Lecture Notes-Multiple Antennas for MIMO Communications - Channel Correlation

Hon Tat Hui Multiple Antennas for MIMO Communications - Channel Correlation

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1 IntroductionThe performance of a multiple-input multiple-output (MIMO) is critically dependent on the availability of independent multiple channels. It is well known that channel correlation will downgrade the performance of a MIMO system, especially its capacity. Channel correlation is a measure of similarity or likeliness between the channels. In the extreme case that if the channels are fully correlated, then the MIMO system will have no difference from a single-antenna communication system.

Multiple Antennas for MIMO Communications - Channel Correlation


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The capacity of a MIMO system not only depends on the number of channels (N M), but also depends on the correlation between the channels. In general, the greater the channel correlation, the smaller is the channel capacity. The channel correlation of a MIMO system is mainly due to two components:(1) spatial correlation(2) antenna mutual coupling.

2 Types of channel correlation


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2.1 Spatial correlationIn a practical multipath wireless communication environment, the wireless channels are not independent from each other but due to scatterings in the propagation paths, the channels are related to each other with different degrees. This kind of correlation is called spatial correlation. For a given channel matrix H, the spatial correlation between the channels are defined as:

*

, * *

, 1,2, ,

, 1,2, ,ij pq

ij pq

ij ij pq pq

E h h i j Np q ME h h E h h

(1)


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The spatial correlation depends on the multipath signal environment. Multipath signals tend to leave the transmitter in a range of angular directions (called angles of departure, AOD) rather than a single angular direction. This is the same for the multipath signals arriving at the receiver (called angles of arrival, AOA). Usually, the spatial correlation increases when AOD and AOA are reduced and vice versa.

Scatterers

Transmitting array

x

y

= AOD

Scatterers

Receiving array

y

x = AOA


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Example 1Find the spatial correlation, 11,21, of the channels h11 and h21of a MIMO system with N = 2 and M = 1. All the antennas are dipole antennas. The channels are random with a Gaussian distribution (zero mean and unit variance). Assume that the AOA at the receiver is 360° on the plane (H-plane) perpendicular to the dipole antennas and the radiation patterns of the dipole antennas are omni-directional. Furthermore, assume that the incident fields at the receiver are polarizationmatched.

h11

h21Vin

Vo1

Vo2

dr


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SolutionsAs there is only one transmitting antenna, the AOD is not relevant for the calculation of the spatial correlation.We define a channel as the open-circuit voltage Vo developed at a receiving antenna to the excitation voltage Vin at a transmitting antenna. Therefore,

1 211 21, o o

in in

V Vh hV V

Note that Vo1 and Vo2 are random complex numbers because the channels h11 and h21 are random. However, Vin is deterministic. Thus the correlation coefficient 11,21 can be written as:


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11 21 01 02

11,21

11 11 21 21 01 01 02 02

* *

* * * *

E h h E V V

E h h E h h E V V E V V

As the AOA at the receiver is 360° on the H-plane and the incident field is polarization matched to the dipole antennas, the multipath signals at the receiving antennas are as illustrated on the next page. Note that although the far fieldscome from the same scatterers (aligned in a circular form), the far fields received by dipole 1 and dipole 2 have a phase difference between because their spatial locations are not the same. Hence we denote them by E1 and E2, respectively.


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Scatterers in the far-field region of the receiver

plane waves from the

transmitter

Receiving dipoles(top view)

E1, E2


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2

01 02 10 0

2

20 0

1*

*1

m

m

E V V E z dzdI

z dzdI

I E

I E

Therefore the open-circuit voltages Vo1 and Vo2 can be expressed as:

where I(z) is the current distribution on a dipole antennas when it is in the transmission, E1() and E2() are the incident fields on the receiving dipole antennas. Note that E1() and E2() are random complex Gaussian numbers due to the random nature of the channels.


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01 02

2 2

1 220 0 0 0

2

1 220 0 0

22 cos

020 0 0

0

*

1 * *

1 * *

1 * r

m

m

jkd

m

r

E V V

I z I z dzdz E E d E dI

I z I z dzdz E E E dI

I z I z dzdz E E e dICJ kd

1 0E E cos2 0

rjkdE E e Therefore,

Costant

2

cos0

0

12

rjkdJ kd e d

E0 = path gain from transmitter to receiver (a Gaussian random number with each scatterer)


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where C is a complex constant with the expression:

020 0

1 **m

C I z I z dzdz E EI

01 01 02 02* *E V V E V V C

By a similar derivation procedure, we can find:

01 02

011,21 0

01 01 02 02

*

* *r

r

E V V CJ kd J kdCCE V V E V V

Hence the correlation coefficient is then:


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Example 2Similar to Example 10 but now find the spatial correlation, 11,12, of the channels h11 and h12 of a MIMO system with N = 1 and M = 2, i.e., one receiving antenna and two transmitting antennas. Assume that the AOD at the transmitter is 360°.

h11

h12

Vin

Vo

Vin

dt


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Now there is only one receiving antenna, the AOA is not relevant for the calculation of the spatial correlation. The channels are now:

11 12, o o

in in

V Vh hV V

Solutions

Thus the correlation coefficient 11,12 is:

11 12 0 0

11,12

11 11 12 12 0 0 0 0

* *

* * * *

E h h E V V



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2

0 0 10 0

2

20 0

1*

*1

m

m

E V V E z g dzdI

z g dzdI

I e

I e

Scatterers in the far-field region of the transmitter plane waves

travelling to the receiver

transmitting dipoles(top view)

e1, e2

g = path gain from a transmitter scatterer to receiver (a Gaussian random number with each scatterer)

e1,e2 = far fields generated by transmitting antennas


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0 0

2 2

1 220 0 0 0

2

1 220 0 0

22 cos

20 0 0

0

*

1 * * *

1 * * *

1 * t

m

m

jkd

m

t

E V V

I z I z dzdz E ge d g e dI

I z I z dzdz E ge g e dI

I z I z dzdz E g e dIC J kd

1 0e e cos2 0

tjkde e e

0 far field amplitudee

A constant


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Similarly,

0 0 0 0* *E V V E V V C

0 0

11,12 0 11,21

0 0 0 0

*

* * t

E V VJ kd

E V V E V V



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Example 3Similar to Examples 10 and 11 but now find the spatial correlation, 11,22, of the channels h11 and h22 of a MIMO system with N = 2 and M = 2, i.e., two receiving antennas and two transmitting antennas. Assume that the AOD at the transmitter and AOA at the receiver are both 360°.

h11

h22

Vin

Vin

Vo1

Vo2

dtdr


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Now the output voltages at the two receiving antennas Vo1and Vo2 can be expressed in terms of the channels as:

1 11 12

2 21 22

o in in

o in in

V h V h VV h V h V

Solutions

Thus the correlation coefficient 11,22 is:

11 22 011 022

11,22

11 11 22 22 011 011 022 022

* *

* * * *

E h h E V V



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where Vo11 and Vo22 are the partial output voltages at antenna 1 and antenna 2 that are due to signals passed through, respectively, channels h11 and h22. Combining the expressions in Examples 10 and 11, we have:

011 022

2 2

1 10 0 0

2 2

2 20 0 0

*

1

*1

m

m

E V V

E z g d dzdI

z g d dzdI

I e E

I e E


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As it is assumed that all the fields are polarization matched to the antennas (all aligned in the z direction), we have:

011 022

20 0

2 2 2 2

1 2 1 20 0 0 0

20 0

2 222 cos cos

00 0

0 0

*

1 *

* * *

1 *

t r

m

m

jkd jkd

t

E V V

I z I z dzdzI

E ge d g e d E d E d

I z I z dzdzI

E g e d E E e d

C J kd J kd

r


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Similarly,

011 011 022 022* *E V V E V V C

011 022

11,22 0 0 11,12 11,21

011 011 022 022

*

* * t r

E V VJ kd J kd

E V V E V V



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Notes:In a MIMO system with arbitrary numbers of transmitting (M) and receiving (N) dipole antennas and the antenna separations are dt in the transmitter and dr in the receiver, the correlation coefficients can be calculated two-by-two at a time. The general formula is:

, 0 0ij k t rJ kd i j J kd k


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2.1.1 Generation of a channel matrix H with specified spatial correlationIf the channel correlation is known, we can use a method [1] to generate the channel matrix H whose elements will have the required correlation. (1) Suppose H has the following form:

11 12 1

21 22 2

1 2

M

M

N N NM

h h hh h h

h h h

H

(2)


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(2) Form the following vector vec(H) by stacking the column vectors of H one-by-one:

11

1

12

2

1

vec( ) (the dimension of vec( ) is ×1)

N

N

M

NM

h

hh

NMh

h

h

H H

(3)


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(3) Obtain the covariance matrix RH of vec(H):

H =vec( )vec( )HR H H

(4) Find the eigenvalues and eigenvectors of RH.(5) Then the channel matrix H can be expressed as:

1/2vec( )=H VD r

where r (NM1) is a vector containing i.i.d. complex Guassian random numbers with a unit variance and a zero mean, V is the matrix whose column vectors are the eigenvectors of RH, and D is a diagonal matrix whose diagonal elements are the eigenvalues of RH.

(4)

(5)


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(6) Hence once the desired correlation is given (by specifying RH), H can be obtained by (5). The example on next page demonstrates how this is done.


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Example 4Write a Matlab program to obtain the channel matrix of a 33 MIMO system equipped with dipole antennas with antenna separations at the transmitter and receiver being 0.2 and 0.15, respectively. Assume that the channels are Gaussian random channels with a unit variance and a zero mean, and the antenna mutual coupling can be ignored. Hence calculate the channel capacity when the SNR = 20dB.

11 12 13

21 22 23

31 32 33

, 0,1ij

h h hh h h h CNh h h

H

Solutions


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H =vec( )vec( )HR H H

As the channels are Gaussian random number with a unit variance, the covariance matrix RH can be expressed as:

Instead of calculating RH directly using the above formula, it can be generated by a simple method. Since the antennas are dipoles, the channel correlation matrix r at the receiver (with a fix transmitting antenna, for example antenna 1) can be calculated first.

dt = 0.2, dr = 0.15


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11 11 11 21 11 31

21 11 21 21 21 31

31 11 31 21 31 31

0 0

0 0

0 0

* * *

* * *

* * *

1 0.3 0.60.3 1 0.30.6 0.3 1

r

E h h E h h E h h

E h h E h h E h h

E h h E h h E h h

J JJ JJ J

ρ

Then calculate the channel correlation matrix t at the transmitter (with a fix receiving antenna, for example antenna 1).


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11 11 11 12 11 13

0 0

12 11 12 21 12 31 0 0

0 0

13 11 13 21 13 31

* * *1 0.4 0.8

* * * 0.4 1 0.40.8 0.4 1* * *

t

E h h E h h E h hJ J

E h h E h h E h h J JJ J

E h h E h h E h h

ρ

Then it can be shown that RH is the Kronecker product of tand r. That is,

H = t rR ρ ρ

In Matlab, the Kronecker product is obtained by the command “kron(t,r)”.


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The Matlab codes are shown below (filename: correlated_H):clear all;

M=3; % number of transmit antennasN=3; % number of receive antennas

k=2*pi;dr=0.15 %lambdadt=0.20 %lambda

%-----------spatial channel correlations generation

for i=1:N;for j=1:N;pr(i,j)=bessel(0,k*dr*abs(j-i));end;end;


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for i=1:M;for j=1:M;pt(i,j)=bessel(0,k*dt*abs(j-i));end;end;

RH=kron(pt,pr);

[V,D] = eig(RH);G=V*sqrt(D);

%-----------channel matrix generationsnrdB=20;snr=10^(snrdB/10);

for n=1:5000;

r=sqrt(0.5)*(randn(N,M)+1j*randn(N,M));


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for j=1:M;for i=1:N;vec_r(i+(j-1)*N)=r(i,j);end;end;

vec_H=G*vec_r';

for j=1:M;for i=1:N;H(i,j)=vec_H(i+(j-1)*N);end;end;

%-----------capacity calculationC(n)=log2(real(det(eye(N)+snr/M*(H'*H))));end;

cdfplot(C)Average_C=mean(C)


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The average capacity is found to be 12.3 bits/s/Hz. The cdfof C is shown below.

4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

C (bits/s/Hz)

cdf(C

)


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2.2 Antenna mutual couplingFor MIMO systems, except the spatial correlation will contribute to the channel correlation, antenna mutual coupling will also contribute [2], [3]. In the transmitter antenna array, antenna mutual coupling causes the input signals being coupled into neighbouring antennas. This effect can be represented by a mutual coupling impedance matrix Zt (see Lecture Notes on “Mutual Coupling in Antenna Arrays):

1t sv Z v

where vs is the input voltage vector with mutual coupling not taken into account, v is the input voltage vector when mutual coupling is taken into account, and Zt is given by:

(6)


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11 12 1

21 22 2

1 2

1

1

1

N

L L L

N

L L Lt

N N NN

L L L

Z Z ZZ Z Z

Z Z ZZ Z Z

Z Z ZZ Z Z

Z

Similarly, for the output signals, they are also modified by the antenna mutual coupling effect in the receiving antenna arrays. The actual output voltage vector vo is related to the uncoupled output signal vector vu as:

(7)


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1o r u

v Z vwhere Zr is the mutual impedance matrix containing the receiving mutual impedances (see Lecture Notes on “Mutual Coupling in Antenna Arrays):

12 1

21 2

1 2

1

1

1

Nt t

L LN

t t

r L L

N Nt t

L L

Z ZZ Z

Z ZZ Z

Z ZZ Z

Z

(8)

(9)


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In (8), vo and vu are terminal voltage vectors across the antenna terminal loads. If the uncoupled output voltages refer to the open-circuit voltages, then vu is related to the open-circuit voltage vector voc as:

Lu oc

in L

ZZ Z

v v (10)

In (10), it is assumed that all the antenna elements have the same internal impedance Zin and terminal impedance ZL. Eq. (8) then becomes:

1Lo r oc

in L

ZZ Z

v Z v (11)


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Combining (6) and (11), we have signal model for a MIMO system under both spatial correlation and antenna mutual coupling as:

1 1

oc n

Lo r t s n

in L

ZZ Z

v Hv v

v Z HZ v v

where vn is the vector of noise voltages which are assumed to be not affected by antenna mutual coupling.

(12)


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Example 5Re-do Example 13 but now take the antenna mutual coupling into account. It is given that the mutual impedance between two transmitting antennas are:dt = 0.2, Z12 = 25.91-j15.34 , Z21 = 25.28-j15.78 dt = 0.4, Z12 = -0.90-j20.30 , Z21 = -1.42-j20.11 The mutual impedance between two receiving antennas are:dr = 0.15, Zt

12 = 17.73-j2.75 , Zt21 = 17.48-j2.94

dr = 0.30, Zt12 = 8.29-j10.44 , Zt

21 = 7.96-j10.51 The internal impedance of the dipole antennas is:Zin = 39.00+j7.17 The terminal load impedance of the dipole antennas is:ZL = 50


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SolutionsN = M = 3dt = 0.2, dr = 0.15

1.78 0.14 0.52 - 0.31 -0.02 - 0.410.51- 0.32 1.78 0.14 0.51- 0.32-0.03 - 0.40 0.52 - 0.31 1.78 0.14

t

j j jj j jj j j

Z

1 0.35 - 0.05 0.17 - 0.210.35 - 0.06 1 0.35 - 0.060.16 - 0.21 0.35 - 0.05 1

r

j jj jj j

Z


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The Matlab codes are shown below (filename: mu_correlated_H):

clear all;

M=3; % number of transmit antennasN=3; % number of receive antennas

k=2*pi;dr=0.15 %lambdadt=0.2 %lambda

%-----------spatial channel correlations generation

for i=1:N;for j=1:N;pr(i,j)=bessel(0,k*dr*abs(j-i));end;end;


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for i=1:M;for j=1:M;pt(i,j)=bessel(0,k*dt*abs(j-i));end;end;

RH=kron(pt,pr);

[V,D] = eig(RH);G=V*sqrt(D);

%--receiving and transmitting mutual impedance matrixes creationzin=39.00+1j*7.17;zl=50;


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z12=25.9059+1j*(-15.3365);z21=25.2796+1j*(-15.7831);

z13=-0.8920+1j*(-20.3036);z31=-1.4192+1j*(-20.1113);

zt12=17.73449488+1j*(-2.74569212);zt21=17.47727875+1j*(-2.94131405);

zt13=8.28960286+1j*(-10.43902986);zt31=7.96114038+1j*(-10.50848904);

zt=[1+zin/zl z12/zl z13/zlz21/zl 1+zin/zl z21/zlz31/zl z12/zl 1+zin/zl]


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zr=[1 zt12/zl zt13/zlzt21/zl 1 zt21/zlzt31/zl zt12/zl 1]

%-----------channel matrix generationsnrdB=20;snr=10^(snrdB/10);

for n=1:5000;

r=sqrt(0.5)*(randn(N,M)+1j*randn(N,M));

for j=1:M;for i=1:N;vec_r(i+(j-1)*N)=r(i,j);end;end;


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vec_H=G*vec_r';

for j=1:M;for i=1:N;H(i,j)=vec_H(i+(j-1)*N);end;end;

H=(zl/(zin+zl))*inv(zr)*r*inv(zt);

%-----------capacity calculationC(n)=log2(real(det(eye(N)+snr/M*(H*H'))));end;

cdfplot(C)Average_C=mean(C)


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The average capacity is found to be 9.2 bits/s/Hz. The cdf of C is shown below.

2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

C (bits/s/Hz)

cdf(C

)


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References:

[1] J. W. Wallace and M. A. Jensen, “Modeling the indoor MIMO wireless channel,” IEEE Transactions on Antennas and Propagation, vol. 50, no. 5, pp. 591-599, 2002.

[2] R. Janaswamy, “Effect of element mutual coupling on the capacity of fixed length linear arrays,” IEEE Antennas and Wireless Propagation Letters, vol. 1, pp. 157-160, 2002.

[3] H. T. Hui, "Influence of antenna characteristics on MIMO systems with compact monopole arrays," IEEE Antennas and Wireless Propagation Letters, vol. 8, pp. 133-136, 2009.

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Lecture Notes-Multiple Antennas for MIMO Communications - Channel Correlation