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4 Diversity Order, Diversity Gain and Array Gain

4 Diversity Order, Diversity Gain and Array Gain

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Digital Communications



1/γ N






Diversity gain


Bit error probability

Bit error probability



1/(Gaγ )N




γ (dB)

(a) Diversity gain and diversity order d

Ga > 1





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



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


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)).


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



← Loss in diversity gain









3 dB


Increase in the value of

maximum eigenvalue →







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).


Digital Communications















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


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



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


= log2 e e

1 Nγ c

where E1 x =

1 −r


Nγ c

Nγ c


E1 1 Nγ c

e − t t dt is defined by


(D.115). For the uncorrelated case (ρ = 0), the

ergodic spectral efficiency is found using

(D.74): [4]


Diversity and Combining Techniques

Mean spectral efficiency (bps/Hz)














γc (dB)



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


N −1 γ Nc

log2 1 + γ




N −1

e − x x N −1 log2 1 + γ c x dx


N −1

= log2 e



k = 0 k −γ c


e1 γc E1 1 γ c +


i −1

− γc




For 0 < ρ < 1, the mean spectral efficiency is

found using (D.73) and (12.46):

Eη =


log2 1 + γ






= log2 e


ln 1 + λk γ c x exp − x dx



π k e1

= log2 e



exp −

λk γ c

λk γ c

λk γ c

E1 1 λk γ c



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


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)



N = 2, ρ = 0

N = 3, ρ = 0

N = 3, ρ = 0.8

N = 3, ρ = 1







Figure 12.19





Effect of Correlation on the MeanSpectral Efficiency.

Mean spectral efficiency (bps/Hz)






ρ = 0, γc = 10 dB

ρ = 1, γc = 10 dB

ρ = 0, γc = 15 dB

ρ = 1, γc = 15 dB








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


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


Prob (spectrum efficiency < ηth)

Diversity and Combining Techniques





N = 2, ρ = 0

N = 3, ρ = 0

N = 3, ρ = 0.7

N = 3, ρ = 0.9

N = 3, ρ = 1









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


while, for ρ = 0, it is given by

Prob η < 2 = 1 − e − 3


1 + 3 10 = 0 037 3 7


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



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)






Figure 12.22


N = 2, ρ = 0

N = 3, ρ = 0

N = 3, ρ = 0.7

N = 3, ρ = 0.9

N = 3, ρ = 1







Cdf of the Spectral Efficiency For an N-Antenna SIMO System Versus Mean SNR For ηth = 2.

2% outage spectrum efficiency (bps/Hz)



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

















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


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.


Digital Communications

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.


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


Diversity and Combining Techniques






(a) Pre-detection SC







(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


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:


, XN

Z = max X1 , X2 ,

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


When the receive antennas are located sufficiently far from each other, then Xi’s are independent and the cdf of Z reduces to



Prob Xi ≤ z =

Fz z =


FXi z




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10log(γ/γc) (dB)




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 =


Fz z =




fXi z


FXj z

j = 1, j i


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]


FXi z =

fXi u du





2πσ i

EZ =

u − γi

exp −

2 σ 2i


z − γi

du = 1− Q




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



Fz z =


= 1−Q

z− γ i


z− γ c




γi = γc, σi = σ

i = 1, 2, …,N


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 − ρ 2, 1

N =2


N =3


exp − 0 5α2 1 + x2


1 + x2



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

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