4 Diversity Order, Diversity Gain and Array Gain
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10–1
10–2
1/γ N
1/γ
d=N
10–3
10–4
d=1
Diversity gain
10–5
Bit error probability
Bit error probability
10–1
10–2
1/(Gaγ )N
10–3
10–4
10–5
γ (dB)
(a) Diversity gain and diversity order d
Ga > 1
d=N
1/γ
N
d=N
Array gain
γ (dB)
(b) Array gain Ga
Figure 12.15 Effect of Diversity Order, Diversity Gain and Array Gain on the BEP in Fading Channels with
N Independently Fading Diversity Branches.
Pe γ
∞ logγ
d = − lim
γ
(12.55)
The diversity order denotes the slope of the
BEP curve versus SNR γ in dB as γ ∞.
Figure 12.15 provides a pictorial description
of diversity order, diversity gain and array gain
for diversity reception via N independently fading channels. A receiver equipped with N
antennas will have a diversity order of N and
the BEP curve will decrease as 1 γ N , that is
faster than for N = 1. For a SISO system in a
Rayleigh fading channel, the BEP curves
shown in Figures 11.42-46 for various modulation schemes all have unity diversity order, that
is, BEP 1 γ. Figure 12.14 shows that the
BEP at the output of a N-element receiving
ULA achieves a diversity order d = N, when
its elements are sufficiently apart from each
other. Figure 12.13 shows that the diversity
order may also be expressed in terms of the
slope of the outage probability curve. Note that
the diversity order is defined for large values of
the mean SNR. Since the slope of the BEP
curve at finite SNR values is not the same as
its slope at infinite SNR (the diversity order),
one needs to be cautious about using the concept of diversity order as a performance measure for SNR levels encountered in practice.
The diversity gain of a SIMO system is
defined, for a given BEP level, as the decrease
in the SNR in dB compared to a SISO system
(N = 1) (see Figure 12.15a). For a BEP level
of 10−3 in Figure 12.14, a single antenna
receiver requires γ c = 23 96dB, but the use of
a two element array reduces the required
SNR level to γ c = 11 09dB, hence a diversity
gain of 12.87 dB. The diversity gain becomes
higher at lower BEP levels. One may also
observe from Figure 12.14 that increased correlation has a negative impact on both the
diversity gain and the diversity order at finite
SNR values. For example, the curve for N = 3,
ρ = 1 in Figure 12.14 has the same diversity
order as for N = 1 at finite SNR values and
shows only 4.78 dB better performance compared to N = 1. However, the diversity gain for
N = 3, ρ = 0 is 17.41 dB.
Outage and BEP curves also depend on the
value of the array gain, which is defined as
the increase in the average channel SNR, usually in dB, with N. An increase in the array gain
shifts the BEP curve to smaller SNR values (see
Figure 12.15b)). For example, consider two
receivers, each equipped with N-element
ULAs. When they operate with N independently fading channels with mean SNRs γ and
Ga γ (Ga > 1), the slope (diversity orders) of
both BEP curves will be the same 1 γ N and
1 Ga γ N , hence no diversity order advantage
to each other. However, the shift between the
two curves is simply due to the array gain
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Diversity and Combining Techniques
(see Figure 12.15b). For example, Figure 12.14
shows that curve for N = 3, ρ = 1 has an array
gain Ga = 3 in contrast with Ga = 1 for N = 1.
However, they both have unity diversity order
(see (12.45)). The 4.78 dB shift between them is
evidently due to the difference in their array
gains. For example, a receiving array with
N uncorrelated elements has N identical eigenvalues λi = 1, i = 1, 2, ,N; hence Ga = 1, d = N
for ρ = 0 (see (12.42)). For ρ = 1,
λ1 = N,λi = 0, i = 1, 2, 3, ,N; hence Ga = N
and d = 1 (see (12.45)). For 0 < ρ < 1, the N
non-zero eigenvalues will be different from
each other with Ga = 1, d = N (see (12.46)).
12.4.1
Tradeoff Between the
Maximum Eigenvalue
and the Diversity Gain.
We already observed from (12.41), (12.44) and
(12.47) that the mean SNR at the output of a
SIMO system is increased by a factor of N,
independently of the correlation between the
array elements. For ρ = 0, all eigenvalues have
equal means γ c but, for ρ = 1, the only nonzero
eigenvalue has a mean value of Nγ c . For 0 <
ρ < 1, there are N distinct eigenvalues each with
a different mean level. However, their sum is
the same as for uncorrelated and fully correlated arrays. There are some operational modes,
such as MIMO beamforming, which exploit
only the largest eigenvalue for minimizing
the BEP. The number of antennas and the
degree of correlation between them determine
the value of the maximum eigenvalue λ1 in
(12.52) for a SIMO system. For example, we
observe from Figure 12.11 that λ1 increases
with increasing values of the correlation coefficient. For example, higher values of λ1 lead
to longer ranges and lower BEPs in communication systems. On the contrary, increased
correlation decreases the diversity gain and
hence degrades the BEP performance for finite
values of λ1 γ c (see Figure 12.14). The tradeoff
between these two conflicting consequences of
correlation between the signals of a receiving
array may be observed in Figure 12.16, where
the loss in the diversity gain and the increase in
the value of the maximum eigenvalue are shown
as a function of the correlation coefficient. The
loss in the diversity gain becomes significant for
0
6
← Loss in diversity gain
−2
N=4
−4
N=3
dB
−6
−8
−10
3 dB
N=2
Increase in the value of
maximum eigenvalue →
0
0.2
0.4
0.6
0.8
0
Correlation coefficient ρ
Figure 12.16 The Variation of the Maximum Eigenvalue and the Loss in the Diversity Gain with the
Correlation Coefficient at a BEP = 10−4 For a Receiving ULA with N = 2, 3 and 4 Elements. For ρ = 0,
SNR = 16.28 dB is required to achieve a BEP = 10−4 for N = 2; SNR = 10.31 dB for N = 3; and SNR =
7.15 dB for N = 4 (see Figure 12.14).
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N=4
N=3
N=2
dB
0
–5
–10
0
0.2
0.4
0.6
0.8
1
Correlation coefficient ρ
Figure 12.17 The Difference Between the Increase in the Maximum Eigenvalue and the Loss in the
Diversity Gain at a BEP level of 10−4 For N = 2, 3, 4. Note that the increase in the maximum eigenvalue
offsets the losses in the diversity gain for ρ < 0.84 for N = 2; ρ < 0.9 for N = 3, and ρ < 0.92 for N = 4.
ρ > 0.8 while the maximum eigenvalue shows
almost a linear increase with ρ. Figure 12.17
shows the difference between the increase in
the maximum eigenvalue and the loss in diversity gain as a function of the correlation coefficient. As long as ρ < 0.8, a few dB additional
gain can be realized by the system by purposely
inserting correlation between the received signals, for example, by controlling the distance
between array elements. This also alleviates
the restriction on the separation between the
array elements, that is, the array elements can
be placed closer to each other.
efficiency of a MISO system with N array elements is given by
η≜C B = log2 1 + γ , γ = γ c
whh H w H
ww H
(12.56)
where the instantaneous value of γ is given
by (12.36).
The ergodic (mean) spectral efficiency may
be written as
E η = E log2 1 + γ =
∞
log2 1 + γ f γ dγ
0
(12.57)
12.5 Ergodic and Outage Capacity
in Fading Channels
We already considered the fading effects on the
BEP and the outage probability. Another performance measure is related to the maximum
or minimum data rate that can be supported
by a system in a fading environment, that is,
ergodic and outage capacities. Here, we prefer
to work with the spectral efficiency, which is
defined as the capacity normalized by the transmission bandwidth. The instantaneous spectral
For the fully correlated case (ρ = 1), the mean
spectral efficiency is found using (D.73):
Eη =
∞
log2 1 + γ
0
= log2 e e
1 Nγ c
∞
where E1 x =
1 −r
e
Nγ c
Nγ c
dγ
(12.58)
E1 1 Nγ c
e − t t dt is defined by
x
(D.115). For the uncorrelated case (ρ = 0), the
ergodic spectral efficiency is found using
(D.74): [4]
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Diversity and Combining Techniques
Mean spectral efficiency (bps/Hz)
8
N=1
N=2
N=3
N=4
AWGN
6
4
2
0
0
5
10
γc (dB)
15
20
Figure 12.18 Spectral Efficiency of a SIMO System with N Antennas For Identical Mean Branch SNR’s in
a Rayleigh Fading Channel.
Eη =
∞
1 γ N −1 − γ
e
N −1 γ Nc
log2 1 + γ
0
=
∞
1
N −1
dγ
e − x x N −1 log2 1 + γ c x dx
0
N −1
= log2 e
γc
1
k = 0 k −γ c
k
e1 γc E1 1 γ c +
k
i −1
− γc
i
i=1
(12.59)
For 0 < ρ < 1, the mean spectral efficiency is
found using (D.73) and (12.46):
Eη =
∞
N
log2 1 + γ
0
πk
k=1
N
πk
= log2 e
k=1
∞
ln 1 + λk γ c x exp − x dx
0
N
π k e1
= log2 e
1
γ
dγ
exp −
λk γ c
λk γ c
λk γ c
E1 1 λk γ c
k=1
(12.60)
where π k is defined by (12.46). Figure 12.18
shows that the spectral efficiency in bps/Hz
increases with mean SNR and N, as expected.
Also note that the spectral efficiency in a fading
channel is lower than in the AWGN channel
unless diversity reception and combining is
used. For finite values of the mean SNR, the
slope of the spectral efficiency curve is practically the same for all values of N. However,
spectral efficiency increases with N, implying
that the use of SIMO systems increases the
throughput.
Figure 12.19 shows the variation of the ergodic spectral efficiency with mean SNR for
various values of the correlation coefficient
for N = 3. The results for N = 1 and 2 are also
shown for comparison purposes. Note that the
correlation effects are not as strong as in outage
and bit error probabilities. At a given spectral
efficiency value, the performance difference
for ρ = 0 and ρ = 1 is less than 2 dB for N = 3.
In other terms, the spectral efficiency for ρ = 1
is approximately 12% less than that for ρ = 0
at a given value of the mean SNR. Nevertheless,
the ergodic spectral efficiency is still higher
than that for ρ = 0, N = 2.
Figure 12.20 shows the variation of the
mean spectral efficiency with the number of
antennas for ρ = 0 and ρ = 1. The increase in
the spectral efficiency with N shows similar
trends for uncorrelated and perfectly correlated cases.
The cdf of the ergodic capacity, that is, the
probability that the ergodic capacity is below a
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Mean spectral efficiency (bps/Hz)
8
N=1
N = 2, ρ = 0
N = 3, ρ = 0
N = 3, ρ = 0.8
N = 3, ρ = 1
6
4
2
0
0
5
Figure 12.19
10
γc(dB)
15
20
Effect of Correlation on the MeanSpectral Efficiency.
Mean spectral efficiency (bps/Hz)
9
8
7
6
5
ρ = 0, γc = 10 dB
ρ = 1, γc = 10 dB
ρ = 0, γc = 15 dB
ρ = 1, γc = 15 dB
4
3
2
4
6
8
10
Number of antennas, N
Figure 12.20 The Mean Spectral Efficiency (bps/Hz) as a Function of the Number of Antennas For
γ c = 10,15 dB and ρ = 0, 1.
threshold normalized capacity level ηth = Cth B,
may be formulated as follows:
Prob η < ηth = Prob log2 1 + γ < ηth
= Prob γ < 2ηth − 1 = F 2ηth − 1
(12.61)
where F(γ) denotes the cdf of γ and is given by
(12.53). Figure 12.21 shows the correlation
effects on the cdf of the spectrum efficiency.
The outage probability decreases with
increasing values of the number of antennas,
as expected. However, strong correlations
between the channel gains may offset this
decrease. For example, let N = 2, γ c = 10 dB
and ηth = 2. For ρ = 1, the probability that the
spectral efficiency is below ηth is found using
(12.53) and (12.61) as
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Prob (spectrum efficiency < ηth)
Diversity and Combining Techniques
0.8
0.6
0.4
N=1
N = 2, ρ = 0
N = 3, ρ = 0
N = 3, ρ = 0.7
N = 3, ρ = 0.9
N = 3, ρ = 1
0.2
0
0
2
4
ηth
6
8
Figure 12.21 Cdf of the Spectral Efficiency For a SIMO System with N Antennas For 10 dB Mean
Channel SNR.
Prob η < 2 = 1− exp − 3 20 = 0 139 13 9
(12.62)
while, for ρ = 0, it is given by
Prob η < 2 = 1 − e − 3
10
1 + 3 10 = 0 037 3 7
(12.63)
Hence, for ρ = 1, the spectrum efficiency will
be higher than 2 for 86.1% of the time. However, it is higher than ηth = 2 during 96.3% of
the time for ρ = 0. In other words, the outage
probability for the spectrum efficiency is
increased from 3.7% to 13.9% as the correlation coefficient ρ changes from 0 to 1.
Figure 12.22 shows the variation of the probability that the spectrum efficiency is less than
ηth = 2 as a function of γ c . The outage performance of the spectrum efficiency is evidently
improved as the mean SNR increases. The
curve for ρ = 1, N = 3 is shifted by 4.78 dB
compared to that for N = 1 due to the 3-fold
increase in the array gain, as we already
observed in Figure 12.13 and Figure 12.14.
In some applications, it may be more desirable to use outage capacity rather than the ergodic capacity. Outage capacity is defined as the
capacity that remains below a certain threshold
capacity level, Cth, for a given time percentage
p %. In other words, the capacity will higher
than Cth for (1-p) % of the time. Let us now
determine the probability that the capacity is
below a threshold level. Figure 12.23 shows
the variation of the 2% outage spectral efficiency ηth = Cth B for N = 1, 2 and 3. The
degradation in the outage spectrum efficiency
is observed to be relatively low for ρ < 0.8.
12.5.1 Multiplexing Gain
Multiplexing gain is defined as the slope of the
ergodic channel capacity (spectrum efficiency)
with SNR:
C γ
∞ log γ
r = lim
γ
(12.64)
where C γ denotes the ergodic (average) channel capacity at the mean SNR level γ. Multiplexing gain provides a measure of the
increase in the maximum data rate that can be
transmitted by the system with the number of
antennas on transmit and/or receive side. In
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Prob (spectrum efficiency < ηth)
1
0.1
0.01
0.001
0.0001
Figure 12.22
N=1
N = 2, ρ = 0
N = 3, ρ = 0
N = 3, ρ = 0.7
N = 3, ρ = 0.9
N = 3, ρ = 1
0
5
10
γc(dB)
15
20
Cdf of the Spectral Efficiency For an N-Antenna SIMO System Versus Mean SNR For ηth = 2.
2% outage spectrum efficiency (bps/Hz)
8
N=1
N = 2, ρ = 0
N = 2, ρ = 0.9
N = 2, ρ = 1
N = 3, ρ = 0
N = 3, ρ = 0.9
N = 3, ρ = 1
7
6
5
4
3
2
1
0
0
5
10
15
20
25
30
γc(dB)
Figure 12.23 The Spectrum Efficiency that Falls Below the Values on the Ordinate For Only 2% of the
Time. The spectrum efficiency is higher than the values shown on the ordinate for 98% of the time.
other words, it gives the number of parallel
channels that is used for transmission in the
presence of multiple antennas at the transmit
and/or receive side. One may observe from
Figure 12.18 that the multiplexing gain is given
by the slope of the bandwidth efficiency as the
SNR approaches infinity. The conditions under
which the multiplexing gain changes will be
discussed in Chapter 13.
12.6 Diversity and Combining
Diversity and combining techniques offer
significant performance improvements in fading
channels in terms of array, diversity and multiplexing gains. However, they require the availability of a number of transmission paths, which
carry the same message, have approximately the
same mean power levels, and preferably exhibit
Diversity and Combining Techniques
independent fading statistics. Since the probability of simultaneous fading of independent channels is much lower than that of a single channel,
or correlated multiple channels, the availability
of a number of independent diversity channels
allows more reliable data transmission with
lower error probabilities. In other terms, diversity mitigates fading. Properly combined diversity signals at the receiver lead to mean output
SNRs higher than that of any of the individual
channels (array gain) and pdf’s with lower tails,
leading to decreased outage and bit error probabilities (diversity gain) (see Example 11.19).
Diversity may be categorized as macroscopic
and microscopic diversity. Macroscopic diversity
mitigates long-term (in the order of ms) shadow
fading, which is caused by hills, mountains,
and buildings in the propagation path. It results
in random variation of the local mean power of
the received signal, which is usually characterized by log-normal pdf whose standard deviation
depends on the terrain topology and the propagation distance. Shadowing is mitigated by macroscopic diversity provided by physically separated
antennas for transmission and/or reception.
Microscopic diversity, which is used to mitigate the effects of short-term (in the order of μs)
fading, is usually implemented as frequency,
time or space diversity. Space diversity may
be realized mainly as antenna diversity, angular
diversity or polarization diversity. In antenna
diversity, multiple antennas are used to
receive/transmit signals. Antenna spacings
must be adjusted so that the faded signals incident on each receive antenna become independent from each other. Antenna diversity
becomes more effective if reception conditions
depend on the location, and/or the diversity
channels are time variant. In angle diversity,
the signals received from different angles of
arrival are resolved and combined at the
receiver. Angle diversity requires a number of
directional antennas, each selecting signals
arriving from a narrow ranges of angles with
low correlations between them.
Polarization diversity is a special case
of space (antenna) diversity and is based on
the assumption that wireless signals with
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orthogonal polarizations (vertical/horizontal,
RHCP/LHCP) undergo independent fading
in wireless channels. Signals simultaneously
transmitted from two orthogonal polarizations
are resolved and combined at the receiver. If
the propagation loss is appreciably different
at these polarizations, then the transmit power
may be divided accordingly so as to realize equal
powers in two diversity branches. This scheme
requires the isolation of the two polarizations
from each other, since signals suffer depolarization in the wireless channel due to scattering
and/or antennas can not isolate them perfectly.
Transmit and receive antennas should therefore
be designed with low depolarization characteristics. Monopole and patch antennas are commonly used in polarization diversity systems.
Frequency diversity implies the transmission
of a signal with different carrier frequencies separated from each other by at least the channel
coherence bandwidth. Frequency diversity is
feasible when the channel undergoes frequency
selective fading. The major drawbacks of the
frequency diversity include the requirement
for larger transmission bandwidths due to multiple frequency channels, and the generation and
combining of signals of different carriers with
the same phases (for coherent signalling).
In time diversity, the same information bearing signal is transmitted at different time slots
that are separated by at least the channel coherence time. Time or delay diversity is feasible
when fading is time-selective. In slow fading
channels, cyclic delay diversity may be more
effective. Delay diversity requires information
storage at the transmitter and the receiver. In
frequency selective channels, the multipath signals arriving to the receiver with delays longer
than the symbol duration, can be treated as
diversity signals. As shown in Section 11.5.2,
turning multipath propagation into its advantage, a Rake receiver uses MRC to cophase
and sum the powers of multipath components
in proportion to their respective SNR ratios.
Synchronization is an important design factor
for a MRC system, which operates coherently
and, hence, requires channel state information
(CSI) at the receiver.
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In the receive diversity (SIMO) systems, the
replicas of a signal transmitted by a single
transmit antenna are received by an ULA and
combined at the receiver (see Figure 12.10). Combining the received signals may be implemented
at pre-detection or post-detection stages. In SIMO
systems with pre-detection combining, the signals
received by the array elements are combined at the
receiver front-end and a single receiver processes
the combined signals. However, in post-detection
combining, separate receivers are required for
each channel since signals are combined at the
demodulator output. The performance difference
between pre-detection and post-detection
combining is observed to be negligibly small
for ideal coherent detection. However, a slight
performance difference is observed for differentially coherent detection.
In transmit diversity (MISO) systems, the
transmit power is divided between the array
elements. Consequently, signals transmitted
by each branch have lower transmit powers,
compared to receive diversity systems where
the transmit power is allocated to a single transmit antenna. Therefore, the transmit diversity
has potentially lower performance compared
to the receive diversity. However, if the channel
state information (CSI) is made available to the
transmitter, then the transmit power may be
divided between transmit antennas so as to
level the SNR’s received by each element of
the receiver ULA. The CSI, which includes
the gain, the phase and the delay of each channel, may be provided to the transmit side using
feedback (closed-loop), training information
but no feedback (open-loop), and blind
schemes. This approach, which is called as
maximal ratio transmission (MRT), provides
a performance identical to MRC.
12.6.1
Combining Techniques for
SIMO Systems
There are various techniques for combining
diversity branches. In selection combining
(SC), for each symbol duration, the branch with
the highest SNR is selected and signals of other
branches are ignored. Switched combining and
switch and stay combining (SSC) are special
versions of the SC. [5]
As given by (12.31), the output of a linear combiner consists of a weighted sum of signals
received by each branch. The weights are used
to co-phase the received signals; hence linear
combination is a coherent process and requires
CSI at the receiver. The receiver can acquire
the CSI by using a recovery circuit at its input;
hence this process is not as difficult as in MISO
systems. Maximal ratio combiner (MRC) and
equal-gain combiner (EGC) are examples linear-combining techniques. In MRC, signals
received by diversity branches are co-phased
and are weighted in proportion to their received
signal levels; hence the MRC output consists of
the sum of branch SNR’s as in (11.213) and
(12.40). Therefore, MRC requires the estimation
of phases and amplitudes of signals received by
each diversity branch. For receivers operating
in frequency-selective fading channels, for
example, Rake receiver, the delays are also
required. MRC gives the best possible performance among the diversity combining techniques
and provides highest protection against fading.
In EGC, received signals in diversity
branches are co-phased and added with equal
weights. Therefore, the EGC output, which
consists of the sum of co-phased signal envelopes, requires the estimation of only the channel phases. In frequency-selective channels,
estimation of the signal delays in each diversity
branch is also required.
MRC emphasizes signals with higher SNRs
by choosing the weights to be complex conjugate of the channel gain (see (12.39)). Therefore, it is not optimum in the presence of
strong interference, since higher signal levels
are due to interference. Optimum combining
(OC) is based on minimizing the mean square
error between a training sequence and the combiner output, or maximizing signal-to-noise
and interference power ratio (SINR). OC
reduces to MRC in the absence of interference.
Square-law combining (SLC) is commonly
used in channels where CSI is not available,
for example, for noncoherent detection
of orthogonal modulations. SLC suffers
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Diversity and Combining Techniques
Front-end
Selector
Detector
Data
Front-end
(a) Pre-detection SC
Front-end
Detector
Data
Selector
Front-end
Detector
(b) Post-detection SC
Figure 12.24 Pre-Detection and Post-Detection SC Receiver with Two Antennas For a SIMO System.
noncoherent combining losses, since branch
signals and noises are combined together, and
performs worse than linear combining
techniques.
All these combining techniques have advantages and drawbacks and different areas of
applications. Their performances differ in terms
of outage and bit error probabilities as well as
the output SNR.
if selection is made at the front-end (predetection diversity) but two receivers are necessary when the signal with highest SNR is
selected at the demodulator output (postdetection diversity) (see Figure 12.24).
Consider a N-branch SC system with a single
transmitter and N receivers. The selected SNR
Z is the strongest of the SNRs Xi, i = 1, 2, …, N
of the N diversity branches:
(12.65)
, XN
Z = max X1 , X2 ,
12.6.1.1
Since the cdf of Z is defined as the probability that Z ≤ z during a symbol period, all branch
SNRs should be less than or equal to z during
the same symbol period:
Selection Combining
SC is often used for macroscopic diversity by
BS to increase the coverage area in cellular radio
systems against shadowing. For example, a second BS installed in a tunnel may be help user
receivers, suffering heavy losses due to shadowing by tunnel, to select between the two signals
with the highest SNR. Macroscopic diversity
makes use of a number of antennas that are well
separated from each other so that the received
signals undergo independent shadowing. At
each symbol duration, the selection combiner
at a receiver selects the received signal with
the largest SNR. A single receiver is required
Fz z = Prob Z ≤ z
= Prob X1 ≤ z, X2 ≤ z,
, XN ≤ z
(12.66)
When the receive antennas are located sufficiently far from each other, then Xi’s are independent and the cdf of Z reduces to
N
N
Prob Xi ≤ z =
Fz z =
i=1
FXi z
i=1
(12.67)
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N=1
N=2
N=3
N=4
N=6
FZ(γ)
0.1
0.01
0.001
–25
–20
–15
–10
–5
10log(γ/γc) (dB)
0
5
10
Figure 12.25 The Cdf of the Output SNR of a SIMO System with N-branch SC in Log-Normal Shadowing
For 8 dB Shadowing Variance.
The pdf of Z then becomes
fz z =
d
Fz z =
dz
N
N
fXi z
i=1
FXj z
j = 1, j i
(12.68)
where fXi z denotes the pdf of Xi.
If the signals undergo log-normal shadowing,
then the CDF of the ith branch may be written as
(N = 1) is already depicted in Figure 11.28 for
various values of the shadowing variance.
Figure 12.25 clearly shows the improvement
provided by SC on the outage probability for
log-normal shadowing.
The mean SNR value of the correlated SC
output in a log-normal shadowing channel is
given by: [6]
z
FXi z =
fXi u du
−∞
z
=
1
2πσ i
EZ =
u − γi
exp −
2 σ 2i
2
z − γi
du = 1− Q
σi
−∞
(12.69)
where γ i and σ 2i denote, respectively, the mean
SNR and the variance of shadowing, both
expressed in dB. Using (12.67), the CDF of output SNR for N branch diversity may be written as
N
1−Q
Fz z =
i=1
= 1−Q
z− γ i
σi
z− γ c
σ
N
,
γi = γc, σi = σ
i = 1, 2, …,N
(12.70)
The outage probability in a log-normal
shadowing environment with no diversity
2γ c 1 − Q σ
1−ρ 2
3γ c 1 − Q σ
1−ρ 2
− 6T σ
T α, β =
1
2π
1 − ρ 2, 1
N =2
3
N =3
β
exp − 0 5α2 1 + x2
dx
1 + x2
0
(12.71)
where shadowing variance and mean branch
SNRs are assumed to be identical γ c = γ i ,
σ = σ i , i = 1, 2. Here, E Z γ c denotes the
increase in the mean SNR due to diversity.
Figure 12.26 shows the normalized output
SNR for SC with N = 2 and 3 branches as a
function of the correlation coefficient between
branches. The output SNR was observed to
become higher for higher shadowing variances
and decreases rapidly for ρ > 0.8. Figure 12.9
and (12.28) show that the ρ = 0.8 corresponds