IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 10, OCTOBER 2001
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Optimum Selection Diversity for BPSK Signals in Rayleigh Fading Channels
Young Gil Kim and Sang Wu Kim, Senior Menber, IEEE
Abstract—We propose a new selection diversity scheme, called
-selection diversity, that selects the branch providing the
largest magnitude of log-likelihood ratio (LLR). The LLR for
BPSK signals in fading channels is found to be proportional to
the product of the fading amplitude and the matched filter output
-selection diversity
after phase compensation. The proposed
scheme is shown to be optimal in the sense of minimizing the
bit error rate (BER), and outperform existing selection diversity
schemes. We also propose a suboptimal selection diversity scheme,
called
-selection diversity, that does not require a phase compensation in the selection process, thereby significantly reducing
-seimplementation complexity. We show that the proposed
lection and
-selection diversity schemes exhibit significant
power gains over existing selection diversity schemes in Rayleigh
fading channels.
Index Terms—BPSK signaling, log-likelihood ratio, Rayleigh
fading, selection diversity.
I. INTRODUCTION
D
IVERSITY is a powerful communication technique that
reduces the depth of the fades and/or the fade duration
by supplying the receiver with multiple replicas of the transmitted signal that have passed over independently fading channels [1], [2]. The simplest form of diversity combining is the
selection diversity. The conventional selection diversity scheme
selects, among the diversity branches, the branch providing
the largest signal-to-noise ratio (or largest fading amplitude)
[1]. We will call this -selection diversity. Another selection diversity scheme proposed by Neasmith and Beaulieu selects the
branch providing the largest magnitude of matched filter output
[3]. We will call this -selection diversity.
In this letter, we propose a new selection diversity scheme,
-selection diversity that selects the branch providing
called
the largest magnitude of log-likelihood ratio (LLR). We show
that the magnitude of LLR for BPSK signals in fading channels is proportional to the product of the fading amplitude and
the magnitude of matched filter output . The magnitude of
LLR represents the reliability of hard decision. The proposed
-selection diversity scheme is shown to be optimal among
all selection diversity schemes in the sense of minimizing the
bit error rate (BER). We also propose a suboptimal selection
-selection diversity, that requires
diversity scheme, called
no phase compensation in the selection process and thus substantially reduces implementation complexity. We show that the
Paper approved by R. Kohno, the Editor for Spread Spectrum Theory and
Applications of the IEEE Communications Society. Manuscript received May
8, 2000; revised December 6, 2000. This paper was presented in part at the IEEE
Sixth International Symposium on Spread Spectrum Techniques and Applications, Parsippany, NJ, September 6–8, 2000.
The authors are with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Taejeon 305–701 Korea
(e-mail: swkim@san.kaist.ac.kr; ygkim@bada.kaist.ac.kr).
Publisher Item Identifier S 0090-6778(01)09114-0.
proposed
-selection and
-selection diversity schemes
outperform -selection and -selection diversity schemes. We
-selection diversity provides the
also show that the proposed
same BER provided by the maximal ratio combining (MRC)
for dual diversity system, and is only 0.3 dB inferior to the
MRC when the number of diversity branches is four, in Rayleigh
-selection diversity provides
fading channels. The proposed
power gains of 0.5 dB and 1.7 dB over -selection diversity and
-selection diversity, respectively, when the number of diversity
branches is four, and even a higher power gain is obtained with
a larger diversity order.
This letter consists of five sections. In Section II, we describe
-selection and
the system model. In Section III, we propose
-selection diversity schemes, and derive the BER of BPSK
-selection diversity in frequency-nonselective
signals with
Rayleigh fading channels. In Section IV, numerical results are
presented. Finally, conclusions are provided in Section V.
II. SYSTEM MODEL
We consider binary phase shift keying (BPSK) with coherent
detection in slow frequency-nonselective Rayleigh fading
channels with additive white Gaussian noise (AWGN). We
assume that there are independent branches for combining.
The low-pass equivalent received signal at the th branch before
phase compensation is
(1)
where
fading amplitude in the th branch;
fading phase in the th branch;
or
with a priori probability
;
additive complex Gaussian noise in the th branch.
Note that is a nonnegative real number.
The low-pass equivalent received signal at the th branch after
phase compensation is
(2)
. We assume that
and
for all , where
is the energy per informa-
where
tion bit.
III. NEW SELECTION DIVERSITY SCHEMES
In this section, we propose optimal and suboptimal selection
diversity schemes that select the branch based on the magnitude
in the th branch is given by
of LLR. The LLR
0090–6778/01$10.00 © 2001 IEEE
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 10, OCTOBER 2001
(3)
in (3) is the hard decision value (i.e., decide that
The sign of
was transmitted if
, and decide that
was
represents
transmitted, otherwise), and the magnitude of
the reliability of the hard decision. Our new selection diver-selection diversity, selects the branch providing
sity, called
. In Appendix A, we prove that the proposed
the largest
-selection diversity minimizes the BER among all selection
diversity schemes.
Note that
-level quantization (or
bits) for
, the required
register sizes for storing the decision statistics
for the MRC and
for
-selection, where
is
bits and
the selected branch index, are
bits, respectively. Also, the MRC requires
summations, whereas the proposed
-selection requires
comparisons: comparison of
and
can be
from
and reading only the sign
done by subtracting
bit after the subtraction. Therefore, the MRC incurs a larger
-selection. This
computational delay than the proposed
-bit
is because the borrow propagation delay in the
s for the
subtraction operations involved in comparing
-selection is smaller than the carry propagation
proposed
-bit summation operations
delay in the
involved in calculating the decision statistic
for the MRC. The register size and computational delay are
further increased when the Viterbi decoding is performed.
with
In Appendix B, we show that the average BER
-selection diversity in Rayleigh fading channels is given by
(4)
.
where the approximation in (4) is accurate for
Based on the relationship in (4), we propose another new
-selection diversity,
selection diversity scheme, called
. Note that
that selects the branch providing the largest
-selection diversity does not require phase information
in the branch selection process, and
thus substantially reduces implementation complexity.
For the MRC, which provides the lowest BER among all
combining schemes for fading channels with additive Gaussian
noise, the combiner output is [1]
(6)
where
and
(5)
was transmitted if
, and otherwise,
We decide that
was transmitted. It should be noted that the maxdecide
,
dominates the sign of .
imum of
, the sign of
is determined
Particularly for
and
. This implies that the proby the maximum of
-selection diversity yields the same BER provided by
posed
.
the MRC for
The MRC requires the fading phase information
and
phase compensations before
combining, as indicated in (5). Similarly, -selection diversity
-selection diversity require phase information and
and
phase compensations before selection. But -selection diversity
-selection diversity do not require phase information
and
in the selection process. In fact,
-selection diversity
requires phase compensation only in one branch selected,
thereby significantly reduces implementation complexity,
-selection
particularly for large . Also, the proposed
diversity scheme has an advantage over the MRC in terms of
computational delay and hardware complexity. Assuming an
Note that if
is substituted in (6), then we get
which is the well-known average BER of BPSK signaling in
Rayleigh fading channels [1].
IV. NUMERICAL RESULTS
Fig. 1 is a plot of the average BER versus
for several
diversity schemes in frequency nonselective Rayleigh fading
-selection diversity is evaluated by
channel. The BER with
-selection
computer simulation. We find that the proposed
-selection diversity schemes outperform -selection diand
,
-selection diversity and -selection diversity. For
versity provides power gains of 2 dB and 3.1 dB over -selection diversity and -selection diversity, respectively, and is
-seleconly 0.3 dB inferior to the MRC. We also find that
tion diversity provides power gains of 1.7 dB over -selection
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 10, OCTOBER 2001
Fig. 1.
BER versus E
=N
1717
for L = 1; 2; 4, and 8.
diversity, and 0.5 dB over -selection diversity for
power gain increases as increases.
. The
where
V. CONCLUSION
We proposed optimal (
-selection) and suboptimal
-selection) selection diversity schemes that select the
(
branch based on the magnitude of the LLR. The LLR is found
to be proportional to the product of the fading amplitude and
the matched filter output after phase compensation. We showed
-selection diversity scheme minimizes
that the proposed
the BER, and outperforms existing selection diversity schemes.
The suboptimal selection diversity scheme does not require
phase compensation in the selection process, thereby signif, the
icantly reduces implementation complexity. For
-selection diversity and
-selection diversity
proposed
provide power gains of 2 and 0.5 dB over -selection diversity,
and 3.1 and 1.7 dB over -selection diversity, respectively, in
frequency nonselective Rayleigh fading channels. The power
gain increases as increases.
APPENDIX A
PROOF OF THE OPTIMUM SELECTION DIVERSITY
In Appendix A, we prove that the proposed
-selection diversity minimizes the average BER among selection diversity
schemes. Let be the transmitted BPSK symbol and is the
is given by
demodulated symbol. Then, the average BER
and
is the joint probability density function of
It follows from (8) that is minimized by maximizing
for all and , where
.
th branch selected
th branch selected
th branch selected
(9)
(10)
th branch selected
and
th branch selected
. Note that equality
is minimized, by selecting the
in (10) is achieved, and thus
. In what
branch providing the maximum
is a monotonically
follows, we show that
increasing function of
because
(7)
(8)
(11)
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If
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 10, OCTOBER 2001
, then
where we use the fact that
and
. Thus
and independency between
(19)
(12)
Since
If
, then
(13)
(20)
Using (3) and
where we use the equality [4]
if
(14)
if
For the derivation of
, we calculate
.
Thus
(15)
. Therewhich is a monotonically increasing function of
in (8) is minimized by selecting the
fore, the average BER
.
branch providing the largest
APPENDIX B
DERIVATION OF (6)
Here we derive (6) using the method in [3]. Without loss of
was transmitted. Then
generality, we assume that
where
. Since
and
s are indepen-
dent
th branch selected
first branch selected
(16)
(17)
(21)
(18)
Using (19)–(21) in (18), we get (6).
Let
, and
. Then
REFERENCES
[1] S. Benedetto and E. Biglieri, Principles of Digital Transmission With
Wireless Applications. New York: Kluwer Academic/Plenum, 1999.
[2] M. Schwartz, W. R. Bennett, and S. Stein, Communication Systems and
Techniques. New York: McGraw-Hill, 1966.
[3] E. A. Neasmith and N. C. Beaulieu, “New results on selection diversity,”
IEEE Trans. Commun., vol. 46, pp. 695–704, May 1998.
[4] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, corrected and enlarged ed. New York: Academic, 1980, p. 307.