Multi-Input Multi-Output Communication
MSc Laboratory Experiment
Multi-Input Multi-Output (MIMO) Communication
Academic supervisor: Dr. Cong Ling
Aims:
The main aim of this experiment is to give the students the concept of multi-input
multi-output (MIMO) communication. Moreover, it will give them the idea of channel
capacity, fading channel, QAM modulation, MIMO detection, space-time coding and
computer simulation of MIMO communication. All the experiments will be
performed using MATLAB.
Introduction:
MIMO is one of the key enabling technologies of broadband wireless. It makes use of
multiple antennas in communication. Multi-antenna is not new; it has been used in
diversity reception for decades. Traditionally, fading in wireless channels was
regarded as a nuisance, which was combated by means of diversity, namely, receiving
many copies of a signal from different antennas. In contrast, fading is now seen as a
beneficial factor in MIMO communication. In MIMO communication, antennas are
usually far spaced so as to achieve independent fading. It in fact takes advantage of
independence naturally provided by fading to create multiple spatial channels in
parallel. Hence, higher channel capacity is obtained over the same bandwidth, thereby
higher spectral efficiency.
MIMO Setup:
Fig. 1. Block diagram of a MIMO system.
Fig. 1 shows an nR-by-nT MIMO system. The nR-dimensional received signal y can
be written as
T
n
y H x n (1)
where x the nT-dimensional complex transmitted vector, H is the nR-by-nT complex
Gaussian channel matrix with i.i.d. entries, and n is the nR-dimensional i.i.d. complex
Gaussian noise vector. Each entry of H or n has unit variance. The transmitted signal
x satisfies the unit power constraint so that represents the average signal to noise
ration (SNR) at each receive antenna. Precise knowledge about H is assumed at the
receiver, which can be acquired by sending pilots through the channel.
x1
y1
xnT
ynR
nnR
Capacity:
Now we assume nR = nT = n. Conditioned on a particular channel realization H, the
instantaneous MIMO capacity is given by
2
log (bps/Hz )
H
T
C
n
H
I H H (2)
where | . | denotes matrix determinant.
Assume the channel exhibits sufficiently fast fading so that channel matrices H are
statistically independent over the time. For such ergodic channels, the Shannon
capacity is equal to the statistical average
ergodicC E C H H . (3)
0 5 10 15 20 25 30
0
5
10
15
20
25
30
35
SNR (dB)
C
a
p
a
c
it
y
(
b
p
s
/H
z
)
n
T
= n
R
= 1, 2, 3, 4
Fig. 2. MIMO capacity. Solid: MIMO fading channel; dotted: AWGN channel.
Fig. 2 contrasts the MIMO capacity to the capacity
2
log (1 ) of a single-antenna
complex AWGN channel. It is clear that the MIMO capacity can easily surpass that of
a Gaussian channel. The MIMO capacity actually increases linearly with the antenna
number. Hence fading is indeed a beneficial factor. This is in contrast to the common
intuition that fading impairs the performance. In fact, the fading penalty is
insignificant even for the single antenna case, as shown by the bottom two curves in
Fig. 2.
Exercise 1: It is difficult to derive the closed-form expression for MIMO capacity (3).
To generate a graph like Fig. 2, you can use Monte Carlo simulation, i.e., you
generate a large number (e.g., 10000) of random matrices H, and take the average of
CH. Plot a graph of MIMO capacity for nR = nT = 5, 6, …, 10, for SNR from 0 to 30
dB.
Note: In communications, SNR is usually measured in decibel (dB). One has the
formula
10
(dB) 10 log .
Transmission and Reception:
The incoming date are split into n substreams and simultaneously transmitted
through n antennas, as shown in Fig. 1. The receiver has to detection the transmitted
data. There are many ways to detection. Here we study a few of them.
The standard maximum-likelihood (ML) detector is given by
2
ˆ arg m in
C
T
n
x
x y H x (4)
where C denotes the transmitter constellation and ||.|| denotes the Euclidean norm.
However, it is slow for a large number of antennas.
Some suboptimal decoding strategies have been used in MIMO detection. The most
obvious strategy is zero-forcing (ZF)
1
ˆ
T
Q
n
x H y (5)
where Q{.} denotes the quantization rule. A well known drawback of ZF is the effect
of noise amplification, since the channel matrix H might be ill-conditioned.
The performance of ZF can be improved by taking into account the knowledge of
noise variance, which gives rise to minimum mean-square error (MMSE) detection
1
ˆ
H H
T T
Q
n n
x H H I H y . (6)
Fig. 3 shows the bit error rate of a 4-by-4 MIMO system with 64 QAM modulation
with different detectors. Gray mapping is used for QAM. It is a rule mapping bits to
QAM symbols and has the feature that neighboring symbols only differ in one bit.
This is to minimize the bit error rate when a symbol is in error.
15 20 25 30 35 40 45 50
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
SNR (dB)
B
it
E
rr
o
r
R
a
te
n
T
= n
R
= 4
ZF
ML Decoding
Fig. 3. Bit error rate of a 4-by-4 MIMO system with 64 QAM modulation
Exercise 2: Simulate a 2-by-2 MIMO system with QPSK modulation with Gray
mapping, and produce the performance curves of ML, ZF and MMSE detection like
Fig. 3, for SNR from 0 to 35 dB.
Space-Time Coding:
In the previous section, data are uncoded. Space-time codes can be used to improve
the performance of MIMO. There are many types of space-time codes. Here we
restrict to the famous Alamouti code for 2 antennas. Its code matrix is given by
*
1 2
*
2 1
x x
x x
X
where x1 and x2 are information symbols and * denotes complex conjugation. That is,
in the first time slot x1 and x2 are respectively sent through Antenna 1 and Antenna 2,
while in the second time slot –x2
*
and x1
*
are respectively sent through Antenna 1 and
Antenna 2. It can be seen that X is a unitary matrix.
In this case, the received signal is also a 2 2 matrix
T
n
Y H X N (7)
The ML detection rule reads
1 2
2
( , )
ˆ arg m in
x x C
T
F
n
x Y H X (8)
where the Frobenius norm of a matrix is defined as
2 2
,, i jF i j
a A .
Exercise 3: Simulate the Alamouti code with QPSK modulation and Gray mapping,
and produce the performance curve of ML detection for SNR from 0 to 35 dB.
Compare it with Exercise 2.
Remark 1: The ML detector (8) requires exhaustive search. In fact, it can be
simplified to symbol-by-symbol detection, thanks to orthogonality of code matrix X.
If you are interested, please refer to [1, Chap. 5] for details.
Remark 2: For Exercises 2 and 3, it is recommended to generate about 300 random
matrices H, each carrying 1,000,000 bits.
References:
[1] Paulraj A., Nabar R. and Gore D., Introduction to Space-Time Wireless
Communications, Cambridge Univ. Press, May 2003.
[2] Biglieri, E., MIMO Wireless Communications, Cambridge University Press, 2007.