable paths, ,
which may vary with location as well as time. From implementation point of view, having the receiver complexity dependent on a characteristic of the physical channel is undesirable.
Generalized selection combining (GSC) scheme selects an
arbitrary (fixed) number of diversity branches and combines
them. The conventional way of selecting diversity branches is
to select branches with the largest instantaneous SNR (or
fading amplitude) [2]-[6]. We will call this -GSC.
In this paper, we propose a new GSC scheme that selects
branches based on the magnitude of log-likelihood ratio
(LLR), which, for binary phase shift keying (BPSK) signals,
is proportional to the product of the fading amplitude and the
matched filter output. We will call this -GSC. The motivation of considering LLR in selecting diversity branches is
that the hard decision based on the sign of LLR minimizes the
error probability and the magnitude of LLR provides the reliability of the hard decision. We derive the BER for the case
of and for a given . In [7], the BER of -GSC
for (selection combining) is derived. We show that
the proposed GSC provides a significant power gain over the
-GSC.
This paper consists of five sections. In Section 2, we describe the system model. In Section 3, we propose a new GSC
scheme and analyze the BER in a Rayleigh fading channel.
In Section 4, numerical results are presented. Finally, conclusions are given in Section 5.
2 System Model
We consider BPSK signals with coherent detection in slow
frequency-nonselective Rayleigh fading channels with additive white Gaussian noise (AWGN). We assume that there are
diversity branches available for combining. The received
low-pass equivalent signal at the th diversity branch is
(1)
branch,
where is the fading amplitude at the th diversity is the fading phase at the th diversity branch, is or
with a priori probability , and is the AWGN at
the th diversity branch with a two-sided power spectral density of . We assume that . The received lowpass equivalent signal at the th diversity branch after phase
compensation, (matched filter output), is
Re Re (2)
(3)
3 New Generalized Selection Combining
We now consider a new GSC scheme that selects diversity
branches based on the magnitude of LLR. The LLR in the
th diversity branch is
(4)
(5)
0-7803-7206-9/01/$17.00 © 2001 IEEE
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(6)
(7)
Since is proportional to the magnitude of LLR, i.e. the
reliability of hard decision, we propose to select diversity
branches with the largest . The decision statistic for the
proposed -GSC, , is therefore
(8)
where is an ordered set
such that , and . We decide that was transmitted if ,
and otherwise, decide was transmitted. When and , the proposed GSC becomes the optimum selection diversity [7] and the MRC, respectively. The decision
statistic for the conventional GSC, called -GSC, is
5 Conclusion
We proposed a new GSC scheme that selects diversity
branches out of ( ) diversity branches based on the magnitude of LLR, which, for BPSK signals, is proportional to the
product of the fading amplitude and the matched filter output.
We derived the BER for for a given , and showed the
BER for is identical to that for . The proposed
-GSC is shown to provide a significant power gain over
conventional GSC (-GSC) which selects diversity branches
based on instatanous SNR (or fading amplitude). The proposed -GSC for is found to be only dB inferior
to the MRC ( ), when .
References
[1] G. L. Stueber, Principles of Mobile Communication. Kluwer
Academic Publishers, 1996.
(9)
[2] M. Alouini and M. K. Simon, “An MGF-based performance
analysis of generalized selection combining over Rayleigh fading channels,” IEEE Trans. on Commun., vol. 48, pp. 401-415,
March 2000.
When , the proposed GSC becomes the optimum
selection diversity which yields the BER of [7]
[3] T. Eng, N. Kong and L. B. Milstein, “Comparison of diversity
combining techniques for Rayleigh-fading channels,” IEEE
Trans. on Commun., vol. 44, pp. 1117-1129, Sept. 1996.
[4] N. Kong and L. B. Milstein, “Average SNR of a generalized
diversity selection combining scheme,” IEEE Commun. Lett.,
vol. 3, pp. 57-59, March 1999.
where , , and . The BER of -GSC for
is the same as that of -GSC for , because
the sign of does not change by adding to . In other words, the sign of decision statistic
for is always the same as that of for since . The BER of -GSC
for is derived in Appendix A. The BER of -GSC
for is hard to derive.
[5] M. Z. Win and J. H. Winters, “Analysis of hybrid
selection/maximal-ratio combining in Rayleigh fading,” IEEE
Trans. on Commun., vol. 47, pp. 1773-1776, Dec. 1999.
[6] L. Yue, “Analysis of generalized selection combining techniques,” Proc. of VTC’2000-Spring, Tokyo, pp. 1191-1195.
[7] Y. G. Kim and S. W. Kim, “Optimum selection diversity
for BPSK signals in Rayleigh fading channels,” Proc. of
ISSSTA’2000, Parsippany, NJ, USA, pp. 50-52.
4 Numerical Results
Figure 1 is a plot of the average BER versus for several
GSC schemes in Rayleigh fading channels. We find that the
proposed -GSC outperforms -GSC. For and , the proposed -GSC provides a power gain of 2.8 dB
over the -GSC. For and , the proposed GSC provides a power gain of 4 dB over the -GSC. For and , the proposed -GSC provides a power gain
of 1.5 dB over the -GSC. The proposed -GSC for is only 0.3 dB inferior to the MRC ( ). The power
gain increases as increases. In Table 1, we compare the
derived BER (11) in the Appendix with the simulated BER for
and . We can verify that the derived BER (11) is
consistent with the simulated BER.
[8] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series
and Products. Academic Press, corrected and enlarged edition,
p. 307, 1980.
[9] H. A. David, Order Statistics. John Wiley & Sons, Inc., 1981.
A
Derivation of the BER for the Proposed -GSC
In this Appendix, we derive the BER of the proposed GSC for and . The BER of -GSC for in derived in [7], and the BER for is identical to that
1206
for . Assuming that and are transmitted
with equal probability, the BER is given by
where .
Let , , be such that . Then,
There are four possible error events given
1.
2.
3.
transmitted (10)
A.1 Derivation of EE1 since the event and
event " or # occurs, where
transmitted:
and (EE1);
, , and (EE2);
, , , and # (when , , and From order statistics [9],
(11)
where we use the equality [8]
where
and
(14)
if where and are the pdf of
and the joint pdf of , respectively. Thus,
! ! ! (13)
Their cumulative distribution function (cdf) ! is given by
(18)
and
(12)
Since ’s are i. i. d., their probability density function (pdf)
is given by
! where , , , and are derived in the following subsections. Let be . Then, assuming was transmitted,
can occur if disjoint
" (when , , and Since the error events EE1, EE2, EE3, and EE4 are mutually
exclusive,
! ).
, , , and (EE4).
(17)
);
(EE3);
4.
, be an ordered
. Assuming that
Let , where
set such that was transmitted,
(19)
(20)
and
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(16)
(15)
otherwise,
$
$ $ (21)
A.2 Derivation of Assuming that
EE2 since the event can occur
if disjoint event " and # occurs, where
and
# (when , , , , and ).
and
" (when , , , , and );
where
$ $ $
was transmitted,
# (when , , , and ).
EE3 since the event can occur if disjoint event " or # occurs,
" (when , , , and );
Assuming that
was transmitted,
A.3 Derivation of ! ! , , , and
From order statistics, the joint pdf of
is given by [9]:
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$
$
0.1
$ 0.01
0.001
A.4 Derivation of Assuming that
0.0001
BER
was transmitted,
EE4 since the event can occur if disjoint event " or # occurs,
a-GSC m=1
a-GSC m=3
|ar|-GSC m=1
|ar|-GSC m=3
MRC
L=4
1e-05
1e-06
L=8
1e-07
1e-08
-2
-4
0
2
4
6
Figure 1: BER vs.
Eb/N0
8
10
12
14
16
18
for , and .
where
" , , , , and );
(when
Table 1: Verification of (11) by computer simulation ( ,
)
(dB)
# , , , , and ).
(when
$
$ $ $ $ and
Simulated BER
0.0569535
0.0280276
0.0114178
0.00382825
0.00105776
0.000254563
5.3285e-05
BER using (11)
0.0569617
0.0279826
0.0113838
0.00380854
0.00105993
0.000251923
5.28678e-05
-4
-2
0
2
4
6
8
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