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2498 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 5, SEPTEMBER 2005
On the Amount of Fading in MIMO
Diversity Systems
Bengt Holter, Student Member, IEEE, and Geir Egil Øien, Member, IEEE
Abstract—In this paper, a closed-form expression is presented
for the amount of fading (AF) experienced at the output of a
space–time block-coded multiple-input–multiple-output (MIMO)
diversity system operating on identically distributed spatially cor-
related Nakagami-mfading channels. For the separable correla-
tion model (Kronecker model), the AF is presented for identically
distributed Rayleigh fading channels and different types of an-
tenna correlation models. For an MIMO diversity system based on
the Kronecker model, it is shown, by capitalizing on recent results
in IEEE Transactions on Communications, Vol. 51, No. 8 (August
2003), p. 1389, that the average symbol error rate (SER) at high
signal-to-noise ratio (SNR) may be directly expressed in terms of
the AF, when a constant correlation model is assumed.
Index Terms—Amount of fading, coding gain, diversity, fading
channels, multiple-input–multiple-output (MIMO).
I. INTRODUCTION
SEVERAL performance measures may be employed to
characterize the behavior of wireless communication sys-
tems operating on fading channels. For systems utilizing spatial
diversity techniques, it is of interest to employ measures that
can capture and quantify the improvement on system perfor-
mance caused by reducing the fading-induced fluctuations of
the received signal. Commonly encountered measures in this
respect are the average symbol error rate (SER) or the average
bit error rate (BER). The diversity order of a spatial diversity
system is usually determined by the slope of the average SER
curve at high signal-to-noise ratios (SNRs), whereas different
levels of correlation between the diversity branches are visible
as shifted versions of the SER curve, relative to a benchmark
SER curve. However, as noted in [2], the average error rate
may, in some cases, be difficult to evaluate analytically, since
it requires statistical averaging of the conditional error rate over
the statistics of the fading. A more simple, yet effective, way of
quantifying the severity of fading (and the effect of correlation)
can be obtained by using a measure directly related to the
moments of the fading distribution itself.
In [3], Charash introduced the notion of amount of fading
(AF) to quantify the severity of fading experienced for a partic-
ular channel model. In terms of the probability density function
(pdf) of the instantaneous fading amplitude α=|h|of a single
Manuscript received November 24, 2003; revised July 28, 2004 and August
10, 2004; accepted August 26, 2004. The editor coordinating the review of this
paper and approving it for publication is L. Hanzo.
The authors are with the Department of Electronics and Telecommunica-
tions, Norwegian University of Science and Technology, NO-7491 Trondheim,
Norway (e-mail: bholter@iet.ntnu.no; oien@iet.ntnu.no).
Digital Object Identifier 10.1109/TWC.2005.853832
complex fading channel h, the AF is defined by [3, eq. (2)],
[4, eq. (2.5)]
AF = Var{α2}
(E{α2})2(1)
with E{·} and Var{·} denoting the statistical average and
variance, respectively. For a single Nakagami-mfading channel
[5], AF = 1/m. Hence, for large values of m[line-of-sight
(LOS)], the fading channel will approach a nonfading additive
white Gaussian noise (AWGN) channel. For a single Rayleigh
fading channel (m=1),AF = 1.
Although the AF originally was defined and applied to quan-
tify the severity of fading experienced at the output of a single
fading channel, it is, in this paper, employed to quantify the
degree of fading experienced at the output of a multiple-input
multiple-output (MIMO) system. The benefit of transmitting
from multiple antennas in a wireless system may either be
utilized to improve the diversity order (high-reliability solution)
or to improve the capacity (high-rate solution). These two trans-
mission strategies are commonly denoted as MIMO diversity
and spatial multiplexing, respectively [6]. In this study, an
MIMO-diversity system is considered, where transmit diversity
is realized by utilizing space–time block coding (STBC)1at the
transmitter [7], [8].
There are some examples in the literature of how the AF
can be applied as a performance measure in communication
systems employing spatial diversity. In [2], closed-form expres-
sions for the AF are obtained for three dual-branch diversity-
combining techniques in the presence of log-normal fading,
namely, maximum-ratio combining (MRC) [4, Sec. 9.2], selec-
tion combining (SC) [4, Sec. 9.7], and switch-and-stay com-
bining (SSC) [4, Sec. 9.8]. In [9], the square root of the AF
was applied to the combiner output to assess the effectiveness
of a hybrid-selection/MRC diversity-combining scheme in the
presence of Rayleigh fading.2
In this paper, the AF is presented for an MIMO diversity
system operating on identically distributed spatially correlated
Nakagami-mfading channels. The necessary moments are
derived by utilizing a recent and compact representation of the
characteristic function (CF) of the (instantaneous) combined
fading power at the output of an MIMO diversity system [10].
1Space-time block codes are designed to achieve the maximum diversity
order for a given number of transmit and receive antennas, subject to the
constraint of having a simple decoding algorithm.
2The square root of the AF is equal to a statistical measure known as the
coefficient of variation.
1536-1276/$20.00 © 2005 IEEE
HOLTER AND ØIEN: ON THE AMOUNT OF FADING IN MIMO DIVERSITY SYSTEMS 2499
Subsequently, it is assumed that the correlation properties at
the transmitter is independent of the correlation properties at
the receiver. Under this assumption, the channel correlation
matrix can be written as a Kronecker product [11, Ch. 9] of
a transmit correlation matrix and a receive correlation matrix
[12], [13]. In the literature, this model is frequently denoted as
the Kronecker model. The Kronecker model has been validated
for non-LOS (NLOS) scenarios [14], [15], but the accuracy of
the model has recently been questioned, at least for antennas
with high spatial resolution (large antenna arrays) [16]–[18].
Since all the measurement campaigns referenced in this paper
(both supporting and questioning the Kronecker model) have
been conducted in NLOS scenarios (typically modeled with
Rayleigh fading channels), the AF expressions based on the
Kronecker model are, in this paper, limited to systems operating
on Rayleigh fading channels.
The remainder of this paper is organized as follows.
Section II presents statistical information of the combined
fading power in an MIMO diversity system. In Section III, this
information is used to obtain a compact closed-form expression
for the AF. The general result (valid for identically distributed
Nakagami-mfading channels) is then simplified by introducing
the Kronecker model and by incorporating different types of
antenna correlation models. In Section IV, it is shown that by
capitalizing on recent results in [1], the average SER at high
SNR may be expressed in terms of the AF, when a constant
correlation model is assumed at both ends of the MIMO link.
Numerical results are presented in Section V, and the main
results of the paper are summarized in Section VI.
II. STATISTICS OF THE COMBINED FADING POWER
To obtain an expression for the AF, the statistics of either the
combined SNR or combined fading power must be known. For
a flat-fading MIMO diversity system with nTtransmit antennas
and nRreceive antennas, the combined SNR (per symbol) may
be written as [19]
γc=PTH2
F
σ2nT
=PT
σ2nT
nR
i=1
nT
j=1
α2
ij (2)
where H2
Frepresents the combined fading power (squared
Frobenius norm of the channel matrix H[11]), and α2
ij =|hij |2
represents the fading power of a single narrowband channel
between the jth transmitter and the ith receiver. From (2), it
can be seen that the statistics of γcis governed by the statistics
of H2
F. For simplicity, just a single subscript nwill be
used to distinguish between the entries of the channel matrix,3
and the squared Frobenius norm may then be expressed as
H2
F=N
n=1 α2
n, where N=nT·nRdenotes the maximum
number of channels in the MIMO channel. Viewing all the
fading amplitudes in the set {αn}N
n=1 as identically distributed
3n=n(i, j)=nT·(i−1) + jfor i∈[1,2,...,n
R]and j∈
[1,2,...,n
T].
Nakagami-mrandom variables (RVs) [5], the pdf of a single
fading amplitude αnis equal to [5]
p(αn)=2mmα2m−1
n
ΩmΓ(m)·e−mα2
n
Ω(3)
where Γ(·)denotes the gamma function,4Ω=E{α2
n}denotes
the average fading power, and mis the Nakagami fading
parameter.
When the instantaneous fading amplitude αnis distributed
according to (3), the instantaneous fading power α2
nwill follow
a gamma distribution5with shape parameter mand scale pa-
rameter Ω/m. In short, α2
n∼G(m, (Ω/m)). Hence, H2
F
represents a sum of Nidentically distributed possibly corre-
lated gamma variates.
III. AMOUNT OF FADING
TheCFofH2
F
∆
=α2
c, representing a sum of Nidentically
distributed correlated gamma variates, may be written com-
pactly as [10]
Φα2
c(w)=|IN×N−wRH|−m
=|IN×N−wΛ|−m
=
N
n=1
(1 −wλn)−m(4)
where wis the variable of the transform domain, |·|denotes
the determinant operator, IN×Nis the identity matrix of size
N×N, and Λis the diagonal eigenvalue matrix of the (com-
plex) channel correlation matrix RH=E{vec(H)vec(H)H},
containing the set of eigenvalues {λn}N
n=1.6Since RHis a
Hermitian matrix, the existence of Λis guaranteed by the
spectral theorem [11, Th. 6.2].7The moments of an RV can
be determined from its CF, according to [21, eq. (2-1-74)]
Eα2q
c=1
q·dqΦα2
c(w)
dwqw=0
.(5)
4Γ(z)=∞
0tz−1e−tdt, R(z)>0[20].
5Y=α2
nfollows a gamma distribution with shape parameter a>0
and scale parameter b>0, when the pdf of Yis given by pY(y)=
[ya−1e−y/b]/[baΓ(a)]. The short-hand notation Y∼G(a, b)is used to
denote that Yfollows a gamma distribution with shape parameter aand scale
parameter b.
6The superscript (·)Hdenotes the Hermitian transpose operator, and the
vec(·)operator stacks the individual columns of the argument matrix on
top of each other, i.e., represents the matrix as a single-column vector [11,
Sec. 9.3].
7In addition, this means that the algebraic multiplicity of eigenvalues in
RH(the number of eigenvalues) is equal to the geometric multiplicity of
eigenvalues (the size of the nullspace spanned by the eigenvectors). Thus, if
RHis an N×NHermitian matrix of rank r<N (rank deficient), exactly
N−rof the eigenvalues of RHare equal to zero. This means that the effective
number of terms contributing to the product in (4) (i.e., product terms different
from 1), in general, is equal to r. For full-rank matrices, r=N. In this paper,
it is assumed that RHis full rank.
2500 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 5, SEPTEMBER 2005
Using logarithmic derivation [22], the first- and second-order
derivatives of (4) may be expressed as
Φ
α2
c(w)
=N
n=1
mλn
1−wλn·Φα2
c(w)(6)
Φ
α2
c(w)
=
N
n=1
mλn
1−wλn2
−
N
n=1
mλ2
n
(1−wλn)2
·Φα2
c(w).(7)
Using (5), the mean and variance of α2
cmay be identified as
Eα2
c=m·
N
n=1
λn(8)
Var α2
c=m·
N
n=1
λ2
n.(9)
According to (1), the AF may now be expressed as AF =
(N
n=1 λ2
n/m ·(N
n=1 λn)2). For normalized average power
on all channels, i.e., E{α2
n}=Ω=1for all n∈[1,2,...,N],
the channel correlation matrix RHwill contain just 1s on the
main diagonal. Since the sum of eigenvalues equals the sum of
diagonal entries in a matrix [23, Ch. 5], the AF for identically
distributed Nakagami-mfading channels may be written as
AF = tr(Λ2)
m·N2(10)
where tr(·)is the matrix trace operator [11].
In the following, it is assumed that the correlation between
the antennas at the transmitter is independent of the corre-
lation between the antennas at the receiver. This separability
assumption (Kronecker model) has been validated by some
authors [14], [15], and has become quite popular, due to its
analytical tractability. However, note that some authors recently
have questioned its accuracy [16]–[18].
All the measurement campaigns on the Kronecker model
referenced in this paper (both supporting and questioning the
Kronecker model) have been conducted in NLOS scenarios.
As a result, we have chosen to limit the analysis based on
the Kronecker model to identically distributed Rayleigh fading
channels. According to Appendix A, the AF may then be
written as
AF = nT
j=1 tj2nR
i=1 ri2
N2(11)
where ·
2denotes the squared Euclidean vector norm, and
the vectors tjand ridenote rows jand iof the transmit and
receive correlation matrices, respectively. Next, it is shown that
the result in (11) may be simplified even further, when specific
types of antenna correlation models are taken into account. For
simplicity, results are presented for identical correlation models
at the transmitter and the receiver. Results for other scenarios
(i.e., nonidentical correlation models at the transmitter and the
receiver) may also easily be obtained, although not presented in
this paper.
1) Constant Correlation: For a constant correlation model,
applicable for an array of three antennas placed on an equilat-
eral triangle or for closely spaced antennas other than linear
arrays [24], the correlation matrix RYof size nY×nYat the
transmitter/receiver may be written as
RY=
1y··· y
y∗1··· y
.
.
..
.
.....
.
.
y∗y∗··· 1
(12)
where the subscript Y∈{T,R}is used to distinguish between
the transmit correlation matrix RTand the receive correlation
matrix RR,y∈{t, r}denotes the complex transmit/receive
correlation coefficient, and y∗denotes the complex conjugate of
y. For constant correlation matrices at each end of the MIMO
link, the AF (denoted AFcon) is expressible as
AFcon =1+|t|2(nT−1)
nT·1+|r|2(nR−1)
nR
.(13)
An alternative expression to (13) is obtained by noting that
the fading amplitude (envelope) correlation coefficient ρenv
and the fading power correlation coefficient ρpow may be as-
sumed equal for all practical purposes [5], [25]. The correlation
between fading envelopes can also be approximated by the
squared amplitude of the complex correlations in RH, referred
to as power correlation coefficients [25], [26, eq. (35)]. Hence,
ρenv
ij ≈|RH(i, j)|2=ρpow
ij , where RH(i, j)represents a single
entry of the matrix at the ith row and jth column. As a result,
(13) may also be written as
AFcon =AF
Tx ·AFRx
=[1+ρt(nT−1)]
nT·[1+ρr(nR−1)]
nR
(14)
where ρt=|t|2and ρr=|r|2represent the transmit and receive
power correlation coefficients, respectively.
2) Circular Correlation: A circular correlation model may
apply to antennas lying on a circle, or four antennas placed on a
square [24]. The correlation matrix RYcan then be written as
RY=
1y2y3··· ynY
y∗
nY1y2··· ynY−1
y∗
nY−1y∗
nY1··· ynY−2
.
.
..
.
..
.
.....
.
.
y∗
2y∗
3y∗
4··· 1
(15)
where y1=1. Since every correlation matrix is Hermitian, it
implies that y2=y∗
nY,y3=y∗
nY−1,.... The AF with circular
correlation at each end of the MIMO link (denoted AFcir) can
be expressed as
AFcir =nT
j=1 |tj|2
nT·nR
i=1 |ri|2
nR
.(16)
HOLTER AND ØIEN: ON THE AMOUNT OF FADING IN MIMO DIVERSITY SYSTEMS 2501
3) Exponential Correlation: An exponential correlation
model may apply to an equispaced linear array of antenna
elements [4]. The correlation matrix RYcan be written as
RY=
1y··· ynY−1
y∗1··· ynY−2
.
.
..
.
.....
.
.
(y∗)nY−1(y∗)nY−2··· 1
.(17)
For exponential correlation on each side of the MIMO link, the
AF (denoted AFexp) can be expressed as
AFexp =1+2nT−1
j=1 1−j
nT|t|2j
nT
×1+2nR−1
i=1 1−i
nR|r|2i
nR
.(18)
In [4], numerical results of the average SER and outage
probability in an MRC system show that the constant corre-
lation model suffers only a minor performance degradation
when compared to the exponential correlation model, but the
performance difference is more noticeable for a large number
of diversity paths and for high correlation between the paths.
As a result, the constant correlation model may be employed as
a worst case correlation scenario, since the impact of correlation
on system performance for other correlation models, typically,
will be less severe. Hence, among the correlation profiles used
in this paper, the impact of fading correlation is most severe
(highest AF) for the constant correlation model.
IV. AFcon AND ITS RELATION TO THE AVERAGE
SER AT HIGH SNR
In this section, it is shown that for a constant correlation
matrix at either side of an MIMO diversity system operating
on identically distributed spatially correlated Rayleigh fading
channels, the average SER at high SNR may be expressed in
terms of AFcon. To this end, we will be invoking some recent
results by Wang and Giannakis [1].
1) Approximate SER: In [1], the average SER PEof an
uncoded (or coded) system at high SNR is approximated by
the expression
PE≈(Gc·γ)−Gd(19)
where Gcrepresents a coding gain, and Gdrepresents the
diversity order. The diversity order determines the slope of the
average SER curve versus the received average SNR γat high
SNR in a log–log scale, whereas the coding gain (in decibel)
determines the shift of the curve in SNR relative to a benchmark
SER curve given by (γ−Gd).
Capitalizing on results in [24], the moment-generating func-
tion (MGF) Mγc(s)of the combined SNR γcof an MIMO
diversity system with identically distributed channels may be
expressed as
Mγc(s)=
N
n=1 1−sγ
nT
λn−1
(20)
where sis the variable of the transform domain, the set
{λn}N
n=1 denotes the eigenvalues of an N×Npower corre-
lation matrix Cdefined by
C=
1√ρ12 ··· √ρ1N
√ρ21 1··· √ρ2N
.
.
..
.
.....
.
.
√ρN1√ρN2··· 1
(21)
and ρij denotes the power correlation coefficient between the
instantaneous SNR received on channels iand j, respectively.8
When decoupled correlation properties are assumed, the power
correlation matrix Cmay be written as a Kronecker product
C=CTx ⊗CRx of the transmit and receive power correlation
matrices, respectively.
Applying [1, Prop. 3] to the MGF in (20), the approximate
(yet accurate) expression for the average SER PEat high SNR
may be written as9
PE≈2N−1bΓN+1
2
√πΓ(N+1) ·1
kN
(22)
where b= [det(CTx ⊗CRx)]−1(nT/γ)N, and kis a fixed
code-dependent positive constant [1]. Using (19), the MIMO
diversity system may then be characterized by the following
diversity order and coding gain at high SNR
Gd=N(23)
Gc=k2N−1pΓN+1
2
√πΓ(N+1) −1
N
(24)
where p= [det(CTx)nR·det(CRx )nT]−1·nN
T. Here, the fol-
lowing matrix identity has been utilized [11, eq. (9.8)]:
det(CTx ⊗CRx) = det(CTx )nR·det(CRx)nT.(25)
For uncorrelated channels, the determinant of the Kronecker
product in (25) is equal to 1, and thus, as pointed out
in [1], correlation increases PEby a factor [det(CTx)nR·
8The parameter ρij represents a power correlation coefficient since ρij =
Cov(γi,γ
j)/Var ( γi)Var(γj)= Cov(α2
i,α
2
j)/Var ( α2
i)Var(α2
j),
where α2
idenotes the instantaneous fading power received on channel i,
α2
jdenotes the instantaneous fading power received on channel j,andthe
expressions Cov(·,·)and Var ( ·)denote the covariance and variance of its
arguments, respectively.
9This result differs from the result presented in [1, eq. 10], since the MGF of
the combined SNR has been utilized and not the MGF of the fading power as
used in [1]. Hence, the average SNR γis, in this paper, included in the factor b,
whereas, in [1], it is not.
2502 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 5, SEPTEMBER 2005
Fig. 1. Numerical comparison of closed-form expressions for the determinant of a constant correlation matrix R. Left panel: det(R)=(1−x)L−1(1 +
x(L−1)). Right panel: det(R)=betacdf(1−x, L −1,2).
det(CRx)nT]−1≥1. A general increase in the number of trans-
mit antennas will also increase PEfor power-limited systems,
as long as the channel matrix is assumed unknown at the
transmitter, since the available power at the transmitter must
be divided between the transmit antennas. As a consequence,
a penalty in error performance will be incurred [7]. If the
channel matrix is known prior to transmission, beam forming
could be utilized, and no error-performance penalty would be
incurred.
If the correlation level is increased, a corresponding increase
in the AF will be experienced at the diversity combiner output.
Our goal now is to express PEin (22) in terms of AF, to directly
relate this performance measure to the effect of correlation on
diversity system performance. In the following, an alternative
expression for the determinant of a constant correlation matrix
is utilized to express (22) in terms of AFcon.
When both CRx and CTx are constant correlation matrices,
their determinants may be written in closed form as [27]
det(CRx)=(1−√ρr)nR−1(1 + √ρr(nR−1)) (26)
det(CTx)=(1−√ρt)nT−1(1 + √ρt(nT−1)) (27)
where both ρrand ρtare real numbers taking on values be-
tween zero and one. Since normalized average power has been
assumed on all channels, we have observed (see Fig. 1) that the
cumulative distribution function (cdf) of a beta distributed RV
[28] can be used as an alternative closed-form expression for
the determinants in (26) and (27). Hence, we may view √ρrand
√ρtas beta distributed RVs (see Appendix B), which implies
det(CRx)=I1−nR·AFRx −1
nR−1;nR−1,2
∆
= betacdf (AFRx)(28)
det(CTx)=I1−nT·AFTx −1
nT−1;nT−1,2
∆
= betacdf (AFTx)(29)
where I(·;·,·)denotes the regularized beta function.10 Upon
inserting (28) and (29) into (22), the average SER may now be
expressed as
PE≈2N−1ΓN+1
2
√πΓ(N+1) ·nT
kγ N
·[betacdf(AFTx)]−nR·[betacdf(AFRx)]−nT(30)
valid for nT≥2and nR≥2.
2) Exact SER: To evaluate the result in (30), it is compared
to an exact SER expression. As an example, a binary phase-shift
10http://mathworld.wolfram.com
HOLTER AND ØIEN: ON THE AMOUNT OF FADING IN MIMO DIVERSITY SYSTEMS 2503
Fig. 2. AFcon (in decibel) as a function of the power correlation coefficient ρ=ρt=ρrfor SIMO, MISO, and MIMO diversity systems (nT×nR)operating
on identically distributed Rayleigh fading channels: (a) (1 ×nR);(b)(2 ×nR);(c)(3 ×nR);(d)(4 ×nR).
keying (BPSK) modulation scheme will be utilized. Using a
general M-ary PSK (M-PSK) modulation scheme as a starting
point, an expression for a BPSK modulation scheme is later
obtained by letting M=2. According to [4, eq. (5.67)], the
average SER performance of an M-PSK modulation scheme
over a fading channel is expressible as
PE=1
π
(M−1)π
M
0Mγ−gpsk
sin2θdθ(31)
where Mis equal to the number of symbols in the sig-
nal constellation, gpsk =sin
2(π/M), and Mγ(·)denotes the
MGF of the received SNR. For an MIMO diversity system,
the MGF of the combined SNR is given by (20), but due to the
multiplicative property of eigenvalues in a Kronecker product,
the expression in (20) may also be written as
Mγc(s)=
nR
i=1
nT
j=1 1−sγ
nT
λiλj−1
(32)
where the sets {λj}nT
j=1 and {λi}nR
i=1 denote the eigenvalues of
the transmit and receive power correlation matrices CTx and
CRx, respectively.11 Upon inserting (32) into (31), replacing
11For nT=1, the MGF in (32) reduces the MGF in [4, eq. (9.173)], valid
for a single-input multiple-output (SIMO) system.
the variable sin (32) with the factor −gpsk/sin2θ, and assum-
ing that both CTx and CRx are constant correlation matrices,
the eigenvalues of the two matrices can be expressed in closed
form [27]. After some manipulation, (31) reduces to
PE=1
π
u
0
[1 + gλt,1λr,1]−(nR−1)(nT−1)
×[1 + gλr,1λt,2]−(nR−1)
×[1 + gλt,1λr,2]−(nT−1)
×[1 + gλt,2λr,2]−1dθ(33)
where u=[(M−1)π]/M ,λt,1=1−√ρt,λr,1=1−√ρr,
λt,2=1+√ρt(nT−1),λr,2=1+√ρr(nR−1), and g=
γgpsk/nTsin2θ. By letting M=2 (BPSK), the exact SER
expression in (33) can be compared to the approximate SER
expression in (30) with k=2[1].
In [1], the coding gain Gcwas defined as the (left) shift of
the average SER curve relative to the benchmark curve (γ−Gd).
To quantify the impact of increased AF in Gcfor a fixed
modulation scheme, the benchmark curve utilized in this paper
is given by the approximate average SER curve at high SNR
realized with uncorrelated channels. For correlated channels,
the average SER curve is then visible as a (right) shifted version
2504 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 5, SEPTEMBER 2005
Fig. 3. Top panel: Exact average SER (dashed lines) and approximate average SER at high SNR (solid lines) for a 3 ×3 MIMO diversity system operating on
identically distributed Rayleigh fading channels (constant correlation models at each end of the MIMO link). A BPSK modulation scheme is utilized. Bottom
panel: Relative (right) shift (in decibel) of the SER benchmark curve presented in the top panel as a function of ρrwhen ρt=0. Using the top xaxis as a reference,
the AF realized by the 3 ×3 MIMO diversity system is compared (in percentage) to the AF of a single Rayleigh fading channel (100% reduction represents the
nonfading AWGN channel).
of the benchmark curve, and the relative shift represents the loss
in coding gain due to correlation between the diversity branches
(increased AF).12 The relative loss in coding gain ∆Gc(in
decibel) may be expressed as
∆Gc=10·nR·log [det(CTx)] + nT·log [det(CRx)]
N
(34)
or equivalently, see (35) at the bottom of the next page, which
is valid when nT≥2and nR≥2.
V. N UMERICAL RESULTS
In this section, some numerical examples of the results
derived in this paper are presented. Since the impact of corre-
lation is most noticeable for the constant correlation model, the
results are limited to MIMO systems operating on identically
distributed Rayleigh fading channels with constant correlation
models at each end of the MIMO link. Related results for other
correlation models will typically be less severe.
12Note that with the current benchmark definition, the placement of the
benchmark will differ depending on the size of the MIMO diversity system
(total number of antennas).
In Fig. 2, AFcon in (14) is depicted as a function of the power
correlation coefficient ρ=ρt=ρrfor both SIMO, multiple-
input single-output (MISO), and MIMO diversity systems. By
comparing the various panels in the figure, it can be seen that
AFcon is progressively reduced when the number of anten-
nas is increased either at the transmitter or the receiver. As
expected, the reduction is largest for uncorrelated antennas.
In Fig. 3 and Fig. 4, the approximate average SER in (30)
is compared to the exact result in (33) for a 3 ×3MIMO
diversity system and a BPSK modulation scheme. In Fig. 3
(top panel), it can be seen that approximate average SER curve
for correlated channels at high SNR can be obtained as a
(right) shifted version of the approximate average SER curve
for uncorrelated channels (benchmark). This is visualized for
ρr=0.8. In the bottom panel, ∆Gcis depicted as a function of
both ρr(lower xaxis) and AFcon (upper xaxis). Using the
lower xaxis as reference, ρr=0.8amounts to a right shift
of −5 dB from the benchmark curve, which is in agreement
with the observed difference of the curves in the top panel.
According to the upper xaxis, the AF is reduced by 71%,
compared to a single Rayleigh fading channel, when ρr=0.8
(and ρt=0). For uncorrelated channels (ρt=ρr=0),a3×3
MIMO diversity system reduces the AF by 89%, compared to
a single Rayleigh fading channel, which is equivalent to ninth-
order diversity.
HOLTER AND ØIEN: ON THE AMOUNT OF FADING IN MIMO DIVERSITY SYSTEMS 2505
Fig. 4. Top panel: Exact average SER (dashed lines) and approximate average SER at high SNR (solid lines) for a 3 ×3 MIMO diversity system operating on
identically distributed Rayleigh fading channels (constant correlation models at each end of the MIMO link). A BPSK modulation scheme is utilized. Bottom
panel: Relative (right) shift (in decibel) of the SER benchmark curve presented in the top panel as a function of ρrwhen ρt=0.5. Using the top xaxis as
reference, the AF realized by the 3 ×3 MIMO diversity system is compared (in percentage) to the AF of a single Rayleigh fading channel (100% reduction
represents the nonfading AWGN channel).
In Fig. 4 (top panel), the same set of curves as depicted
in Fig. 3 (top panel) are presented, but this time, when ρt=
0.5. Once again, excellent agreement between the exact and
approximate curves at high SNR is observed. From Fig. 4
(bottom panel), it can be seen that ∆Gchas increased from −5
to −7.3 dB when ρr=0.8, due to ρt=0.5at the transmitter.
This corresponds to a 42% reduction in the AF compared to a
single Rayleigh fading channel.
VI. CONCLUSION
A closed-form expression for the AF in a multiple-input
multiple-output (MIMO) diversity system operating on identi-
cally distributed spatially correlated Nakagami-mfading chan-
nels has been presented. For the Kronecker model, the amount
of fading (AF) has been presented for identically distributed
Rayleigh fading channels and different correlation models.
Capitalizing on recent results in [1], it has been shown that the
approximate average symbol error rate (SER) at high signal-
to-noise ratio (SNR) for an MIMO diversity system based on
the Kronecker model can be expressed as a function of the AF,
when a constant correlation model is assumed.
APPENDIX A
PROOF OF THE AMOUNT OF FADING
EXPRESSION IN (11)
Using a mathematical model assuming that the transmit
and receive correlation properties are decoupled in an MIMO
diversity system operating on identically distributed spatially
correlated Nakagami-mchannels, the AF may be expressed as
AF = nT
j=1 tj2nR
i=1 ri2
m·N2(36)
where ·
2denotes the squared Euclidean vector norm, mis
the common fading parameter of all the channels, nTdenotes
the number of transmit antennas, nRdenotes the number of
∆Gc=10·nR·log [betacdf(AFTx )] + nT·log [betacdf (AFRx)]
N(35)
2506 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 5, SEPTEMBER 2005
receive antennas, and N=nT·nR. The vectors tjand ri
denote rows jand iof the transmit and receive correlation
matrices, respectively.
Proof: Let Λrepresent the diagonal eigenvalue matrix
of the complex correlation matrix RH. Assuming decoupled
correlation properties at the receiver and transmitter, the cor-
relation matrix RHmay be written as RH=RTx ⊗RRx,
where the symbol ⊗denotes the Kronecker product, and the
matrices RTx and RRx denote the decoupled transmit and
receive correlation matrices, respectively. Due to the multiplica-
tive properties of the eigenvalues of matrices in a Kronecker
product [11, Th. 9.1], the nominator in (10) may be decomposed
into the product tr(Λ2
Tx)tr(Λ2
Rx), where ΛTx and ΛRx are the
diagonal eigenvalue matrices of RTx and RRx, respectively.
The following chain of equalities can then be obtained for the
transmitter part
tr Λ2
Tx=trR2
Tx=
nT
j=1
(RTxRTx )jj =
nT
j=1
nT
k=1
tjk ·tkj
(i)
=
nT
j=1
nT
k=1
tjk ·t∗
jk =
nT
j=1 tj2(37)
where tjk denotes a single entry at the jth row and kth column
of RTx, and tjdenotes the jth row of RTx. The equality (i)
is true, since a correlation matrix is Hermitian symmetric. A
similar chain of equalities can be obtained for the receiver part,
resulting in
tr Λ2
Rx=trR2
Rx=
nR
i=1 ri2(38)
where ridenotes the ith row of RRx.
APPENDIX B
ANALTERNATIVE EXPRESSION FOR THE DETERMINANT
OF A CONSTANT CORRELATION MAT R I X
A constant correlation matrix Rof size L×Lis called an
Lth order intraclass correlation matrix if it has the following
structure [4]
R=
abb··· b
bab··· b
bba··· b
.
.
..
.
..
.
.....
.
.
bbb··· a
(39)
with b≥−a/(L−1). By normalizing this matrix, 1s are ob-
tained on the main diagonal and the factor b/a,offthemain
diagonal. Denoting x=b/a, and assuming that x∈[0,1],a
closed-form expression for the determinant of Rmay be written
as [27]
det(R)=(1−x)L−1(1 + x(L−1)) .(40)
Due to the normalized main diagonal, and the fact that the
variable xis confined to the finite interval range x∈[0,1],
we have observed (depicted in Fig. 1) that the cdf of a beta-
distributed RV can be used in an alternative expression for the
determinant given in (40). The alternative expression may be
derived as follows.
The pdf of a beta-distributed RV with free parameters α>0
and β>0is given by [28]
betapdf (x)= Γ(α+β)
Γ(α)Γ(β)(1 −x)β−1xα−1.(41)
The cdf can then be expressed as
betacdf(x;α, β)=
x
0
Γ(α+β)
Γ(α)Γ(β)(1 −u)β−1uα−1du. (42)
Evaluating the function f(x, α, β)=1−betacdf(x;α, β)
when α=2, the following result is obtained
f(x, 2,β)=1−betacdf(x;2,β)
=1−
x
0
Γ(β+2)
Γ(β)(1 −u)β−1udu
=1−Γ(β+2)
Γ(β)1−(1 −x)β(1 + βx)
β(β+1)
=(1−x)β(1 + βx).(43)
By comparison, (43) and (40) represent identical expressions
by selecting β=L−1. Hence, the determinant of a constant
correlation matrix Rcan be written as
det(R)=1−betacdf(x;2,L−1).(44)
Since the parameter βmust be larger than zero, this expression
is valid only when L≥2. The cdf of a beta-distributed RV
is equal to the regularized beta function I(·;·,·),10 and the
determinant can also be written as
det(R)=1−betacdf(x;2,L−1)
=1−I(x;2,L −1)
=I(1 −x;L−1,2) (45)
where I(x;α, β)=1−I(1 −x;β,α).13 Hence
det(R) = betacdf(1 −x;L−1,2).(46)
13http://functions.wolfram.com
HOLTER AND ØIEN: ON THE AMOUNT OF FADING IN MIMO DIVERSITY SYSTEMS 2507
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Bengt Holter (S’01) was born in Lillehammer, Nor-
way, on October 21, 1971. He received the B.Sc.
and M.Sc. degrees in electrical engineering from the
Oslo College of Engineering, Oslo, Norway, and the
Norwegian University of Science and Technology
(NTNU), Trondheim, Norway, in 1995 and 1997,
respectively, and is currently working toward the
Ph.D. degree in electrical engineering at NTNU.
Since January 1998, he has been a Research
Scientist at The Foundation for Scientific and In-
dustrial Research Information and Communication
Technology (SINTEF ICT), Trondheim, Norway. His current research interests
are in the areas of spatial and multiuser diversity techniques for wireless
communications.
Geir Egil Øien (S’89–M’01) was born in Trond-
heim, Norway, on 1965. He received the M.Sc. and
Ph.D. degrees from the Norwegian Institute of Tech-
nology (NTH), Trondheim, Norway, in 1989 and
1993, respectively.
From 1994 to 1996, he was an Associate Professor
at Stavanger University College, Stavanger, Norway.
In 1996, he joined The Norwegian University of
Science and Technology, Trondheim, Norway, as an
Associate Professor and in 2001 was promoted to
Full Professor. His current research interests include
the analysis and design of bandwidth-efficient channel coding and modula-
tion schemes for fading channels; wireless channel analysis, estimation, and
prediction; analysis and characterization of spatial diversity and multiple-
input-multiple-output (MIMO) systems; and orthogonal frequency division
multiplexing (OFDM)-based wireless systems.
