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Limits On Wireless Communication In Fading
Environment Using Multiple Antennas
Presented By Fabian Rozario
ECE Department
Paper By G.J. Foschini and M.J. Gans
OutlineIntroduction.Mathematical model.Capacity formulas.Lower bound on capacity.Capacity improvement.Comparison of various systems.Min-Max strategy.Summary.
IntroductionWe 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 fadingModel 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 TheoryUse Shannon capacity formula.
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.
Roughly for n antennas increase is n bits per 3db
increase in SNR.
)1(log* 2 SNRBC
)||.1(log 22 HC
Mathematical modelFocus on single point to point channel.
where
How does receiver diversity affect capacity
Noise remains same but output signal is linear
combination of diff antennas.
This is maximal ratio combiner.
)(.)ˆ/()(
)()(*)()(2/1 thPPtg
tvtstgtr
)||.1(log0
22
Rn
i
HC
Capacities: Matrix channel is RayleighRandom 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 preamble.
No Diversity case: nt=nr=1
|H| replaced by Chi squared variate with 2 degree of freedom.
].1[log 222 C
Capacities: ContdReceiver Diversity case: nt=1, nr=n
Transmit Diversity case: nt=n, nr=1
Combined Transmit and Receiver Diversity: nt=nr
Cycling using one transmitted at a time:
].1[log 222 nC
])./(1[log 222 nTnC
T
RT
n
nnkkTnC
)1(
222 ])./(1[log
T
R
n
iinTnC
1
222 ].1[log)./1(
Lower Bound On CapacityEmploys unitarily equivalent rectangular matrices, here H is
unitarily equivalent to m*n matrix.
Where are chi squared variables with j degree freedom.
Final result: contribution of the form L+Q
where
and Q is positive, negative term are cancelled out by
positive Q
hence C>L with probability 1.
22 , jj yx
m
nmjjXL
)1(
22 )1(
jTj
jTj
ynY
xnX
22/1
2
22/1
2
)/(
)/(
Capacity Derivation:One spatially cycled transmitting antenna/symbol:
Channel capacity defined in terms of mutual information
between input and output.
Where ε- entropy
Tn
iT
outcomeinoiseoutcomeioutputn
outputinputI1
)]|()|([.1
)/(
Capacity improvement: CCDF
2 antenna case 4 antenna case
Capacity per dimension:
Comparison of systems
Min-Max communication systemWhen multiple antennas are used the other antennas
will add noise.
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
SummaryWe 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 Tx.
See the use of min-max strategy.
This application is useful for indoor wireless LAN.
Thank You
