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Combinatorial Channel Signature Modulation for Wireless ad-hoc Networks Robert J. Piechocki Dino Sejdinovic Merchant Venturers School of Engineering Gatsby Computational Neuroscience Unit University of Bristol University College London Woodland Rd, Bristol, BS8 1UB, UK 17 Queen Square, London, WC1N 3AR, UK Email: [email protected] Email: [email protected] 2 1 Abstract—In this paper we introduce a novel modulation beneficial feature is the ability to achieve a true duplex, i.e. 0 2 and multiplexing method which facilitates highly efficient and allusersinthenetworkcantransmitandreceivesignalsatthe simultaneouscommunicationbetweenmultipleterminalsinwire- same frequency and in the same time slot. n less ad-hoc networks. We term this method Combinatorial a Channel Signature Modulation (CCSM). The CCSM method is The CCSM method is inspired by a cross-layer scheme for J particularly efficient in situations where communicating nodes wireless peer-to-peer mutual broadcast considered by Zhang 6 operate in highly time dispersive environments. This is all and Guo in [1]. In this paper each node is assigned a code- 2 achievedwithaminimalMAClayeroverhead,sinceallusersare book of on-off signalling codewords, such that every possible allowed to transmit and receive at thesame time/frequency(full message corresponds to a single codeword. However, the ] simultaneousduplex).TheCCSMmethodhasitsrootsinsparse T scheme by Zhang and Guo is suitable only “for the situation modelling and the receiver is based on compressive sampling I techniques. Towards this end, we develop a new low complexity wherebroadcastmessagesconsistofarelativelysmallnumber . s algorithmtermedGroupSubspacePursuit.Ouranalysissuggests of bits”. Namely, the size of the sparse recovery problem c that CCSM at least doubles the throughput when compared to which needs to be solved is exponential in the length of the [ the state-of-the art. message. Our scheme overcomes this limitation by encoding 1 the message in a combination of the codeword span, i.e., in a v I. INTRODUCTION choice of l out of L codewords in the codeword span, where 8 Time dispersion has traditionally posed a very challenging l ≪L. Such representation of useful information results in a 0 6 problem for communications systems. Typical examples of significant reduction of the computationalcomplexity1, as the 5 highlytime dispersivechannelsinclude wireless systems with number of possible messages is expressed through a number . large bandwidth, power line communication (e.g. for Smart of all possible combinations, which is L . This, in turn, 1 l 0 Grids),underwaterchannelsetc. The currentlyfavouredstate- renders our scheme practical for broadca(cid:0)st(cid:1)ing much longer 2 of-the-artsolution is typified by OFDM and SC-FDE systems messages. Moreover, in CCSM additional information can be 1 (e.g., 4G mobile systems, WiFi). Other existing solutions encoded in the choice of the weights assigned to a particular : v include: equalisation in single carrier receivers (e.g., 2G mo- combination of the codeword span. In addition, the scheme i bile systems) and rake receivers for CDMA (e.g., 3G mobile of [1] cannot cope with time dispersive environments. Our X systems). In all those techniques time dispersion represents a scheme, in contrast, thrives on dispersive nature of wireless r a hindrance to a larger or smaller extent. The system described systems, by adapting the sparse recovery problem to the herethrivesonthedispersivenatureofcommunicationschan- channel signatures. nels and turns it into an advantage. Combinatorial modulation constructions have been pre- MACLayer coordinationis anothersource of inefficiencies viously considered in optical communication systems. A in communications systems. The MAC protocol regulates throughput efficient version of pulse-position modulation how competing users access a shared resource (e.g. a radio (PPM) signalling scheme is called multipulse or combinato- channel). In a standard solution only a single user can oc- rial PPM (MPPM) [9], [10], [11]. However, MPPM applies cupy a shared resource; otherwise a “collision” occurs. The such informationrepresentationdirectlyin time domainusing mostimportantMACprotocolsincludeCSMA/CA (e.g.IEEE single pulses. The MPPM signalling is inherently sensitive 802.11x)or(slotted)Aloha.TheDS-CDMAsystemsomewhat relaxes this constraint by allowing a group of synchronised 1In the set-up by Zhang and Guo, the size of the sparse vector to be recovered is L·N, where L is the number of all possible messages, and users to transmit at the same time and in the same frequency N isthe number odusers. Thismeans that each message has logL nats of (in the same cell). However, synchronisation is very difficult information. Ontheotherhand,thesamesizeoftheproblem inourscheme to achieveinan ad-hocnetwork.TheCCSM methoddoesnot resultsinthemessagelengthoflog L natsofinformationforappropriately l require a complicated MAC layer coordination mechanism. chosen l ≪ L. If, for example, l = L1/2, the standard bounds on the binomialcoefficientsyieldlog L =(cid:0)O(cid:1)(L1/2logL1/2).AssumingNfixed, The CCSM allows all users to transmit signals at the same l thesparserecoveryproblemsizeisnowonlyquadraticinthenumberofnats time, therefore no coordination is needed. Another highly ofinformation permessage. (cid:0) (cid:1) Transmitted codeword by user 2 Received waveform by Codeword user 2 (from user 1) span Transmitted codeword by user 1 Modified codeword Channel span Impulse Response Received waveform by user 2 Figure1. Simpleexampleofacodebookandconstructionofthetransmitted Figure2. Simpleexampleofareceiver codebook. signal. are constructed from very short bursts of digital modulation to multipath interference, time dispersion and multiple access signals.Weemphasize,itisnotthedigitalsignalwhichcarries interference (MAI) [12]. Whereas MPPM signalling typically usefulinformation-theinformationrateisthesamenomatter uses a maximum-likelihood receiver [11], which involves an what modulation (BPSK, QPSK, 16-QAM etc) we choose to optimisation problem over the set of all binary sequences of constructthe waveforms. It is the choice of the l-combination length L havingweight l, which becomesintractable evenfor of the codeword span and of the associated weights which moderate values of L and l, the CCSM method utilizes fast carries the information. reconstruction methods based on sparse recovery solvers [2], The transmitted waveform is propagated in a dispersive [3] found in the field of compressed sensing [5], [4]. channel (depicted as a green line) and received as a convo- II. SIGNALING MODULATIONAND CODEBOOK DESIGN lution of the two (black line). The implicit assumption here is that the channel can be modelled as a linear time invariant A. System Overview channel(FIRfilter). Suchanassumptionisa commonplacein To improve the clarity of presentation we describe our the literature and in practice. system using toy examples in baseband signaling. However, The CCSM method relies on the linearity property of thesystemisequallyapplicabletopass-bandsignaling,which, convolution. The receiver reconstructs a modified codeword in fact, we use in the following sections. span – blue waveforms in Figure 2, where each waveform Each of the users constructs its transmitted signal using in the original codeword span is convolved with the channel a codeword span known to all intended receivers. Figure 1 signature. The task for the receiver is to estimate which l depicts an example of the codeword span with L = 6. The waveformswere used by the transmitter. The whole detection message to be transmitted is encoded in an l-combination of process can be performed efficiently using sparse recovery thecodewordspan,i.e.,inachoiceofloutofLcodewordsin solvers. The transmitted waveform is essentially a sequence the codeword span, where l≪L. Note that there are L = of on-off duty cycles, where for most of the time there are l (cid:16) (cid:17) L! suchcombinations.Specifically,thetransmittedsignal silentperiods(“offcycles”).Eachuser utilisesits“offcycles” l!(L−l)! isaweightedsumofthechosenwaveforms.Inbase-band,the to receive signals from the other users. In the “on cycles”, weights could be points in Amplitude Shift Keying (ASK) however, the user cannot receive the signal, which represents modulatione.g. {+1,−1}. In the providedexample in Figure an erasure in the codebook. This is depicted in Figure 2 as 1, l = 2 waveforms are chosen: first and third (depicted in thedotedboxes.Onlynon-erasedportionsofthecodebookare red).Bothweightshappentobe+1.Thetransmittedwaveform used in the detection process. Technically, with this scheme is the sum of the two (brown line). The information rate of the Rx/Tx chainsdo notoperatesimultaneously.Furthermore, this signaling scheme is thus R= 1 log L +lq bits/s, theexplicitassumptionisthatthenodesoperatefastswitching W 2 l where W is the time duration of the(cid:16)wavef(cid:16)orm(cid:17)s in s(cid:17)econds, (at the symbol rate) between Rx/Tx, which is indeed possible and 2q is the size of the alphabet of weights. with the current RF technology. Thisparticularconstructionofconstituentwaveforms(code- B. Constant Weight Codes word span), combinatorial construction of the transmitted signal and the fact that l ≪ L all play a crucial part since The CCSM requires a non-linear encoding operation. The they allow very efficient decoding, MAC-less user coordina- process of mapping the information vectors at each user tion and full duplex operation for each user. A key feature to a unique l-combination of the codeword span can be of the constituent waveforms is sparsity i.e. the waveforms viewed as constant (hamming) weight coding (CWC). The inforZmationvectorZZi(cid:143)F2k(cid:14)lq tcrhaannsnmeilt tbeer t0w aenend i,1:k i,k(cid:14)1:k(cid:14)lq receiver i x time slots 0 * hi...,0 self-hcahti ,iainnel* xi encoder i ImCaWCppCer bi wmIeaigpwhpienrg +-11 xx ssii,,14 6 transxmiitted xxii(cid:16)(cid:14)11 ** hhii..,,ii(cid:14)(cid:16)11 ad6ditiv~ze i cehraasnunreel ~yi inrteemsrfeeolrfve-xenrcie cc(cid:214)(cid:214).0.. dweemigahpipnegr ZZ...0 ci on-odficf tsioignnaarlyling codeword xN * h.i,N noise resscpooalvvrseeyerr yi c(cid:214).ii.(cid:16)(cid:14).11 IIw(cid:16)(cid:16)11 Z..ii.(cid:14)(cid:16)11 c(cid:214) C Z N CWC N v(cid:214) demapper encoder i (cid:16)i decoder i Figure3. CCSMencoder atthei-thuser. Figure4. CCSMdecoderforati-thuser. problem of efficient encoding and decoding constant-weight parameters. In the sequel, we will consider the following vectors received significant interest in the literature. There construction: all columns of S have equal number of non- are practical algorithms of computational complexity linear i zero entries, set to M , and non-zero entries are selected in the length L of constant weight vectors, which are based L on lexicographic ordering and enumeration [6]. However, uniformlyatrandom(cid:4)from(cid:5)apredefinedconstellation,e.g,from the set {+1,−1}. Moreover, every two columns in S have the approach particularly suitable for our system is that of i disjoint support. This way, as the transmitted codeword x [7], as its complexity is quadratic in the weight of constant i is formed as a weighted sum of exactly l columns in S , weight vectors, which fares favourably in comparison to the i the transmitted codeword will have exactly l· M non-zero enumerationapproachinthecasewherel ≪L.In[7],authors L pursue geometric representation of information vectors in an entries, implying that every user i will have ex(cid:4)act(cid:5)ly l· ML l-dimensionalEuclideanspaceandestablishbijectivemapsby on-slotsandwilluseitsM˜ =M−l· ML off-slotstolist(cid:4)ento(cid:5) dissecting certain polytopes in this space. the incomingsignalsof other users. A(cid:4)no(cid:5)therway to construct a signalling dictionary would be to apply a regular Gallager C. CCSM Encoder construction,whichwasoriginallydevelopedforLDPCcodes Consider a network of N +1 users denoted 0,1,...,N, (cf., e.g., Ch. VI of [13] and references therein). each of which has a k + lq bit message to transmit to all Figure 3 depicts a CCSM encoder at user i. The encoding othersthroughawirelessmediumusingthesamesinglecarrier three-step procedure is summarized below2: frequency. Denote by M the number of transmissions, and 1) User i encodes b :=φ (ω ) using a CWC code. i C i,1:k by ωi ∈ Fk2+lq the message at user i. It is assumed that 2) Further lq bits are encoded on non-zero entries users are equipped with an encoder, which constitutes of in b , i.e., c := φ (ω ,b ) = i i w i,k+1:k+lq i bijective maps φC and φw. The first map, φC : Fk2 → C, φw(ωi,k+1:k+lq,φC(ωi,1:k)). This is based on a mapsk-bit binarywordsinto an (L,l)constantweightbinary bijective map that assigns a different complex number code C ⊆ {c ∈ FL2 : wH(c) = l}. The second map to each binary sequence of length q, which can be φw : Fl2q × C → CL assigns complex-numbered values to thought of as a QAM modulation with 2q constellation the non-zero entries in a constant weight binary codeword points. from C. For simplicity, we may assume that C consists of 3) User i transmits x = S c , where the matrix-vector i i i allpossible L constantweightcodewords,in whichcase we multiplication S c is performed over C. l i i can take k =(cid:0) ⌊(cid:1)log L ⌋. Each user i is assigned a signaling 2 l dictionary Si = (si(cid:0),1,(cid:1)si,2,...,si,L), where each si,j ∈ CM D. CCSM Decoder is a sparse column vector. (Columns of the matrix S can be i Figure4 depictsa CCSM decoderatuseri.TheCCSM de- thought of as sampled waveforms constituting the codeword coder receives a superposition of all signals from all intended span in Fig. 1.) Each user hasperfectknowledgeof all N+1 transmitters, i.e., users j 6=i. As aforementioned,the receiver signaling dictionaries. Furthermore, each user i has a perfect does not receive the signal in on-cycles (when it transmits), knowledgeofthechannelimpulseresponsesh ∈CM ofthe i,j which is represented by the erasure channel. Upon removing channelbetweenusersj andi,andofitsownchannelimpulse the self interferencecomponents,theCCSM decoderemploys response h ∈ CM, which we refer to as a “self-channel”. i,i a sparse recovery solver. (“Self-channel” can be thought of as a “radar return”, and its Specifically, the recovery at node i proceeds as follows: role is explained in the description of the CCSM decoder.) We remarkthatthesignallingdictionarySi atuseri canbe 2Throughout the paper, for a,b ∈ N, a ≤ b, a : b denotes the set judiciously optimized to suit the preferred choice of system {a,a+1,...,b},andforavectorx,andsetofindicesA,xA:=(xa)a∈A. 1) Defineanerasurepatternvectorase =∼1(x ),where i i 1(υ) = 0 if υ = 0 and 1(υ) = 1 otherwise. Define (ωˆ ,bˆ ) = φ−1(ˆc ), an erasure matrix E , produced from I identity j,k+1:k+lq j w j i M,M matrix, by removing rows where corresponding ei has ωˆj,1:k = φ−C1(bˆj). zero entry. Denote the number of rows in E by M˜. i 2) User i using off-duty cycles receives: III. SPARSE RECOVERY FORCCSM We recallthateachuseri isrequiredto solvethesparse re- coveryproblem(4)inordertocorrectlydetectthetransmitted N y˜i =Ei hi,j ∗Sjcj +˜zi+Ei(hi,i∗Sici), messages. This is a non-convex and intractable optimization j=X0,j6=i problem. However, in the spirit of the compressed sensing   (1) framework, one can apply a convex relaxation, by replacing where ”∗” symboldenotesconvolutiontruncatedto M the L0 norm with the L1 norm: time slots and˜z represents the additive Gaussian noise i over M time slots. vˆ = argminky −A v k 3) Since each user switches into reception mode in be- −i v−i i −i −i 2 tween transmitting short bursts, there would be echoes s.t. kc k = l, for allj 6=i. (5) j 1 of its own transmitted signal in the received signal We will refer to the convex relaxation in 5 as Group Basis (self interference). However, all users know their own Pursuit (GBP). Furthermore, one can employ an even simpler transmitted signal and can therefore subtract the term form of the convexrelaxation, i.e., a standard embodimentof E (h ∗S c )ineq.(1)aslongastheyknowthe“self i i,i i i the LASSO/Basis Pursuit (BP): channel”. yi =y˜i−Ei(hi,i∗Sici)=A−iv−i+zi, (2) vˆ−i = argmv−ini kyi−A−iv−ik2 s.t. kv k ≤ lN, (6) where zi =Ei˜zi, v−i is the NL-columnvectorformed −i 1 ib.ye.,cvo−ncia=tenact⊤0in|cg⊤1v|e.r.t.ic|cal⊤il−y1|cc0⊤i+, 1c|1.,.....,|cc⊤Ni−⊤1,acni+d1A,..−.ciNis, wbuhtecreanthbeegreonufoprscterducatuftreerosfonlvoinn-gze(r6o).entriesinv−iisomitted, an M˜ ×NL(cid:2) matrix, given by: (cid:3) Another method to solve our original problem (4), is to employagreedyiterativesparserecoveryalgorithm.Anumber of such algorithms have appeared in the literature including A = E h ∗S |h ∗S |··· (3) Compressive Sampling Matching Pursuit (CoSaMP) [2] and −i i i,0 0 i,1 1 (cid:20) Subspace Pursuit (SP) [3]. These algorithmscan be enhanced to take into account the additional group structure of the |h ∗S |h ∗S |···|h ∗S . i,i−1 i−1 i,i+1 i+1 i,N N (cid:21) unknown vector, which is imposed by our system set-up. Namely, in addition to the unknown vector v having lN Note thatthematrixA canbe calculatedoffline,as it −i −i non-zero entries, each of its N subvectors c of length L, depends only on the channelimpulse responses and the j has exactly l non-zero entries. In Algorithm 1, we present signaling dictionaries. Therefore,it needs to be updated the modification of Subspace Pursuit, which we name Group only when the channel impulse response changes. Subspace Pursuit (GSP). For simplicity and without loss of 4) User i needs to solve the following problem to detect generality, we present the GSP as applied to the sparse the desired signal: recoveryproblemat user i=0. The GSP is a low complexity method,whichhas computationalcomplexityof LeastSquare estimator of size lN, and is vastly more computationally vˆ = argminky −A v k −i v−i i −i −i 2 efficient than convex optimisation based methods, including s.t. kc k = l, for allj 6=i, (4) Group Basis Pursuit (GBP) and Basis Pursuit (BP). j 0 Figure5depictsperformanceofthethreesparsesolversfor This is a non-convex optimisation problem. However, groupCSset-up.In thisstudythereare (N+1)=10groups, we note that exactly Nl out of NL entries in v andin eachgroupl =4 outofL=32elementsarenonzero. −i are non zero, hence its sparsity level is by the initial This investigation was performed for three under sampling assumption l ≪1. This set-upis foundin compressive ratios for each of those reconstructionmethods. For example, L sensing (CS) problems, and thus one can apply a range BP-100signifiestheBasisPursuitsolverona ComplexGaus- of efficient sparse recovery solvers available in the sian dense measurementmatrix with size 100×320(i.e. 31% literature to find an approximate solution to eq. (4), undersamplingratio).The non-zeroelementsin the unknown which we discuss in the next Section. vectoraredrawnfromaQPSKmodulationset.Theerrorevent 5) Finally, user i decodes the messages for all j 6=i: is defined as any symbol error in the group. For low under Algorithm 1 Group Subspace Pursuit (GSP). 100 • Input.A waveform y0 ∈CM˜ at user 0, received during the off-duty cycles, with the self-interference component removed, CIR/Signaling 10−1 matrixA−0∈CM˜×NL,CCSMparameters Landl. • Output.Vectorvˆ−0 consisting ofN subvectors cj oflengthL,each havingexactly lnon-zeroentries. 10−2 12)) IImndaietgninatiiltfiuzyde..eFSionerttehrae0chj=-tjhy=0L,-ts1u,=b.-.v1.e,,cTNto0r,=oseftØAU.∗−j0trot−th1e,li.ein.,dices largest in Pr(error)10−3 M=150 M=175 M=200 Uj ∈ argmax |hrt−1,a−0,wi|: 10−4 M=225 W ( M=250 wX∈W W⊂[(j−1)L+1 : jL],|W|=l . 10−50 2 4 6 8 10 12 14 16 18 20 ) SNR [dB] 3) Merge. Put the old and new columns into one set: U = Tt−1 ∪ Nj=1Uj . Fraitgeu:reIn6t.hisPcearsfeortmhearnecearoef(tNhe+pr1o)po=sed5museetrhsodsiminulttearnmesouosflymbersosaadgceasetrionrgr 4) (cid:16)ESstimate. S(cid:17)olvetheleast-squares problem onthechosencolumn-set: messagesusingCCSMwithL=64,l=12. vU′ = argmvin A−0,Uv−rt−1 2 v′ = 0 (cid:13) (cid:13) [1:N]\U (cid:13) (cid:13) sampling ratios, Group Basis Pursuit performsbest. However, 5) Prune. Retain the l coefficients largest in magnitude in each L-sub- for moderate and larger values, our Group Subspace Pursuit vectorofv′,i.e., is almost the same. Therefore, given its low complexity, we apply GSP to analyse the CCSM performance in the sequel. Uj ∈ argmax vw′ : W (wX∈W(cid:12) (cid:12) IV. NUMERICAL RESULTS (cid:12) (cid:12) W⊂[(j−1)L+1 : jL],|W|=l . In this section we report numerical results of the proposed ) method and quantitative comparison with the state-of-the-art. toobtainthesupportestimate Tt= Nj=1Uj. We considera multi-user wireless network with N+1 nodes, 6) Iterate. Findthet-thestimateandupdatetheresidual: whereall usersare within radiorangeof each other. Allusers S vt,Tt = argmvin A−0,Ttv−rt−1 2 attemptto broadcasta message to all othernodes.We assume vt,[1:N]\Tt = 0 (cid:13)(cid:13) (cid:13)(cid:13) a very dispersive channel, modelled by an FIR filter with rt = y0−A−0vt 32 taps, with exponentially decaying profile. Moreover, we assume that each pair of nodes has an independent channel. Sett←t+1andrepeat(2)-(6)untilstoppingcriterion holds. We set L = 64, l = 12, and use QPSK signalling (q = 2), 7) Output.Returnvˆ−0=vt. i.e., each message contains log L +lq =65 bits. Figure 2 l 6 depicts the performanceofjthe p(cid:0)ro(cid:1)pkosedmethod for 5 users in terms of message error rate (MER) as a function of signal- to-noiseratio,forvariousvaluesofthenumberM ofavailable 100 symbolintervals.TheMERisanempiricalprobabilityestimate ofa failureoccurringin themessage delivery.We remarkthat the values of MER could be further decreased by the use of 10−1 outer coding. TofurtherassesstheperformanceoftheCCSMmethod,we Pr(error) 10−2 BBBPPP−−−123000000 ocofmwphaartewitosuladchbieevtehdetbhersotuhgyhppoutthteoticthalestohlruotuiognhsp,uctoenssttirmucatteeds GSP−100 using the state-of-the-art in an idealised scenario. As before, GSP−200 GSP−300 weassumethatthetransmissionoccurrsoveratimedispersive 10−3 GBP−300 channel, modelled by an FIR filter with 32 taps, but, in order GBP−200 GBP−100 to make a fair comparison to MAC protocols below, without any additive noise. Achieved throughputof CCSM in bits per 10−4 5 10 15 20 25 30 symbol interval, given by (N +1)·65/M , where M is 1/σn2 [dB] theminimumnumberofsymbolintervalsamtiwnhichnomemssinage errorsoccurredin at least 100,000simulation trials, is plotted Figure5. PerformancecomparisonofBasisPursuit/Lasso(BP),GroupBasis Pursuit(GBP)andGroupSubspacePursuit(GSP). in Fig. 7. We note that the throughput performance of the CCSM is insensitive to the number of users in the network. First hypothetical system we consider exploits a central 2 controlling mechanism that closely coordinates transmissions between all users, using a TDMA channel access. To avoid val]1.8 interference the total transmission time would be divided er nt equally into N +1 non overlapping slots. Each user would ol i1.6 broadcast its message to all other users in its designated mb CSMA/CA with 20% GI y slot, and receive messages from all other users in remaining s/s1.4 fully centralized system with 20% GI N slots. To cope with the dispersive channel nature, such bit CCSM systemwouldneedtouseFDE/OFDM.AtypicalFDE/OFDM ut [in 1.2 system requires a guard interval (cyclic prefix) of about 20% p h g 1 slot duration. However, in reality, additional guard intervals u o would be needed, and close coordination between nodes thr0.8 implies additional overheads. When compared even to this 5 10 15 20 25 idealised and highly impractical system, our method offers number of users a better throughput, as each message transmission requires 65/2 =41symbolintervals,whichresultsinthethroughput Figure7. Throughput comparison ofthe CCSMandtheidealised versions 0.8 l m ofCSMA/CA andfullycentralized TDMAwithguardintervals. of 65/41 = 1.58 bits per symbol interval regardless of the number of users in the network. However, in most cases, such a central controlling mech- state-of-the art. However, the presented performance gains anism would be unavailable, and the second, more realistic, of CCSM are conservative, since we have opted for a low benchmarking system we consider is based instead on dis- complexity detection method. Further performance gains can tributed coordination function (DCF) and CSMA/CA [14], be achieved by employing sparse recovery methods which more specifically on DCF as used in IEEE 802.11b MAC would capitalise on the discrete nature of the unknown signal in broadcasting mode. Such system relies on the randomised vector. deferment of transmissions in order to avoid collisions on a shared wireless medium. Since we assume that all users are REFERENCES withinradiorangeofeachother,thereisnoinefficiencyresult- [1] Lei Zhang and Dongning Guo “Wireless Peer-to-Peer Mutual ing from hidden/exposed terminals, thus we employ only the Broadcast via Sparse Recovery” preprint available from: basic access mechanism of CSMA/CA protocol. In addition, http://arxiv.org/abs/1101.0294. [2] D. Needell and J. A. 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