Generalized Selection Combining Based on the
Log-Likelihood Ratio
Sang Wu Kim
Young Gil Kim
Marvin K. Simon†
Korea Adv. Inst. of Sci. and Tech.
Taejon 305-701, S. Korea
sangkim@ieee.org
University of Seoul
Seoul 130-743, S. Korea
ygkim@ieee.org
Jet Propulsion Laboratory
Pasadena, CA 91109-8099
marvin.k.simon@jpl.nasa.gov
Abstract- We propose a generalized selection combining (GSC)
scheme for binary signaling in which M diversity branches providing
the largest magnitude of log-likelihood ratio (LLR) are selected and
combined. The bit error probability provided by LLR-based GSC
serves as a lower bound on the bit error probability provided by
any GSC techniques. We also propose a suboptimal GSC based
on a noncoherent envelope detection. We derive the bit error
probability with LLR-based and envelope-based GSC techniques and
examine their power gains over the conventional SNR-based GSC
technique. We show that the bit error probability with maximum
ratio combining or square-law combining of L branches is identical
to that with LLR-based GSC of L/2 branches.
Keywords- Generalized selection combining, log-likelihood
ratio, diversity, envelope detection, Rayleigh fading channel.
I. I NTRODUCTION
Diversity combining is an efficient method for improving
the performance of digital communication systems over fading
channels [1]. The optimal combining technique is maximal
ratio combining (MRC) which maximizes the SNR of the combined signals at the expense of implementation complexity.
The conventional selection combining (SC) technique selects
the path (diversity branch) providing the largest SNR. SC is
the simplest method because only the best diversity branch
is selected for further processing. However, SC provides a
lower SNR gain than MRC, because the receiver does not
fully exploit the available diversity offered by the channel.
Recently, a hybrid of MRC and SC, called generalized
selection combining (GSC), has been proposed and analyzed
[2]-[6]. GSC selects a subset of all diversity branches and
combines the signals in the subset using the MRC rule. MRC
is sensitive to channel estimation errors which tend to be
more important when the instantaneous SNR is low. With
GSC, the weak signals which are prone to these errors are
excluded in the combining. In the context of spread-spectrum
communication with RAKE reception, the complexity of MRC
depends on the number of resolvable paths available which
may vary with location as well as time. From an implementation point of view, having the receiver complexity dependent
on a characteristic of the physical channel is undesirable.
† Currently a Visiting Scholar in the Department of Electrical Engineering
at University of California, Los Angeles (UCLA), Los Angeles, CA 90095.
The conventional GSC rule selects the M out of L diversity
branches providing the largest instantaneous SNR (or fading
amplitude). This will be called SNR-based GSC. In this paper,
we propose a new GSC scheme for binary signaling that
selects M diversity branches based on the magnitude of the
log-likelihood ratio (LLR) and will be called LLR-based GSC.
The motivation for using LLR in selecting diversity branches
is that the magnitude of the LLR provides the reliability of
hard decision and the LLR-based hard decision minimizes the
bit error probability. As such, the bit error probability provided
by LLR-based GSC serves as a lower bound on the bit error
probability provided by any other GSC scheme. In [7], LLRbased GSC for M = 1 has been analyzed. In this paper, we
generalize the result in [7] to M > 1 and present the optimum
GSC rule that minimizes the bit error probability. We show that
the bit error probability with MRC or square-law combining
(SLC) of L branches is identical to that with LLR-based GSC
of L/2 branches. We also present a simple, but suboptimum,
GSC technique based on noncoherent envelope detection and
refer to it as envelope-based GSC.
This paper consists of seven sections. In Section II, we
describe the system model. In Section III, we present the
LLR-based GSC. In Section IV, we present a suboptimum
GSC based on envelope detection. In Section V, we present
the LLR-based GSC for noncoherent binary frequency-shift
keying (BFSK) signaling. In Section VI, numerical results are
discussed and the conclusion is given in Section VII.
II. S YSTEM M ODEL
We consider selecting M signals from L (≥ M ) independent diversity branches and combining them on a symbolby-symbol basis. The modulation is chosen to be binary
phase-shift-keying (BPSK). The GSC rule for BFSK will
be presented in Section V. The channel is characterized by
slow, flat Rayleigh fading and additive white Gaussian noise
(AWGN). The received low-pass equivalent signal from the
ith diversity branch can be expressed as
yi = hi x + ni
(1)
where hi is the channel complex gain on the ith diversity
branch, and ni is a complex Gaussian noise with mean zero
and variance N0 /2 per dimension. We assume that |hi | is
Rayleigh distributed with E[|hi |2 ] = 1 and that the phase of
0-7803-7802-4/03/$17.00 © 2003 IEEE
2789
hi is uniformly
distributed
over [0,2π]. The transmitted signal
√
√
x is + Es or − Es with equal probability. We assume that
the channel gains and the received signals at different diversity
branches are independent.
III. LLR- BASED GSC
In this section, we present LLR-based GSC which is optimal
in the sense of minimizing the bit error probability. We
first consider selecting two branches, say i and j, among L
branches. Then, the log-likelihood ratio Λi,j for x is given by
√
P (x = + Es |hi , yi , hj , yj )
√
Λi,j = ln
(2)
P (x = − Es |hi , yi , hj , yj )
√
P (yi , yj |x = + Es , hi , hj )
√
(3)
= ln
P (yi , yj |x = − Es , hi , hj )
=
=
√
2
√
2
e−(|yi −hi Es | +|yj −hj Es | )/N0
√
√
ln
−(|yi +hi Es |2 +|yj +hj Es |2 )/N0
√e
4 Es
Re{h∗i yi + h∗j yj }
N0
(4)
(5)
where Re{h∗i yi + h∗j yj } = (|hi |2 + |hj |2 )x + Re{h∗i ni +
error
h∗j nj }. The optimum decision√rule that minimizes the bit √
probability is to decide x = + Es if Λi,j > 0 and x = − Es
if Λi,j < 0. The sign of Λi,j is the hard decision value and the
magnitude of Λi,j represents the reliability of hard decision.
Then, assuming that diversity branches i and j are selected,
the bit error probability Pe (i, j) is given by
1
Pe (i, j) =
.
(6)
1 + e|Λi,j |
A derivation of (6) is provided in Appendix A. Since Pe (i, j)
decreases with increasing |Λi,j |, the optimum selection rule
that minimizes the bit error probability is to select the pair
of branches providing the largest |Λi,j |. In general, for M (≤
L/2) branches the optimum selection rule is to select those
that provide the largest |Λi1 ,i2 ,...,iM | where
√
4 Es
Λi1 ,i2 ,...,iM =
Re{h∗i1 yi1 + h∗i2 yi2 + ... + h∗iM yiM } (7)
N0
is the LLR given that diversity
L branches i1 , i2 , ..., iM are
hypotheses.
Then, the opselected. This involves a test of M
√
timum decision rule is to
decide
x
=
+
E
if
Λ
>0
s
i
1 ,i2 ,...,iM
√
and, otherwise, x = − Es .
For the special case of M = L (MRC), the average error
probability is given by
1
(8)
Pe,M RC = EΛM RC
1 + e|ΛM RC |
where
ΛM RC =
√
L
4 Es
i=1
N0
Re{h∗i yi }.
(9)
It can be shown that the bit error probability with MRC of L
branches is identical to that with LLR-based GSC of M = L/2
branches. This follows from the fact that the sign of the MRC
output is identical to that of the LLR-based GSC output. A
proof is given in Appendix B.
In determining the appropriate M (≥ L/2) branches upon
which to make a final data decision (by taking the sign
of its
L
LLRs
LLR), it turns out that instead of having to compute M
and choose the largest, one merely needs to divide the group
of L branches into two arbitrarily selected complementary
subgroups of M and L − M branches (i.e. the groups have
no overlapping members), make a single comparison of the
magnitudes of the LLR for these two subgroups and choose
the larger of the two for the data decision. By doing so we get
equivalent performance to MRC of L branches. This is proved
in Appendix C.
It should be noted that averaging Λi,j in (5) over ni and nj
and taking the absolute value yields 4(|hi |2 + |hj |2 )Es /N0 ,
which is the sum of SNR’s at diversity branches i and j.
Therefore, SNR-based GSC takes the average noise power
into account. However, LLR-based GSC exploits the noise
term Re{h∗i ni + h∗j nj } in (5) by selecting the branches for
which the signs of x and Re{h∗i ni + h∗j nj } are identical
and |hi |2 + |hj |2 is large, thereby providing the largest LLR
magnitude. As a result, the performance is governed by the
peak, as opposed to the mean, channel condition. Thus, the
bit error probability resulting from LLR-based GSC serves as
a lower bound on the bit error probability of any generalized
selection combining rule.
IV. E NVELOPE - BASED GSC
In this section, we present a simple suboptimal generalized selection combining rule based on noncoherent envelope
detection. Suppose branches i and j were selected. Then, it
follows from (5) that
√
4 Es
|Λi,j | =
|Re{h∗i yi + h∗j yj }|
(10)
N0
√
4 Es
[|Re{h∗i yi }| + |Re{h∗j yj }|] (11)
≤
N0
√
4 Es
[|hi ||yi | + |hj ||yj |].
(12)
≤
N0
Since envelope detection of the received RF signal yields |yi |,
we propose selecting the M diversity branches that maximize
M
i=1 |hi ||yi |. We call this envelope-based GSC. If we let
{|h(1) ||y(1) |, |h(2) ||y(2) |, ..., |h(L) ||y(L) |} denote an ordered set
such that |h(1) ||y(1) | ≥ |h(2) ||y(2) | ≥ ... ≥ |h(L) ||y(L) |, then
the log-likelihood ratio ΛM,env for deciding x is
ΛM,env
√
M
4 Es =
Re{h∗(i) y(i) }
N0 i=1
(13)
which√leads to a simpler test than that based on (7). We decide
that Es√was transmitted if ΛM,env > 0, and otherwise,
decide − Es was transmitted. Then, it follows from (6) and
(13) that the bit error probability with envelope-based GSC is
given by
1
Pe,env = EΛM,env
.
(14)
1 + e|ΛM,env |
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V. N ONCOHERENT BFSK S IGNALING
In this section, we present the generalized selection combining rule for BFSK signals with noncoherent detection. Let
sl (t) = A cos(2πfl t), l = 1, 2
(15)
be the transmitted signal, where A is the signal amplitude and
fl is the lth tone frequency. We assume that the tone frequencies f1 and f2 are chosen such that the signals {sj (t), j =
1, 2} are noncoherently orthogonal, i.e., |f2 − f1 | = 1/T .
Then, the received signal ri (t) at the ith branch given that
sj (t) is transmitted is
ri (t) = |hi |A cos(2πfj t+θi )+ni (t), i = 1, ..., L, 0 ≤ t ≤ T
(16)
where |hi | is Rayleigh distributed fading at the ith branch and
θi is uniformly distributed over [0, 2π]. We assume that |hi |
and θi are independent for all i, and are also independent of
ni (t). Then, the mth energy detector output yim at the ith
branch, given that sl (t) is transmitted, is
2
2 T
yim =
ri (t) cos(2πfm t)dt
(17)
T 0
2
2 T
+
ri (t) sin(2πfm t)dt
(18)
T 0
 2

A2 T

|h
|
cos
θ
+
n

i
i
c,ij
2


2
=
(19)
A2 T

+
, m=l

2 |hi | sin θi + ns,ij



m = l,
(nc,im )2 + (ns,im )2 ,
where
nc,im
ns,im
2 T
ni (t) cos(2πfm t)dt,
T
0 T
2
=
ni (t) sin(2πfm t)dt
T 0
=
(20)
(21)
are independent Gaussian random variables each with mean
zero and variance N0 /2 for all i and m. Since nc,im and ns,im
are independent, yi1 and yi2 are independent. Also, since |hi |
is Rayleigh distributed and θi is uniformly distributed over
[0, 2π], |hi | cos θi is a Gaussian random variable with mean
zero and variance E[|hi |2 ]/2. Thus, the conditional probability
density function (pdf) of yim , Pyim (y|l), given that sl (t) is
transmitted, is
1 −y/2σ2
1,
e
m=l
2σ12
(22)
Pyim (y|l) =
1 −y/2σ02
e
, m = l
2σ 2
0
where
2σn2
=
Ēs + N0 ,
N0 ,
n=1
n=0
(23)
and Ēs = E[|hi |2 ]A2 T /2 is the average received symbol
energy. Therefore, the conditional joint pdf of yi = yi1 , yi2
given sl , l = 1, 2, transmitted is given by
1
yi1
yi2
exp − 2 − 2
P (yi |s1 ) =
4σ02 σ12
2σ1
2σ0
1
yi1
yi2
exp − 2 − 2
P (yi |s2 ) =
4σ02 σ12
2σ0
2σ1
(24)
(25)
Then, the log-likelihood ratio Λi,j , given branches i and j
selected, is given by
Λi,j
=
=
=
=
P (s2 |yi , yj )
(26)
P (s1 |yi , yj )
P (yi , yj |s2 )
ln
(27)
P (yi , yj |s1 )
P (yi |s2 )P (yj |s2 )
(28)
ln
P (y |s )P (y |s )
i 1 j 1
1 1
1
− 2 [(yi1 − yi0 ) + (yj1 − yj0 )].(29)
2 σ02
σ1
ln
Therefore, the optimum selection rule that minimizes the bit
error probability is to select the pair of branches providing
the largest |Yi + Yj |, where Yi = yi1 − yi0 . In general, for
M branches the optimum selection rule is to select those that
provide the largest |Yl1 + Yl2 + ... + YlM |. It can be shown that
the bit error probability with SLC of L branches is identical
to that with LLR-based GSC of L/2 branches. This follows
L
from the fact that the SLC output is i=1 Yi whose sign is
identical to that of the L/2-GSC output. The latter can be
proved in a similar way to that given in Appendix B.
VI. N UMERICAL R ESULTS AND D ISCUSSIONS
Figure 1 is a plot of the average bit error probability
versus Ēs /N0 for several GSC schemes with M =2 and L =
4, 8, where Ēs is the average received energy per diversity
branch and is equal to Es assuming that E[|hi |2 ] = 1.
The performance curves in Figure 1 are computed assuming
an independent identically distributed (i.i.d.) slow Rayleigh
fading model. We find that LLR-based GSC and envelopebased GSC provide a power gain of 1.2 ∼ 2.7 dB and
0.7 ∼ 1.4 dB, respectively, over SNR-based GSC for M =2
and L = 4 ∼ 8. We also find that the bit error probability
with MRC of four branches (L = 4) is identical to that with
LLR-based GSC of two branches in accordance with the proof
in Appendix B.
Figure 2 is a plot of the average bit error probability versus
Ēs /N0 with the LLR-GSC scheme. We find that the diversity
order (slope) depends on L, and the SNR gain increases as M
is increased. For a given L, the receiver complexity depends
largely on M . In multiple antenna systems, for example, the
number of RF chains and A/D converters, where much of the
hardware and power consumption lies, is equal to M . We find
that the combination M = 1, L = 8 provides a lower bit error
probability than the combination M = 3, L = 6 does. This
indicates that diversity gain is more important than SNR gain
in reducing the bit error probability.
Figure 3 is a plot of the average bit error probability versus
Ēs /N0 for noncoherent BFSK with M =2 and L = 4, 8. We
2791
find that the bit error probability with SLC of four branches
is identical to that with LLR-based GSC of two branches. For
L=8, LLR-based GSC provides a power gain of 1 dB over
SNR-based GSC and is only 0.4 dB away from SLC.
We compare now the complexity of various combining
schemes. For Env-GSC of M out of L branches, denoted
(L, M ) Env-GSC, we need magnitudes of channel gains and
L envelope detectors for branch selection and M (≤ L) RF
chains for combining, whereas for MRC of L branches,
denoted L-MRC, we need both magnitudes and phases of
channel gains and L RF chains for selection and combining.
Figure 4 shows that (8,1) Env-GSC provides a lower Pe
(average bit error probability) than 5-MRC for Pe ≤ 10−3 and
(8,2) Env-GSC provides a lower Pe than 6-MRC. Since the
envelope detector is much simpler than the RF chain and the
phase estimator is not needed for Env-GSC, Env-GSC provides
a significant saving in complexity over MRC.
But the LLR-GSC provides a minor saving in complexity
over MRC; the final combiner for LLR-GSC need combine
M instead of L branches. However, LLR-GSC presents the
performance limit that can be achieved by any GSC schemes
and guides the way toward a better GSC that fills the gap
between LLR-GSC and Env-GSC.
Proof:
It follows from (16) and P (x = +1|R) + P (x = −1|R) = 1
that
1
(32)
P (x = +1|R) =
1 + e−Λ(R)
and
P (x = −1|R) =
Pe (R) = P (x̂ = x|R)
(34)
= P (x̂ = 1, x = −1|R) + P (x̂ = −1, x = 1|R)
where x̂ is the detector output. If Λ(R) > 0, i.e. x̂ = 1, then
since P (A, B|R) ≤ P (A|R) or P (B|R),
Pe (R) ≤ P (x = −1|R) + P (x̂ = −1|R)
=
where
1
1 + e|Λ(R)|
P (x = +1|R)
Λ(R) = ln
P (x = −1|R)
Pe (R) = 1 − P (x̂ = x|R)
= 1 − P (x̂ = 1, x = 1|R)
is the log-likelihood ratio (LLR). Moreover, the relationship
in (15) is true for any binary signals in any channel.
(37)
− P (x̂ = −1, x = −1|R)
≥ 1 − P (x = 1|R) − P (x̂ = −1|R)
(38)
(39)
=
(41)
=P (Λ(R)<0)=0
=
1
1−
1 + e−Λ(R)
1
.
1 + eΛ(R)
(40)
Since, for Λ(R) > 0, Pe (R) is upper and lower bounded by
the same quantity, then
Pe (R) =
1
.
1 + eΛ(R)
(42)
If Λ(R) < 0, we can similarly show that
(30)
(31)
(36)
Also,
In this appendix, we show that the bit error probability,
Pe (R), with MAP (optimum) detection for a received observation R can be expressed as
Pe (R) =
(35)
=P (Λ(R)<0)=0
1
.
1 + eΛ(R)
Pe (R) =
1
.
1 + e−Λ(R)
(43)
Pe (R) =
1
.
1 + e|Λ(R)|
(44)
As a result,
A PPENDIX A
(33)
By definition,
VII. C ONCLUSION
We presented the optimum generalized selection combining
scheme for binary signaling that combines M out of L (≥
M ) diversity branches based on the magnitude of the loglikelihood ratio (LLR). For M = 2 and L = 4 ∼ 8, the
power gain provided by LLR-based GSC over SNR-based
GSC is 1.2 ∼ 2.7 dB for BPSK signaling in Rayleigh flat
fading channels. It is shown that the bit error probability for
BPSK signaling with MRC of L channels is identical to that
with LLR-based GSC of L/2 channels. We also presented a
simple, but suboptimum, GSC technique based on noncoherent
envelope detection that provides a gain of 0.7 ∼ 1.4 dB over
SNR-based GSC for M =2 and L = 4 ∼ 8. For noncoherent
BFSK signaling, the bit error probability with SLC of L
branches is identical to that with LLR-based GSC of L/2
branches and LLR-based GSC provides a power gain of 1
dB over SNR-based GSC when M = 2 and L = 8.
1
.
1 + eΛ(R)
A PPENDIX B
In this Appendix, we prove that the bit error probability
with MRC of L diversity branches is identical to that with
LLR-based GSC of L/2 diversity branches, or equivalently
the sign of the MRC output is identical to that of the LLRbased GSC output. We prove this by contradiction. Consider
for the moment L = 4. Let
Yi = Re{h∗i yi }.
2792
(45)
Suppose that the sign of Y1 + Y2 + Y3 + Y4 is different from
that of Yi∗ + Yj ∗ , where
{i∗ , j ∗ } = arg max |Yi + Yj |.
i,j
(46)
Then, there can be two possibilities: 1) Yi∗ + Yj ∗ > 0 and
Y1 +Y2 +Y3 +Y4 < 0 2) Yi∗ +Yj ∗ < 0 and Y1 +Y2 +Y3 +Y4 >
0.
1) If Yi∗ + Yj ∗ > 0 and Y1 + Y2 + Y3 + Y4 < 0, then
Yp + Yq < −(Yi∗ + Yj ∗ ) < 0,
[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.
[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,” IEEE Tr. on
Commun., Vol. 49, pp.1715-1718, Oct. 2001.
(47)
where i∗ , j ∗ , p, q are distinct. Thus,
|Yp + Yq | > |Yi∗ + Yj ∗ |.
(48)
This implies that {i∗ , j ∗ } = arg maxi,j |Yi + Yj | and thus
contradicts the fact that i∗ and j ∗ were selected.
2) A similar contradiction occurs.
The above argument can be extended to L diversity branches
in a straightforward way.
A PPENDIX C
Lemma: Consider two real quantities A and B. Then, if |A| >
|B|, it follows that sgn(A) = sgn(A + B). If |A| < |B|, then
sgn(B) = sgn(A + B). This is easy to show.
Suppose now that we select an arbitrary combination of M
and L − M branches, where M ≥ L/2, from the total
of L branches and compute its LLR. We denote this by
Λi1 ,i2 ,...iM , where i1 , i2 , ...iM now refers to a particular group
of M indices, one example of which might simply be the
first M branches, i.e. i1 = 1, i2 = 2, ...im = M . Next
compute the LLR for the remaining L − M branches and
denote this by Λi1 ,i2 ,...iL−M , where, for the above example,
we would have i1 = M + 1, i2 = M + 2, ...iL−M = L.
Now let A = Λi1 ,i2 ,...iM and B = Λi1 ,i2 ,...iL−M . Clearly
then A + B = Λi1 ,i2 ,...iM + Λi1 ,i2 ,...iL−M = ΛM RC . Thus,
if |Λi1 ,i2 ,...iM | > |Λi1 ,i2 ,...iL−M |, then from the above lemma
a binary data decision based on sgn(Λi1 ,i2 ,...iM ) is identical
to a data decision based on sgn(ΛM RC ) and vice versa if
|Λi1 ,i2 ,...iL−M | > |Λi1 ,i2 ,...iM |. Thus, we conclude that we
get equivalent performance to MRC by dividing the group
of L branches into two arbitrarily selected complementary
subgroups of M and L − M branches, making a single
comparison of the magnitudes of the LLR for these two
subgroups and choosing the larger of the two for the data
decision.
Fig. 1. Average bit error probability Pe versus Ēs /N0 for several GSC
schemes; M = 2, BPSK.
R EFERENCES
[1] W.C.Jakes, Microwave Mobile Communications, Wiley, New
York, 1974.
[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.
[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.
Fig. 2.
BPSK.
2793
Average bit error probability Pe versus Ēs /N0 with LLR-GSC;
Fig. 3. Average bit error probability Pe versus Ēs /N0 ; M = 2, noncoherent
BFSK.
Fig. 4.
Average bit error probability Pe versus Ēs /N0 ; BFSK.
2794