Optimum Receive Antenna Selection
Minimizing Error Probability
Sang Wu Kim∗ Eun Yong Kim†
Abstract— The optimum receive antenna selection combining
rule that minimizes the bit error probability is presented. It
is derived from a general relationship between the bit error
probability and the log-likelihood ratio (LLR), and selects the
receive antenna providing the largest LLR magnitude. The
optimum selection combining rule is applied to single transmit
(Tx) antenna and space-time block code (STBC) systems, with NR
receive (Rx) antennas, and the bit error probability is derived for
BPSK signaling in Rayleigh flat fading channels. A suboptimum
selection combining rule based on noncoherent envelope detection
is also presented and analyzed. For STBC systems, the optimum
generalized selection combining in which M (≤ NR ) Rx antennas
providing the largest LLR magnitude are selected and combined
is presented.
Index Terms— Selection combining, log-likelihood ratio, diversity, space-time block code, multiple antenna, Rayleigh fading
channel.
I. I NTRODUCTION
The radio link is characterized by multi-path fading which
causes the link quality to vary with time, frequency, and
space. Diversity techniques are known to be very effective in
mitigating multi-path fading, and multiple-transmit multiplereceive antennas are known to provide an increased diversity
order [1],[2]. However, the implementation of such multipletransmit multiple-receive antenna systems requires multiple
RF chains and A/D converters, where much of the hardware
and power consumption lies. A promising approach for reducing hardware complexity and power consumption, while
retaining a reasonably high performance, is to employ some
form of selection combining [3],[4],[5].
In this paper, we present the optimum receive antenna
selection combining rule that minimizes the bit error probability in interference-limited systems. The optimum selection
combining rule, derived from a general relationship between
the bit error probability and the log-likelihood ratio (LLR),
is to select the antenna providing the largest LLR magnitude.
We apply this rule to a single transmit (Tx) antenna system
and space-time block code (STBC) system [6],[7], with NR
receive (Rx) antennas, and derive the bit error probability
for BPSK signaling in Rayleigh flat fading channels. Also,
we present a suboptimum selection combining rule based on
a noncoherent envelope detection. We compare the bit error
probabilities of optimum and suboptimum selection combining rules with those of traditional combining rules such as
signal-to-interference ratio (SIR)-based selection combining
and maximum ratio combining (MRC) [3],[8]. We find that
∗ Sang
† Eun
Wu Kim sangkim@ieee.org
Yong Kim eykim@bada.kaist.ac.kr
0-7803-7700-1/03/$17.00 (C) 2003 IEEE
the optimum selection combining provides a gain of 3-3.5dB
over the SIR-based selection combining, and is only 0.30.6dB away from MRC that requires multiple RF chains. The
envelope-based suboptimum selection combining provides a
gain of 1.8-2.1dB over SIR-based selection combining, when
selecting one out of four Rx antennas. For STBC systems, we
present the optimum and suboptimum generalized selection
combining rules in which M (≤ NR ) Rx antennas providing
the largest LLR magnitude are selected and combined.
The remainder of this paper is organized as follows. In
Section II, we present the system model. In Section III, we
present a general relationship between the bit error probability
and LLR, and present the optimum selection combining rule
that minimizes the bit error probability. In Section IV, we
consider a wireless channel with one Tx antenna and NR Rx
antennas, and derive the bit error probability with optimum and
suboptimum antenna selection combining rules. In Section V,
we consider a STBC system with two Tx antennas and NR
Rx antennas and derive the bit error probability with optimum
and suboptimum antenna selection rules. In Section VI, we
consider a STBC system with two Tx antennas and NR Rx
antennas and present optimum and suboptimum generalized
selection combining rules. In Section VII, numerical results are
provided along with discussions. In Section VIII, conclusion
is made.
II. S YSTEM M ODEL
In this paper, we consider the reverse link (mobile to base)
of a digital mobile radio communication system. The system is
assumed to be interference-limited. The base station consists
of an NR element antenna array, and the antenna element
is selected on a symbol-by-symbol basis. The modulation is
chosen to be binary phase-shift-keying (BPSK) with coherent
detection in two channel models. In Section IV, we consider a
wireless communication system with one Tx antenna and NR
Rx antennas (Fig.1). The received low-pass equivalent signal
yi , given the ith Rx antenna selected, is
y i = hi x +
K
hk,i xk
(1)
k=1
where hi and hk,i are channel gains between the Tx antenna
and the ith Rx antenna for the desired and the kth interfering
users, respectively, and K is the number of interfering users.
We assume that |hi | and |hk,i | are Rayleigh distributed with
2
2
E{|hi | } = E{|hi,k | } = 1 and that the phases of hi and hk,i
are uniformly distributed over [0,2π]. x and xk , representing
the desired and the kth interfering signals, respectively, are
441
√
√
+ Es or − Es with probability 1/2, where Es is the transmit
energy per symbol. We assume that the interference term in
(1), denoted hereafter as ni , has a Gaussian distribution with
mean zero and variance I0 /2 per dimension. The Gaussian
assumption follows from the Cramer’s central limit theorem
[10].
In Sections V and VI, we consider a space-time block code
(STBC) system with two transmit antennas and NR receive
antennas (Fig.2). The received low-pass equivalent signals r1i
and r2i from the ith antenna at time t and t + T , respectively,
are given by [6]
r1i
= h1i x1 + h2i x2 + n1i
(2)
r2i
=
(3)
−h1i x∗2
+
h2i x∗1
+ n2i
where hji is the channel gain between jth Tx antenna and
ith Rx antenna, and n1i and n2i , i = 1, 2, ..., N are i.i.d.
Gaussian random variables with mean
I0 /2
zero and variance
per dimension. x1 and x2 are + Es /2 or − Es /2 with
probability 1/2. Then, the combiner outputs y1i and y2i are
given by
y1i
y2i
∗
= h∗1i r1i + h2i r2i
= (|h1i |2 + |h2i |2 )x1 + h∗1i n1i + h2i n∗2i
∗
= h∗2i r1i − h1i r2i
= (|h1i |2 + |h2i |2 )x2 − h1i n∗2i + h∗2i n1i .
(4)
(5)
(6)
(7)
We assume that the channel is quasi-static flat fading, i.e.
the channel gains {hi } and {hji } remain constant over several
symbols. We also assume that the channel gains are i.i.d., and
are known at the receiver.
selection combining rule that minimizes the bit error probability is to select the antenna providing the largest |Λi |. Then,
the minimum average bit error probability is
1
Pe,opt = EΛmax
(12)
1 + eΛmax
where, by definition,
Λmax = max |Λi |.
1≤i≤NR
(13)
It should be noted that averaging Λi in (10) over ni and
taking the absolute value yields 4|hi |2 Es /I0 , which is the SIR
at diversity branch i. Therefore, selecting the antenna providing the largest SIR or equivalently the largest |hi |2 (assuming
that the average interference power is a constant across all
antennas) takes the average interference power into account.
However, the optimum selection combining rule exploits the
interference term Re{h∗i ni } in (10) by selecting the antenna
for which signs of x and Re{h∗i ni } are identical and |hi |
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. The bit error probability resulting
from the optimum selection combining provides a lower bound
on the bit error probability of any selection combining rule.
In contrast
NRwith the maximum ratio combining (MRC) that
Λi , the optimum selection combining selects the
yields i=1
best component among NR components used in MRC. In what
follows, we apply the optimum selection combining rule to two
wireless communication systems.
IV. S INGLE T X A NTENNA
We consider a wireless communication system with one Tx
antenna and NR Rx antennas (Fig.1). The channel output given
the ith antenna is selected is given by
III. O PTIMUM S ELECTION C OMBINING
yi = hi x + ni .
Consider a channel where the channel output yi , given the
ith Rx antenna is selected, is given by
Then, it follows from (10) that the LLR, given the ith Rx
antenna selected, is
√
4 Es
Λi =
[|hi |2 x + Re{h∗i ni }].
(15)
I0
If we let
√
Es
Zi =
||hi |2 x + Re{h∗i ni }|
(16)
I0
then the pdf of Zi can be shown to be (derived in Appendix
II)
√
√
1
−1
−1
fZi (z) = e−2( 1+γ +1)z + e−2( 1+γ −1)z
γ2 + γ
(17)
where
yi = hi x + ni ,
(8)
gain√between the Tx antenna and ith
where hi is the channel
√
Rx antenna, x is + Es or − Es with probability 1/2, and ni
is a white Gaussian noise with mean zero and variance I0 /2
per dimension. Then, the log-likelihood ratio (LLR) Λi for x
is given by
√
P (x = + Es |hi , yi )
√
Λi = ln
(9)
P (x = − Es |hi , yi )
√
4 Es
=
Re{h∗i yi }
.
(10)
I0
γ
=|hi |2 x+Re{h∗
n }
i i
The sign of Λi is the hard decision value, and the magnitude
of Λi represents the reliability of hard decision. In general,
the bit error probability Pe,i and the LLR Λi is related by
Pe,i =
1
1 + e|Λi |
(11)
A derivation of (11) is provided in Appendix I. Since Pe,i
decreases with increasing |Λi |, the optimum receive antenna
= E[|hi |2 ]Es /I0
= Ēs /I0
(14)
(18)
(19)
is the average received signal-to-interference ratio (SIR) per
diversity branch. The cdf of Zi is then given by
z
fZi (u)du
(20)
FZi (z)=
0
√
√
−1
−1
1 − e−2( 1+γ +1)z 1 − e−2( 1+γ −1)z
=
+
.(21)
2(γ + 1 + γ 2 + γ) 2(γ + 1 − γ 2 + γ)
442
If we let
Zmax = max {Zi }
1≤i≤NR
(22)
and assume that Z1 , Z2 , ..., ZNR are i.i.d., then the pdf of
Zmax is given by
fZmax (z) = NR [FZi (z)]NR −1 fZi (z).
(23)
Therefore, it follows from (12) and (23) and the relationship
Λmax = 4Zmax that the bit error probability with the optimum
antenna selection is given by
∞
1
Pe,opt =
fZ (z)dz.
(24)
1
+
e4z max
0
Envelope-based Selection
In this subsection we present a simple suboptimum selection
combining rule based on noncoherent envelope detection.
First, we note that
|Re{h∗i yi }|
≤ |h∗i yi |
= |hi ||yi |.
(26)
SIR-based Selection
Since the SIR, given that the ith antenna is selected, is
|hi |2 Es /I0 , the antenna providing the largest SIR is the one
providing the largest |hi |2 . If we let
1≤i≤NR
(29)
then the bit error probability, when the antenna providing the
largest SIR is selected, is given by
∞ Pe,snr =
Q( 2gEs /I0 )fGmax (g)dg
(30)
0
where fGmax (g) is the pdf of Gmax given by
fGmax (g) = NR (1 − e−g )NR −1 e−g ,
and γ is the average received SIR per diversity branch.
V. S PACE -T IME B LOCK C ODE
In this section, we consider a space-time block code (STBC)
with two Tx antennas and NR Rx antennas (Fig.2). The
baseband combiner output y1i for data x1 , given the ith Rx
antenna is selected, is
(25)
Since noncoherent envelope detection of the received RF signal yields |yi |, we propose a suboptimum selection combining
rule that selects the antenna element providing the maximum
of {|h1 ||y1 |, |h2 ||y2 |, ... , |hNR ||yNR |}. It will be called
envelope-based selection combining. If we let {|h(1) ||y(1) |,
|h(2) ||y(2) |, ... , |h(NR ) ||y(NR ) |} be an ordered set such that
|h(1) ||y(1) | ≥ |h(2) ||y(2) | ≥ ... ≥ |h(NR ) ||y(NR ) |, then the
log-likelihood ratio Λenv for deciding x with envelope-based
selection combining is
√
4 Es
Λenv =
Re{h∗(1) y(1) }.
(27)
I0
√
We decide that E√
s was transmitted if Λenv > 0, and
otherwise, decide − Es was transmitted. Then, it follows
from (11) and (27) that the average bit error probability with
the envelope-based selection combining is given by
1
Pe,env = EΛenv
.
(28)
1 + e|Λenv |
Gmax = max {|hi |2 }
Maximum Ratio Combining
The bit error probability with maximum ratio combining of
NR independent paths is given by [8]
N
R −1 NR − 1 + k
Pe,mrc = [(1 − µ)/2]NR
[(1 + µ)/2]k
k
k=0
(32)
where, by definition,
γ
µ=
(33)
1+γ
(31)
y1i = (|h1i |2 + |h2i |2 )x1 + η1i
(34)
η1i = h∗1i n1i + h2i n∗2i
(35)
where
is a conditional Gaussian random variable with mean zero
and variance (|h1i |2 + |h2i |2 )I0 /2 per dimension. Then, for
a given {hji }, y1i is a Gaussian random variable with mean
(|h1i |2 + |h2i |2 )x1 and variance (|h1i |2 + |h2i |2 )I0 /2 per
dimension. Therefore, the log-likelihood ratio (LLR) Λi for
data x1 1 , given {hji } and y1i , is
P (x1 = + Es /2|{hji }, y1i )
Λi = ln
(36)
P (x1 = − Es /2|{hji }, y1i )
4 Es /2
=
Re{y1i }
(37)
I0
=Ai x1 +Re{η1i }
where, by definition,
Ai =
2
j=1
|hji |2 .
(38)
If we let
√
2Es
Zi =
|(|h1i |2 + |h2i |2 )x1 + Re{h∗1i n1i + h2i n∗2i }| (39)
I0
then the pdf of Zi can be shown to be (derived in Appendix
III)
1
fZi (z) = (1 + z 1 + 2/γ)
3
γ(γ + 2)
√
√
−z( 1+2/γ+1)
[e
+ e−z( 1+2/γ−1) ]. (40)
If we assume that Z1 , Z2 , ..., ZNR are i.i.d., then the pdf of
Zmax = max1≤i≤NR {Zi } is given by
fZmax (z) = NR [FZi (z)]NR −1 fZi (z),
(41)
where FZi (z) is the cdf of Zi . Therefore, it follows from (12)
and (41) and the relationship Λmax = 2Zmax that the bit error
assuming that {|hi |2 } are i.i.d. with pdf f|hi |2 (g) = e−g .
1 The
443
LLR for other information symbols is exactly the same.
probability with the optimum antenna selection rule is given
by
∞
1
Pe,opt =
fZ (z)dz.
(42)
1 + e2z max
0
Envelope-based Selection
In this subsection we present a simple suboptimum selection
combining rule based on noncoherent envelope detection. We
note that
4 Es /2
|Λi | =
|Re{y1i }|
(43)
I
0
4 Es /2
≤
|y1i |.
(44)
I0
Since combining the received RF signals and detecting the
envelope of the combined signal yields |y1i | and |y2i |, we
propose a suboptimum selection combining rule that selects
the antenna element providing the maximum of {|y1i |}, i =
1, 2, ..., NR . It will be called envelope-based selection combining. Then, the log-likelihood ratio Λenv for deciding x with
envelope-based selection combining is
4 Es /2
Λenv =
Re{y(1) }
(45)
I0
where y(1) is the baseband combiner output associated with
the antenna element providing
√ the maximum of {|y1i |}, i =
1, 2, ..., NR . We decide that
Es was transmitted if Λenv > 0,
√
and otherwise, decide − Es was transmitted. Then, it follows
from (11) and (45) that the average bit error probability with
envelope-based selection combining is
1
Pe,env = EΛenv
.
(46)
1 + e|Λenv |
SIR-based Selection
The SIR, given the ith Rx antenna selected, is Ai Es /(2I0 ).
Therefore, the antenna providing the largest SIR is the one
providing the largest Ai . If we let
Amax = max {Ai },
1≤i≤NR
(47)
then the bit error probability with SIR-based selection is given
by
∞ Pe,snr =
Q( 2aEs /(2I0 ))fAmax (a)da,
(48)
0
where fAmax (a) is the pdf of Amax given by
fAmax (a) = NR [1 − e−a
1
l=0
a1−l ]NR −1 · ae−a
(49)
assuming that {Ai }’s are i.i.d. with fAi (a) = ae−a , a ≥ 0.
Maximum Ratio Combining
STBC with two Tx antennas and NR Rx antennas provides
a diversity order of 2NR . Therefore, the bit error probability
with MRC at the receiver is given by (32) with NR replaced
by 2NR and γ replaced by γ/2 because Es /2 is the energy
per transmit antenna.
VI. O PTIMUM G ENERALIZED S ELECTION C OMBINING
In this section we consider the optimum generalized selection combining (GSC) for STBC systems. The optimum GSC
for single Tx antenna case is analyzed in [11]. We consider
selecting M out of NR Rx antennas and combining those M
signals. The baseband combiner output y1i1 ,i2 ,···,iM for data
x1 , given Rx antennas i1 , i2 , · · · , iM are selected, is
y1i1 ,i2 ,···,iM =
=
M
m=1
M
m=1
where
y1im
(|h1im |2 + |h2im |2 )x1 + η1im (50)
η1im = h∗1im n1im + h2im n∗2im .
(51)
Then, for a given {hji }, y1i1 ,i2 ,···,iM is a Gaussian random
M
variable with mean m=1 (|h1im |2 + |h2im |2 )x1 and variance
M
2
2
m=1 (|h1im | + |h2im | )I0 /2 per dimension. Therefore, the
LLR Λi1 ,i2 ,···,iM for x1 , given {hji }, y1i1 ,i2 ,···,iM , is
P (x1 = + Es /2|{hji }, y1i1 ,i2 ,···,iM )
Λi1 ,i2 ,···,iM = ln
(52)
P (x1 = − Es /2|{hji }, y1i1 ,i2 ,···,iM )
√
4 Es
=
Re{y1i1 ,i2 ,···,iM }
(53)
I
√0
4 Es
=
[A1i1 ,i2 ,···,iM x1
I0
+ Re{η1i1 + η1i2 + . . . + η1iM }] (54)
where, by definition,
A1i1 ,i2 ,···,iM =
M
[|h1im |2 + |h2im |2 ].
(55)
m=1
It follows from (11) that the optimum selection combining
rule that minimizes the bit error probability is to select those
antennas that provide the largest |Λi1 ,i2 ,···,iM |. It can be shown
that the bit error probability with the optimum GSC of M =
NR
2 antennas is identical to that of MRC, since the sign of the
NR
2 optimum GSC output is the same as that of MRC.
Envelope-based GSC
In this subsection we present a simple suboptimum generalized selection combining rule based on noncoherent envelope
detection. We note that
√
4 Es
|Λi1 ,i2 ,···,iM | =
|Re{y1i1 ,i2 ,···,iM }|
(56)
I0
√
M
4 Es ≤
|Re{y1im }|
(57)
I0 m=1
√
M
4 Es ≤
|y1im |.
(58)
I0 m=1
Since combining the received RF signals and detecting the
envelope of the combined signal yields |y1im |, we propose a
suboptimum generalized selection
M combining rule that selects
M Rx antennas maximizing m=1 |y1im |. It will be called
envelope-based GSC.
444
SIR-based GSC
Given the Rx. antenna i1 , i2 , · ·
· , iM are selected, the SIR is
M
proportional to A1i1 ,i2 ,···,iM = m=1 [|h1im |2 + |h2im |2 ]. The
conventional GSC rule selects M Rx antennas providing the
largest A1i1 ,i2 ,···,iM . It will be called SIR-based GSC.
VII. N UMERICAL R ESULTS AND D ISCUSSION
In this section, we present numerical results and discussions.
Figure 3 is a plot of average bit error probability versus Ēs /I0
with single transmit antenna and NR Rx antennas, where
Ēs is the average received energy per diversity
branch and
is equal to Es since we have assumed E |hi |2 = 1. The
performance curves are computed assuming an independent
identically distributed (i.i.d.) slow Rayleigh fading model. We
find that the diversity order (slope) depends on the number of
receive antennas. For NR =4, the optimum selection combining
(OSC) provides a gain of 3dB over the SIR-based selection
combining (SIR-SC), and is only 0.3dB away from MRC
that requires multiple (four) RF chains. The envelope-based
selection provides a gain of 1.8dB over SIR-SC. For NR =2,
the error probability of MRC can be achieved by OSC. This
is because the sign of MRC output is identical to that of
LLR-based selection combiner output. We can also notice that
MRC of four channels (requiring four RF chains, NR = 4)
is outperformed by envelope-based combining of one channel
(requiring one RF chain) out of eight antennas (NR = 8).
Figure 4 is a plot of average bit error probability versus
Ēs /I0 with space-time block code employing two Tx antennas
and NR Rx antennas. For NR =4, the OSC provides a gain
of 3.5dB over the SIR-SC, and is 0.6dB away from MRC
that requires four RF chains. The envelope-based selection
combining provides a gain of 2.1dB over SIR-SC. For NR =2,
the error probability of MRC can be achieved by the optimum
selection combining, as in single Tx antenna case.
Figure 5 and 6 are plots of average bit error probability
versus Es /I0 with generalized selection combining of M Rx
antennas and STBC employing two Tx antennas and NR
Rx antennas. We find that the optimum GSC and Env-GSC
provide power gains of 1.6 ∼ 3.1dB and 0.9 ∼ 1.6dB over
SIR-GSC, respectively, for M = 2 and NR = 4 ∼ 8. We also
find that the diversity order (slope) depends on NR , and the
SIR gain increases with increasing M .
and suboptimum selection combining rules with those of traditional combining rules such as SIR-based selection combining
and maximum ratio combining (MRC). For NR = 4, the
optimum selection combining provides a gain of 3-3.5dB over
the SIR-based selection combining, and is only 0.3-0.6dB
away from MRC that requires multiple (four) RF chains. The
envelope-based selection combining provides a gain of 1.82.1dB over the SIR-based selection combining. The power
gain increases with increasing NR . For STBC systems, we
present the optimum and suboptimum generalized selection
combining rules in which M (≤ NR ) Rx antennas providing
the largest LLR magnitude are selected and combined.
A PPENDIX I
In this appendix, we show that the bit error probability,
Pe (R), with MAP (optimum) detection for a received observation R can be expressed
1
Pe (R) =
(59)
1 + e|Λ(R)|
where
P (x = +1|R)
Λ(R) = ln
(60)
P (x = −1|R)
is the log-likelihood ratio (LLR). Moreover, the relationship
in (59) is true for any binary signals in any channel.
Proof:
It follows from (60) and P (x = +1|R) + P (x = −1|R) = 1
that
1
P (x = +1|R) =
(61)
1 + e−Λ(R)
and
1
P (x = −1|R) =
.
(62)
1 + eΛ(R)
By definition,
Pe (R) = P (x̂ = x|R)
(63)
= P (x̂ = 1, x = −1|R)+P (x̂ = −1, x = 1|R)(64)
where x̂ is the detector output. If Λ(R) > 0, i.e. x̂ = 1, then
Pe (R) ≤ P (x = −1|R) + P (x̂ = −1|R)
=P (Λ(R)<0)=0
=
VIII. C ONCLUSION
In this paper we presented the optimum receive antenna
selection combining rule that minimizes the bit error probability. The optimum selection combining rule, derived from a
general relationship between the bit error probability and the
log-likelihood ratio (LLR), is to select the antenna providing
the largest LLR magnitude. We applied the optimum selection
combining rule to single transmit (Tx) antenna and spacetime block code (STBC) systems with NR Rx antennas, and
derived the average bit error probability for BPSK signaling
in a Rayleigh flat fading channel. Also, we presented a
suboptimum selection combining rule based on noncoherent
envelope detection and derived its average bit error probability.
We compared the average bit error probabilities of optimum
(65)
1
.
1 + eΛ(R)
(66)
Also,
Pe (R) = 1 − P (x̂ = x|R)
=
(67)
1 − P (x̂ = 1, x = 1|R)
−P (x̂ = −1, x = −1|R)
≥ 1 − P (x = 1|R) − P (x̂ = −1|R)
(68)
(69)
=P (Λ(R)<0)=0
1
=
.
1 + eΛ(R)
Therefore, if Λ(R) > 0, then
445
Pe (R) =
1
.
1 + eΛ(R)
(70)
(71)
If Λ(R) < 0, we can similarly show that
Pe (R) =
1
.
1 + e−Λ(R)
(72)
Pe (R) =
1
.
1 + e|Λ(R)|
(73)
As a result,
A PPENDIX II
In this Appendix, we derive (17). Let
√
Es
Yi =
(|hi |2 x + Re{h∗i ni }).
I0
(74)
√
Since the pdf
√ of Zi = |Yi | is the same√ whether x = Es
or x = − Es , we will assume x = Es without loss of
generality. Then, given hi , Yi is a Gaussian random variable
with mean |hi |2 γ and variance |hi |2 γ/2, where γ = Es /I0 .
2
Since the pdf of |hi | is Rayleigh, i.e. f|hi | (a) = 2ae−a , the
pdf of Yi is
∞
2
2
2
2
1
fYi (y) =
e−(y−a γ) /(a γ) · 2ae−a da(75)
2
πa γ
0
√
1
−1
= e2y−2|y| 1+γ
(76)
γ(1 + γ)
where we used the equality [9]
∞
1 π −2√ab
−ax2 −b/x2
e
dx =
e
.
2 a
0
The cdf of Zi is
FZi (z) =
=
z
1
2y−2|y|
√
1+γ −1
e
γ(1 + γ)
√
−1
(1 − e−2( 1+γ +1)z )
2(γ + 1 + γ 2 + γ)
√
−1
(1 − e−2( 1+γ −1)z )
+
.
2(γ + 1 − γ 2 + γ)
(77)
=
where we use the equality [9]
∞
√
√
2
2
1 π
x2 e−a/x −bx dx =
(1 + 2 ab)e−2 ab .
3
4 b
0
The cdf of Zi , FZi (z), is
z
FZi (z) =
fYi (y)dy
−z
(85)
(86)
(87)
(88)
√
1 + 2/γ)ey 1+2/γ+y
=
dy
γ(γ + 2)3
−z
√
z
(1 + y 1 + 2/γ)e−y 1+2/γ+y
+
dy (89)
γ(γ + 2)3
0
2c + 1 − [2c + 1 + c(c + 1)z]e−(c+1)z
=
γ(γ + 2)3 [(c + 1)]2
2c − 1 − [2c − 1 + c(c − 1)z]e−(c−1)z
+
(90)
γ(γ + 2)3 [(c − 1)]2
where c = 1 + 2/γ. Therefore, the pdf of Zi is given by
0
(1 − y
dFZi (z)
(91)
dz (1 + z 1 + 2/γ)
=
γ(γ + 2)3
√
√
[e−z( 1+2/γ+1) + e−z( 1+2/γ−1) ]. (92)
fZi (z) =
dy
(78)
−z
R EFERENCES
(79)
Therefore, the pdf of Zi is
fZi (z) =
√
where γ = Es /I0 . If we let b = a then
√ y ∞
y2 1
2
2e
fYi (y) = √
b2 e− 2γ b2 −(1+γ/2)b db
πγ 0
1 + |y| 1 + 2/γ −|y|√1+2/γ+y
=
e
γ(γ + 2)3
dFZi (z)
(80)
dz √
√
−1
−1
e−2( 1+γ +1)z + e−2( 1+γ −1)z
.(81)
γ2 + γ
A PPENDIX III
In this Appendix, we derive (40). Let
√
2Es
Yi =
[(|h1i |2 + |h2i |2 )x1 + Re{h∗1i n1i + h2i n∗2i }] (82)
I0
Since the pdf
1 = + Es /2
of Zi = |Yi | is the same whether x
or x1 = − Es /2, we will assume x1 = + Es /2. Then,
given h1i and h2i , Yi is a Gaussian random variable with mean
Ai Es /I0 and variance Ai Es /I0 , where Ai = |h1i |2 + |h2i |2 .
Since the pdf of Ai , fAi (a), is ae−a , the pdf of Yi is
∞
2
1
√
fYi (y) =
e−(y−γa) /(2γa) fAi (a)da (83)
2πγa
0
∞
√ − y2 1 −(γ/2+1)a
ey
= √
ae 2γ a
da
(84)
2πγ 0
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446
Fig. 1.
Channel model for one Tx and NR Rx antennas.
Fig. 4. Bit error probability versus Ēs /I0 with 2 Tx and NR Rx: Space-time
block code.
Fig. 2. Space-time block code systems with two Tx antennas and NR Rx
antennas.
Fig. 5. Bit error probability versus Ēs /I0 with 2 Tx and NR Rx: Space-time
block code, GSC, M=2
Fig. 3.
Bit error probability versus Ēs /I0 with 1 Tx and NR Rx.
Fig. 6. Bit error probability versus Ēs /I0 with 2 Tx and NR Rx: Space-time
block code, Opt.-GSC
447