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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.