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

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44 MIMO System Technology for Wireless Communications

where V 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 , V 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

× n

matrix, with ones as elements (we refer to this matrix as H(1)).

Also, it is obvious that K

affects the contribution of H

LOS

to H

Rice

. For reasons

of simplicity it is assumed that

= U/4, so e

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 K-factor. Speciﬁcally, the channel with the orthogonal H

LOS

outper-

forms the channel with the degenerate H

LOS

for increasing K.

Rd

12 12+ j

HHH

Rice

()

Rd

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

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

× n

(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

…

] is n

× n

then vec(H) = [h

…h

]isn

× 1.

7.5

LOS

orthogonal-H

LOS

degenarated as a function of K

6.5

5.5

Ergodic capacity(bits/sec/Herz)

4.5

0 2 4 6 8 10 12

16 18 14 20

Ones

Orth

vec R vecHH

()

RHH=

() ()

{}

E vec vec

HRHR=

RWT

4190_book.fm Page 45 Tuesday, February 21, 2006 9:14 AM

46 MIMO System Technology for Wireless Communications

where R

is the reception correlation matrix and R

is the transmission

correlation matrix. Equation 2.42 is derived by 2.40 under the assumption

that matrix R

and R

remain unchanged, regardless of the transmitting and

receiving elements, respectively.

The correlation matrices R

and 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

, 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 (O) and the angle spread ()) of an incoming multipath signal.

TT T

rr r











()

()













rr r

RR R

()



 rr



rd d

()

~

()

exp 23

)

2∆

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

= d

and 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

is the zero-order Bessel function of the ﬁrst kind, , and R

is the signal correlation coefﬁcient between the ith and the kth antenna array

element.

For very small ) and O = 0 Equation 2.46 can be approximated by:

(2.47)

The largest value of ) that maintains the validity of Equation 2.47 is U/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



100

010

001







 



½½



100

010

001







 



½½

RJzik

= 

[( )]

zd= 2UQ

zi k



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

. 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).

nrank nn

==()min(,)H

CDF-Rayleigh-SNR = 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 5 10 15

c bits/sec/Hz

20 25

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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

“equi-powers” 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 Q) 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].

0.9

0.8

0.7

(2, 2)

CDF-corre

ate

anne

-SNR = 10

(4, 4)

0.6

0.5

0.4

0.3

Prob(capacity < c)

0.2

0.1

0 5 10 15 20 25 30

c bits/sec/Hz

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.

0.9

0.8

0.7

(2, 2)

(2, 4)

(4, 2)

(4,4)

CDF-rank

(

Rice

) = 1, SNR = 10, K = 4

0.6

0.5

0.4

0.3

Prob(capacity < c)

0.2

0.1

2 4 6 8 10 12

c bits/sec/Hz

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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 non-negative and equal to the square

FIGURE 2.10

CDFs of capacity for a (2,2) channel for different K-factors.

0.9

0.8

0.7

CDF-SNR = 10

0.6

0.5

0.4

0.3

Prob(capacity < c)

0.2

0.1

2 3 4 5 6 8 9 10 11 7

c bits

sec

(2, 2)

(4, 4)

K = 8

K = 10

K = 12

K = 0

HUDV=

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52 MIMO System Technology for Wireless Communications

root of the eigenvalues of the positive semi-hermitian matrix . Let J

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

= n

When n

= 1 and

= n

When n

= 1,

When

冨

i,j

冨

= 1

We use the known algebraic equality :

(2B.1)

HH UDV UDV UDV VD U UDD U

HH UD U

HHH

HHH HH

= 

()

==¡

== UD U

2 H



log det

IUDU,

we use the equality det detIAB IB+



=+C

log det log (

1 JJJ J

)( ) ( )

log

+  +



()

ij ij

Tr h h h

111===

¨¨¨

()

=  =HH

,, ,jj

===

¨¨

¨¨¨

4190_book.fm Page 52 Tuesday, February 21, 2006 9:14 AM

Theory and Practice of MIMO Wireless Communication Systems 53

When n

= 1, Equation 2B.1 gives:

(2B.2)

However, since n

= 1, and as a result there is only one .

Hence, Equation 2B.2 takes the form:

(2B.3)

When n

= 1, Equation 2B.1 gives:

(2B.4)

Appendix 2C

Minimization of Angular Shift When R



In accordance with Figure 2.4, we denote V the angular shift between the

neighbor elements; this angle is given by the equation:

(2C.1)

where Q is the wavelength of the transmitted signal.

From the ﬁgure above we have:

(2C.2)

substituting Equation 2C.2 into Equation 2C.1:

JJJJ

===

¨¨¨

= ¡ +++

... == n

rank

()HH = 1 J

= n

===

¨¨¨

= ¡ =

VU U



cos

sin

cos

sin

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