MIMOMultiple 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 AhmadPrepared for Advanced Wireless Networks, Spring 2006

Part 1

• An Intuition of SISO MISO and MIMO

• A Look at the Channel Capacity

An IntuitionSISO 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.

fading h1

y1 = h1x1

transmit receive

x1

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

fading h 2

y1 = h1x1

transmit receive

x1

fading h1

y2 = h2x1

An Intuition MISO Multiple Input Single Output

Assume 2 transmitting antennas and 1 receive antenna. There 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)

fading h2

y1 = h1x1+ h2x2

transmit receive

x1

fading h1

x2

y2 = h1x2*+ h2-x1*

Time 1 Time 2

-x1*

x2*Time 1 Time 2

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.

fading h2

y1 = h1x1+ h2x2

transmit receive

x2

x1

y2 = h3x1+ h4x2fading h 3

fading h1

fading h4

y1

y2

Finally Assume there is some white Gaussian Noise, and we have a set of linear equations

y = Hx + w

All 2 degrees of freedom are being utilized in the MIMO case, giving us Spatial Multiplexing.

2

1

2

1

43

21

2

1

w

w

x

x

hh

hh

y

y

A Look at the Channel Capacity

fading h2

transmit receive

x2

x1

fading h 3

fading h1

fading h4

y1

y2

Once again, the time invariant MIMO channel is described by

y = Hx + w

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 = *VU

A Look a the Channel Capacity*H U V

m i n1 2 3. . . n

m i n*

1

n

i i ii

H u v

' *x V x ' *y U y ' *w U w

' ' 'y x w

' ' 'i i i iy x w

U and V are unitary (rotation) matrices.

are the ordered singular values of the matrix H. The SVD can be rewritten as

We then Define

And rewrite the channel y = Hx + w as

or equivalently

Is a diagonal matrix whose elements:

A Look at the Channel Capacity

2i i w h e r e w h a s v a r i a n c e i i iy x w

N N2

n21 n = 1 n = 1

1l o g 1 w h e r e

2

Nn

nn n

EC E x E

' ' 'i i i iy x w

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:

By information theory, we know the noise capacity to be for parallel Gaussian Channels to be

So for the case of MIMO, the spatial dimension plays the role of time. The capacity is now

2

21

1l o g 1

2 2

Nn n

n n

EC

A Look at the Channel CapacitySo what else does this mean? Each eigenvalue

m i n1 2 3. . . 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 multi-paths which interfere with one another to form the fading coefficient.

We can easily show this in the time domain:

( ) ( ) ( ( ) )

( ) ( , ) ( )

( , ) ( ) ( ( ) )

[ ] [ ] [ ] [ ]

i ii

i ii

y t a t x t t

y t h t x t d

h t a t t

y m h m x m w m

The channel coefficients can be modeled as complex Rayleigh fading coefficients. The analysis proceeds then with the following:

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