Air Interface Club Lra Fading Channels
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Transcript of Air Interface Club Lra Fading Channels
Air Interface Club
24 February, 2005 Asif Hamid 1/15
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
Air Interface Club
24 February, 2005 Asif Hamid 2/15
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
Air Interface Club
24 February, 2005 Asif Hamid 3/15
System ModelSystem Model
• Assumptions g[i] : Stationary & Ergodic No Estimation Error No Feedback delay
Air Interface Club
24 February, 2005 Asif Hamid 4/15
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
Air Interface Club
24 February, 2005 Asif Hamid 5/15
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.
Air Interface Club
24 February, 2005 Asif Hamid 6/15
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
Air Interface Club
24 February, 2005 Asif Hamid 7/15
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)()(
Air Interface Club
24 February, 2005 Asif Hamid 8/15
Channel Known at Tx & RxChannel Known at Tx & Rx
0
00
0
11)(
S
S
0
1)()11
(0
dp
0
)(1log)(0
dpBSC
Air Interface Club
24 February, 2005 Asif Hamid 9/15
Channel Known at Tx & RxChannel Known at Tx & Rx
Air Interface Club
24 February, 2005 Asif Hamid 10/15
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(
Air Interface Club
24 February, 2005 Asif Hamid 11/15
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
Air Interface Club
24 February, 2005 Asif Hamid 12/15
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
Air Interface Club
24 February, 2005 Asif Hamid 13/15
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
Air Interface Club
24 February, 2005 Asif Hamid 14/15
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
Air Interface Club
24 February, 2005 Asif Hamid 15/15
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
Air Interface Club
24 February, 2005 Asif Hamid 16/15
Capacity in log-normal FadingCapacity in log-normal Fading
Air Interface Club
24 February, 2005 Asif Hamid 17/15
Capacity in Rayleigh FadingCapacity in Rayleigh Fading
1m
Air Interface Club
24 February, 2005 Asif Hamid 18/15
Capacity in Nakagami FadingCapacity in Nakagami Fading
2m
Air Interface Club
24 February, 2005 Asif Hamid 19/15
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.