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Capacity of Fading Channels Capacity of Fading Channels With Channel Side InformationWith Channel Side Information

Goldsmith, A.J.   Varaiya, P.P.   California Inst. of Technol., Pasadena, CA; IEEE Transactions on Information Theory

Publication Date: Nov 1997On page(s): 1986-1992Volume: 43,   Issue: 6 

203

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OutlineOutline

1. System Model2. Optimal Channel Capacity

• Channel known to Tx & Rx• Channel known to Rx Only

3. Sub-optimal Channel Capacity• Channel Inversion• Truncated Channel Inversion

4. Numerical Results5. Conclusion

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System ModelSystem Model

• Assumptions g[i] : Stationary & Ergodic No Estimation Error No Feedback delay

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OutlineOutline

1. System Model2. Optimal Channel Capacity

• Channel known to Tx & Rx• Channel known to Rx Only

3. Sub-optimal Channel Capacity• Channel Inversion• Truncated Channel Inversion

4. Numerical Results5. Conclusion

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Channel Known at Tx & RxChannel Known at Tx & Rx

Ss

s spCC )(

Channel Capacity

Set of Discrete Memoryless channels

Probability of channel being in state s

J. Wolfowitz, Coding Theorems of Information Theory, 2nd ed. New York: Springer-Verlag, 1964.

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Channel Known at Tx & RxChannel Known at Tx & Rx

sec]/)[1log( bitsBC

AWGN Channel Capacity (Received SNR )

Channel Bandwidth

)][()( ippdefine:

then:

dpCC )(

dpB )()1log( Fading Channel

Capacity

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Channel Known at Tx & RxChannel Known at Tx & Rx

• Transmit Power is allowed to adapt:

Coding Theorem: There exists a coding scheme with average power S that achieves any rate R < C(S) with arbitrarily small probability of error.

SdpS

)()(

dpS

SBSC

S

)()(

1logmax)()(

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Channel Known at Tx & RxChannel Known at Tx & Rx

0

00

0

11)(

S

S

0

1)()11

(0

dp

0

)(1log)(0

dpBSC

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Channel Known at Tx & RxChannel Known at Tx & Rx

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Channel Known only at RxChannel Known only at Rx

• McEliece: has shown that:

provided that: channel variation satisfy a compatibility constraint.

• The Constraint: Channel is i.i.d. (independently identically distributed) Input distribution is same regardless of channel state

R. J. McEliece and W. E. Stark, “Channels with block interference,”IEEE Trans. Inform. Theory, vol. IT-30, pp. 44–53, Jan. 1984.

dpBC )()1log(

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Channel Known only at RxChannel Known only at Rx

• Therefore, fading AWGN channel satisfy the constraint only if fading is i.i.d and constant Transmit Power S.

dpBSC )()1log()(

With iid fading and constant power, the availabilityof channel Information at Transmitter brings no extra capacity benefit. However coder complexity is reduced

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OutlineOutline

1. System Model2. Optimal Channel Capacity

• Channel known to Tx & Rx• Channel known to Rx Only

3. Sub-optimal Channel Capacity• Channel Inversion• Truncated Channel Inversion

4. Numerical Results5. Conclusion

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Sub-optimal (Channel Inversion)Sub-optimal (Channel Inversion)

S

S )(

Constant Received SNR

1)(

dp]/1[

1

E

Channel is no longer a fading channelIt becomes AWGN

)]/1[

11log()(

EBSC

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Sub-optimal (Truncated Channel Sub-optimal (Truncated Channel Inversion)Inversion)

0

00

..0

)(

S

S

dpE

)(1

]/1[

1

00

]/1[

11logmax)(

00 E

BSC

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OutlineOutline

1. System Model2. Optimal Channel Capacity

• Channel known to Tx & Rx• Channel known to Rx Only

3. Sub-optimal Channel Capacity• Channel Inversion• Truncated Channel Inversion

4. Numerical Results5. Conclusion

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Capacity in log-normal FadingCapacity in log-normal Fading

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Capacity in Rayleigh FadingCapacity in Rayleigh Fading

1m

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Capacity in Nakagami FadingCapacity in Nakagami Fading

2m

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ConclusionConclusion

• Capacity of Fading AWGN channel with average power constraint is calculated.

• When Channel is known to both Tx and Rx: Optimal adaptation is water filling for power and variable rate multiplexed coding.

• In correlated fading, adaptive schemes yields higher capacity and lower complexity.

• However iid fading, this gain is not appreciable.• Channel inversion has lowest coding and decodeing

complexity, but suffers large capacity loss under severe fading

• The capacity of all schemes converges to AWGN as fading severity if reduced.