Communications and Network, 2009, 25-34
doi:10.4236/cn.2009.11004 Published Online August 2009 (http://www.scirp.org/journal/cn)
Copyright © 2009 SciRes CN
Approximate Analysis of Power Offset over Spatially
Correlated MIMO Channels
Guangwei YU, Xuzhen WANG
School of Information and Communication Engineering, Beijing University of Posts
and Telecommunications, Beijing, China
Email: {yuguangwei1982, Gxitao}@gmail.com
Abstract: Power offset is zero-order term in the capacity versus signal-to-noise ratio curve. In this paper, ap-
proximate analysis of power offset is presented to describe MIMO system with uniform linear antenna arrays
of fixed length. It is assumed that the number of receive antenna is larger than that of transmit antenna. Spa-
tially Correlated MIMO Channel is approximated by tri-diagonal toeplitz matrix. The determinant of tri-di-
agonal toeplitz matrix, which is fitted by elementary curve, is one of the key factors related to power offset.
Based on the curve fitting, the determinant of tri-diagonal toeplitz matrix is mathematically tractable. Conse-
quently, the expression of local extreme points can be derived to optimize power offset. The simulation re-
sults show that approximation above is accurate in local extreme points of power offset. The proposed ex-
pression of local extreme points is helpful to approach optimal power offset.
Keywords: MIMO, multiplexing gain, power offset, high snr region, toeplitz matrix
1. Introduction
Multiple-input multiple-output (MIMO) system is widely
used in wireless communication to improve the perform-
ance. The spectral efficiency of MIMO channel is much
higher than that over the conventional signal antenna
channel. The research of MIMO includes two different
perspectives: the first one concerns performance analysis
in terms of error performance of practical systems, the
second one concerns the study of channel capacity.
The design of communication schemes was mainly
considered in the former perspectives with the aid of theo-
retical analysis and simulation [1][2], and [3]. For the
latter, important parameters such as diversity gain [4],
multiplexing gain [5][6] (referred to degree of freedom in
other literatures), and power offset [7-9], and [10] were
emphatically analyzed in the high signal-to-noise ratio
(SNR) region. Furthermore, diversity and multiplexing
tradeoff has been proposed in [11][12], and [13], that is to
say diversity and multiplexing tradeoff can be obtained
for a given multiple antenna channel. It is worth pointing
out in [11] that the diversity and multiplexing tradeoff is
essentially the tradeoff between the error probability and
the data rate of a given system.
The multiplexing gain is not sufficient to accurately
characterize the property of MIMO capacity. A more ac-
curate representation of high SNR behavior in SNR-ca-
pacity curve is provided by an affine approximation to
capacity, which includes both the multiplexing gain (i.e.
slope) and power offset (i.e. zero-order term) [7]. High
SNR power offset has been analyzed in [8] over multiple
antenna Ricean channels. It was shown in [8] that the im-
pact of the Ricean factor at high SNR region could be
conveniently quantified through the corresponding power
offset. In [9], high SNR power offset in multiple antenna
communication was derived in detail. Achievable through-
put was compared between the optimal strategy of dirty
paper coding and suboptimal linear precoding techniques
(zero-forcing and block diagonalization) in [10] on appli-
cation of power offset. Hence, power offset is an impor-
tant parameter in multiple antenna communication.
In many practical environments, signal correlation
among the antenna array exists due to the scattering. Fad-
ing correlation and its effect on the capacity of multiele-
ment antenna system has been studied in [14]. Hence,
analysis of spatially correlated MIMO channels has been
another topic in the past few years, which necessitates the
model of correlated channel. A general space-time corre-
lation model for MIMO systems in mobile fading channel
has been presented in [15]. The model in [15] was flexible
and mathematically tractable. In [16], the correlated
model of [15] has been used to investigate the capacity of
spatially correlated MIMO Rayleigh-fading channels.
Recently, application of different antenna arrays in
MIMO system has also been exploited. With the consid-
eration of physical constraints imposed by maximum size
of the antenna array, uniform linear array of fixed length
has been used in [17] to analyze the asymptotic capacity
of MIMO systems.
The uniform linear array of fixed length is also used in
this paper to analyze the power offset of MIMO system
over spatially correlated channel. On application of curve
fitting, the determinant of tri-diagonal toeplitz matrix is
mathematically tractable. Over spatially correlated chan-
26 APPROXIMATE ANALYSIS OF POWER OFFSET OVER SPATIALLY CORRELATED MIMO CHANNELS
nel, the main result of this paper is the derived expression
of local extreme points of power offset, using the uni-
form linear array of fixed length. To the best of our
knowledge, the main result has not been presented in
other literatures.
The rest parts are organized as follows. In Section 2, the
basic definitions of multiplexing gain and power offset are
presented. System model and correlation model are given
in Section 3. Approximate analysis of power offset by
fitting determinant curve of tri-diagonal toeplitz matrix is
put forward in Section 4. Section 5 shows the simulation
results. Finally, a brief conclusion is given in Section 6.
Notation: In the following context, matrices and vectors
are denoted by boldface upper case symbols and boldface
lower case symbols, respectively. The transpose and
Hermitian transpose are denoted by and


T
H
, re-
spectively. The expected value is represented by
E.
is used for the Euclidian norm.
2. Basic Definition
In the high SNR region, the capacity of single-user
MIMO system of coherent reception is given by [18][19]
2
()min(,)log (1
TR
CSNRn nSNRo) (1)
where denote the number of transmit and receive
antenna elements , respectively. is the signal to
noise ratio. The MIMO capacity in high SNR region is a
linear function of , i.e. the stationary slope in
SNR-capacity curve. In other words, any increase in
is immaterial, i.e. without any impact to the
slope. The slope is the so-called maximum multiplexing
gain (degree of freedom).
,
TR
nn
, )
TR
nn
SNR
min( ,)
TR
nn
max (
If is fixed to a given value (generally speaking,
is less than six, expressed as
T
n
T
n
T
n in the following
context), and
R
T
nn, the multiplexing gain equals to a
fixed value without consideration of correlation of chan-
nel response and the number of receive antenna elements.
So power offset is introduced to compare the impact of
channel property with the same value of multiplexing
gain, and (1) is replaced by [9]
2
()(log() )(1
(1)
3
dB
C SNRSSNRLo
SNR
SLo
dB




 


)
(2)
where is multiplexing gain and is power offset
in 3-dB units. When the multiplexing gain
and power offset can be computed by
SL
SNR 
2
()
lim log ()
SNR
CSNR
SSNR

(3)
2
()
limlog ()
SNR
CSNR
LSNR
S





(4)
In the high SNR region, SNR-capacity curve is ap-
proximately determined by multiplexing gain and power
offset. Most channels, having the same multiplexing gain,
may have very different capacities because of various
values of the power offset. Hence, power offset is an im-
portant parameter to describe the capacity behavior in
high SNR region.
3. System Model and Correlation Mode
The system model and channel correlation model are de-
fined in this section. For MIMO system, the general
baseband model is given by
yHxn
(5)
where is the channel response matrix, and
is the input complex signal vector
whose spatial covariance matrix normalized by the energy
per dimension can be expressed as
H
12
[
T
T
n
xx xx]
2
()
1[]
H
T
E
E
n
xx
Φ
x
(6)
while and are the
output vector and additive white Gaussian noise vector,
respectively. Because of normalization and assumption of
isotropic input,
12
[]
R
T
n
yy yy
()tr
12
[
R
T
n
nn n n]
T
n
Φ and , where is
T
n
ΦIn
I
nn
identity matrix.
Kronecker model [16] is used to describe the correla-
tion of channel response. On the assumption of rich scat-
tering environments and having no line of sight, the cor-
related channel response matrix is denoted by
H
TR
ΣΣΣ (7)
where
is Kronecker product, and
T
Σ
R
Σ are cor-
related matrix of transmit and receive antenna, respec-
tively. So the channel response matrix is given by
H
1
2
RT
HΣHΣ1
2
(8)
where is channel response matrix whose elements are
H
independent and identical distribution random variables.
Without loss of generality, we assume that the distance
between transmit antenna elements is large enough to
neglect the correlation at the transmit node. Consequently,
without considering the antenna correlation at the transmit
node, the correlation matrix is identity matrix, and
the receive antenna correlation model
T
Σ
R
Σ is given by
[15][16].
Copyright © 2009 SciRes CN
APPROXIMATE ANALYSIS OF POWER OFFSET OVER SPATIALLY CORRELATED MIMO CHANNELS 27
Copyright © 2009 SciRes CN
222
0
0
44sin()
(, )()
ij ij
R
I
dj d
ij I
 

Σ (9) 02,
1
R
R
R
L
J
n
ap
æö
÷
ç÷
ç÷
ç÷
ç-
èø
 10
L
£ (12)
where
R
L is the fixed length of receive antenna array
normalized by
. In this paper,
R
L is assumed to be in
the range from 1 to 10.
where [0, )

[,)
is the so-called AOA(angle of arrival),

 is average value of AOA, ij
ds normalized
distance between antenna array elements, that is to say
i
ij
ij
dd
he factual distance between i and j
antenna array element,
,
ij
d
In the Figure 1 and 2, power offset approximation
based on tri-diagonal, five-diagonal, and seven-diagonal
toeplitz matrix is obtained versus the number of receive
antenna with various value of
R
L. Furthermore, power
offset over independent channel is also provided as lower
bound for numerical comparison. It can be seen from
Figure 1 and 2 that tri-diagonal toeplitz matrix can be
used to approximate the correlation model at the receive
node.
is t
s wave length, i()
n
I
is
n-order modified Bessel function. It is obvious that
is real symmetric toeplitz matrix.
(, )
RijΣ
On the assumption of isotropic scattering, the correla-
tion matrix at the receive node is simplified to
0
(, )(2)
R
ij
ij JdΣ
(10) 4. Approximate Analysis of Power Offset
where is n order Bessel function, and
can be simplified to
()
n
J(,)
RijΣ
tri-diagonal toeplitz matrix through
the simulation results illustrated by Figure 1 and Figure
2.
Depending on the discussion in Section 3, tri-diagonal
toeplitz matrix can be used to approximate the correla-
tion model at the receive node. In this section, the deter-
minant curve of tri-diagonal toeplitz matrix is fitted by an
elementary function, which simplifies the approximate
analysis of power offset.
1
R
R
R
nn
Σ





 






Expression of power offset for MIMO system over corre-
lated channel is derived in [9], and the expression is given by
(11)
2
2
1
log(ln det())
ln 2
1log det()
H
TR
T
T
T
Ln E
n
n


WΛW
ΛP
(13)
where
2468 10 12 14 16
-10
-8
-6
-4
-2
0
2
4
number of Nr
power offset/dB
power offset LR=5 Nt=2
det erminant
tri-diagonal app.
five-diagonal app.
seven-diagonal app.
no correlation
Figure 1. Approximate analysis of power offset based on 3, 5, and 7 diagonal toeplitz matrix with , 5
R
L
2
T
n
28 APPROXIMATE ANALYSIS OF POWER OFFSET OVER SPATIALLY CORRELATED MIMO CHANNELS
24 6 81012 14 16 1820
-10
-8
-6
-4
-2
0
2
4
number of Nr
power offset/dB
power offset LR=6 Nt=2
determinant
tri-diagonal app.
five-diagonal app.
seven-diagonal app.
no correlation
Figure 2. Approximate analysis of power offset based on 3, 5, and 7 diagonal toeplitz matrix with , 6
R
L
2
T
n
where and
T
Λ
R
Λ are diagonal matrix whose ele-
ments are eigenvalues of and
T
Σ
R
Σ
Φ
)
T
P
, respectively.
is also diagonal matrix whose elements are eigenvalues of
normalized spatial covariance matrix , so is iden-
tity matrix and is constant zero.
P
P
1/n2
g de
T
lo t(Λ
R
Λcan be derived from (11). It is obvious that
R
Λ is
full rank matrix, and then (13) is simplified to
2
log(() ())
TRR
Lnfngn
 (14)
where ()
R
f
n and ()
R
g
n can be described by
(lndet())
()
ln 2
H
R
T
E
fn
n
WW
 (15)
ln det()
()
ln 2
R
R
T
gn
n
 Λ (16)
Because is wishart matrix and nonsingular
with probability 1,
H
WW
()
R
f
n is reduced to [9]
1
0
1
()(log det())
ln 2
1()
ln 2
T
H
Re
T
n
R
l
T
fn E
n
nl
n

 
WW
(17)
where ()
is digamma function and the definition of
digamma function is
1
1
1
()
n
l
nl

  (18)
where
is Euler-Mascheroni constant, and
1
1
limlog 0.5772
n
e
nl
n
l





(19)
When it comes to function ()
R
g
n, the derivation of
()
R
g
n can be expressed as
 
11
() lndet() lndet()
ln 2ln 2
R
RR
TT
gn
nn
 ΛΣ (20)
It can be found from (20) that ()
R
g
n is only deter-
mined by the determinant of
R
Σ, and the determinant of
R
norder tri-d iagonal toeplitz matrix
R
Σ can be com-
puted by [20].


11
22
12
2
2
114 114
214
14 0
det
1
2
14 0
RR
R
R
nn
n
R
R
n
n


 



Σ (21)
With the aid of numerical computation and mathemati-
cal analysis, the curve of the determinant of tri-diagonal
toeplitz matrix can be fit by some tractable elementary
Copyright © 2009 SciRes CN
APPROXIMATE ANALYSIS OF POWER OFFSET OVER SPATIALLY CORRELATED MIMO CHANNELS 29
function. When in (21), det
2
140

R
Σcan be ap-
proximated by the fitted curve as follow:
22
0
()1515 (2)
1
R
R
R
L
hnJ n

  (22)
The approximation between the fitted curves and de-
terminant curves of tri-diagonal toeplitz matrix are illus-
trated in Figure 3 and Figure 4. From Figure 3 and 4, we
can find that the fitting curve can exactly approach the
extreme points of the determinant of tri-diagonal toeplitz
matrix. However, the expression is much simpler. In other
words, the further derivation using the fitted curve is
tractable using well-known mathematical software such
as WOLFRAM MATHEMATICA.
In the following context, power offset is analyzed in
detail. The approximate analysis of power offset is based
on the piecewise function of
R
n. The condition
in (22) is necessary due to , so
the number of receiver antenna
2
15 0
 det 0
R
Σ
R
n can not be large than
. Moreover, it shows from the conclusion in [21]
that the effect of correlation can be neglected when the
normalized length of receive antenna array
41
R
L
R
L
1
RR
nL
is larger
than one wave length, that is to say . Hence,
the considered range of the number of receive antenna is
.
1
RR
Ln 4
R
L 1
With the discussion above, when , the
normalized length of receive antenna array
1
RR
nL
R
L is large
than one wave length and the effect of correlation can be
neglected, that is to say and the power offset
is expressed as
()0
R
gn
1
2
0
1
log( )
ln 2
T
n
T
l
T
Ln n
n
R
l
1
(23)
When it comes to the range of , the
normalized length of receive antenna array
14
RRR
LnL 
R
L
()0
R
gn
is smaller
than one wave length and the effect of correlation can not
be neglected. In other words, . However, for a
given value of
T
n, the first-order derivative (()
R
f
n
()
is
assumed to be continuous) of
R
f
n is obtained by
'
1
0
1
1
0
() 1()
ln 2
1()
ln 2
T
T
n
R
R
l
RT
n
R
l
T
df nnl
dn n
nl
n

 



0

(24)
where 1()
is trigamma function. Hence, it shown
from (24) that ()
R
f
n approximately equal to the con-
stant (approximation accuracy is related to the value
of
C
T
n). The power offset is expressed as
2
log( )
TR
Lngn
C
 (25)
2 4 6 81012 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
number of Nr
determinant
curve fitted of determinant of tri-diagonal toeplitz matrix LR=5
determinant
curve fitted
Figure 3. Approximation of determinant of tri-diagonal toeplitz matrix by curve fitting and 5
R
L
C
opyright © 2009 SciRes CN
30 APPROXIMATE ANALYSIS OF POWER OFFSET OVER SPATIALLY CORRELATED MIMO CHANNELS
051015 20 25
0
0. 1
0. 2
0. 3
0. 4
0. 5
0. 6
0. 7
0. 8
0. 9
1
number of Nr
determinant
curve fitted of determinant of tri-diagonal toeplitz matrix LR=6
determinan t
curve fitted
Figure 4. Approximation of determinant of tri-diagonal toeplitz matrix by curve fitting and
6
R
L
Without loss of generality, the constant can be
neglected in the approximate computation. Hence, the
power offset in (25) is only determined by
C
()
R
g
n
2
. From
(22), the curve of the determinant of tri-diagonal toeplitz
matrix can be fitted by ()1
R
hn 5
 , where
0
2
(
1)
R
R
L
Jn
. The first order derivative (()
R
hn is as-
sumed to be continuous) of ()
R
hn can be computed by
01
2
22
4()(
()1 1
(1)
RR
R
RR
RR
LL
LJ J
dh nnn
dn n

)
R
 (26)
where , that is to say
14
RRR
LnL 
And then, the second-order derivative of ()
R
hn can
be computed by
01
2
23
22 2
1
4
22
00 2
4
22
8()()
()1 1
(1)
2
8()
1
(1)
22 2
4()()(
11
(1)
RR
R
RRR
RR
R
R
R
R
RR
R
RR R
R
LL
LJ J
dhnn n
dn n
L
LJn
n
LL L
LJ JJ
nn n
n

 




 


)
1
R
3
1
22
21
R
L
(27)
Using (27), the second-order derivative in points
,1,2,
i
Pi
  is obtained as follow:
R
n

. In the range above, function
0
2
()
1
R
R
Jn
L in (26) has two zero points, i.e.
1
22.4048
1
R
L

2
122
22.4048
1
( )4.51
:0
R
R
L
RR
n
dhn
Pdn L
 
2
(28)
R
Pn
and 2
25.5201
1
R
R
L
Pn

, and
function 1
2
()
1
R
R
L
Jn
only has one zero point, i.e.
3
23.8317
1
R

2
222
25.5201
1
() 53.75
:0
R
R
L
RR
n
dhn
Pdn L
 
2
(29)
R
L
Pn
. Consequently, ()
R
hn has three
extreme points in the range of 2
21
R
R
L
n
2
322
23.8317
1
() 17.48
:0
R
R
L
RR
n
dhn
Pdn L

2
(30)
2
3
, de-
scribed by .
,1
i
Pi ,2,
With the discussion above, it is obvious that point
and are local maximum points of
1
P
2
P()
R
hn and point
Copyright © 2009 SciRes CN
APPROXIMATE ANALYSIS OF POWER OFFSET OVER SPATIALLY CORRELATED MIMO CHANNELS 31
3
P is local minimum point. Furthermore, it is obvious
from (20) that the extreme points of ()
R
hn is same as
ones of ()
R
g
n. Consequently, in the range of 1
R
L
t
, the extreme points of power offset in (25)
can be approximately computed by
41
R
L
arg mi
arg ma
R
R
n
n


P
R
n
R
n


n
x
,for local minit
,for local maxiin
L
L


mal poin
mum po
(31)
where is defined as follows:
22
1,1 ,1,2,3=
RR
ii
LL
i
PP
pp
ìü
êúéù
ïï
ïï
êúêú
+ +
íý
êúêú
ïï
ïï
ëûêú
îþ
P= (32)
where
x
êú
ëû
and
x
éù
êú denote the maximum integer
smaller than
x
and minimum integer larger than x, re-
spectively. is the extreme points com-
puted above. (31) can be used to approximately compute
the number of receive antenna, in order to achieve the
optimal power offset.
,1,2,3
iPi
5. Simulation Results
In the Figure 1 and 2, with the assumption of
2
T
n
,
power offset is approximately analyzed based on tri-di-
agonal, five-diagonal, and seven-diagonal toeplitz versus
the number of receive antenna, with in Figure 1
and in Figure 2. Furthermore, power offset in
independent channel is also provided as lower bound for
numerical comparison. The approximation trend can be
5
R
L
6
R
L
found from Figure 1 and 2. The tri-diagonal toeplitz ma-
trix can be assumably used to characterize the property of
power offset over spatially correlated channel. Hence,
tri-diagonal toeplitz matrix can be used to approximate
the correlation model at the receive node.
The approximation of fitting the curve of tri-diagonal
toeplitz matrix determinant is illustrated in Figure 3 and
Figure 4 to compare the difference between fitted curves
and determinant curves, with in Figure 3 and
5
R
L
6
R
L
in Figure 4. From Figure 3 and 4, we can find
that the fitting curve can exactly approach the extreme
points of the determinant of tri-diagonal toeplitz matrix.
However, the expression is mathematically tractable.
With the same assumption of
2
T
n, approximation
of power offset is shown in Figure 5 and 6, with 5
R
L
in Figure 5 and 6
R
L
in Figure 6. From Figure 5 and
6, we can find the position of two local minimal points
and one local maximum point. With the condition of
5
R
L
and
T
n2
, the local extreme points computed
by (31) is 9
R
n
for local maximum point and
7,
R
n15

6
for local minimal points. With the condition
of R
L
and
2
T
n
, the local extreme points com-
puted by (31) is 11
R
n
for local maximum point and
8,
R
n16
 for local minimal points. The computation
results based on (31) coincide with the illustration results
in Figure 5 and 6. Hence, all the approximation in this
paper is reasonable, including approximation of correla-
tion model and approximation of determinant of tri-di-
agonal toeplitz matrix.
24 6 81012 1416 18 20
-10
-8
-6
-4
-2
0
2
4
number of Nr
power offset/dB
er offsetLR=5 Nt=2
pow
determ inant
curve fitted
no correlation
Figure 5. Power offset comparation between accurate and approximate determinant of toeplitz matrix with , 5
R
L
2
T
n
C
opyright © 2009 SciRes CN
32 APPROXIMATE ANALYSIS OF POWER OFFSET OVER SPATIALLY CORRELATED MIMO CHANNELS
05 10 15 20 25
-12
-10
-8
-6
-4
-2
0
2
4
number of Nr
power offset/dB
power offsetLR=6 Nt=2
det erminant
curve fitted
no correlation
Figure 6. Power offset comparation between accurate and approximate determinant of toeplitz matrix with ,
6
R
L
2
T
n
Similar results are presented in Figure 7 and 8, with
the assumption of and in Figure 7
while in Figure 8. With the condition of
4
T
n5
R
L
8
R
L5
R
L
and
4
T
9
n
R
, the local extreme points computed by (31)
is for local maximum point and for
local minimal points. With the condition of
n7,
R
R
L
15
8
n
and
, the local extreme points computed by (31) is
for local maximum point and for
local minimal points. The computation results based on (31)
coincide with the illustration results in Figure 7 and 8.
T
n
n
4
14
R10,22
R
n
Consequently, it is presented in Figure 5 and 7 that the
effect caused by the number
T
n can be neglected. In
other words, for small value of
T
n and in the range of
number of receive antenna , the
assumption of
14
RRR
LnL 1
()
R
f
nC
is tenable with neglectable
loss of precision. The extreme points of power offset is
only determined by ()
R
g
n.
468 1012 141618
-12
-10
-8
-6
-4
-2
0
2
4
6
8
Nr
power offset/dB
power offset LR=5 Nt=4
determ i nant
curve fitted
no correlation
Figure 7. Power offset comparation between accurate and approximate determinant of toeplitz matrix with ,
5
R
L
4
T
n
Copyright © 2009 SciRes CN
APPROXIMATE ANALYSIS OF POWER OFFSET OVER SPATIALLY CORRELATED MIMO CHANNELS 33
510 1520 25 30
-12
-10
-8
-6
-4
-2
0
2
4
6
8
Nr
power offset/dB
power offset LR=8 Nt=4
determin ant
curve fitted
no correlation
Figure 8. Power offset comparation between accurate and approximate determinant of toeplitz matrix with , 8
R
L
4
T
n
6. Conclusions
Approximate analysis of power offset over spatially cor-
related channel is proposed to optimize the number of
receive antenna with antenna array of fixed length. An-
tenna correlation matrix is approximated by tri-diagonal
toeplitz matrix. The determinant of tri-diagonal toeplitz
matrix is reduced to simple style by curve fitting with
neglectable loss of precision. Based on the approxima-
tion above, the expression of local extreme points of
power offset is derived. Simulation results shows that the
derived expression is accurate in the local extreme points
of power offset, that is to say, all the approximation in
this paper is reasonable.
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