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Multiple Antennas for
MIMO Communications - Basic Theory
1 Introduction
The multiple-input multiple-output (MIMO) technology
(Fig. 1) is a breakthrough in wireless communication
system 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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Multiple Antennas for MIMO Communications - Basic Theory
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Fig. 1. A 33 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 Model
Consider 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.
y1
Space-time
processor
y2
xM
Space-time
processor
yN
hNM
Fig. 4. The channel model of a typical MIMO system.
The received signal vector y can be expressed in terms of
the channel matrix H as:
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y  Hx  n
(1)
where the symbols are:
Hon Tat Hui
 y1 
y 
2

  received signal vector
y
 
y 
 N
(2)
 x1 
x 
x   2   transmitted signal vector
  
x 
 M
(3)
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 h11 h12
h
h22
21
H

 
h
 N 1 hN 2
 h1M 
 h2 M 
  channel matrix
  
 hNM 
(4)
 n1 
n 
(5)
n   2   noise vector
  
n 
 M
Thereafter, we study the transmit power PT constrained
MIMO systems only, i.e., PT  C for some fixed C. We
can also write:
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PT  x x  x1  x2    xM  C
H
2
2
2
(6)
2.1 The Covariance Matrices
The covariance matrices of the transmitted signals and
received signals are:
R xx  E xx
H
R yy  E yy H


 E Hxx H H H   E nn H 
(7)
The covariance matrices are important parameters to
characterize a MIMO communication system.
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The traces of Rxx and Ryy give the total powers of the
transmitted and received signals, respectively. The offdiagonal elements of Rxx and Ryy give the correlations
between the signals at different antenna elements.
Consider a symbol period of time Ts, for the transmitted
signals, it is usually made such that:
R xx  E xx H   I M
Then within Ts,
(8)
R yy  E Hxx H H H   E nn H 
 HE xx H  H H  R nn
 HH H  R nn
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where Rnn is the noise covariance matrix. In (8) and (9),
we have assumed that the channels are stable within Ts.
Thus over a longer period of time (>> Ts), the average
received signal covariance matrix is:
R yy  E HH H   R nn
(10)
From (10), it can be seen that the received signal power is
determined by the channel covariance matrix E{HHH}
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
3.1 For SISO Systems
The channel capacity C of a single-input single-output
(SISO) system is given by [3]:
C  B log 2 1  S N  bit/s
(11)
where B (in Hz) is the channel bandwidth, S (in Watt) is
the 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.
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Example 1
A black-and-white TV screen picture may be considered as
composed of approximately 3105 picture elements. Assume
that each picture element has 10 brightness levels each being
equally 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 = 3.323105 = 9.96105 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:
R  30  9.96 105
 29.9 10 bit/s
6
This maximum information rate is the channel capacity C
for TV broadcast. That is,
R  29.9  10  C  B log 2 1  S N 
6
Therefore the bandwidth B can be calculated as:
C
29.9  106
B

 3  106  3 MHz
log 2 1  S N  log 2 1  1000 
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3.2 For MIMO Systems
For a MIMO system, the calculation of the capacity is
more complicated due to the determination of the signalto-noise ratio S/N.
Consider a MIMO system with a channel matrix H
(NM) as below:
y  Hx  n
(12)
By the singular value decomposition (SVD) theorem [4],
any N  M matrix H can be written as:
H  UDV H
(13)
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
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unitary matrix. That is, UUH = UHU = IN and VVH =
VHV = IM. The diagonal elements of D are called the
singular values of H and they are the non-negative square
roots of the eigenvalues  of the following equation:
 HH H  x   x, if N  M
 H
 H H  x   x, if N  M
(14)
where x is the N  1 eigenvector associated with .
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Example 2
Find the SVD for the following matrix (with N < M):
 2 5 1 4
H   4 3 2 2


6 3 1 2
Solutions
2
 2 5 1 4 
5
H


HH  4 3 2 2 

 1
6 3 1 2 
4
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4 6
 46 33 36

3 3 
  33 33 39

2 1 
36 39 50

2 2
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The eigenvalues of HHH are:
1 = 115.5900, 2 = 12.4511, 3 = 0.9588
Therefore,
 115.5900

0
D

0

0
12.4511
0
0

0
0

0.9588 0 
0
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
H
H  UDV   0.5258

0.6277
10.7513
 0

 0
0.1955 
-0.1749 -0.8324  

-0.5806 0.5185 
0
0
0
3.5286
0
0 

0
0.9792 0
0.7951
0.6527 -0.7349 0.1759 -0.0538 
 0.5888 0.4843 0.0363 0.6461 


 0.2096 -0.0384 -0.9711 -0.1077 
 0.4282 0.4731 0.1573 -0.7537 


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Now putting (13) into (12), we have,
y  UDV H x  n
(15)
Consider the following transformations:
 y  U H y

H

x  V x
n  U H n

(16)
Eq. (15) can be transformed as:
U H y  U H UDV H x  U H n
H

y  DV x  n
(17)
y  Dx  n
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The system in (17) is called the equivalent MIMO
system of (12). Note that:
R yy
H
H
H
H


 E y y   E U yy U  U R yy U
R xx  E xxH   E V H xx H V  V H R xx V
(18)
R nn  E nnH   E U H nn H U  U H R nn U
So that:
tr R yy   tr R yy 
tr R xx   tr R xx 
(19)
tr R nn   tr R nn 
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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 the
channel 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 , i = 1,
2, , N.
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The number of nonzero eigenvalues of matrix HHH is
equal to the rank of matrix H, denoted by r. This means
that we can expand (17) as:
yi  i xi  ni, for i  1,2, r
yi  0  ni,
for i  r ,2, N
(20)
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
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receiving antennas. This situation is illustrated below:
x’1
y’1
x’2
y’2
x’N
N
y’N
x’N+1
x’M
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 NM 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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x’1
y’1
x’2
y’2
x’M
M
y’M
y’M+1
y’N
Fig. 6. The equivalent MIMO system with M < 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,
r
 Pyi 
C  B log 2 1  2 
(21)
i 1
  
where B (in Hz) is the channel bandwidth, Pyi (in Watt) is
the 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.
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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 transmitting
antennas. 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:
P
Pyi  i
M
(22)
where P is the total transmitting power.
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Therefore, (21) can be written as:
r
P
P 



C  B  log2 1  i
 B log2  1  i
2
2
M 
M 

i 1
i 1 
r
(23)
The eigenvalue i in (23) can be expressed in terms of the
matrix HHH or HHH in (14) and (23) can be re-written as
(see details of derivation in [5], pp. 7-8):
P


H 
B
log
det
I

,
if
N

M
HH


N
2
2

M


C
 B log det  I  P H H H  , if N  M
M
2

M 2


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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:
Pr  P  Ploss
(25)
where Ploss is the average path loss from the transmitter to
the receiver. Then (24) can be re-written as:


Pr HH H 
 B log2 det  I N  M  2 P  , if N  M


loss 
C
H

P
H
H
 B log det I  r
, if N  M
M
2
2



M  Ploss 

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Or, in terms of the SNR at the receiving antennas , we
have:


 HH H 
 B log2 det  I N  M P  , if N  M


loss 
C
H

 B log det I   H H  , if N  M
2
 M M P 


loss 
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(B)
EE6832
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 
Pi      , i  1,2,, r  rank(H)
i 

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where if Pi in (28) is negative, it will be set to zero. The
parameter  in (28) is determined by satisfying the
transmitting power constraint:
r
P   Pi
(29)
i 1
With the transmitting powers in (28), the received powers
in (22) is then:
Pyi  i Pi   i    2 
(30)
The channel capacity is then obtained as:
1

C  B log 2 1  2  i    2  
 

i 1
r
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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 the
average 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 all
transmitting antennas. r = rank (H) = 1, 1 = 1. Therefore,
P
Py1  1  P
M
r
 Pyi 
P

C  B log 2 1  2   B log 2 1  2 
  
i 1
  
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Example 4
Find the channel capacity of a MIMO system with N = M = 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
1
1
H
1
1

Hon Tat Hui
1 1 1
1 1 1
 , r = rank(H)  1, 1  42  16
1 1 1
1 1 1
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Without CSI, the transmitter transmits power equally over all
transmitting antennas. Therefore,
P
Py1  1  4 P
M
 Pyi 
4P 

C  B log 2 1  2   B log 2 1  2 
  
i 1
  
r
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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
1 1 1 1
1 1 1 1
 , r = rank(H)  1, 1  42  16
H
1 1 1 1
1 1 1 1


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:
P
Py1  1  16 P
1
The capacity will then be:
 Pyi 
16 P 

C  B  log2 1  2   B log2 1  2 
 

i 1
  
r
Note the capacity in this example is much larger than the one
in Example 4, due to the availability of the CSI, i.e, H.
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Example 6
Find the channel capacity of a MIMO system with N = M = 4
and
1
0
H
0
0

0 0 0
1 0 0

0 1 0
0 0 1 
Assume that the total transmitting power = P and the noise
power at the receiver = 2. The transmitter has no knowledge
of the channels.
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Solutions
EE6832
r = rank( H )  4
1  2  3  4  1
Without channel knowledge, the transmitter transmits equally
over all transmitting antennas. Therefore,
P P
Py1  Py2  Py3  Py4  1  
M 4
 Pyi 
C  B log 2 1  2 
i 1
  
P 

 4 B log 2 1  2 
 4 
r
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Multiple Antennas for MIMO Communications - Basic Theory
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Example 7
Find the channel capacity of a MIMO system with N = 4, M =
1, and
1
1
H 
1
1

Assume that the total transmitting power = P and the noise
power at the receiver = 2. The transmitter has no knowledge
of the channels.
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Multiple Antennas for MIMO Communications - Basic Theory
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Solutions
r = rank( H )  1, 1  4
Without channel knowledge,
P
Py1  1   4 P
M
 Pyi 
C  B log 2 1  2 
i 1
  
4P 

 B log 2 1  2 
  
r
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Multiple Antennas for MIMO Communications - Basic Theory
NUS/ECE
EE6832
Example 8
Find the channel capacity of a MIMO system with N = 1, M =
4, and
H  1 1 1 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
 Pyi 
C  B log 2 1  2 
i 1
  
P

 B log 2 1  2 
  
r
r = rank(H)  1, 1  4
P
Py1  1  P
M
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Multiple Antennas for MIMO Communications - Basic Theory
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EE6832
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
 r11
r
H   21
 r31
r
 41
r12
r13
r22
r23
r32
r33
r42
r43
r14 
r24 

r34 
r44 
rij  aij  jbij (i, j  1,,4) where aij , bij
are real random Gaussian numbers
with E aij   E bij   0,
and Var aij   Var bij   1 2
rij (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.
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Multiple Antennas for MIMO Communications - Basic Theory
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EE6832
Normal distribution
Solutions
rij  aij  jbij (i, j  1,,4), aij , bij    0,1 2 
E rij   E aij   jE bij   0
 
Var rij   E rij
2
 E rij   E aij2   E bij2   1 2  1 2  1
2
Using (27) with N = M = 4,  = 20 dB, Ploss = 1, we have
100 H 

C  log2 det  I 4 
H H  bits/s/Hz
4


Normalized by
bandwidth B
Using Matlab, we can find
E C  E log2 det  I 4  25  H H H   22.1709 bits/s/Hz
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Multiple Antennas for MIMO Communications - Basic Theory
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EE6832
We can further plot the cdf of C as follows:
1
0.9
0.8
cdf (C)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
5
10
15
20
25
30
35
40
C (bits/s/Hz)
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Multiple Antennas for MIMO Communications - Basic Theory
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The Matlab codes are shown below (filename: mimo_iid.m):
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 runs
H=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)
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Multiple Antennas for MIMO Communications - Basic Theory
NUS/ECE
EE6832
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.
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Multiple Antennas for MIMO Communications - Basic Theory