combinatorial channel signature modulation for wireless networkssejdinov/talks/pdf/2012-02-08... ·...
TRANSCRIPT
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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
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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?
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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?
![Page 4: Combinatorial Channel Signature Modulation for Wireless Networkssejdinov/talks/pdf/2012-02-08... · 2020-01-23 · Combinatorial Channel Signature Modulation for Wireless Networks](https://reader034.vdocument.in/reader034/viewer/2022042416/5f31a4ad257e585b5b735667/html5/thumbnails/4.jpg)
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
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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
![Page 6: Combinatorial Channel Signature Modulation for Wireless Networkssejdinov/talks/pdf/2012-02-08... · 2020-01-23 · Combinatorial Channel Signature Modulation for Wireless Networks](https://reader034.vdocument.in/reader034/viewer/2022042416/5f31a4ad257e585b5b735667/html5/thumbnails/6.jpg)
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
![Page 7: Combinatorial Channel Signature Modulation for Wireless Networkssejdinov/talks/pdf/2012-02-08... · 2020-01-23 · Combinatorial Channel Signature Modulation for Wireless Networks](https://reader034.vdocument.in/reader034/viewer/2022042416/5f31a4ad257e585b5b735667/html5/thumbnails/7.jpg)
Overview
CS for multiterminal communications
Combinatorial encoder
Sparse recovery solver
Simulation results
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Outline
CS for multiterminal communications
Combinatorial encoder
Sparse recovery solver
Simulation results
![Page 9: Combinatorial Channel Signature Modulation for Wireless Networkssejdinov/talks/pdf/2012-02-08... · 2020-01-23 · Combinatorial Channel Signature Modulation for Wireless Networks](https://reader034.vdocument.in/reader034/viewer/2022042416/5f31a4ad257e585b5b735667/html5/thumbnails/9.jpg)
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)
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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)
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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
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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
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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
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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
· · ·
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Outline
CS for multiterminal communications
Combinatorial encoder
Sparse recovery solver
Simulation results
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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
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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.
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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.
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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.
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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)
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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)
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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
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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
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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.
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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.
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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)
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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
)
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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
)
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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
)
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Outline
CS for multiterminal communications
Combinatorial encoder
Sparse recovery solver
Simulation results
![Page 31: Combinatorial Channel Signature Modulation for Wireless Networkssejdinov/talks/pdf/2012-02-08... · 2020-01-23 · Combinatorial Channel Signature Modulation for Wireless Networks](https://reader034.vdocument.in/reader034/viewer/2022042416/5f31a4ad257e585b5b735667/html5/thumbnails/31.jpg)
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)
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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
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Sparse recovery solver
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.
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Sparse recovery solver
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.
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Sparse recovery solver
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.
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Sparse recovery solver
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)
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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))
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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.
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Outline
CS for multiterminal communications
Combinatorial encoder
Sparse recovery solver
Simulation results
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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
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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.
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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
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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
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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
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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
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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
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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.
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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)
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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
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Step 1: brick to triangular prism
B3 = B2 × [2/3, 1]→ T2 × [2/3, 1]
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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
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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)