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24 MIMO System Technology for Wireless Communications

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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

, 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 error-free

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

error-free transmission. On the contrary, if the transmission rate is more than

bps, then the received signal, regardless of the robustness of the employed

code, will involve bit errors.

=  +

log

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Theory and Practice of MIMO Wireless Communication Systems

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

elements at the transmitter and an

antenna array with n

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

). The MIMO channel can then be described by the n

) 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 Y

. The input-output notation of the MIMO

system can now be expressed by the following equation:

(2.3)

where denotes convolution, s

) is a n

1 vector corresponding to the n

transmitted signals, y

) is a n

1 vector corresponding to the n

received

signals and u

) is the additive white noise.

FIGURE 2.1

The MIMO channel.

Channel

H(,)

(,) (,) (,)

,, ,

YY Y

ht ht h t

11 12 1



ttht h t

hth

)(,) (,)

(,) (

22 2





YYY,) (,)

tht





yH su() ( ,) () ()tttt=  +Y



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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

channel matrix, is the covariance matrix of the

transmitted vector s, H

is the transpose conjugate of the H

matrix and p

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

and n

accordingly, while D

is a non-negative diagonal matrix of dimensions n

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

= min(n

, n

) 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)

rHsu

YYY

tr p

()

max log det

()R

IHRH

nnn

= min( , )

H 



HUDV=

sVx

= 

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Theory and Practice of MIMO Wireless Communication Systems

(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 J

are the singular values of matrix H

according to the transformation

that took place above. Equation 2.11 implies that the initial (n

, n

) MIMO

system has been transformed into n

= min(n

, n

) 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.

yUr

= 

nUu

= 

yUr

yUHsUu

yUHVx

YYY

= ¡

=+ ¡

(.),(.)27 29

++ ¡

yUUDVVxn

YYY

(.)26

UU I

VV I

yDxn

YYY

yxn

YYY

n = min(n

)

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