Lecture Notes-Multiple Antennas for MIMO Communications - Basic Theory

8/13/2019 Lecture Notes-Multiple Antennas for MIMO Communications - Basic Theory

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Hon Tat Hui Multiple Antennas for MIMO Communications - Basic Theory

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Multiple Antennas forMIMO Communications - Basic Theory

1 Introduction

The multiple-input multiple-output (MIMO) technology

(Fig. 1) is a breakthrough in wireless communicationsystem design. It uses the spatial dimension (provided by

the multiple antennas at the transmitter and the receiver) to

combat the multipath fading effect. Fig. 2 shows the

dramatic increase in transmission data rate with the

increase in the number of transmitting and receiving

antennas M of a MIMO system.




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Fig. 1. A 33 MIMO system [1].




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Fig. 2. The average data rate versus SNR with different number of antennas

M in a MIMO system. The channel bandwidth is 100 kHz [2].




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2 Channel and Signal ModelConsider a typical MIMO system shown in Fig 3 below.

Fig. 3. A typical MIMO system including the signal processing subsystems.




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The wireless channel part is extracted below.

The received signal vector y can be expressed in terms of

the channel matrix H as:

Fig. 4. The channel model of a typical MIMO system.

Space-time

processor

Space-time

processor

y N

y2

y1

x M

h NM




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y Hx n

1

2

1

2

received signal vector

transmitted signal vector

N

M

y y

y

x

x

x

y

x

where the symbols are:(1)

(2)

(3)




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11 12 1

21 22 2

1 2

1

2

channel matrix

noise vector

M

M

N N NM

M

h h hh h h

h h h

n

n

n

H

n

Thereafter, we study the transmit power PT constrained

MIMO systems only, i.e., PT C for some fixed C . We

can also write:

(5)

(4)




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2 2 21 2

H T M P x x x C x x (6)

The covariance matrices of the transmitted signals andreceived signals are:

H

xx

H

yy

H H H

E

E

E E

R xx

R yy

Hxx H nn

2.1 The Covariance Matrices

The covariance matrices are important parameters to

characterize a MIMO communication system.

(7)




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The traces ofR

xx andR

yy give the total powers of thetransmitted and received signals, respectively. The off-

diagonal elements of R xx and R yy give the correlations

between the signals at different antenna elements.

Consider a symbol period of time T s, for the transmitted

signals, it is usually made such that:

H

xx M E R xx I

H H H

yy

H H

nn

H

nn

E E

E

R Hxx H nn

H xx H R

HH R

Then within T s,(8)

(9)




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where Rnn

is the noise covariance matrix. In (8) and (9),

we have assumed that the channels are stable within T s.

Thus over a longer period of time (>> T s), the average

received signal covariance matrix is:

H

yy nn E R HH R (10)

From (10), it can be seen that the received signal power is

determined by the channel covariance matrix E {HH H }

and the noise covariance matrix Rnn. As Rnn is

determined by the environment and cannot be changed,

we can manipulate or select an H to optimize the channel

output SNR (so as the capacity) of the MIMO system.




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3 Channel Capacity

The channel capacity C of a single-input single-output

(SISO) system is given by [3]:

2log 1 bit/sC B S N

where B (in Hz) is the channel bandwidth, S (in Watt) isthe signal power, and N (in Watt) is the noise power.

Both S and N are measured at the output of the channel.

The channel capacity is a measure of the maximum rate

that information (in bits) can be transmitted through the

channel with an arbitrarily small error after using a

certain coding method.

(11)

3.1 For SISO Systems




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Example 1

A black-and-white TV screen picture may be considered as

composed of approximately 3105 picture elements. Assume

that each picture element has 10 brightness levels each beingequally likely to occur. TV signals are transmitted at 30

picture frames per second. The signal-to-noise ratio at the

TV is required to be at least 30 dB. What is the required

channel bandwidth for TV broadcast?

Solutions

Information per picture element = log210 = 3.32 bits

Information per picture frame = 3.323105 = 9.96105 bits




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As 30 picture frames are transmitting per second, therefore

the maximum information rate, R, for the TV transmission is

then:5

6

30 9.96 10

29.9 10 bit/s

R

This maximum information rate is the channel capacity C

for TV broadcast. That is, 6

229.9 10 log 1 R C B S N

Therefore the bandwidth B can be calculated as:

66

2 2

29.9 103 10 3 MHz

log 1 log 1 1000

C B

S N




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For a MIMO system, the calculation of the capacity is

more complicated due to the determination of the signal-

to-noise ratio S / N .

Consider a MIMO system with a channel matrix H

( N M ) as below:

3.2 For MIMO Systems

y Hx n

By the singular value decomposition (SVD) theorem [4],

any N M matrix H can be written as:

H H UDV

where D is an N M a diagonal matrix with non-negative

elements, U is an N N unitary matrix, and V is a M M

(12)

(13)




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Example 2

Find the SVD for the following matrix (with N < M ):

2 5 1 4

4 3 2 2

6 3 1 2

H

Solutions

2 4 62 5 1 4 46 33 36

5 3 34 3 2 2 33 33 391 2 1

6 3 1 2 36 39 504 2 2

H

HH




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The eigenvalues of HH H are:

1 = 115.5900, 2 = 12.4511, 3 = 0.9588

Therefore,

115.5900 0 0 0

0 12.4511 0 0

0 0 0.9588 0

D

By using Matlab with the command: [U,S,V]=svd(H), we

can find the SVD of H as:




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0.5741 0.7951 0.1955

0.5258 -0.1749 -0.8324

0.6277 -0.5806 0.5185

10.7513 0 0 0

0 3.5286 0 0

0 0 0.9792 00.6527 -0.7349 0.1759 -0.0538

0.5888 0.4843 0.0363 0.6461 0.

H

H UDV

2096 -0.0384 -0.9711 -0.1077

0.4282 0.4731 0.1573 -0.7537

H




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H y UDV x n

Consider the following transformations:

H

H

H

y U y

x V x

n U n

(15)

H H H H

H

U y U UDV x U ny DV x n

y Dx n

Eq. (15) can be transformed as:

(16)

(17)

Now putting (13) into (12), we have,




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The system in (17) is called the equivalent MIMO

system of (12). Note that:

H H H H

y y yy

H H H H

x x xx

H H H H

n n nn

E E

E E

E E

R y y U yy U U R U

R x x V xx V V R V

R n n U nn U U R U

So that:

y y yy

x x xx

n n nn

tr tr

tr tr

tr tr

R R

R R

R R

(18)

(19)




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This means that the equivalent MIMO system has the

same total input power, total output power and total noise

power as the actual MIMO system in (12). The output

SNR of the equivalent MIMO system is thus the same as

the actual MIMO system. This in turn means that thechannel capacity of the equivalent MIMO system is the

same as that of the actual MIMO system because capacity

is a function of the output SNR.

Now the system in (17) has its channels all decoupled.

The N channels are parallel to each other, with channel

gains given by the diagonal elements of D, i.e., , i = 1,

2, , N .i




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The number of nonzero eigenvalues of matrix HH H is

equal to the rank of matrix H, denoted by r . This means

that we can expand (17) as:

, for 1,2,

0 , for ,2,

i i i i

i i

y x n i r

y n i r N

We note that if the MIMO system has more transmitting

antennas than the receiving antennas ( M > N ), than H is a

horizontal matrix with a maximum rank = N . According

to (20), the maximum number of uncoupled equivalent

MIMO channels is N (< M ). The remaining M - N

transmitting antennas will become redundant with no

(20)




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receiving antennas. This situation is illustrated below: x’1

x’2

x’ N

x’ N +1

x’1

x’ M

y’1

y’2

y’ N

N

Fig. 5. The equivalent MIMO system with M > N .




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On the other hand, if the MIMO system has more

receiving antennas than the transmitting antennas ( M < N ),

than H is a vertical matrix with a maximum rank = M .

According to (20), the maximum number of uncoupled

equivalent MIMO channels is M (< N ). The remaining N - M receiving antennas will become redundant with no

received signals. This is illustrated on next page.

In general, for an N M MIMO system, the maximum

number of uncoupled equivalent channels is min( N , M ).




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Fig. 6. The equivalent MIMO system with M < N .

x’1

x’2

x’ M M

y’1

y’2

y’ M

y’ M +1

y’ N




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As the channels of the equivalent MIMO system in (17)

are uncoupled and parallel, the channel capacity of (17)can be calculated by a summation of the individual

capacities of the parallel channels. That is,

2 21

log 1 ir

y

i

PC B

(21)

where B (in Hz) is the channel bandwidth, (in Watt) isthe power received at the ith receiving antenna, 2 (in

Watt) is the noise power at the ith receiving antenna, and

r is the rank of H. In order to related the received power

to the channel parameters, we need to classify a MIMO

system according to the availability of the channel

knowledge to the transmitter or receiver.

i yP




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(A) Channel state information (CSI) known to the

receiver only

As the transmitter does not know the CSI, its best strategy

is to transmit power equally from all its transmittingantennas. For the equivalent MIMO system in (17), this

can be done by making all the elements of x’ to have the

same power. Under this situation, the received power is

then calculated as:

i y i

P

P

where P is the total transmitting power.

(22)




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Therefore, (21) can be written as:

2 22 21 1

log 1 log 1r r

i i

i i

P PC B B

M M

The eigenvalue i in (23) can be expressed in terms of the

matrix HH H or H H H in (14) and (23) can be re-written as

(see details of derivation in [5], pp. 7-8):

(23)

2 2

2 2

log det , if

log det , if

H

N

H

M

P B I N M

M C P

B I N M M

HH

H H

(24)




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The total transmitting power P in (24) may not be easily

known. If the average received powers Pr at each of the

receiving antennas are the same, we have:

2 2

2 2

log det , if

log det , if

H

r N

loss

H

r M

loss

P B I N M

M P

C P B I N M

M P

HH

H H

r lossP P P where Ploss is the average path loss from the transmitter to

the receiver. Then (24) can be re-written as:

(25)

(26)




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2

2

log det , if

log det , if

H

N

loss

H

M loss

B I N M M PC

B I N M M P

HH

H H

Or, in terms of the SNR at the receiving antennas , we

have:

(27)




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(B) Channel state information (CSI) known to both

the transmitter and receiver

If the transmitter knows the CSI, i.e., the channel matrix

H, its best strategy is to transmit more powers along those

channels whose channel gains are larger and to transmit

less powers or along those channels with a smaller

channel gain. This is called the water-filling principle.

Under this condition, the transmitting power Pi for the ith

channel in the equivalent MIMO system in (17) is given

by (see details of derivation in [5], pp. 45-46):

2

, 1,2, , rank( )i

i

P i r

H (28)




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3.3 Random channels

When the channels are random in nature, the channel

capacity is a random number. The most popular random

channel model is the Gaussian channel model H whose

channel matrix elements are all complex Gaussian

random numbers with a mean and a variance 2. Note

that the channel capacity expression is same as for the

deterministic channel case except that C becomes a

random number. Because the capacity is a random

number, a pdf and cdf of C can be obtained. Instead of

finding the instantaneous C , it is more often to find theaverage channel capacity E {C }.




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Example 3

Find the channel capacity of a MIMO system with N = M = 1

and H = h = 1. Assume that the total transmitting power = P

and the noise power at the receiver = 2. The transmitter has

no knowledge of the channels.

Solutions

Without CSI, the transmitter transmits power equally over alltransmitting antennas. r = rank (H) = 1, 1 = 1. Therefore,

2 22 21

log 1 log 1i

r y

i

P PC B B

1 1 y

PP P




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Example 4


and hij = 1, (i = 1,2,…,4, j = 1,2,…,4). Assume that the total

transmitting power = P and the noise power at the receiver =

2. The transmitter has no knowledge of the channels.

Solutions

2

1

1 1 1 1

1 1 1 1, = rank( ) 1, 4 16

1 1 1 1

1 1 1 1

r

H H




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1 1 4 y

P

P P

2 22 21

4log 1 log 1

i

r y

i

P PC B B

Without CSI, the transmitter transmits power equally over all

transmitting antennas. Therefore,




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Example 5

The conditions are same as those in Example 4 but the

transmitter now knows the channel matrix H perfectly. Find

the channel capacity.

Solutions

2

1

1 1 1 1

1 1 1 1, = rank( ) 1, 4 16

1 1 1 1

1 1 1 1

r

H H

With knowledge of H, the transmitter can transmit power

along only one channel, i.e., the channel with eigenvalue 1.




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The received power will then be:

1 1 161

y

PP P

The capacity will then be:

2 22 21

16log 1 log 1i

r y

i

P PC B B

Note the capacity in this example is much larger than the onein Example 4, due to the availability of the CSI, i.e, H.

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Example 6


and

Assume that the total transmitting power = P and the noise

power at the receiver = 2. The transmitter has no knowledge

of the channels.

1 0 0 00 1 0 0

0 0 1 0

0 0 0 1

H

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Solutions

1 2 3 4

= rank( ) 4

1

r

H

2 2

1

2 2

log 1

4 log 14

i

r y

i

PC B

P B

1 2 3 4 1 4 y y y y

P PP P P P

Without channel knowledge, the transmitter transmits equallyover all transmitting antennas. Therefore,

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Example 7

Find the channel capacity of a MIMO system with N = 4, M =

1, and



of the channels.

11

1

1

H

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Solutions

1= rank( ) 1, 4r H

2 21

2 2

log 1

4log 1

i

r y

i

PC B

P B

1 1 4 y

PP P

Without channel knowledge,

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Example 8

Find the channel capacity of a MIMO system with N = 1, M =

4, and



of the channels.

1 1 1 1

H

Solutions

1

1

1

= rank( ) 1, 4

y

r

PP P

H 2 2

1

2 2

log 1

log 1

i

r y

i

PC B

P B

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Example 9

Find the average channel capacity of a MIMO system with N

= M = 4 and the channel matrix H is a random matrix with

r ij (i, j = 1, …, 4) are random complex numbers with a mean

equal to zero and a variance equal to one. Assume that the

transmitter has no knowledge of the channels. The SNR at

the receiving antennas is = 20 dB and there is no path loss

such that Ploss = 1.

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

r r r r

r r r r

r r r r

r r r r

H

( , 1, ,4) where ,are real random Gaussian numbers

with 0,

and 1 2

ij ij ij ij ij

ij ij

ij ij

r a jb i j a b

E a E b

Var a Var b

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Hon Tat Hui Multiple Antennas for MIMO Communications - Basic Theory45

Solutions

Using (27) with N = M = 4, = 20 dB, Ploss = 1, we have

2 4

100log det bits/s/Hz

4

H C I

H H

22 2 2

( , 1, ,4), , 0,1 2

0

1 2 1 2 1

ij ij ij ij ij

ij ij ij

ij ij ij ij ij

r a jb i j a b

E r E a jE b

Var r E r E r E a E b

Normal distribution

Using Matlab, we can find

2 4log det 25 22.1709 bits/s/Hz H E C E I H H

Normalized by bandwidth B

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We can further plot the cdf of C as follows:

5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

C (bits/s/Hz)

c d f ( C )

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clear all;

M=4; % number of transmitting antennas

N=4; % number of receiving antennas

snrdB=20; % SNR

snr=10^(snrdB/10); % SNR in numerical value

for n=1:5000; % number of runsH=sqrt(0.5)*(randn(N,M)+1j*randn(N,M)); % channel matrix

C(n)=log2(real(det(eye(N)+snr/M*(H’*H)))); % random capacity

end;

cdfplot(C)

Average_capacity=mean(C)

The Matlab codes are shown below (filename: mimo_iid.m):

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References:

[1] D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, “From theory to practice: an overview of MIMO space–time coded wireless systems,” IEEE

Journal on Selected Areas in Communications, vol. 21, no. 3, pp. 281-302,

2003.

[2] E. Biglieri, R. Calderban, A. Constantinides, A. Goldsmith, A. Paulraj, and H.V. Poor, MIMO Wireless Communications, Cambridge University Press, 2007.

[3] F. G. Stremler, Introduction to Communication Systems, Addison-Wesley,

1982.

[4] R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1985.

[5] Branka Vucetic and Jinhong Yuan, Space-Time Coding, John Wiley & Sons

Ltd, 2003.

Lecture Notes-Multiple Antennas for MIMO Communications - Basic Theory

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