Combinatorial Channel Signature Modulation for Wireless Networkssejdinov/talks/pdf/2012-02-08... ·...

Combinatorial Channel Signature Modulationfor Wireless Networks

Dino Sejdinovic (Gatsby Unit, UCL)joint work with Rob Piechocki (CCR, Bristol)

Signal Processing and Communications Laboratory, University of Cambridge

February 8, 2012

Problem outline

I wireless mutual broadcast withN + 1 half-duplex nodes

I each node has a k-bit message totransmit to all others

I Can all nodes transmit at thesame frequency without elaboratetime scheduling?

Introduction

CCSM (Combinatorial Channel Signature Modulation):

I sparse, combinatorial representation of messages

I compressed sensing based decoding

I minimal MAC Layer coordination: access to a sharedchannel

I no need for collision detection/avoidance scheme(CSMA/CA)

I robust to time dispersion

I no need for guard intervals to eliminate ISI

Overview

CS for multiterminal communications

Combinatorial encoder

Sparse recovery solver

Simulation results

Outline




Simulation results

Compressed sensing

I Compressed sensing (Candes, Romberg and Tao 2006;Donoho 2006) combines sampling and compression stepsinto one - into taking random linear projections.

sample compress

N

K<<N

Kstore/transmit

load/receive

decompress

K N

f

f

project

Mstore/transmit

load/receive

reconstruct

M N

f

f

M<<N

projections ≈ number of non-zeros × log (size of the vector)

Embracing the additive nature of wireless

I (Zhang and Guo 2010): each node is assigned a (known)dictionary of (sparse) on-o� signalling codewords, eachcodeword corresponding to a single message

I Own transmissions seen as erasures in dictionary

I The dictionary size exponential in the number of bits k in amessage - sparse recovery problem of size 2kN

1

1

1

1

transmitted codewords

1

1

1

2krecipient transmits

Combinations of the codebook elements?

Receivedwaveformby user 2

ChannelImpulseResponse

Codewordspan

Transmitted codewordby user 1

I Idea: Transmit (weighted) sums of a �xed number l ofon-o� signalling codewords.

I the choice of the l-combination of the dictionary elementscarries information

I Requires e�cient encoding of messages into l-combinations:constant weight coding (l out of L codes)

I Need l� L for sparse recovery

Reduction of complexity

I (Zhang and Guo, 2010): k = log2 L - sparse recovery problemof size 2kN

I CCSM: k ≈ log2(Ll

)= O(Lα log2 L), for l = Lα, 0 < α < 1

- sparse recovery problem of size∼ k1/αN

Encoder

transmitted

codeword

ix

C w

information vector

ki :1, lqkki :1,

lqk

i

2F

ib

CWC

mapper

weighing

mapper

ic

x

x

+1

-1

time slots

1,is

4,is

on-off signalling

dictionary

encoder i

φC : 00000 7→ 010100000

φC : 00001 7→ 000101000

· · ·

Outline




Simulation results

Constant weight coding

I Combinatorial representation of information

I using l-combinations of the set of L elements, i.e., length-Lbinary vectors with exactly l ones.

I l� L (sparse)

De�nition

An (L, l)-constant weight code is a set of length-L binaryvectors with Hamming weight l:

C ⊆ {c ∈ FL2 : wH(c) = l}.

I single-bit error/single-type errordetection

I two-out-of-�ve barcode

Constant weight coding

I Set of possible messages: S = {0, 1 . . . ,K − 1}. UsuallyK = 2k, and we identify S ↔ Fk2.

I An (L, l)-constant weight code is C ⊆ {c ∈ FL2 : wH(c) = l}.I Goal: construct a bijective map φC : S → C

I Enumeration approaches:

I (Schalkwijk 1972), (Cover 1973): lexicographic ordering ofcodewords, requires registers of length O(L).

I (Ramabadran 1990): arithmetic coding, computationalcomplexity O(L).

I (Knuth 1986): complementation method, computationalcomplexity O(L) - but much faster in practice. Works onlyfor balanced codes: l = bL/2c.

I Can we do better when l� L?

I (Tian, Vaishampayan, Sloane 2009): embed both S and Cinto Rl and establish bijective maps by dissecting certainpolytopes in Rl.

Embedding C into Rl

I Given a constant-weight codeword c, de�neφ(c) = (y1L , . . . ,

ylL ) ∈ Rl, with yi:= position of the i-th 1 in

c.

I for L = 5, l = 2, φ(01010) = (2/5, 4/5).

I φ(C) is a discrete subset of the convex hull Tl of the points:

(0, 0, . . . 0, 0)(0, 0, . . . 0, 1)...

... . . ....

...(0, 1, . . . 1, 1)(1, 1, . . . 1, 1)

C embedded in a tetrahedron

x

y

(0,0)

(0,1) (1,1)

T2

xy

z

T3

(0,0,0)

(0,0,1)

(0,1,1) (1,1,1)

I T2 is the right triangle with vertices (0, 0), (0, 1), (1, 1)

I V ol(Tl) =1l! (the unit cube can be split into l! tetrahedra

congruent to Tl)

The case l = 2

−0.5 0 0.5 1 1.5−0.5

0

0.5

1

1.5

n=5

(2/5,4/5)=φ(01010)

(1/5,2/5)=φ(11000)

−0.5 0 0.5 1 1.5−0.5

0

0.5

1

1.5

n=50

Assume: information is a brick

I Brick Bl ⊂ Rl is a hyper-rectangle

Bl := [0, 1]×[12, 1]×[2

3, 1]×· · ·×[ l − 1

l, 1]

I Represent messages s ∈ {0, 1, . . . ,K − 1},K ≤

(Ll

)as integer l-tuple

(b(s)1 , b

(s)2 , . . . , b

(s)l ), s.t.

(b(s)1n

,b(s)2n

, . . . ,b(s)ln

) ∈ Bl: quotient andremainder

I V ol(Bl) =1l! = V ol(Tl)

x

y

(0,1/2)

(0,1)(1,1)

B2

(1,1/2)

x

y

z

1

1/2

1/3

B3

DissectionsHilbert's third problem:For any two polyhedra of the same volume, is it

possible to dissect one into a �nite number of pieces

that can be rearranged to give the other?

I In l = 2 dimensions (Bolyai-Gerwien 1833): yes

�scissor-equivalence�

I In d ≥ 3 dimensions (Dehn, 1902): no - polyhedra musthave equal Dehn invariants.

Encoding

x

y

z

1

1/2

1/3

B3

xy

z

T3

(0,0,0)

(0,0,1)

(0,1,1) (1,1,1)

I messages S = {0, 1 . . . ,K − 1} ←→ Bl ⊂ Rl diss.←→

Tl ⊂ Rl

←→ constant weight code CI (Tian, Vaishampayan, Sloane 2009) give an explicitrecursive dissection of Bl into Tl computable in O(l2)

Rate

I CWC rate: R(C) = 1L log2 |C| ≤ 1

L log2(Ll

)= O(l log2 LL )→ 0,

L→∞, when l� L.

CCSM rate 6= CWC rate

I CCSM rate: NM

(log2

(Ll

)+ lq

), where

I M is the time duration of the waveforms (number of rows inthe CS problem + number of own transmissions)

I Typical CS results: it su�ces to take M to be the numberof non-zeros × log (size of the vector)

I M = O(lN log2(LN))⇒ CCSM rate = O( log2 Llog2 L+log2N

)

Outline




Simulation results

Channel signatures

Receivedwaveformby user 2

ChannelImpulseResponse

Codewordspan

Transmitted codewordby user 1

Transmittedcodewordby user 2

Modifiedcodewordspan

Received waveform byuser 2 (from user 1)

I Each user convolves codebook elements with the channelimpulse response

I Can subtract echoes of its own transmissions(self-interference remover)

Decoder

.

.

.

1ix

0x

1ix

.

.

.

Nx

*

*

*

*

channel between

transmitter 0 and

receiver i

0,ih

1, iih

1, iih

Ni ,h

ii,hix

*

erasure

channel

encoder i

self-

interference

remover

ixiy~

sparse

recovery

solver

iy

decoder i

iv

...

...

0c

1ˆic

1ˆic

Nc

1

w

1

C

0

1i

1i

N

...

...

self-channel

at i

weighing

demapper

CWC

demapper

additive

noise

iz~

xi = Sici

yi = Ei

N∑j=0

hi,j ∗ Sjcj + zi

−Ei (hi,i ∗ Sici) = A−iv−i + zi


yi = Ei

N∑j=0

hi,j ∗ Sjcj + zi

−Ei (hi,i ∗ Sici) = A−iv−i + zi

I User i needs to solve the following problem to detect thedesired signal:

v−i = argminv−i

‖yi −A−iv−i‖2s.t. ‖cj‖0 = l, for all j 6= i,

I A non-convex optimisation problem.

I Exactly Nl out of NL entries in v−i are non-zero - butsparsity level is: l

L � 1I Can use any of the myriad of CS decoding algorithms.


I Basis Pursuit/LASSO/convex relaxation (Tibshirani 1996),(Chen, Donoho, and Saunders 1998)

I LARS / homotopy (Efron et al, 2004)

I Greedy iterative methods:

I Compressive Sampling Matching Pursuit - CoSaMP(Needell and Tropp 2009)

I Subspace Pursuit - SP (Dai and Milenkovic 2009)

Subspace Pursuit

Greedily searches for the support set S such that y is closest tospan(AS).

1. Initialise. Set S to the Nl columns that maximize |〈ai, y〉|2. Identify further candidates. Set S ′ to the Nl columns that

maximize |〈ai, y − proj(y, span(AS)〉|3. Merge and Prune. Set S to the Nl columns from S ∪ S ′

with largest magnitudes in A+S∪S′y

4. Iterate (2)-(3) until the stopping criterion holds.

yyr

span(AS)

proj(y, span(AS))

Group Subspace Pursuit

Sparse vector has additional structure: each of N subvectors hasexactly l non-zero entries.

1. Initialise. Set S to the union of sets of l columns withineach group of L that maximize |〈ai, y〉|

2. Identify further candidates. Set S ′ to the union of sets of lcolumns within each group of L that maximize|〈ai, y − proj(y, span(AS)〉|

3. Merge and Prune. Set S to the union of sets of l columnswithin each group of L with largest magnitudes in A+

S∪S′y

4. Iterate (2)-(3) until the stopping criterion holds.

Outline




Simulation results

Group Subspace Pursuit (2)

5 10 15 20 25 3010

−4

10−3

10−2

10−1

100

1/σn

2 [dB]

Pr(

err

or)

BP−100

BP−200

BP−300

GSP−100

GSP−200

GSP−300

GBP−300

GBP−200

GBP−100

Figure: Performance comparison of Basis Pursuit/Lasso (BP), GroupBasis Pursuit (GBP) and Group Subspace Pursuit (GSP). N = 10,l = 4, L = 32

CCSM Performance

0 2 4 6 8 10 12 14 16 18 2010

−5

10−4

10−3

10−2

10−1

100

SNR [dB]

Pr(

err

or)

M=150

M=175

M=200

M=225

M=250

Figure: Performance of the proposed method in terms of messageerror rate: In this case there are (N + 1) = 5 users simultaneouslybroadcasting messages using CCSM with L = 64, l = 12.

Comparison to CSMA/CA and TDMA (1)1. TDMA

I central controlling mechanismI time divided equally - no collisions, performanceindependent of the number of nodes

I guard interval (cyclic pre�x) of 20% slot duration

2. CSMA/CAI randomised deferment of transmissions in order to avoidcollisions; contention window: 16-1024 symbol intervals

I no symbol intervals wasted on distributed or shortinterframe space (DIFS/SIFS), propagation delay, physicalor MAC message headers and ACK responses

I a single message in each transmission queueI guard interval (cyclic pre�x) of 20% slot duration

randomized deferment time

Comparison to CSMA/CA and TDMA (2)

5 10 15 20 25

0.8

1

1.2

1.4

1.6

1.8

2

number of users

thro

ug

hp

ut

[in

bits/s

ym

bo

l in

terv

al]

CSMA/CA with 20% GI

fully centralized system with 20% GI

CCSM

CCSM: the minimum number of symbol intervals at which no messageerrors occurred in at least 100,000 simulation trials

L = 64, l = 12, q = 2 (QPSK)

k =⌊log2

(Ll

)⌋+ lq = 65

Concluding remarks

I a novel decentralized modulation and multiplexing methodfor wireless networks

I e�ective time/frequency duplexI minimal MACI inherent robustness to time dispersion

I low computational complexity which takes advantage of

I combinatorial representation of messagesI sparse recovery detection

I signi�cant throughput improvement in comparison tocollision avoidance schemes

References

R. Piechocki and D. Sejdinovic, �Combinatorial Channel SignatureModulation for Wireless ad-hoc Networks� preprint available from:arxiv.org/abs/1201.5608v1 (to appear in Proc. IEEE Int. Conf. on

Communications ICC 2012).

L. Zhang and D. Guo, �Wireless Peer-to-Peer Mutual Broadcast via SparseRecovery� preprint available from: arxiv.org/abs/1101.0294, in Proc. IEEEInt. Symp. Inform. Theory ISIT 2011.

W. Dai and O. Milenkovic, �Subspace pursuit for compressive sensing signalreconstruction,� IEEE Trans. Inform. Theory, vol. 55, pp. 2230� 2249, 2009.

C. Tian, V. Vaishampayan and N. J. A. Sloane, �A coding algorithm forconstant weight vectors: a geometric approach based on dissections,� IEEETrans. Inform. Theory, vol. 55, pp. 1051�1060, 2009.

G. Bianchi, �Performance Analysis of the IEEE 802.11 DistributedCoordination Function�, IEEE Journal on Selected Areas in

Communications, vol. 18, pp. 535�547, 2000.

Signalling dictionary

I One construction of on-o� signalling dictionary:

I all columns of Si have equal number of non-zero entries, setto⌊ML

⌋I every two columns in Si have disjoint supportI non-zero entries selected uniformly at random from somediscrete constellation

I transmitted codeword xi has exactly l ·⌊ML

⌋non-zero entries

(on-slots)

I M =M − l ·⌊ML

⌋o�-slots used to listen to the incoming signals

I Can also use LDPC / regular Gallager constructions (Baron,Sarvotham, Baraniuk 2010)

Assume: information is a brick

I Brick Bl ⊂ Rl is a hyper-rectangleBl := [0, 1]× [12 , 1]× [23 , 1]× · · · × [ l−1l , 1]

I Let l = 2, and L even. Then one can uniquely representmessage indices s ∈ {0, 1, . . . ,K − 1} (where K ≤ L(L−1)

2 )

as integer pairs (b(s)1 , b

(s)2 ), where

0 ≤ α =

⌊s

L/2

⌋≤ L− 1

0 ≤ β = s− L

2α ≤ L

2− 1

I De�ne b(s)1 = α, b

(s)2 = β + L

2 .

I It follows that

{(b(s)1L ,

b(s)2L )

}K−1s=0

⊂ Bl

Step 1: brick to triangular prism

B3 = B2 × [2/3, 1]→ T2 × [2/3, 1]

Step 2: triangular prism to tetrahedron

I inductive dissection for general w:

1. Bw = Bw−1 × [w−1w , 1]→ Tw−1 × [w−1

w , 1]2. Tw−1 × [w−1

w , 1]→ Tw

Step 2: triangular prism to tetrahedron

I inductive dissection for general w:

1. Bw = Bw−1 × [w−1w , 1]→ Tw−1 × [w−1

w , 1]2. Tw−1 × [w−1

w , 1]→ Tw

I Some loss in rate due to rounding (when n is in the range100-1000, and w =

√n, the loss is 1-2 bits/block)

Combinatorial Channel Signature Modulation for Wireless Networkssejdinov/talks/pdf/2012-02-08... ·...

Documents

Transcript of Combinatorial Channel Signature Modulation for Wireless Networkssejdinov/talks/pdf/2012-02-08... ·...