# Limits On Wireless Communication In Fading Environment Using by pptfiles

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```									     Limits On Wireless
Communication In Fading
Environment Using Multiple
Antennas
Paper By
G.J. Foschini and M.J. Gans
Presented By
Fabian Rozario
ECE Department
Outline
 Introduction.
 Mathematical model.
 Capacity formulas.
 Lower bound on capacity.
 Capacity improvement.
 Comparison of various systems.
 Min-Max strategy.
 Summary.
Introduction
 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.
Mathematical model
 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
nR
C  log 2(1   . | H |2 )
i 0

 Noise remains same but output signal is linear combination of diff
antennas.

 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

preamble.

 No Diversity case: nt=nr=1
C  log 2[1   . 2 ]
2

|H| replaced by Chi squared variate with 2 degree of freedom.
Capacities: Contd
 Receiver Diversity case: nt=1, nr=n

C  log 2[1   . 2 n ]
2

 Transmit Diversity case: nt=n, nr=1

C  log 2[1  (  / nT ).  2 n ]
2

 Combined Transmit and Receiver Diversity: nt=nr
nT
C         log 2[1  (  / nT ). 2 k ]
k  nT ( nR 1)
2

 Cycling using one transmitted at a time:
nT
C  (1 / nT ). log 2[1   . 2 nR i ]
2

i 1
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.
j     j

 Final result: contribution of the form L+Q
m
L        (1  X 2 j )
2

j  m ( n 1)

X 2 j  (  / nT )1/ 2 x2 j
where
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.
Capacity Derivation:
 One spatially cycled transmitting antenna/symbol:
 Channel capacity defined in terms of mutual information between

input and output.

1 nT
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
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
Summary
 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
Tx.

 See the use of min-max strategy.

 This application is useful for indoor wireless LAN.
Thank You

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