MIMO
Multiple Input Multiple Output
Communications
“On the Capacity of Radio Communication Systems with Diversity in a Rayleigh
Fading Environment”
IEEE Journal on Selected Areas in Communication
VOL. SAC-5, NO. 5, JUNE 1987
© Omar Ahmad
Prepared for Advanced Wireless Networks, Spring 2006
Part 1
• An Intuition of SISO MISO and MIMO
• A Look at the Channel Capacity
An Intuition
SISO Single Input Single Output
Disclaimer: This Intuition is incomplete with respect to
how communication signals are actually analyzed
Forget about noise for now and the frequency domain transformation. Assume
we have an antenna, which transmits a signal x at a frequency f. As the signal propagates
through an environment, the signal is faded, which is modeled as a multiplicative coefficient h.
The received signal y will be hx.
x1
y 1 = h 1x 1
fading h1
transmit
receive
An Intuition
SIMO Single Input Multiple Output
Now assume we have two receiving antennas. There will be two received signals y1 and
y2 with different fading coefficients h1 and h2. The effect upon the signal x for a given
path (from a transmit antenna to a receive antenna) is called a channel.
The channel capacity has not increased
The multiple receive antennas can help us
get a stronger signal through diversity
y2 = h2x1
x1
fading h1
transmit
y1 = h1x1
receive
An Intuition
MISO Multiple Input Single Output
Assume 2 transmitting antennas and 1 receive antenna. There
Time 1
x2
-x1*
Time 1
x1
Time 2
Time 2
x2*
will be one received signal y1 (sum of x1h1 and x2h2). In order to
separate x1 and x2 we will need to also transmit, at a different
time, -x1* and x2*.
The channel capacity has not really increased because we
still have to transmit -x1* and x2* at time 2. (Alamouti scheme)
y1 = h1x1+ h2x2
y2 = h1x2*+ h2-x1*
transmit
receive
An Intuition
MIMO Multiple Input Multiple Output
With 2 transmitting antennas and 2 receiving antennas, we actually add a degree
of freedom! Its quite simple and intuitive. However, in this simple model, we are
assuming that the h coefficients of fading are independent, and uncorrelated. If they are
correlated, we will have a hard time finding an approximation for the inverse of H. In
practical terms, this means that we cannot recover x1 and x2.
x1
y1
y1 = h1x1+ h2x2
y2 = h3x1+ h4x2
x2
y2
h2 x1 w1

h4 x2 w2
Finally Assume there is some white Gaussian
Noise, and we have a set of linear equations
y = Hx + w
fading h4
transmit
h1

y2 h3
y1
receive
All 2 degrees of freedom are being utilized in
the MIMO case, giving us Spatial Multiplexing.
A Look at the Channel Capacity
x1
y1
Once again, the time invariant MIMO channel
is described by
y = Hx + w
x2
y2
fading h4
transmit
receive
H, the channel matrix, is assumed to be
constant, and known to both transmitter and
receiver. From basic linear algebra, every
linear transformation (i.e., H applied to x) can
be decomposed into a rotation, scale, and
another rotation (SVD)
H=
UV *
A Look a the Channel Capacity
H U V*
U and V are unitary (rotation) matrices.

Is a diagonal matrix whose elements:
1  2  3. . .  n
mi n
are the ordered singular values of the matrix H.The SVD can be rewritten as
nm i n
H   i ui vi*
i 1
We then Define
x'  V * x
y'  U * y
w'  U *w
And rewrite the channel y = Hx + w as
y'   x'  w'
or equivalently
yi'  i xi'  wi'
A Look at the Channel Capacity
yi'  i xi'  wi'
This expression looks VERY similar to something we should know how to
calculate the channel capacity of very easily! That is, Parallel Additive Gaussian
Channels where the channels are separated by time:
yi  xi  wi w h e r e wi h a s v a r i a n c e  i2
By information theory, we know the noise capacity to be for parallel Gaussian
Channels to be
 E 
1
C   l o g 1  n2  w h e r e
n 1 2
 n 
N
N
N
 E  x
2
n
n
n=1
E
n=1
So for the case of MIMO, the spatial dimension plays the role of time. The
capacity is now
 n2 En 
1
C   l o g 1 
2 
2
2

n 1
n 

N
A Look at the Channel Capacity
So what else does this mean? Each eigenvalue
1  2  3. . .  n
mi n
Corresponds to an eigenmode of the channel (also called
an eigen-channel) Each non-zero eigen-channel can
support a data stream;
thus, the capacity of MIMO depends upon
the rank of the channel matrix!
Part 2
Multipath Fading
Multipath Fading
Each entry in the Channel matrix is actually a sum of different multipaths which interfere with one another to form the fading coefficient.
We can easily show this in the time domain:
y (t )   ai (t ) x(t   i (t ) )
i

y (t ) 

h( , t ) x(t   )d

h( , t )   ai (t ) (   i (t ) )
i
The channel coefficients can be modeled as complex Rayleigh
fading coefficients. The analysis proceeds then with the following:
y[m]  h[m]x[m]  w[m]
Multipath Fading
• There should be a significant number of
multipaths for each of the coefficients
• The energy should be equally spread out
• If there are very few or no paths in some
of the directions, then H will be correlated
• The antennas should be properly spaced
otherwise H will be correlated
Conclusions
• MIMO adds a full degree of freedom
• Think of it as a dimensionality extension to
existing techniques of time and frequency
• The more entropy in the fading
environment, the more “richly” scattered,
and less likely for zero eigenvalues
• Rayleigh fading is a reasonable estimate