Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally...

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Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar University of Illinois at Urbana-Champaign November 5, 2004

Transcript of Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally...

Page 1: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

Universally Tradeoff Optimal Codes forFading Channels

Pramod Viswanath

Joint work withSaurabh Tavildar

University of Illinois at Urbana-Champaign

November 5, 2004

Page 2: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

Parallel Fading Channel

h3

h2

h1

y3 = h3x3 + w3

y2 = h2x2 + w2

y1 = h1x1 + w1

x3

x2

x1

Page 3: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

Parallel Channel Model

h3

h2

h1

y3 = h3x3 + w3

y2 = h2x2 + w2

y1 = h1x1 + w1

x3

x2

x1

¦ Time Diversity:coding over time

Page 4: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 5: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 6: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 7: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 8: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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.

Page 9: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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.

Page 10: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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)

Page 11: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

Main Result

¦ Setting:

– coherent communication over short block length at high SNR

– measure performance in terms of tradeoff curve

Page 12: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 13: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 14: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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)

Page 15: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 16: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 17: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

)+

Page 18: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 19: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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)

Page 20: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 21: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

Repetition Coding

¦ theidentitypermutation

� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �

Identity

Permutation

� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � �

��������������

¦ L = 2, product distance= SNR2

SNR2r

¦ We want: product distance> SNR2

SNRr

Page 22: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

Key Property of Permutations

� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � �

��������������

f

¦ Two adjacent points should be mapped as far apart as possible

Page 23: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 24: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

An Example

¦ 16-point Permutation Code

� �

� �

� �

� �

� �

¦ Minimum Product Distance: 0.4

Page 25: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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.

Page 26: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 27: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 28: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

Effect of the Fading Channel

¦ Considerbinaryrepresentation of integersa andb

– requiren = r2 log2 SNR bits

¦ Additive Gaussian noise very likely to move within neighboring

integers

Page 29: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 30: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 31: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 32: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

Fading MIMO Channel

y1

x3

x2

y2

x1

¦ y = Hx + w

¦ Entries ofH have a joint distribution.

– slow fading, coherent

Page 33: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

Fading MIMO Channel

y1

x3

x2

y2

x1

¦ y = Hx + w

¦ Focus here

Short block-length communication at highSNR

Page 34: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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)

Page 35: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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)

Page 36: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

Universal Tradeoff Optimality

¦ A code isuniversallytradeoff optimal if it is tradeoff optimal for

everychannel distribution

Page 37: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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)

Page 38: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 39: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 40: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

Universal Codes

¦ Needrotational invariance

XX∗ = SNR I

– full rate orthogonal designs are universal

Page 41: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 42: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

Main Results

¦ Setting:

– coherent communication over short block length at high SNR

– universal tradeoff performance over arestricted classof channels

Page 43: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 44: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 45: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 46: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 47: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 48: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

2× 2 i.i.d. Rayleigh channel

2 Multiplexing Rate1

Diversity

Optimal tradeoff

1

4 nr = 2, nt = 2

Page 49: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

2× 2 i.i.d. Rayleigh channel

Diversity

D-BLAST with successive cancellation

1

Optimal tradeoff

Multiplexing Rate2

4

1

nr = 2, nt = 2

Page 50: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 51: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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.

Page 52: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 53: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

3× 2 i.i.d. Rayleigh channel

2 Multiplexing Rate1

Diversity

Optimal tradeoff

2

6 nr = 2, nt = 3

Page 54: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

3× 2 i.i.d. Rayleigh channel

Diversity

D-BLAST with successive cancellation

1

Optimal tradeoff

Multiplexing Rate2

6

nr = 2, nt = 3

2

Page 55: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 56: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 57: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

nt × 2 i.i.d. Rayleigh channel

2 Multiplexing Rate1

Diversity

Optimal tradeoff

nt − 1

2ntnr = 2

Page 58: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

nt × 2 i.i.d. Rayleigh channel

D-BLAST with successive cancellation

Diversity

1

Optimal tradeoff

Multiplexing Rate2

nr = 2

2nt

nt − 1

Page 59: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 60: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

V-Blast with ML decoding

Independentsignaling

ML-Decoder

x2

x1

¦ Send QAM independently across antennas (space only code)

Page 61: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 62: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 63: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

3× 2 i.i.d. Rayleigh channel

Time-space code

6

2

Optimal tradeoff

Diversity

1 Multiplexing Rate2

nr = 2, nt = 3

Page 64: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 65: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

nt × 2 i.i.d. Rayleigh channel

Time-space code

Optimal tradeoff

Diversity

1 Multiplexing Rate2

nr = 22nt

nt − 1

Page 66: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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

Page 67: Universally Tradeoff Optimal Codes for Fading Channelspramodv/talks/ucb_1.pdf · Universally Tradeoff Optimal Codes for Fading Channels Pramod Viswanath Joint work with Saurabh Tavildar

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.