1
13
Layered MIMO systems and Interference Suppression
13.1
ML Detection in Layered MIMO systems
Consider a MIMO system, where nT symbols are simultaneously transmitted at
each time slot through nT transmit antennas:
r1
.
.
.
= [h1 , h2 , , hn ]
T
x1

1
11
Space-time trellis codes
At time t, the received signal at antenna j is given by
j
rt
nT
=
Es
ht xi + nj ,
t
j,i t
i = 1, , nT ; j = 1, , nR ; t = 1, , N
i=1
where nj NC (0, 2 ), ht NC (h , 1), and
t
j,i
j,i
11.1
E[|xi |2 ] = 1.
t
Tarokhs criteria (

Fig. 5-1
E. G. Larsson and P. Stoica, Space-time block coding for wireless communications, Cambridge 2003
Fig. 5-2 Ergodic capacity of an nT = nR =2 MIMO channel
A. Paulraj, R. Nabar, and D. Gore, Introduction to space-time wireless communications, Cambri

1
10
Pairwise Error Probabilities
One of the most powerful analytical tools in studying the performance of coded
communications over AWGN or fading channels is the pairwise error probability.
The pairwise error probability depends on the Euclidean dista

1
9
Beamforming
9.1
Rx Beamforming (SIMO): Maximum SNR
The received signal is
r = hR x + n
where r = (r1 , r2 , rnR )T , hR = (h1 , h2 , , hnR )T , n NC (0, 2 I), and the
total transmission power is given by E[|x|]2 = P .
At the receiver, a receive beam

1
7
Space-time block coding
7.1
7.1.1
Alamouti Coding [2,4]
Alamouti coding
The transmission matrix of Alamouti code is
x1 x
2
X2 =
x2 x1
Note that the symbol rate (code rate) is one.
Assume symbols x1 and x2 are transmitted, where x1 , x2 S. The rece

1
5
Capacity of MIMO systems
5.1
5.1.1
Deterministic MIMO channels
Review of Information Theory
For a continuous random vector z of length n, the dierential entropy is dened
as follows:
H(z) = E log2
1
pz (z)
=
log2
1
pz (z)dz
pz (z)
Recall that if z NC

Fig. 13-1. H-BLAST (V-BLAST)
G. J. Foschini et. al, Analysis and performance of some basic space-time architecture, IEEE JSAC, vol. 21, pp. 303-320, Apr. 2003.
(a)
(b)
Fig. 13-2. D-BLAST
G. J. Foschini et. al, Analysis and performance of some basic space-

1
6
Error probability analysis
6.1
Diversity gain and coding gain
In the high-SNR region, the average probability of a detection error over a fading
channel usually behaves as:
P (error) (Gc )Gd (SN R)Gd
1. Coding gain: The constant Gc is referred to as

1
4
MIMO System Model
4.1
4.1.1
Frequency at MIMO channel model
Spatially white MIMO channel model
Consider a MIMO channel with nT transmit and nR receive antennas.
Let hj,i denote the channel gain from transmit antenna i to receive antenna j.
The chann

1
3
Fading channel models
3.1
Frequency at and frequency selective fading
Delay spread is caused by reection and scattering of a microwave.
Let k denote the kth delay and P (k ) the received power at k . Then the mean
excess delay is dened to be
=
k k P

Fig. 7-1 Performance of Alamouti coding, BPSK
S. M. Alamouti, A simple transmit diversity technique for wireless communications, IEEE JSAC, pp. 1451-1458, Oct. 1998.
Fig. 7-2 OSTBCs: BPSK for X2; 4-PSK for X3 (rate-halving); 4-PSK for X4 (rate-halving)
V.

1
2
Review of probability theory, optimum decision,
and linear algebra
2.1
2.1.1
Probability theory
Multivariate Gaussian Random variables
Real Gaussian random vector:
A real vector x = (x1 , , xn )T is called Gaussian with mean E[x] = x and
covariance m

1
1
Introductory Examples
1.1
SISO
The received signal is
r = hx + n
where the total transmission power is P , i.e., E[|x|2 ] = P and the channel h
is modelled as a complex Gaussian RV with zero mean and variance 0.5 per
dimension. The noise n is modelle

Fig. 4-1 Local Scattering
E. G. Larsson and P. Stoica, Space-time block coding for wireless communications, Cambridge 2003 (page 13)
Fig. 4-2 Effect of correlated fading
E. G. Larsson and P. Stoica, Space-time block coding for wireless communications, Cam