mimocapacity ln

Space-Time Coding 1'

&

$

%

Space-Time Coding

Hamid Jafarkhani

University of California, Irvine


&

$

%

Capacity of MIMO Channels

Copyright c© 2005 by Hamid Jafarkhani


&

$

%

MIMO Channels

• There are N transmit and M receive antennas.

• At each time, N signals are transmitted at the same time each

from a different transmit antenna.

• These signals have the same transmission period.

• Signals transmitted from different antennas undergo

independent fades.

• The received signal at each receive antenna is a linear

superposition of the transmitted signals perturbed by noise.


&

$

%

1@

@@

@

BBBBBBBBBBBB

2��

��

AAAAAAAA

N-1��

��

α1,1 α1,2 · · · α1,M

α2,1 α2,2 · · · α2,M

...

αN−1,1 αN−1,2 · · · αN−1,M

��

��

N��

��

αN,1 αN,2 · · · αN,M

1

2

...

M

Figure 1: A multiple-input multiple-output (MIMO) channel


&

$

%

Transmission Model

• At time t, if Ct,n is transmitted from antenna n, the signal

rt,m, received at antenna m is given by

rt,m =

N∑

n=1

αn,mCt,n + ηt,m,

where αn,m is the path gain from transmit antenna n to receive

antenna m. We assume a flat fading and quasi-static channel.

• Usually, αn,m is an independent complex Gaussian random

variable with variance 0.5 per real dimension and ηt,m is a

zero-mean complex Gaussian random variable with variance

N/(2 γ) per complex dimension.


&

$

%

Transmission Model

r = C.H + N (1)

where r is the T × M received matrix, C is the T × N transmit

matrix, N is the T × M noise matrix and

H =

α1,1 α1,2 · · · α1,M

α2,1 α2,2 · · · α2,M

αN,1 αN,2 · · · αN,M


&

$

%

Transmission Rate and Diversity

• If different path gains in a MIMO channel fade independently,

it is very unlikely that the channel matrix H in Equation (1)

becomes singular.

• One can multiply both sides of Equation (1) by the inverse of

H to recover the transmitted signals.

• Therefore, one can increase the number of transmitted

independent information symbols by the number of transmit

antennas.

• However, such a system does not provide any diversity when

the number of receive antenna is the same as the number of

transmit antennas (a diversity of M − N + 1 is possible if

M > N).


&

$

%

Capacity of a deterministic MIMO Channel

• The mutual information between input and output for the

given realization H = H is

I(C; r|H = H) = log2

(

det[

IM + (γ/N)HH · KC · H])

• The capacity is the maximum of the above mutual information

over all inputs satisfying Tr(KC) ≤ P

C = maxTr(KC)≤N

log2

(

det[

IM + (γ/N)HH · KC · H])

bits/sec/Hz


&

$

%

Capacity of a deterministic MIMO Channel

• When the channel is not known at the transmitter, we assume

an equal power, KC = IN that results in

CEP = log2

(

det[

IM + (γ/N)HH · H])

bits/sec/Hz


&

$

%

Capacity of a random MIMO Channel

• Every realization of the channel has some probability attached

to it through the statistical model of H.

• Since the channel matrix H is random in nature, the capacity

can be considered as a random variable

C = log2

(

det[

IM + (γ/N)HH · H])

bits/sec/Hz


&

$

%

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Capacity

Com

plem

enta

ry C

DF

N=1,M=1N=1,M=2N=2,M=1N=2,M=2

Figure 2: The complementary cumulative distribution function

(CDF) of Shannon capacity; γ = 10.


&

$

%


• In special cases, the capacity can be written in terms of

chi-square random variables

• A chi-square random variable with 2k degrees of freedom using

our normalization is

fχk(x) =

e−xxk−1

[k − 1]!, x > 0.


&

$

%

Capacity of SISO Channels

• For the case of one transmit and one receive antenna, the

capacity formula becomes

C = log2(1 + γ |α|2)

where α = α1,1 is the fade factor of the SISO channel.

• This capacity coincides with the standard Shannon capacity of

a Gaussian channel for a given value of α.

• Since α is a complex Gaussian random variable, |α|2 is a

chi-square random variable with two degrees of freedom or

equivalently an exponential random variable with unit mean as

follows:

f|α|2(x) = e−x, x > 0.


&

$

%

Capacity of SIMO Channels

• For N = 1 transmit antenna (SIMO)

C = log2(det[IM + γ HH · H)] = log2(1 + γ H.HH) =

log2(1 + γ

M∑

m=1

|α1,m|2).

• Assuming independent Rayleigh fading, the capacity is

C = log2(1 + γ.Xr)

where Xr is a chi-square random variable with 2M degrees of

freedom.


&

$

%

Capacity of MISO Channels

• For M = 1 receive antenna (MISO)

C = log2(1 + (γ/N).Xt)

where Xt is a chi-square random variable with 2N degrees of

freedom.


&

$

%


• For the case of N ≥ M , a lower bound on the capacity in terms

of chi-square random variables is

C >N

∑

k=N−M+1

log2(1 + (γ/N).χk)

• For the special case of N = M , we denote this lower bound by

CN

CN =N

∑

k=1

log2(1 + (γ/N).χk)

• The capacity increases linearly as a function of N as N → ∞


&

$

%

Ergodic Capacity

• The ergodic capacity of a MIMO channel is

CE = E[

log2

(

det{

IM + (γ/N)HH · H})]

• For a large number of transmit antennas and a fixed number of

receive antennas, using the law of large numbers,

HH · H

N−→ IM , almost surely

• The ergodic capacity is M log2(1 + γ) for large N


&

$

%

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

SNR (dB)

Erg

odic

Cap

acity

(bi

ts/s

ec/H

z)N=1,M=1N=2,M=1N=1,M=2N=2,M=2

Figure 3: Ergodic capacity for N transmit and M receive antennas.


&

$

%

Outage Capacity

• The outage capacity Cout is a value that the random variable C

is smaller than only with a probability Pout (outage

probability).

• Similarly, one can fix the outage capacity and find the outage

probability (the probability that the capacity random variable

is smaller than the outage capacity).

Pout = P (C < Cout).


&

$

%

Outage Capacity (cont.)

• The importance of the outage probability is that if one wants

to transmit Cout bits/cycle, the capacity of the channel is less

than Cout with probability Pout.

• For a stationary channel, if we transmit a large number of

frames, such a transmission fails for Pout times the total

number of frames.

• The value of the outage capacity Cout guarantees that it is

possible to transmit Cout bits/cycle with a probability of

1 − Pout.

• Picking a higher outage probability for a fixed received signal

to noise ratio results in a larger outage capacity.


&

$

%

Outage Capacity (cont.)

• For N = 1 transmit and M receive antennas

Pout = P

(

Xr <2Cout − 1

γ

)

where Xr is a chi-square random variable with 2M degrees of

freedom.

• For M = 1 receive and N transmit antennas

Pout = P

(

Xt < N2Cout − 1

γ

)

where Xt is a chi-square random variable with 2N degrees of

freedom.


&

$

%

10 11 12 13 14 15 16 17 18 19 20

10−4

10−3

10−2

10−1

SNR (dB)

Out

age

Pro

babi

lity

2 bits/sec/HZ

N=2N=3N=4M=2M=3M=4

Figure 4: Cout=2 bits/sec/Hz; N transmit antennas and one receive

antenna. Also, one transmit antenna and M receive antennas.


&

$

%

MIMO Capacity Conclusions

• The capacity increases by the minimum of the number of

transmit and receive antennas.

• One can show that at high SNRs the capacity of a MIMO

channel in terms of SNR can be described as

C ≈ min{N, M} log2(γ/N) +

min{N,M}∑

k=|N−M |+1

log2(χk)

where χk is a chi-square random variable with 2k degrees of

freedom.

• Therefore, 3 dB increase in SNR results in min{N, M} extra

bits of capacity at high SNRs.

• How to exploit this capacity?

mimocapacity ln

Documents