Tsoulos George (ред.) MIMO System Technology for Wireless Communications

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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

is the power allocated to the kth subchannel and is its power

gain. We notice that according to the singular value decomposition algorithm

, k = 1, 2, …, n are the eigenvalues of the HH

matrix, which are always

non-negative. Furthermore, regardless of the power allocation algorithm

used, p

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 non-preferential, which implies that the n

components of the transmitted signal are independent and equi-powered at

the transmit antennas. Hence, the power allocated to each of the n

subchan-

nels is p

= p/n

. Applying the last expression to Equation 2.5 gives:

(2.13)

(2.14)

Equation 2.14 can be produced from Equation 2.13 as described in greater

detail in Appendix 2A.

()

log

1 J



log det

IHH

log

1 J

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

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 L

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



log

Es=

{}

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

< n

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 upper-triangular, respectively.

(2.18)

(2.19)

where A, B are square matrices and Q is a unitary matrix.

det

DD RR D R

()

v +

()



 



det detIAB IBA+

()

det detIQAQ IA+

()

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 non-negative deﬁnite matrix.

Since the channel is submitted to Rayleigh fading, the channel matrix H

is given by H

, which is referred to as spatially white matrix. The elements

of H

can be modeled as zero mean circularly symmetric complex Gaussian

(ZMCSCG) random variables. The H

has particular statistical properties

that can be found in [3, 12].

We consider the transformation H

= QR, where Q is a unitary and R is

an upper-triangular 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 chi-squared distributed with 2n

, 2(n

– 2 + 1), …,

2(n

– n

+ 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

indepen-

dent subchannels with power gains that follow the chi-square distribution

with 2n

, 2n

– 2, …, 2(n

– n

+ 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

inde-

pendent subchannels, with power gains chi-squared distributed and with

degrees of freedom 2(n

+ n

– 1), 2(n

+ n

– 3), …, 2(n

– n

+ 1). The difference

of the mean values of the upper and lower bounds is less than 1b/s/Hz.

det

()









=+log det log det

IHH IQRRR Q

IRR



(. )

log det

219

««



v +



(. )

log

217









v +

log





f ++

log

,,2

 



»»

 1

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

= n

= 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 J

= 1, 2, …, n) and, as a result, it is maximized when J

= J

= a/n, (k, i = 1, 2,

…, n). The last equation reveals that the HH

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 low-powered device

• when there is a long-range 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 J

HH H H I

an==()/

HH I

an= ()/



¡ =  +log det log

1II

ij,

1= HH I

nCn p



¡ =  +

()

log det log

1II

()J

1/ 

log

()J

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

non-zero 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

= 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

= 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.

nnn

==min( , ) 1

SIMO

=+

()

log

1 J

1= J

= n

Cpn

SIMO r

=+

()

log

nnn

==min( , ) 1

MISO

=+

log

1 J

= n

MISO

()

log

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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.

CEI=

{}

log det

IHH

log

I LJ

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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.

Ergo

ic capacity

5 10 15

SNR

(

)

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 K-factor 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 q h indicates

a non-fading 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 q h [3]. Using the strong law of large numbers [14] we get:

(2.35)

Therefore, the Rayleigh channel capacity bound is given by:

when n q h (2.36)

log det

IHH

nnn

as n

HH I

()qqh

Cppn

q +

()



log det log

1II  +

()

¡log

1 p

Cn pq +

()

log

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 K

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’

HHH

Rice LOS Rayleigh

LOS





jjn







()

() 



()2



4190_book.fm Page 43 Tuesday, February 21, 2006 9:14 AM