Limits On Wireless Communication In Fading Environment Using Multiple Antennas Presented By Fabian...

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