Limits On Wireless
Communication In Fading
Environment Using Multiple
G.J. Foschini and M.J. Gans
Lower bound on capacity.
Comparison of various systems.
We make the following assumptions
Receiver knows channel characteristics but not transmitter.
Ie: fast feedback link required otherwise.
We allow changes to propagation environment to be slow in
time scale compared to burst rate.
Model used is Rayleigh fading.
Use information theory to find out increase in bit/cycle
compared to no of antennas used.
Introduction: Rayleigh fading
Model useful when no LOS path exists.
Zero mean Gaussian process.
Can be used to model ionospheric and tropospheric scatters.
If relative motion exists between TX and RX: fading is correlated
and varying in time.
We can decorrelate path losses by using antennas separated by
λ/2 on a rectangular lattice.
This belongs to small scale fading.
Introduction: Information Theory
Use Shannon capacity formula.
C B * log 2(1 SNR)
We get capacity in terms of bits/second.
In our application we can get the increase in bps/Hz for given no
of TX and RX.
C log 2(1 . | H | ) 2
Roughly for n antennas increase is n bits per 3db increase in SNR.
Focus on single point to point channel.
r (t ) g (t ) * s(t ) v(t )
ˆ 1/ 2
where g (t ) ( P / P) .h(t )
How does receiver diversity affect capacity
C log 2(1 . | H |2 )
Noise remains same but output signal is linear combination of diff
This is maximal ratio combiner.
Capacities: Matrix channel is Rayleigh
Random channel model(|H|) is treated as Rayleigh channel model
with zero mean, unit variance, complex.
H matrix is assumed to be measured at receiver using training
No Diversity case: nt=nr=1
C log 2[1 . 2 ]
|H| replaced by Chi squared variate with 2 degree of freedom.
Receiver Diversity case: nt=1, nr=n
C log 2[1 . 2 n ]
Transmit Diversity case: nt=n, nr=1
C log 2[1 ( / nT ). 2 n ]
Combined Transmit and Receiver Diversity: nt=nr
C log 2[1 ( / nT ). 2 k ]
k nT ( nR 1)
Cycling using one transmitted at a time:
C (1 / nT ). log 2[1 . 2 nR i ]
Lower Bound On Capacity
Employs unitarily equivalent rectangular matrices, here H is
unitarily equivalent to m*n matrix.
Where x 2 , y 2 are chi squared variables with j degree freedom.
Final result: contribution of the form L+Q
L (1 X 2 j )
j m ( n 1)
X 2 j ( / nT )1/ 2 x2 j
Y2 j ( / nT )1/ 2 y2 j
and Q is positive, negative term are cancelled out by positive Q
hence C>L with probability 1.
One spatially cycled transmitting antenna/symbol:
Channel capacity defined in terms of mutual information between
input and output.
I (input / output) .[ (output | i outcome) (noise | i outcome)]
nT i 1
Where ε- entropy
Capacity improvement: CCDF
2 antenna case 4 antenna case
Capacity per dimension:
Comparison of systems
Min-Max communication system
When multiple antennas are used the other antennas will add
Detectors have optimal combining.
Detect 1st signal component using optimal combining and treat
2nd component as noise.
After 1st is detected subtract that from received signal vector and
extract 2nd signal by optimal combining.
2nd component affected by thermal noise as 1st already removed.
Same procedure for second detector.
Min Max performance
We were able to analyze receiver and transmitter diversity.
Conclude that increase in bit rate is n bits/cycle for n antennas for
each 3db increase in SNR.
Compare various combinations of systems with different no of Rx and
See the use of min-max strategy.
This application is useful for indoor wireless LAN.