29
2
Theory and Practice of MIMO Wireless
Communication Systems
Dimitra Zarbouti, George Tsoulos, and Dimitra Kaklamani
CONTENTS
Summary................................................................................................................30
2.1 Shannon’s Capacity Formula.....................................................................30
2.2 Extended Capacity Formula for MIMO Channels.................................31
2.2.1 General Capacity Formula.............................................................31
2.2.2 Transformation of the MIMO Channel into n
SISO
Subchannels......................................................................................32
2.2.3 No CSI at the Transmitter..............................................................34
2.2.4 CSI at the Transmitter.....................................................................35
2.2.5 Channel Estimation at the Transmitter........................................35
2.3 Remarks on the Extended Shannon Capacity Formula........................36
2.3.1 Bounds on MIMO Capacity..........................................................36
2.3.2 Capacity of Orthogonal Channels................................................38
2.3.3 Effective Degrees of Freedom.......................................................38
2.4 Capacity of SIMO — MISO Channels......................................................39
2.5 Stochastic Channels.....................................................................................40
2.5.1 Ergodic Capacity.............................................................................40
2.5.2 Outage Capacity..............................................................................41
2.6 MIMO Capacity with Rice and Rayleigh Channels..............................41
2.6.1 MIMO Channel Matrix for Rayleigh Propagation
Conditions........................................................................................42
2.6.2 MIMO Channel Matrix for Ricean Propagation Conditions...43
2.6.3 Channel Matrix with Spatial Fading Correlation......................45
2.7 Simulations...................................................................................................47
2.7.1 MIMO Capacity for a Rayleigh Channel without Spatial
Fading Correlation..........................................................................48
2.7.2 MIMO Capacity for a Rayleigh Channel with Spatial
Fading Correlation..........................................................................49
2.7.3 MIMO Capacity for a Ricean Channel........................................50
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30 MIMO System Technology for Wireless Communications
Appendix 2A.........................................................................................................51
Appendix 2B..........................................................................................................52
Appendix 2C..........................................................................................................53
References...............................................................................................................54
Summary
This chapter introduces the principles of MIMO systems employing the
necessary mathematical analysis to consider the achieved capacity perfor
mance. In this context, ﬂat fading across time and frequency is considered,
and the Rayleigh model is employed for describing the wireless channel.
Furthermore, the case of spatial selective fading is examined by considering
LOS propagation, with the Ricean model.
The mathematical representation of the MIMO system is performed
through a complex matrix, which depends on the scenario considered each
time (i.e., ﬂat or selective spatial fading). The capacity achieved by the MIMO
channel in all the above cases is studied with the use of the Shannon extended
capacity formula. The capacity performance results, developed from the
simulations performed, are related to the number of the multiple antenna
elements that the Rx and the Tx are equipped with, the distance between
them and the degree of correlation evidenced.
2.1 Shannon’s Capacity Formula
Shannon’s capacity formula approximated theoretically the maximum
achievable transmission rate for a given channel with bandwidth B, trans
mitted signal power P and single side noise spectrum N
o
, based on the
assumption that the channel is white Gaussian (i.e., fading and interference
effects are not considered explicitly).
(2.1)
In practice, this is considered to be a SISO scenario (single input, single
output) and Equation 2.1 gives an upper limit for the achieved errorfree
SISO transmission rate. If the transmission rate is less than C
bits/sec (bps),
then an appropriate coding scheme exists that could lead to reliable and
errorfree transmission. On the contrary, if the transmission rate is more than
C
bps, then the received signal, regardless of the robustness of the employed
code, will involve bit errors.
C B
P
N B
o
log
2
1
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Theory and Practice of MIMO Wireless Communication Systems
31
2.2 Extended Capacity Formula for MIMO Channels
For the case of multiple antennas at both the receiver and the transmitter
ends (Figure 2.1), the channel exhibits multiple inputs and multiple outputs
and its capacity can be estimated by the extended Shannon’s capacity for
mula, as described below.
2.2.1 General Capacity Formula
We consider an antenna array with n
t
elements at the transmitter and an
antenna array with n
r
elements at the receiver. The impulse response of the
channel between the j
th transmitter element and the i
th receiver element is
denoted as h
i,j
(
,t
). The MIMO channel can then be described by the n
r
n
t
H
(
,t
) matrix:
(2.2)
The matrix elements are complex numbers that correspond to the attenu
ation and phase shift that the wireless channel introduces to the signal
reaching the receiver with delay
. The inputoutput notation of the MIMO
system can now be expressed by the following equation:
(2.3)
where denotes convolution, s
(t
) is a n
t
1 vector corresponding to the n
t
transmitted signals, y
(t
) is a n
r
1 vector corresponding to the n
r
received
signals and u
(t
) is the additive white noise.
FIGURE 2.1
The MIMO channel.
Channel
H
Tx
n
t
n
r
Rx
H(,)
(,) (,) (,)
(,
,,,
,
t
h t h t h t
h
n
t
1 1 1 2 1
2 1
t
t h t h t
h t h
n
n n
t
r r
) (,) (,)
(,) (
,,
,,
2 2 2
1 2
,) (,)
,
t h t
M n
R t
y H s u( ) (,) ( ) ( )t t t t
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32 MIMO System Technology for Wireless Communications
If we assume that the transmitted signal bandwidth is narrow enough that
the channel response can be treated as ﬂat across frequency, then the discrete
time description corresponding to Equation 2.3 is
(2.4)
The capacity of a MIMO channel was proved in [1, 4] that can be estimated
by the following equation:
(2.5)
where H
is the n
r
n
t
channel matrix,is the covariance matrix of the
transmitted vector s, H
H
is the transpose conjugate of the H
matrix and p
is
the maximum normalized transmit power. Equation 2.5 is the result of
extended theoretical calculations, and its practical use is not obvious. Never
theless, we can perform linear transformations at both the transmitter and
receiver converting the MIMO channel to SISO subchannels
(given that the channel is linear) and, hence, reach more insightful results.
These transformations can be found in [1] and are brieﬂy described in the
following section.
2.2.2 Transformation of the MIMO Channel into n SISO Subchannels
Every matrix can be decomposed accordingly to its singular val
ues. Suppose that for the aforementioned channel matrix this transformation
is given by Equation 2.6.
(2.6)
where the matrices U
, V
are unitaries of dimensions n
r
n
r
and n
t
n
t
accordingly, while D
is a nonnegative diagonal matrix of dimensions n
r
n
t
.
The diagonal elements of matrix D
are the singular values of the channel
matrix H
. The algorithm of singular value decomposition that provides the
above transformation can be found in [2].
The operations that lead to the linear transformation of the channel into
n
= min(n
r
, n
t
) SISO subchannels are described as follows: First, the trans
mitter multiplies the signal to be transmitted x
with the matrix V
, the receiver
multiplies the received signal r
and noise with the conjugate transpose of
the matrix U
. The above are presented in Equation 2.7 through Equation 2.9.
(2.7)
r Hs u
C
tr p
ss
H
ss
max log det
( )R
I HR H
2
R
ss
n n n
r t
min(,)
H
n n
r t
H UDV
H
s V x
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Theory and Practice of MIMO Wireless Communication Systems
33
(2.8)
(2.9)
Substituting Equation 2.4 into Equation 2.8 gives:
Since U
and V
are unitary matrices, they satisfy,and
hence:
(2.10)
Each component of the received vector y
can be written as:
(2.11)
where
k
are the singular values of matrix H
according to the transformation
that took place above. Equation 2.11 implies that the initial (n
r
, n
t
) MIMO
system has been transformed into n
= min(n
r
, n
t
) SISO subchannels, as illus
trated in Figure 2.2.
The above analysis of a Multiple Antenna Element (MEA) system capacity
is presented in [1]. It was proven in [1] that the total capacity of n SISO
FIGURE 2.2
Conversion of the MIMO channel into n
SISO subchannels.
y U r
H
n U u
H
y U r
y U Hs U u
y U HVx
H
H H
H
(.),(.)2 7 2 9
n
y U UDV Vx n
(.)2 6
H H
U U I
H
n
r
V V I
H
n
t
y Dx n
y x n
k
k
k k
Tx
1
2
n = min( n
r
,n
t
)
n
t
Rx
n
r
4190_book.fm Page 33 Tuesday, February 21, 2006 9:14 AM
34 MIMO System Technology for Wireless Communications
subchannels is the sum of the individual capacities and as a result the total
MIMO capacity is:
(2.12)
where p
k
is the power allocated to the kth subchannel and is its power
gain. We notice that according to the singular value decomposition algorithm
2
k
,k = 1, 2, …, n are the eigenvalues of the HH
H
matrix, which are always
nonnegative. Furthermore, regardless of the power allocation algorithm
used, p
k
must satisfy
because of the wanted power constraint.
At this point, there are two cases of particular interest that need further
consideration: the knowledge (or not) by the transmitter of the Channel State
Information (CSI). These are described in the following sections.
2.2.3 No CSI at the Transmitter
Considering Equation 2.12, we notice that the achieved capacity depends on
the algorithm used for allocating power to each subchannel. The theoretical
analysis assumes the channel state known at the receiver. This assumption
stands correct since the receiver usually performs tracking methods in order
to obtain CSI, however the same consideration does not apply to the transmitter.
When the channel is not known at the transmitter, the transmitting signal
s is chosen to be statistically nonpreferential, which implies that the n
t
components of the transmitted signal are independent and equipowered at
the transmit antennas. Hence, the power allocated to each of the n
t
subchan
nels is p
k
= p/n
t
. Applying the last expression to Equation 2.5 gives:
(2.13)
or
(2.14)
Equation 2.14 can be produced from Equation 2.13 as described in greater
detail in Appendix 2A.
C p
k k
k
n
log
2
2
1
1
k
2
p p
k
k
n
1
C
p
n
t
H
log det
2
I HH
C
p
n
t
k
k
n
log
2
2
1
1
4190_book.fm Page 34 Tuesday, February 21, 2006 9:14 AM
Theory and Practice of MIMO Wireless Communication Systems 35
2.2.4 CSI at the Transmitter
In cases in which the transmitter has knowledge of the channel, it can
perform optimum combining methods during the power allocation process.
In that way, the SISO subchannel that contributes to the information transfer
the most is supplied with more power.
One method to calculate the optimum power allocation to the n subchan
nels is to employ the waterpouring algorithm (a detailed discussion of this
algorithm can be found in [3]).
Considering the assumption of CSI at the transmitter, we can proceed to
the following capacity formula.
(2.15)
The difference between Equation 2.14 and Equation 2.15 is the coefﬁcient
k
that corresponds to the amount of power that is assigned to the kth
subchannel. This coefﬁcient is given by:
(2.16)
and satisﬁes the constraint
.
The goal with the waterpouring algorithm is to ﬁnd the optimum
k
that
maximizes the capacity given in Equation 2.15.
2.2.5 Channel Estimation at the Transmitter
As mentioned earlier, the CSI is not usually available at the transmitter. In
order for the transmitter to obtain the CSI, two basic methods are used: the
ﬁrst is based on feedback and the second on the reciprocity principle.
In the ﬁrst method the forward channel is calculated by the receiver and
information is sent back to the transmitter through the reverse channel. This
method does not function properly if the channel is changing fast. In that
case, in order for the transmitter to get the right CSI, more frequent estima
tion and feedback are needed. As a result, the overhead for the reverse
channel becomes prohibitive. According to the reciprocity principle, the
forward and reverse channels are identical when the time, frequency and
antenna locations are the same. Based on this principle the transmitter may
use the CSI obtained by the reverse link for the forward link. The main
C
p
n
k
t
k
k
n
log
2
2
1
1
k k
E s
2
k
k
n
t
n
1
4190_book.fm Page 35 Tuesday, February 21, 2006 9:14 AM
36 MIMO System Technology for Wireless Communications
problem with this method emerges when frequency duplex schemes are
employed.
2.3 Remarks on the Extended Shannon Capacity Formula
In this section we will use mathematical tools in order to derive the theoretical
upper and lower bounds of MIMO capacity. The algebraic expressions used,
as well as the assumptions considered here, are summarized in Section 2.3.1.
In Section 2.3.3 we introduce the Effective Degrees of Freedom (EDoF),
which we will use for the simulation justiﬁcation in Section 2.7.
2.3.1 Bounds on MIMO Capacity
The lower and upper bounds of MIMO capacity were ﬁrst derived in [1].
We proceed with a short description of those bounds. Four basic assumptions
are considered in the following, summarized here for simplicity.
• The transmitter has no previous knowledge of the channel.
• The parallel subchannels produced by the decomposition of the
MIMO channel are independent.
• The wireless channel is submitted to Rayleigh fading.
• The transmitter antenna array elements are less than the receiver’s
antenna elements (n
t
< n
r
).
In addition, we cite four mathematical expressions that will be used for
deriving the wanted capacity bounds.
(2.17)
where matrices D and R are diagonal and uppertriangular, respectively.
(2.18)
(2.19)
where A, B are square matrices and Q is a unitary matrix.
det
,,
DD RR D R
H H
2 2
det detI AB I BA
det detI QAQ I A
H
4190_book.fm Page 36 Tuesday, February 21, 2006 9:14 AM
Theory and Practice of MIMO Wireless Communication Systems 37
(2.20)
where X is a nonnegative deﬁnite matrix.
Since the channel is submitted to Rayleigh fading, the channel matrix H
is given by H
W
, which is referred to as spatially white matrix. The elements
of H
W
can be modeled as zero mean circularly symmetric complex Gaussian
(ZMCSCG) random variables.The H
W
has particular statistical properties
that can be found in [3, 12].
We consider the transformation H
W
= QR, where Q is a unitary and R is
an uppertriangular matrix. This tranformation is referred to as Householder
Transformation [5]. According to this transformation the elements of R above
the main diagonal are statistically independent, while the magnitude of the
main diagonal entries are chisquared distributed with 2n
r
, 2(n
r
– 2 + 1), …,
2(n
r
– n
t
+ 1) degrees of freedom.
Using Equation 2.13:
(2.21)
Equation 2.21 corresponds to the lower capacity bound and practically
shows that this bound is deﬁned by the sum of the capacities of n
t
indepen
dent subchannels with power gains that follow the chisquare distribution
with 2n
r
, 2n
r
– 2, …, 2(n
r
– n
t
+ 1) degrees of freedom.
In order to ﬁnd the upper bound of the capacity, Equation 2.13 is used
again:
(2.22)
The upper bound of the capacity is the sum of the capacities of n
t
inde
pendent subchannels, with power gains chisquared distributed and with
degrees of freedom 2(n
r
+ n
t
– 1), 2(n
r
+ n
t
– 3), …, 2(n
r
– n
t
+ 1). The difference
of the mean values of the upper and lower bounds is less than 1b/s/Hz.
det
,
X X
C
p
n
p
n
t
H
t
log det log det
2 2
I HH I Q
RRR Q
I RR
H H
t
H
C
p
n
(.)
log det
2 19
2
(.)
,
log
2 17
2
2
1
p
n
R
t
C
p
n
R
t
n
t
log
,2
2
1
1
C
p
n
R R
t
m
m
n
t
log
,,2
2 2
1
1
1
n
t
4190_book.fm Page 37 Tuesday, February 21, 2006 9:14 AM
38 MIMO System Technology for Wireless Communications
2.3.2 Capacity of Orthogonal Channels
It is interesting to study the case where the capacity of the MIMO channel
is maximized. We consider the simple case of n
r
= n
t
= n, along with a ﬁxed
total power transfer through the SISO subchannels (i.e.,, where a
is a constant). The capacity in Equation 2.14 is concave in the variables
2
k
(k
= 1, 2, …, n) and, as a result, it is maximized when
2
k
=
2
i
= a/n, (k, i = 1, 2,
…, n). The last equation reveals that the HH
H
matrix has n equal eigenvalues.
Hence, H must be an orthogoal matrix, i.e.,. Substitut
ing into Equation 2.13:
(2.23)
If, the matrix H satisﬁes, hence, Equation 2.13 becomes:
(2.24)
The last equation indicates that the capacity of an orthogonal MIMO channel
is n times the capacity of the SISO channel.
2.3.3 Effective Degrees of Freedom
Based on Equation 2.14, we can assume that in high SNR regime, capacity
can increase linearly with n. Speciﬁcally, for high SNR regime
Equation 2.14 becomes:
(2.25)
However, this assumption is not always conﬁrmed. For some subchannels
is much smaller than one, and as a result the information transferred
by these channels is nearly zero. This phenomenon is present in at least three
cases:
• when the transmission is serviced through a lowpowered device
• when there is a longrange communication application
• when there is strong fading correlation between subchannels
In the last case, the fading induced to a certain subchannel may cause the
minimization of its corresponding
2
k
.
k
n
k
a
1
2
HH H H I
H H
n
a n ( )/
HH I
H
n
a n ( )/
C
pa
n
C n
pa
n
n
log det log
2
2
2
1I I
22
H
i j,
2
1 HH I
H
n
n
C
p
n
n C n p
n
log det log
2 2
1I I
( )
k
p n
2
1/
C
p
n
k
k
n
log
2
2
1
( )
k
p n
2
/
4190_book.fm Page 38 Tuesday, February 21, 2006 9:14 AM
Theory and Practice of MIMO Wireless Communication Systems 39
In practice, the aforementioned cases are very likely to happen, therefore,
the concept of EDoF is introduced. Intuitively, the EDoF value corresponds
to the number of subchannels that actually contribute to the information
transfer. In a more mathematical approach, the EDoF value indicates the
nonzero singular values of the channel matrix H. A more detailed descrip
tion of the EDoF can be found in [1].
2.4 Capacity of SIMO — MISO Channels
Single input, multiple output (SIMO) and multiple input, single output
(MISO) channels are special cases of MIMO channels. In this paragraph we
discuss the capacity formulas for the case of SIMO and MISO channels.
For a SIMO channel n
t
= 1, so; hence, the CSI at the
transmitter does not affect the SIMO channel capacity:
(2.26)
If we consider then (the proof can be found in Appendix 2B).
Hence Equation 2.26 becomes:
(2.27)
For a MISO channel n
r
= 1 and. With no CSI at the
transmitter, the capacity formula can be expressed as:
(2.28)
If we make the same assumption as earlier and consider that then
(the proof is cited in Appendix 2B); hence, Equation 2.28 becomes:
(2.29)
Comparing Equation 2.29 and Equation 2.27 we can see that C
SIMO
> C
MISO
.
This is because the transmitter, as opposed to the receiver, cannot exploit
the antenna array gain since it has no CSI and, as a result, cannot retrieve the
receiver’s direction.
n n n
r t
min(,) 1
C p
SIMO
log
2 1
2
1
h
i
2
1
1
2
n
r
C p n
SIMO r
log
2
1
n n n
r t
min(,) 1
C
p
n
MISO
t
log
2 1
2
1
h
i
2
1
1
2
n
t
C p
MISO
log
2
1
4190_book.fm Page 39 Tuesday, February 21, 2006 9:14 AM
40 MIMO System Technology for Wireless Communications
2.5 Stochastic Channels
In order to use the aforementioned capacity formulas, it is necessary to obtain
the channel matrix expression. There are many spatial channel models that
are used for this purpose ([6–11]).
The simulations that take place in Section 2.7 consider a stochastic channel
approach. Speciﬁcally, the Rayleigh and the Rice models are used. The
description of these models is cited below. Consequently, under the stochas
tic channel consideration, the capacity achieved becomes a random variable,
and in order to study its behavior, we use stochastic quantities, as described
below.
2.5.1 Ergodic Capacity
The ergodic capacity of a MIMO channel is the ensemble average of the
information rate over the distribution of the elements of the channel matrix
H [3], and it is given by:
(2.30)
When there is no CSI at the transmitter, we can substitute Equation 2.13
into Equation 2.30, so the ergodic capacity is given by
(2.31)
Whereas with CSI at the transmitter we use Equation 2.15, and the ergodic
capacity is given by:
(2.32)
Figure 2.3 illustrates the ergodic capacity for different antenna conﬁgurations
as a function of the SNR, when the channel is unknown at the transmitter.
As expected, the ergodic capacity increases with SNR. In addition, the
ergodic capacity of a SIMO channel appears to be greater than the ergodic
capacity of a MISO channel. The reason for this behavior, as previously
explained, lies in the fact that the transmitter cannot exploit the antenna
array gain since it has no CSI.
C E I
C E
p
n
t
H
log det
2
I HH
C E
p
n
t
k k
k
n
log
2
2
1
I
4190_book.fm Page 40 Tuesday, February 21, 2006 9:14 AM
Theory and Practice of MIMO Wireless Communication Systems 41
2.5.2 Outage Capacity
The outage capacity quantiﬁes the level of capacity performance guaranteed
with a certain level of reliability [3, 12]. For q% outage capacity C
out,q
, indicates
the maximum capacity level the system can achieve with probability
(100 – q)%. For stochastic channels we can observe that there is always a
possibility of outage for a given MIMO system realization, regardless of the
wanted rate. Hence, there is a tradeoff between the system’s outage capacity
and the achieved information rate.
2.6 MIMO Capacity with Rice and Rayleigh Channels
In this section we discuss the capacity expressions for the cases of Rayleigh
and Ricean channels, as well as when spatial fading correlation is induced
to the signal due to the limited distance between the array elements (the trans
mitter is considered blind for the discussion below, i.e., does not have CSI).
When the wireless environment is characterized by strong multipath activ
ity, then the number of paths between the transmitter and receiver allows
the use of the central limit theorem [13] and the envelope of the received
signal follows the Rayleigh distribution. However, in cases that the location
FIGURE 2.3
Ergodic capacity as a function of SNR and number of elements.
50
45
40
35
30
25
20
15
Ergo
d
ic capacity
10
5
0
5 10 15
SNR
(
dB
)
20 25
(4, 4)
Rayleigh channel, ergobic capacity as function of SNR
(2, 2)
(1, 4)
(2, 4)
(4, 4)
(1, 1)
4190_book.fm Page 41 Tuesday, February 21, 2006 9:14 AM
42 MIMO System Technology for Wireless Communications
of buildings leads to the street waveguide propagation phenomenon, and
in areas near the base station where a line of sight (LOS) component may
dominate, the Ricean distribution is more suitable. The receiver in that sce
nario “sees” a dominant signal component along with lower power compo
nents caused by multipath. The dominant component that reaches the
receiver may not be the result of LOS propagation, e.g., the dominant com
ponent may be the mean value of strong multipaths caused by large scatter
ers. The Ricean Kfactor of the channel is deﬁned as the ratio of the powers
of the dominant and the fading components [13].
(2.33)
Obviously, K = 0 indicates a Rayleigh fading channel while K indicates
a nonfading one.
2.6.1 MIMO Channel Matrix for Rayleigh Propagation Conditions
The channel matrix H in Equation 2.31 depends on the channel model.
Speciﬁcally, in cases where the wireless channel is submitted to Rayleigh
fading and the array antennas do not introduce additional correlation to the
transmitted/received signal, the channel matrix becomes spatially white.
The ergodic capacity formula under the assumption of Rayleigh channels
and equal power allocation is (following the analysis in Section 2.5.1):
(2.34)
Equation 2.34 is used for the simulations concerning the Rayleigh channel
that will be shown in the next section.
Under the assumption of, it would be interesting to study the
case of n [3]. Using the strong law of large numbers [14] we get:
(2.35)
Therefore, the Rayleigh channel capacity bound is given by:
when n (2.36)
K
A
2
2
2
C E
p
n
t
H
log det
2
I H H
W W
n n n
r t
1
n
as n
H
n
H H I
W W
( )
C p p n
n
n n
n
log det log
2 2
1I I
log
2
1 p
C n p
log
2
1
4190_book.fm Page 42 Tuesday, February 21, 2006 9:14 AM
Theory and Practice of MIMO Wireless Communication Systems 43
Considering the last expression, two things can be noticed:
• Capacity does not depend on the nature of the channel matrix, as it
increases linearly with n for a ﬁxed SNR.
• Every 3 dB increase of SNR corresponds to an n bits/sec/Hz increase
in capacity.
2.6.2 MIMO Channel Matrix for Ricean Propagation Conditions
In the presence of a dominant component between the transmitter and the
receiver, the wireless channel can be modeled as the sum of a constant and
a variable component caused by scattering [3, 15].
(2.37)
In Equation 2.37 H
Rice
is the MIMO channel matrix, H
Rayleigh
is the MIMO
matrix corresponding to the variable component, H
LOS
is the MIMO matrix
corresponding to the constant signal component, K is the Ricean–K factor
and
0
is the phase shift of the signal when propagating from a transmitting
antenna element to the corresponding receiving antenna element.
The H
Rayleigh
matrix is spatially white, and its structure was described
earlier in this chapter. H
LOS
can be derived by the procedure described in
the following (for more details see [15]).
The general conﬁguration of a multiple transmitting and receiving antenna
array is illustrated in Figure 2.4.
In Figure 2.4, R is the distance between the transmitter and the receiver
and d is the interelement distance. The matrix H
LOS
is given by:
(2.38)
FIGURE 2.4
Geometry of a Tx and an Rx linear antenna array.
Tx Rx R
R’
d
d
H H H
Rice LOS Rayleigh
K
K
e
K
j
1
1
1
0
H
LOS
1
1
1
1
e e
e
e
e e
j j n
j
j
j n
t
r
( )
( )
j n
r
( )2
1
4190_book.fm Page 43 Tuesday, February 21, 2006 9:14 AM
44 MIMO System Technology for Wireless Communications
where is the angle corresponding to phase shift between the neighbor array
elements.
In order to simplify the analysis, the distance between the receiving and
the transmitting antennas is assumed substantially larger than the distance
between the antenna elements. So, under the assumption of, is
minimized to the point that it can be omitted from the matrix in Equation
2.38 (see Appendix 2C for the proof). In that case H
LOS
is given by an n
r
n
t
matrix, with ones as elements (we refer to this matrix as H(1)).
Also, it is obvious that
0
affects the contribution of H
LOS
to H
Rice
. For reasons
of simplicity it is assumed that
0
= /4, so e
j
0
=
As a result, the real and the imaginary parts of the H
Rice
elements are inﬂu
enced in the same manner. After some manipulation, Equation 2.37 becomes:
(2.39)
Equation 2.39 is used to produce the simulation results shown in the next
section.
The assumptions made regarding the Ricean channel analysis can be sum
marized as follows:
• The dominant component is considered to be caused by LOS prop
agation.
• The distance between the transmitter and the receiver is considered
substantially larger than the interelement distance.
Although these assumptions might not always be valid, the results indicate
the effect of the dominant component on the MIMO system capacity, gener
ally. In cases that the dominant signal component is caused by directional
multipath propagation, this component is time varying, and hence, the above
analysis cannot be applied.
However, the case can be found in multibase operations [3]. In these
scenarios the transmit/receive antenna elements are cited in different base
stations. The matrix that describes the constant component of the Ricean
channel, in that case, is orthogonal. In Figure 2.5 is illustrated the ergodic
capacity of a Ricean channel when the H
LOS
is orthogonal and when H
LOS
=
H(1).
Apparently, Figure 2.5 shows that the form of the H
LOS
matrix, which
represents the ﬁxed channel component, inﬂuences the capacity for large
values of Kfactor. Speciﬁcally, the channel with the orthogonal H
LOS
outper
forms the channel with the degenerate H
LOS
for increasing K.
R d
1 2 1 2 j
H H H
Rice
1
2
+j
1
2
1
K
K K
w
1
1
1
R d
4190_book.fm Page 44 Tuesday, February 21, 2006 9:14 AM
Theory and Practice of MIMO Wireless Communication Systems 45
2.6.3 Channel Matrix with Spatial Fading Correlation
The Rayleigh channel assumes ﬂat fading in the space, time, and frequency
domains. However, the signal’s components arriving at the receiver may
experience correlation due to the limited distance of the antenna elements.
As a result, the use of H
W
as the channel matrix is inappropriate. In order
to include the correlation effect the following equation is used [1, 3]:
(2.40)
where vec(H), denotes a vector* made by the columns of H, and R is the
covariance matrix of the channel of dimension n
r
n
t
n
r
n
t
.
(2.41)
The analysis can be simpliﬁed with the use of the channel matrix of Equation
2.42.
(2.42)
FIGURE 2.5
Ergodic capacity of a 2 2 Ricean channel when the H
LOS
is orthogonal, and H
LOS
= H(1).
* If H = [h
1
h
2
…
h
n
t
] is n
r
n
t
then vec(H) = [h
1
T
h
2
T
…h
T
n
t
] is n
r
n
t
1.
7.5
H
LOS
orthogonalH
LOS
degenarated as a function of K
7
6.5
6
5.5
Ergodic capacity(bits/sec/Herz)
5
4.5
0 2 4 6 8 10 12
K
16 18 14 20
Ones
Orth
vec R vecH H
1
2
W
R H H
E vec vec
H
H R H R
R W T
1
2
1
2
4190_book.fm Page 45 Tuesday, February 21, 2006 9:14 AM
46 MIMO System Technology for Wireless Communications
where R
R
is the reception correlation matrix and R
T
is the transmission
correlation matrix. Equation 2.42 is derived by 2.40 under the assumption
that matrix R
R
and R
T
remain unchanged, regardless of the transmitting and
receiving elements, respectively.
The correlation matrices R
T
and R
R
are calculated using two different mod
els. The ﬁrst (used for the simulations), calculates these matrices as a function
of the distance between the receiving and transmitting elements [16]. A short
description follows assuming that the R
T
, R
R
matrices have the form:
(2.43)
where r is the fading correlation between two adjacent antenna elements
and it is approximated by:
(2.44)
is the angular spread and d is the distance in wavelengths between the
antenna elements (Figure 2.6). In order to simplify the procedure we can
make the following assumptions concerning the model:
FIGURE 2.6
The mean angle of arrival () and the angle spread () of an incoming multipath signal.
R
T
T T T
n
t T
T T T
T
r r r
r r
r r r
r
r
t
1
1
1
4 1
4 4
2
( )
TT
n
T T
R
R R
t
r r
r r
( )
1 4
4
2
1
1
R
r
r r
r r r
r
r
R
n
R R
R R R
R
R
n
r
r
( )
( )
1
4 4
1
2
2
1
1
r r
R R
4
1
r d d
exp 23
2 2
d
2∆
ϕ
4190_book.fm Page 46 Tuesday, February 21, 2006 9:14 AM
Theory and Practice of MIMO Wireless Communication Systems 47
• For very small r(d) the higher order terms of the above matrices can
be omitted. Hence, the correlation matrices take the form of triagonal
matrices.
• Moreover, if the same interelement distance for both the transmitter
and the receiver is considered (d
r
= d
r
and r
T
= r
R
= r), we can use a
single parameter model that simpliﬁes the capacity calculations.
The ﬁnal form of the matrices used in simulations are:
(2.45)
The second model is described in [17], where the analysis from [18] is
adopted and calculates the correlation matrices via the following formula:
(2.46)
where J
0
is the zeroorder Bessel function of the ﬁrst kind,, and R
ik
is the signal correlation coefﬁcient between the ith and the kth antenna array
element.
For very small and = 0 Equation 2.46 can be approximated by:
(2.47)
The largest value of that maintains the validity of Equation 2.47 is /4.
2.7 Simulations
In this section we present simulation results for the three capacity cases
presented in Section 2.6. The Cumulative Distribution Function (CDF) of the
capacity is produced for the following scenarios:
• Rayleigh channel without spatial fading correlation
• Rice channel with the dominant component caused by LOS propa
gation
R
T
r
r r
r
r
r
1 0 0
1
0 1 0
0 0 1
R
R
r
r r
r
r
r
1 0 0
1
0 1 0
0 0 1
R J z i k
ik
0
[ ( )]
z d 2
R
z i k
z i k
ik
sin( ( ) )
( )
4190_book.fm Page 47 Tuesday, February 21, 2006 9:14 AM
48 MIMO System Technology for Wireless Communications
• Rice channel with the dominant component caused by weak multipath
• Rayleigh channel with spatial fading correlation
2.7.1 MIMO Capacity for a Rayleigh Channel without Spatial
Fading Correlation
The capacity formula used for the simulations of this section is presented in
Equation 2.13. The channel matrix that it is used for capacity calculations is
H
W
. This matrix is full rank and its elements are independent variables that
follow a Gaussian distribution. As a result, the MIMO channel is transformed
into exactly SISO subchannels.
Figure 2.7 indicates that increasing the number of antenna elements leads
to a capacity increase. Especially, we notice that the large capacity increase
involves array antennas at both the transmitter and the receiver. For example,
the (8,1) MIMO channel supports lower capacity gain than the (2,2) MIMO
channel. This is justiﬁed by the MIMO system transformation concept men
tioned earlier. Speciﬁcally, the (8,1) channel gives n = 1 while (2,2) gives n = 2.
The result can be justiﬁed considering the fact that the independent SISO
subchannels that resulted from the MIMO system transformation are respon
sible for the information transfer.
Also, Figure 2.7 indicates that the presence of an antenna array at the
receiver is more important than the presence of the same antenna array at
the transmitter. For example, we notice again that the channel (4,1) presents
FIGURE 2.7
MIMO capacity for a Rayleigh channel with different antenna array elements (SNR=10 dB).
n rank n n
r t
( ) min(,)H
W
1
CDFRayleighSNR = 10
0.9
0.8
0.7
0.6
0.5
0.4
0.3
(1, 1)
(1, 4)
(4, 1)
(1, 8)
(8, 1)
(2, 2)
Prob(capacity < c)
0.2
0.1
0
0 5 10 15
c bits/sec/Hz
20 25
4190_book.fm Page 48 Tuesday, February 21, 2006 9:14 AM
Theory and Practice of MIMO Wireless Communication Systems 49
better capacity behavior than the channel (1,4). The explanation for this lies
in the assumption that the transmitter does not have CSI, and as a result it
“equipowers” the elements regardless of the channel. On the contrary, the
receiver is considered to possess this information, and as a result it may use
its antenna array for optimum combining based on CSI.
2.7.2 MIMO Capacity for a Rayleigh Channel with Spatial
Fading Correlation
In this case the channel matrix is given by Equation 2.42. Figure 2.8 illustrates
the capacity for different antenna conﬁgurations and interelement spacing
distances, d in wavelengths.
Figure 2.8 proves the great effect of correlation to the MIMO channel
capacity. We can easily notice that the uncorrelated channel* (d = 0.5 ) offers
high capacity performance in comparison to rest cases. Speciﬁcally, the
ergodic capacity of the (4,4) uncorrelated channel is about 13 bits/sec/Hz
greater than the fully correlated one. So as the distance between the antenna
elements decreases, the capacity decreases too. The reason lies in the
correlation increase with the decrease of the interelement distance.
Correlation between the transmitted and received signals decreases the
FIGURE 2.8
CDFs of capacity for the Rayleigh MIMO channel with spatial fading correlation.
* We consider the scenario where half wavelength interelement distance introduces low enough
correlation that the fades can be considered independent [1].
1
0.9
0.8
0.7
(2, 2)
CDFcorre
l
ate
d
c
h
anne
l
SNR = 10
(4, 4)
0.6
0.5
0.4
0.3
Prob(capacity < c)
0.2
0.1
0
0 5 10 15 20 25 30
c bits/sec/Hz
35
d = 0.01
d = 0.1
d = 0.2
d = 0.5
4190_book.fm Page 49 Tuesday, February 21, 2006 9:14 AM
50 MIMO System Technology for Wireless Communications
independent propagation paths and, as a result, decreases the information
transferred.
Also, we notice that the (4,4) MIMO channel achieves higher capacity
compared to the (2,2) channel, under any correlation conditions.
2.7.3 MIMO Capacity for a Ricean Channel
The channel matrix that it is used for capacity calculations is given by
Equation 2.39. Figure 2.9 illustrates the CDFs of capacity for different antenna
conﬁgurations and for a constant Ricean factor equal to K = 4. Figure 2.10
illustrates the CDFs of capacity for a (2,2) MIMO channel for different Ricean
factors. The CDF for K = 0 represents the case of Rayleigh fading channel.
Apparently, Figure 2.9 indicates that the use of array antennas at both the
transmitter and receiver improves substantially the capacity. The same result
arose in the case of the simple Rayleigh channel studied in the previous section.
Figure 2.10 proves that the presence of a ﬁxed component can cause great
damage to the MIMO channel capacity performance. We easily see that the
channel capacity decreases when the Ricean factor increases. The value of K
corresponds to the strength of the dominant component. As K increases, the
dominant component appears stronger and the correlation coefﬁcient
increases too. As mentioned before, correlation leads to the limitation of the
independent paths that transfer information and, hence, to lower capacity
gains.
FIGURE 2.9
CDFs of capacity for the Ricean MIMO channel with K = 4.
1
0.9
0.8
0.7
(2, 2)
(2, 4)
(4, 2)
(4,4)
CDFrank
(
H
Rice
) = 1, SNR = 10, K = 4
0.6
0.5
0.4
0.3
Prob(capacity < c)
0.2
0.1
0
2 4 6 8 10 12
c bits/sec/Hz
14
4190_book.fm Page 50 Tuesday, February 21, 2006 9:14 AM
Theory and Practice of MIMO Wireless Communication Systems 51
Finally, Figure 2.10 shows that the capacity decrease, due to the ﬁxed
component reaching the receiver, can be easily compensated by the use of
more elements. Speciﬁcally, we notice that the (4,4) channel with K = 12
outperforms the (2,2) Rayleigh fading channel (K = 0).
Appendix 2A
Capacity of a MIMO Channel Using the Singular Values
of the Channel Matrix H
In Equation 2.13 we presented the MIMO channel capacity with no CSI at
the transmitter. Equation 2.13 can be transformed into Equation 2.14 as
follows:
According to singular value decomposition
(2A.1)
D is a diagonal matrix with elements the singular values of H. The singular
values of a complex matrix are always nonnegative and equal to the square
FIGURE 2.10
CDFs of capacity for a (2,2) channel for different Kfactors.
1
0.9
0.8
0.7
CDFSNR = 10
0.6
0.5
0.4
0.3
Prob(capacity < c)
0.2
0.1
0
2 3 4 5 6 8 9 10 11 7
c bits
/
sec
/
Hz
12
(2, 2)
(4, 4)
K = 8
K = 10
K = 12
K = 0
H UDV
H
4190_book.fm Page 51 Tuesday, February 21, 2006 9:14 AM
52 MIMO System Technology for Wireless Communications
root of the eigenvalues of the positive semihermitian matrix. Let
k
be
the kth singular value of H, hence,will be the kth eigenvalue of.
Using transformation Equation 2A.1 for the matrix we get:
(2A.2)
We replace Equation 2A.2 in Equation 2.13:
(2A.3)
Appendix 2B
Proof That
2
1
= n
r
When n
t
= 1 and
2
1
= n
t
When n
r
= 1,
When h
i,j
= 1
We use the known algebraic equality:
(2B.1)
HH
H
k
2
HH
H
HH
H
HH UDV UDV UDV VD U UDD U
HH UD U
H H H
H
H H H H H
H H
2
UD U
2 H
C
p
n
H
log det
2
2
I UD U,
we use the equality det detI AB I B
AA
I D
C
p
n
p
n
log det log (
2
2
2
1
1
2
2
2 2
2
1 1
1
)( ) ( )
log
p
n
p
n
C
p
n
n
k
k
n
2
1
k
k
n
r
Tr
2
1
HH
H
k
k
n
H
i j i j
j
n
i
n
i
r tr
Tr h h h
2
1 11
HH
,,,
jj
j
n
i
n
k
k
n
i j
j
n
i
n
tr
r tr
h
2
11
2
1
2
11
,
4190_book.fm Page 52 Tuesday, February 21, 2006 9:14 AM
Theory and Practice of MIMO Wireless Communication Systems 53
When n
t
= 1, Equation 2B.1 gives:
(2B.2)
However, since n
t
= 1,and as a result there is only one.
Hence, Equation 2B.2 takes the form:
(2B.3)
When n
r
= 1, Equation 2B.1 gives:
(2B.4)
Appendix 2C
Minimization of Angular Shift When R
d
In accordance with Figure 2.4, we denote the angular shift between the
neighbor elements; this angle is given by the equation:
(2C.1)
where is the wavelength of the transmitted signal.
From the ﬁgure above we have:
(2C.2)
substituting Equation 2C.2 into Equation 2C.1:
k
k
n
i j
ji
n
n
r r
r
h
2
1
2
1
1
1
1
2
2
2 2
,
... n
r
rank
H
( )HH 1
k
2
0
1
2
n
r
k
k
n
i j
j
n
i
t
r t
h n
2
1
2
11
1
1
2
,
2 2
R R
c
f
R R
R R
R
d
R
d
R
d
cos
sin
cos
sin
sin
4190_book.fm Page 53 Tuesday, February 21, 2006 9:14 AM
54 MIMO System Technology for Wireless Communications
Assuming that or then approaches zero and we can use the
approximation:
(2C.4)
Finally, Equation 2C.3 is given by:
(2C.5)
Equation 2C.5 proves that under the aforementioned assumptions may be
omitted from the matrix (Equation 2.38).
References
1.D.S. Shiu, J. Foschini, J. Gans, and J.M Kahn. 2000. “Fading correlation
and its effect on the capacity of multielement antenna system,” IEEE
Transactions on Communications, 48, 502, 2000.
2.http://www.cs.ut.ee/~toomas_1/linalg/lin2/node14.html
3.A. Paulraj, R. Nabar, and D. Gore. 2003. Introduction to SpaceTime Wireless
Communications, Cambridge: Cambridge University Press, Chap. 4.
4.I.E. Telatar, “Capacity of multiantenna Gaussian channels,” Eur. Trans. Telecom.,
Vol. 10, No. 6, Dec. 1999.
5.http://rkb.home.cern.ch/rkb/AN16pp/node123.html
6.B.R. Ertel and P. Cardieri. 1998. “Overview of spatial channel models for an
tenna array communication systems,” IEEE Personal Communication Magazine,
Vol. 5, Feb. 1998, pp. 10–22.
7.G.V. Tsoulos and G.E. Athanasiadou. 2002. “On the application of adaptive
antennas to microcellular environments: radio channel characteristics and
system performance,” IEEE Trans. Veh. Technol., Vol. 51, No. 1, Jan. 2002, pp.
1–16.
8.G.E. Athanasiadou, A.R. Nix, and J.P. McGeehan. 2000. “A microcellular ray
tracing propagation model and evaluation of its narrowband and wideband
predictions,” IEEE J. Select. Areas Commun., Vol. 18, Mar. 2000, pp. 322–335.
2
2 2R R d d d
sin
cos
sin
11 2
2
2
4
2
cos
sin
sin
sin
sin
d
d
22
2
sin
(2C.3)
R d
d R 1
sin
d
R
4
4
1
2 2
d d d
R
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Theory and Practice of MIMO Wireless Communication Systems 55
9.G.E. Athanasiadou and A.R. Nix. 2000. “A novel 3D indoor raytracing prop
agation model: the path generator and evaluation of narrowband and wide
band predictions,” IEEE Trans. Veh. Technol., Vol. 49, July 2000, pp. 1152–1168.
10.3GPP TR 25.996 V6.1.0 (www.3gpp.org).
11.L. Correia, ed. 2001. Wireless Flexible Personalised Communications, New York:
Wiley.
12.G.J. Foschini. 1996. “Layered spacetime architecture for wireless communica
tions in a fading environment when using multielement antennas,” Bell Labs
Tech. J., Autumn 1996, pp. 41–59.
13.J.G. Proakis. 1983. Digital Communications, 4th ed., New York: McGrawHill,
Chap. 2.
14.A. Papoulis. 1984. Probability, Random Variables, and Stochastic Processes, New
York: McGrawHill.
15.M.A. Khalighi, J.M. Brossier, G. Jourdain, and K. Raoof. 2001. “On capacity of
Ricean MIMO channels,” in Proc. 12th IEEE Int. Symp. Personal, Indoor and Mobile
Radio Communications, A150 to A154, Vol. 1, 2001.
16.A. Zelst and J.S. Hammerschmidt. 2002. “A single coefﬁcient spatial correlation
models for multipleinput multiple output (MIMO) radio channels,” in Proc.
URSI XXVIIth General Assembly, 2002.
17.S. Loyka and G.V. Tsoulos. 2002. “Estimating MIMO system performance using
the Correlation matrix approach,” IEEE Commun. Letters, 6, 19, 2002.
18.J. Salz and J.H. Winters. 1994. “Effect of fading correlation on adaptive arrays in
digital mobile radio,” IEEE Trans. Veh. Technol., Vol. 43, Nov. 1994, pp. 1049–1057.
4190_book.fm Page 55 Tuesday, February 21, 2006 9:14 AM
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