Upload
others
View
21
Download
0
Embed Size (px)
Citation preview
Universally Tradeoff Optimal Codes forFading Channels
Pramod Viswanath
Joint work withSaurabh Tavildar
University of Illinois at Urbana-Champaign
November 5, 2004
Parallel Fading Channel
h3
h2
h1
y3 = h3x3 + w3
y2 = h2x2 + w2
y1 = h1x1 + w1
x3
x2
x1
Parallel Channel Model
h3
h2
h1
y3 = h3x3 + w3
y2 = h2x2 + w2
y1 = h1x1 + w1
x3
x2
x1
¦ Time Diversity:coding over time
Parallel Channel Model
h3
h2
h1
y3 = h3x3 + w3
y2 = h2x2 + w2
y1 = h1x1 + w1
x3
x2
x1
¦ Time Diversity:coding over time
¦ Frequency Diversity:coding over OFDM symbols
Parallel Channel Model
h3
h2
h1
y3 = h3x3 + w3
y2 = h2x2 + w2
y1 = h1x1 + w1
x3
x2
x1
¦ Time Diversity:coding over time
¦ Frequency Diversity:coding over OFDM symbols
¦ Antenna Diversity:coding for MIMO channel: D-BLAST
Communication Over Slow Fading Channel
¦ L parallel sub-channels
¦ Slow fading:h1, . . . , hL random, but fixed over time
¦ Correlated fading:h1, . . . , hL jointly distributed
¦ Coherent communication:h1, . . . , hL known to the receiver
Communication Over Slow Fading Channel
¦ L parallel sub-channels
¦ Slow fading:h1, . . . , hL random, but fixed over time
¦ Correlated fading:h1, . . . , hL jointly distributed
¦ Coherent communication:h1, . . . , hL known to the receiver
¦ Focus in this talk:
Short block-length communication at highSNR
Rate and Probability of Error
¦ Tradeoff between rateR, and probability of errorPe
¦ Outage:Given a rate andSNR:
minPx
P ({ h | I(x; y| h) < R })
¦ At high SNR, Pe ≥ probability of outage
¦ Compound channel result:The outage probability can be achieved
with long block-length codes.
¦ Short block lengthspace timecodes - delay diversity, rotated QAM
codes, space time trellis codes, space time turbo codes.
R and Pe : a Coarser Scaling
¦ Coarser question: Zheng and Tse formulation
– Rate= r log(SNR)
– Probability of error= 1SNRd
¦ Givenr, find maximald = d∗(r)
¦ i.i.d. Rayleigh fading MIMO channel
– outage probability achieved by short block length Gaussian code.
Characterization of Channels in Outage
¦ Outage: Input distribution can be taken as i.i.d. Gaussian
outage=
{h |
L∑
i=1
log(1 + |hi|2SNR
)< r log(SNR)
}
¦ Outage condition independent of distribution onh
¦ Outage curve:P (outage) = SNR−dout(r)
¦ P (outage) ≤ Pe ⇒ d∗(r) ≤ dout(r)
Main Result
¦ Setting:
– coherent communication over short block length at high SNR
– measure performance in terms of tradeoff curve
Main Result
¦ Setting:
– coherent communication over short block length at high SNR
– measure performance in terms of tradeoff curve
¦ Main Result:
– Space onlycodesuniversallyachieve theoutagecurve for the
parallel channel
∗ engineering value: code isrobustto channel modeling errors
– Simple deterministic construction ofpermutation codes
Main Result
¦ Setting:
– coherent communication over short block length at high SNR
– measure performance in terms of tradeoff curve
¦ Main Result:
– Space onlycodesuniversallyachieve theoutagecurve for the
parallel channel
∗ engineering value: code isrobustto channel modeling errors
– Simple deterministic construction ofpermutation codes
¦ Useuniversal parallel channel codes as constituents of universal
MIMO channel codes
Smart Union Bound
¦ P(outage) ≤ Pe ≤ P(outage) + P(error|no-outage)
¦ We want thesecond termto decay exponentiallyin SNR
– Look at the union bound
– Each pairwise errorshoulddecay exponentiallyin SNR
¦ Let d = x(1)− x(2) be the difference codeword (space-only)
Pe(x(1) → x(2) | h) ≤ exp
(−
L∑
i=1
|hi|2|di|2)
Code Construction Criterion
¦ Worst case analysisover all realizations not in outage:
minh/∈outage
[L∑
i=1
|hi|2|di|2]
h :L∑
i=1
log(1 + SNR|hi|2
)> r log SNR
Code Construction Criterion
¦ Worst case analysisover all realizations not in outage:
minh/∈outage
[L∑
i=1
|hi|2|di|2]
h :L∑
i=1
log(1 + SNR|hi|2
)> r log SNR
¦ Outage conditionindependentof the distribution ofh
– Code construction isuniversal
– Viewpoint taken by Kose and Wesel 03
¦ Contrast with the traditional analysis:averageover the channel
statistics
Code Construction Criterion
¦ Worst case analysisover all realizations not in outage:
minh/∈outage
[L∑
i=1
|hi|2|di|2]
> 1
h :L∑
i=1
log(1 + SNR|hi|2
)> r log SNR
¦ Relatedto the water-pouring problem
– Constraint function and objective function reversed
|hi|2 =(
1λ− |di|2
)+
Code Construction Criterion
¦ Turns out to be theproduct distance:
|d1|2|d2|2 · · · |dL|2 ≥ SNRL
SNRr
¦ Sameas the traditional criterion for i.i.d. Rayleigh fading
Revisit Code Design Criterion
|d1|2|d2|2 · · · |dL|2 ≥ SNRL
SNRr
¦ Nonzero product distance
– Need each sub-channel to haveall the information
– So, alphabet sizeSNRr
– Can take it to be aQAM (Q) with SNRr points (for each
sub-channel)
Implications on Code Structure
¦ Nonzero product distance
– Need each sub-channel to haveall the information
– So, alphabet sizeSNRr
– Can take it to be aQAM (Q) with SNRr points
¦ Mapping across sub-channels
– Each point in the QAM for any sub-channel should represent the
entire codeword
– So, code isL− 1 permutationsof Q
Repetition Coding
¦ theidentitypermutation
� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �
Identity
Permutation
� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � �
��������������
¦ L = 2, product distance= SNR2
SNR2r
¦ We want: product distance> SNR2
SNRr
Key Property of Permutations
� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � �
��������������
f
¦ Two adjacent points should be mapped as far apart as possible
Random Permutation Codes
¦ Look at theensembleof all permutation codes
– huge number of permutations:(SNRr!)L−1
– average product distance under appropriate measure
¦ There exists a permutation which satisfies the product distance
¦ Conclusion: A permutationcode achievesuniversallythe outage
curve
An Example
¦ 16-point Permutation Code
� �
�
� �
�
� �
� �
� �
�
�
¦ Minimum Product Distance: 0.4
Simple Permutation Codes Exist
¦ Complexity of QAM on a scalar channel
– easy encoding
– easy decoding
– worst casecomplexity very small
¦ Ideal situation:
With L parallel channels, complexity isL times that of QAM
¦ Next: Demonstrate this forL = 2.
2 Sub-Channels
¦ Transmitq ∈ Q andf(q) ∈ Q over the two sub-channels
q =
√SNR
SNRr/2(a + ib), a, b integers
y1 = h1
√SNR
SNRr/2(a + ib) + w1
y2 = h2
√SNR
SNRr/2f(a + ib) + w2
2-sub Channels
¦ Transmitq ∈ Q andf(q) ∈ Q over the two sub-channels
q =
√SNR
SNRr/2(a + ib), a, b integers
y1 = h1
√SNR
SNRr/2(a + ib) + w1
y2 = h2
√SNR
SNRr/2f(a + ib) + w2
¦ Look at permutations (f ) of realandimaginaryvalues
y1 = |h1|√
SNR
SNRr/2(a + ib) + w1
y2 = |h2|√
SNR
SNRr/2(f(a) + if(b)) + w2
Effect of the Fading Channel
¦ Considerbinaryrepresentation of integersa andb
– requiren = r2 log2 SNR bits
¦ Additive Gaussian noise very likely to move within neighboring
integers
Effect of the Fading Channel
¦ Considerbinaryrepresentation of integersa andb
– requiren = r2 log2 SNR bits
¦ Additive Gaussian noise very likely to move within neighboring
integers
¦ Effect of themultiplicativechannel:distort the LSBs
|h1|√
SNR
SNRr/2≈ 2−k1 =⇒ k1 LSBs ofa andb lost
|h2|√
SNR
SNRr/2≈ 2−k2 =⇒ k2 LSBsf(a) andf(b) lost
Effect of the Fading Channel
¦ Considerbinaryrepresentation of integersa andb
– requiren = r2 log2 SNR bits
¦ Additive Gaussian noise very likely to move to neighboring integers
¦ Effect of themultiplicativechannel:distort the LSBs
|h1|√
SNR
SNRr/2≈ 2−k1 =⇒ k1 LSBs ofa andb lost
|h2|√
SNR
SNRr/2≈ 2−k2 =⇒ k2 LSBsf(a) andf(b) lost
¦ No outage condition:
|h1||h2| > SNRr2
SNR=⇒ k1 + k2 ≤ n
Bit Reversal Permutation
¦ Bit reversal: f(a) is bit reversal ofa
¦ Encoding:
– Same complexity as encoding a QAM
¦ Decoding:
– Use first sub-channel to determine MSBs ofa andb
– Use second sub-channel to determine LSBs ofa andb
– No outage condition means you recover all the bits
Fading MIMO Channel
y1
x3
x2
y2
x1
¦ y = Hx + w
¦ Entries ofH have a joint distribution.
– slow fading, coherent
Fading MIMO Channel
y1
x3
x2
y2
x1
¦ y = Hx + w
¦ Focus here
Short block-length communication at highSNR
R and Pe : a Coarser Scaling
¦ Coarser question: Zheng and Tse formulation
– Rate= r log(SNR)
– Probability of error= 1SNRd
¦ Givenr, find maximald = d∗(r)
R and Pe : a Coarser Scaling
¦ Coarser question: Zheng and Tse formulation
– Rate= r log(SNR)
– Probability of error= 1SNRd
¦ Givenr, find maximald = d∗(r)
¦ A code with rater is tradeoff optimal for a channelif the diversity
gain over that channel isd∗(r)
Universal Tradeoff Optimality
¦ A code isuniversallytradeoff optimal if it is tradeoff optimal for
everychannel distribution
Universal Tradeoff Optimality
¦ A code isuniversallytradeoff optimal if it is tradeoff optimal for
everychannel distribution
¦ Compound channel formulation guarantees universal codes – with
long enough block length (KW03)
Universal Tradeoff Optimality
¦ A code isuniversallytradeoff optimal if it is tradeoff optimal for
everychannel distribution
¦ Universal code design criterion (TV04)
λ1λ2...λmin(nr,nt) ≥SNRmin(nr,nt)
SNRr
¦ λ1 ≤ . . . ≤ λnt are eigenvalues ofDD†, D := X(1)−X(2)
¦ Need this criterion for every pair of codewords
A Contrast
¦ Consider a MISO channel:nr = 1
¦ Traditional design criterion for i.i.d. Rayleigh fading (TSC 98)
maximize λ1λ2...λnt (determinant)
¦ Universal design criterion
maximize λ1 (smallest singular value)
– have to protect against theworst casechannel
Universal Codes
¦ Needrotational invariance
XX∗ = SNR I
– full rate orthogonal designs are universal
Universal Codes
¦ Needrotational invariance
XX∗ = SNR I
– full rate orthogonal designs are universal
¦ Nearly rotational invariant:codes based on number theory
– Tilted QAM code (YW03, DV03), Golden code (BRV04), codes
based on cyclic division algebra (Eli04)
– Universally tradeoff optimal, but hard to decode
Main Results
¦ Setting:
– coherent communication over short block length at high SNR
– universal tradeoff performance over arestricted classof channels
Main Result
¦ Setting:
– coherent communication over short block length at high SNR
– universal tradeoff performance over arestricted classof channels
¦ Main Result:
– Universal tradeoff optimal designs based on parallel channel
codes
∗ engineering value: code isrobustto channel modeling errors
∗ Simple encoding and decoding ofpermutation codes
MISO Channel Revisited
y[m] = h∗x[m] + w[m]
¦ Supposeh1, . . . , hnt are i.i.d.
– no antenna is particularly vulnerable
¦ Universality Result:
It is tradeoff optimal to use one transmit antenna at a time
¦ Converts MISO channel into a parallel channel
– can use the simple universal permutation codes
Restricted Class of Channels
H = UΛV∗
¦ V∗ is the Haar measure onSU(nt) andΛ independent ofV
¦ each transmit direction is equally likely
¦ i.i.d. Rayleigh fading belongs to this class
D-BLAST Architecture
¦ D-BLAST scheme for 2 transmit antennas:
X =
0 a2 b2 c2
a1 b1 c1 0
¦ Successive cancellationof streams
– converts MIMO channel to aparallel channel
– can use permutation codes
¦ Almost universally tradeoff optimal
– initialization overhead
D-BLAST with ML decoding
X =
0 a2 b2
a1 b1 0
¦ Consider joint ML decoding of both streams
¦ Claim: Universal tradeoff performance same as that with successive
cancellation
¦ Main result:
Universal tradeoff performance over the restricted class of channels
2× 2 i.i.d. Rayleigh channel
2 Multiplexing Rate1
Diversity
Optimal tradeoff
1
4 nr = 2, nt = 2
2× 2 i.i.d. Rayleigh channel
Diversity
D-BLAST with successive cancellation
1
Optimal tradeoff
Multiplexing Rate2
4
1
nr = 2, nt = 2
2× 2 i.i.d. Rayleigh channel
4
Diversity
D-BLAST with joint decoding
D-BLAST with successive cancellation
1
Optimal tradeoff
Multiplexing Rate2
1
nr = 2, nt = 2
Universality
¦ D-BLAST with joint ML decoding of two streams:
achieves the low rate part of the tradeoff curve over the restricted
class of channels
¦ H = UΛV∗
– V is independent ofΛ and has the Haar measure
– Supposeλ1 ≤ λ2 are the singular values
P[λ2
1 ≤ ε1, λ22 ≤ ε2
] ≈ εk11 εk2
2
with k1 ≤ 1, k2 = 3.
3× 2 i.i.d. Rayleigh Channel
¦ D-BLAST with two streams
X =
0 0 a3 b3
0 a2 b2 0
a1 b1 0 0
3× 2 i.i.d. Rayleigh channel
2 Multiplexing Rate1
Diversity
Optimal tradeoff
2
6 nr = 2, nt = 3
3× 2 i.i.d. Rayleigh channel
Diversity
D-BLAST with successive cancellation
1
Optimal tradeoff
Multiplexing Rate2
6
nr = 2, nt = 3
2
3× 2 i.i.d. Rayleigh channel
D-BLAST with joint ML decoding
Diversity
D-BLAST with successive cancellation
1
Optimal tradeoff
Multiplexing Rate2
6
nr = 2, nt = 3
2
nt × 2 i.i.d. Rayleigh Channel
¦ D-BLAST with two streams
X =
0 0 . . . ant bnt
......
......
...
0 a2 b2 . . . 0
a1 b1 0 . . . 0
nt × 2 i.i.d. Rayleigh channel
2 Multiplexing Rate1
Diversity
Optimal tradeoff
nt − 1
2ntnr = 2
nt × 2 i.i.d. Rayleigh channel
D-BLAST with successive cancellation
Diversity
1
Optimal tradeoff
Multiplexing Rate2
nr = 2
2nt
nt − 1
nt × 2 i.i.d. Rayleigh channel
D-BLAST with successive cancellation
D-BLAST with joint ML decoding
Diversity
1
Optimal tradeoff
Multiplexing Rate2
nr = 2
2nt
nt − 1
V-Blast with ML decoding
Independentsignaling
ML-Decoder
x2
x1
¦ Send QAM independently across antennas (space only code)
2× 2 i.i.d. Rayleigh fading
Space-only V-BLAST
4
1
Optimal tradeoff
Diversity
1 Multiplexing Rate2
nr = 2, nt = 2
¦ space only V-Blastoptimal forr ≥ 1 on2× 2 Rayleigh channel
D-BLAST Time-Space Code
¦ D-BLAST space-timecode:
X =
0 a2 b2
a1 b1 0
¦ D-BLAST time-spacecode:
X∗ =
0 a1
a2 b1
b2 0
3× 2 i.i.d. Rayleigh channel
Time-space code
6
2
Optimal tradeoff
Diversity
1 Multiplexing Rate2
nr = 2, nt = 3
D-BLAST Time-Space Code
¦ D-BLAST space-timeandtime-spacecodes:
X =
0 0 . . . ant bnt
......
......
...
0 a2 b2 . . . 0
a1 b1 0 . . . 0
X∗ =
0 . . . 0 a1
0 . . . a2 b1
. . . . . . b2 0
ant . . .... 0
bnt
... 0 0
nt × 2 i.i.d. Rayleigh channel
Time-space code
Optimal tradeoff
Diversity
1 Multiplexing Rate2
nr = 22nt
nt − 1
Reprise
¦ Considered universal tradeoff optimality
¦ Main results:
¦ Simple permutation codes for the parallel channel
¦ For a restricted class of MIMO channels
Universal tradeoff optimality of D-BLAST with joint ML decoding
Reprise
¦ Considered universal tradeoff optimality
¦ Main results:
¦ Simple permutation codes for the parallel channel
¦ For a restricted class of MIMO channels
Universal tradeoff optimality of D-BLAST with joint ML decoding
¦ Reference:
Chapter 9 ofFundamentals of Wireless Communication,
D. Tse and P. Viswanath, Cambridge University Press, 2005.