Collision Resolution by Exploiting Symbol Misalignment

1 Collision Resolution by Exploiting Symbol Misalignment Lu Lu Dept. of Information Engineering The Chinese University of Hong Kong ll007@ie.cuhk.edu.hk Soung Chang Liew Dept. of Information Engineering The Chinese University of Hong Kong soung@ie.cuhk.edu.hk Shengli Zhang Dept. of Information Engineering The Chinese University of Hong Kong slzhang5@ie.cuhk.edu.hk Abstract —This paper presents CRESM, a novel collision resolution method for decoding collided packets in random-access wireless networks. In a collision, overlapping signals from several sources are received simultaneously at a receiver. CRESM exploits symbol misalignment among the overlapping signals to recover the individual packets. CRESM can b e adopted in 802.11 networks without modification of the transmitter design; only a simple DSP tech nique is needed at the receiver to decode the o verlapping signals. Our simulations indicate that CRESM has better BER performance than the simplistic Successive Interference Cancellation (SIC) technique that t reats interference as n oise, for almost all SNR regi mes. The implication of CRESM for random-access networking is significant: in general, using CR ESM to resolve collisions of up to n packets, network throughput can be boosted by more than n times if the transmitters are allowed to transmit more aggressively in the MAC protocol. Index Terms —multi-packet reception, collision resolution, interference cancellation, 802.11 I. I NTRODUCTION In wireless random-access networks, packet collisions are common. For ex ample, in the popular IEEE 8 02.11 MAC, collisions occur when two or more station s decide to transmit to the access point (AP) simultan eously. At a station, a random backoff count down process is u sed to decide when the station can transmit its packet. The most common cause of c ollisions is when two or more stations simultaneously count down to zero and transmit together. This can happen even when the stations can carrier-sense each other. Colli sions can also happen due to t he h idden-node phenomenon [1], wherein two stations that can not carrier-sense each other transmi t to the AP simultaneously. This paper presents a novel method to recover collided packets. We call our method CRESM (collision resolut ion by exploiting symbol mi salignment). CRESM does not require symbol-level synchronization among the stations. In fact, it thrives on s ymbol mis alignment among t he statio ns, which occurs naturally. A fundamenta l concept underlyi ng CRESM is that collided signals with symbol misalignment can be treated as the output from a virtual convolutional encoder. To the best of our knowledge, this is the first paper to use thi s concept t o extract collided packets b y means of (1) over-sampli ng and (2) an optimal Viterbi-like decodin g algorithm. Related Work Ref. [2] propo ses the disabl ing of the carrier sensing mechanism i n a carrier-sense multiple access (CSMA) network to increase the likelihood of simultaneous transmissions (collisions). Coll ided signals are modeled using higher order constellation maps, and the joint dec oding method requires symbol-level synchronization. Ref. [3] makes use of interference cancellation techniques to resol ve the collisions. CRESM, on the other hand, does not assume symbol alignment. Also, CRESM does not requ ire de-activating carrier sensin g and can be deployed in a CSMA or a non-CSMA random access network. In general, however, we do not advocate the disabling of carrier sensing wh en we can resolve collisions. Although we would want to encourage simul taneous transmissions , it is not clear that disabling carrier sensing altogether i s the best way to do so. Instead, we woul d have better co ntrol over the system using a hig her carrier sens ing threshold or by increasing the transmission probabilities of the stations [4] in a way that is commensurate with the degree of collisions (number of packets in collisions ) that can be dealt with. Ref. [1] focuses on resol ving collisions du e to the hidden-node phenomenon. In Zig-Zag decod ing [1 ], for example, s everal con secutive hidden-node collisions of the same group of packets are used to resolve the collid ed symbol s. In practice, and in particular with the use of RTS/CTS, hidden-node co llisions are no t as common as backoff collisions. Furthermore, resolving hidden-node collisions does not boost the overal l system throughput so much as it solves the unfairness problem induced by hidden nodes. Resolv ing backoff collisions, on the o ther hand, can potentially lead to much higher syst em throughput by allowing the statio ns to attempt to transmit more aggressively. CRESM can be used to deal with both backoff coll isions and hidden-node c ollisions. Recently, there have been inten se research activities on using physical-la yer network coding (PNC) [5, 6] to bo ost wireless network performance. The application domain of PNC is in relay networ ks. Here, we are interested in the more common WLAN scenario in which multiple s tations wan t to transmit to a comm on access point (as the gateway t o the Internet), and that the inter-traffic among the stations i n the WLAN is minimal. Finally, CRESM can be conside red as a meth od for successive interference cancellation (SIC) [7, Ch.6] and multi-user det ection (MUD) [8]. A distinguishing feature of CRESM is that it makes use of over-sampling on the unaligned overlapping packets to a cquire more information on them. 2 II. S YSTEM M ODEL AND B ASIC CRESM A LGORITHM In this s ection we p resent the system model an d the basic CRESM al gorithm. T o ease exposition, we describe CRESM in the terms of two-packet collisions and we assume carrier-phase synchronization between the two packets. Col lisions of more than two packets and CRESM without carrier-phase synchronization will be tre ated in Section IV. Fig. 1. System model for two pa cket collisions A. System Model For a concrete picture, consider a CSMA wireless LAN as shown in Fig. 1. Suppose that both nodes A and B have packets to trans mit to the AP. Nodes A and B first sense the channel to see if the channel is busy . Despite carrier sensing, it is still possible for A an d B to transmit simultaneo usly when their backoff mechanism decides to transmi t together. When th at happens, the transmission s will collide at the AP. We represent a wire less packet by a stream o f d iscrete complex numbers. Specifically, we use complex n umbers x A [ m ] and x B [ m ] to represent the modulated symbols of nodes A and B respectively. The overlapped signal received at the AP under packet collision for an AWGN channel is ( ) ( ) [ ] cos( ) ( ) [ ] cos( ( )) ( ) A A c B B c y t h t x t t h t x t t w t ω ω =     + − ∆ − ∆ +     (1) where ( ) w t is Gaussian white noise with powe r spec tral density 0 ( ) / 2 w S f N = ; h A ( t ) and h B ( t ) are complex n umbers that r epresent the channel attenuations with phase shift from A and B to the A P, re spectively; t     is the largest in teger no larger tha n t ; c ω is the carrier frequen cy; ∆ is relative difference of the times of arrival of the two symbols at the AP. We assume that h A ( t ) and h B ( t ) stay constant throughout a packet duration. We further assume the transmit powers of nodes A and B have been combined into the corresponding h A ( t ) and h B ( t ). Fig. 2. Signals from A and B and the combined signal at the AP In wireless CSMA protocols, ther e is t ypically no collision detection (e.g., 802.11). In the absence of collision detection, once the transmission of a packet begins, it will co ntinue until the whole packet is transmitted, even while a collision is ongoing. An exampl e of a colli sion of two BPSK modulated signals with perfect power control an d carrier phase synchronization in con tinuous time is illustrat ed in Fig. 2. B. Discretization with Over-sa mpling The basic stru cture of the receiver of CRESM is shown in Fig. 3. Over -sampling is used to generate two outputs in one symbol duration T . We as sume normalizati on such that T = 1 and 0 1 < ∆ < throughout thi s paper. The t wo output streams are then mu ltiplexed into one discrete outpu t stream [ ] y m , so that in thi s outpu t stream there are two symbo ls per symbol duratio n, one from A and one from B . Fig. 3. Over-sampling at the re ceiver For the over-samplin g in CRESM, the i ntegral of the traditional receiver [9], which integrat es ove r the whole s ymbol duration, is modified and is now divid ed into two parts: one integral is from time 0 to time ∆ , and the other is from time ∆ to time 1. The two discrete outp uts (pre-MUX) of the receiver in Fig. 2 can be expre ssed as 0 1 2 ( ) cos( ) [ 2 ] [ ] [ 1 ] [ ] 2 ( ) cos( ) [ 2 1 ] [ ] [ ] [ ] 1 c A A B B c A A B B y t t dt y k h x k h x k w k y t t dt y k h x k h x k w k ω ω ∆ ∆ = = + − + ∆ ′ + = = + + − ∆ ∫ ∫ (2) for k =0,1,2,…, where x B [-1]=0; and [ ] w k and [ ] w k ′ are Gaussian noises with variances of 0 2 N ∆ and 0 2( 1 ) N − ∆ respectively. In this and the following sub-sections, we assume perfect power con trol and carrier synchronization to eas e exposition, i.e., h A = h B =1. Then the outputs in (2) can be simplified to (3) [2 ] [ ] [ 1 ] [ ] [2 1 ] [ ] [ ] [ ] A B A B y k x k x k w k y k x k x k w k = + − + ′ + = + + (3) C. Basic Idea of CRESM First of all, we note that technically the two received packets will mos t like ly arriv e at the AP with s ymbol mi salignment ∆ if one does not deliberately try to synchronize the symbol arrival ti mes. CRESM exploits this sym bol misali gnment to resolve collisions. Let us refer to the pack ets from A and B as A X and B X , respectively. The symbols in A X are denoted by [0] [ 1 ] [ 2]... A A A x x x , and the symbols in B X are deno ted by [0] [ 1 ] [ 2]... B B B x x x X A X B Node A Node B AP y ( t ) y [ m ] 2cos ( w c t ) y [2 k ] y [2 k+ 1] M M M M U U U U X X X X 1 ti m e X B ti m e ti m e 1 1 - 1 - 1 2 1 - 2 X A Y 3 Fig. 4. The received signal a t AP when there is no noise: (a) an example of specific symbol values; (b) ov er-sampled decomposition o f overlapped symbols from A and B ; (c) net s uperimposed symbol values re ceived at AP. The net effect of symbo l misalignment is sh own in Fig . 4, assuming the use of BPSK modulation (i.e., we map “0 ” bit to 0 1 j e = and “1” bit to 1 j e π = − ). The effect ive “over-sampled” symbols as perceived at the AP a re given in Fig. 4(c). CRESM makes use of t hese over-sampled symbols to recover the original symbols from A and B . Virtual Encoding Conceptually, CRESM treats [ ] y m as the output from a “virtual” encoder [ ] z m plus noise. Specifically, [2 ] [ ] [ 1 ] [2 1 ] [ ] [ ] A B A B z k x k x k z k x k x k = + − + = + (4) where [ ] A x k and [ ] B x k are the k th symbols of A X and B X respectively. The possible values of the virtual encoder, i.e., th e sum of the original symbo ls from A and B , are as follows: 1 1 2 1 ( 1 ) 0 ( 1 ) 1 0 ( 1 ) ( 1 ) 2 + = + − = − + = − + − = − (5) That is, two symbols in the domain of {1 , -1} are encoded by the virtu al encod er into one symbol with 3 p ossible v alues in the domain of {2, 0, -2}. In the absen ce of noise, the possible values of the received sequence at the AP are the pos sible v alues of the v irtual encoder’s outputs as in (5). While a value of 2 (-2) in the output symbol mean s the original symbols from A and B are 1/1 (-1/-1), a value of 0 means the original symbols from A and B are either 1/-1 or -1/1 as in (6). 2 1 / 1 0 1 / 1 1 / 1 2 1 / 1 or → → − − − → − − (6) Thus, based on one virtual symbol alone, we cannot always recover the original symbo ls from A or B . However, with symbol misalignment and over-sampling receiver, an original symbol is actually mapped to two succ essive virtual symbo ls. Exploiting this “redundanc y”, the A P could recover the orig inal symbols, with the followin g CRESM Algorithm. Successive Decoding CRESM employs a sort of successive decodin g on the virtual symbols to recover the o riginal symbols. The basic idea is illustrated in Fig. 5. Fig. 5. The CRESM successive deco ding For the notation in (4), the decoding process can also be expressed as [ ] [ 2 ] [2 1 ] [ 1 ] [ ] [ 2 1 ] [2 ] [ 1 ] A A B B x k z k z k x k x k z k z k x k = − − + − = + − + − (7) where the current bit is actually decoded from the previous one with the knowledge of the overlapped symbols. In the above paragraphs, we des cribed the basic idea of CRESM i n th e absen ce o f noise. In practice, noise is unavoidable and it can cause decoding error. To improve performance, we should make full use of the information received at the AP. For example, when the virtual symbol is 2 or -2, we do not need to rely on the previo us virtual symbol to recover the original symbols as in (6 ). In Sect ion III , we propose a sort of Viterbi-like decoding algorithm to decode the received signal t o make full use of it s information and to increase our confidence o f correct detection. III. CRESM WITH V ITERBI - LIKE D ECODING A. Virtual Convolution al Encoding We can consider the virtual symbol s as the output of a virtual convolutional enco der as sh own in Fig. 6. The inp ut to the virtual encoder consists of a stream o f symbols which is a multiplexed stream of the original symbols fro m th e two s ource nodes A and B . Specifically, the inpu t stream is denoted by [0] [ 1 ] [2 ] [3 ] [ 4] [5] [0] [0] [ 1 ] [ 1 ] [ 2] [ 2]... A B A B A B v v v v v v x x x x x x = where [0] [ 1 ] [ 2]... A A A x x x is the symb ol stream from node A and [0] [ 1 ] [2]... B B B x x x is the sy mbol stream fr om node B . The virtual encoder in Fig. 6 produces an output stream [0] [ 1 ] [ 2]... [0]( [0] [ 1 ])( [ 1 ] [ 2])... z z z v v v v v = + + . Obviously, they are the virtual symbols arrived at the receiver when there is no noise. To resolve the p acket colli sion, the goal of ou r receiver is to recover the input stream of the vir tual encoder, [0] [ 0] [ 1 ] [ 1 ] [2] [ 2]... A B A B A B x x x x x x , from which the individual streams from A and B can then be ex tracted. Fig. 6. The virtual convolutiona l encoder where D means o ne symbol delay From Fig. 6 we notice that the virtual con volutional coding is the same as that of the convent ional convo lutional c oding except for t he fo llowing subtleties. First, the conv entional -1 1 -1 1 1 -1 1 1 X A X B X A X B 1 -1 -1 1 -1 1 1 -1 1 Y 2 0 0 -2 0 ... ... (a) (b) (c) y [1] y [2] y [3] y [5] y [7] y [4] y [6] 1 -1 1 1 1 -1 2 0 D v z 0 X A 1 -1 -1 1 -1 1 1 -1 1 1 -1 1 1 1 -1 1 2 0 0 0 0 2 ... -2 1 X B Z t 4 convolutional code applies ‘XOR’ operation on the bits, b ut here we apply a rithmetic ad dition. Seco nd, in the conventional system, the output bits are transmitted with the same t ime duration an d therefore the noise is stati stically the same for each of the bits; in our system, the output bits have t wo poss ible durations: ∆ for even bits, and ( 1 ) − ∆ for odd bits, an d consequently the noise leve ls for the bits alternate through the stream, due to the varying noise bandwidth durin g the sampling process. The effect of the al ternating noise can be expressed mathematically as follows: [2 ] [ 2 ] [ ] [2 1 ] [ 2 1 ] [ ] y k z k w k y k z k w k = + ′ + = + + (8) where [ ] w k and [ ] w k ′ are the i.i.d Gaussian noises as introduced before, with variances of 0 2 N ∆ and 0 2( 1 ) N − ∆ respectively. The virtua l encod er in Fig. 6 corresp onds to a convolu tional code with code rate 1. That is t o say there is no coding redundancy at all. The channel coding theory treats convolutional coding redunda ncy as a means for forward error correction (FEC). But since we introduce no coding redundancy here, CRESM its elf can not co rrect bit e rrors during transmission. Howev er if the t ransmitted sign als apply a FEC before modulation, the erro rs can be corrected after CRESM. Treating the symbol in the register (Fig. 6) as having two possible states: ‘1’ and ‘-1’ , the encoding process is given by Fig. 7 , where / i o denotes the input i that trigger s the transition, causing output o to be produced. Let u s assume that the register val ue is initialized to ‘1’. When t he first symbol v [0] arrives, the state co uld go from ‘1’ to ‘-1’ or s tay at ‘1’ depending on whether the this symbol i s -1 or 1. If we further consider the possible t ransitions due to successive inputs over time, we get the virtual convolutional trellis as shown in Fig. 8, which is very similar to t he conventional convolu tional encoding process except that this virtual en coding is the conceptual outcome of simultaneously received symbols from two sources rather than FEC coding on bits. Although the motivation and t he underlying phe nomenon giving ri se to CRESM are totally different from those of convolutional channel co de, we can still apply a Viterbi-like d ecoding m ethod to recover the originals bits in CRESM, as will be illustrated in the following subsecti on. Fig. 7. The CRESM state transition diagram Fig. 8. The virtual encoding pr ocess of the collision shown in Fig. 3(d) B. Viterbi-like Decoding Maximum Likelihood (M L) decoding is optima l in terms of minimizing error prob ability when all input message sequ ences are equally likely. In particular, a ML deco der chooses ( *) m U if ( ) ( *) ( ) over all ( | ) arg max ( | ) m m m U P Y U P Y U = (9) where ( ) m U is a p ossible input sequence and ( ) ( | ) m P Y U is the likelihood funct ion given the received sequence Y . The Viterbi decoding algorit hm, propos ed and analyzed by Viterbi [10] in 1967 , esse ntially performs ML decodin g fo r convolutional code; however, it reduces the computational complexity by taking advantage of the special structure in the code trellis. Omura [1 1] dem onstrated that the Vite rbi algorithm is, in fac t, a n ML decoding metho d. The goal of selecting the optimal ML path can be expressed, equivalently, as choos ing the codeword with the minimum distance metric [9, Ch. 7]. Inspired by Viterbi decoding, we prop ose a Viterbi-like decoding method for our virtual encoding process in Secti on IIIA. The detai ls of our scheme and the original scheme are quite similar an d can be expressed as follows. W ith referen ce to Fig. 7, once a symbol is received, we can calculate the distance from an or iginating sta te to a ne xt s tate as the Euclidean distance d i between the received symbol y i and the corresponding transmitted s ymbol z i for that stat e transition. Different pos sible tran sitions cor respond to different distances . We th en sto re th e cum ulative distances i d ∑ of d ifferent paths that correspond to different sequences of transitio ns. Given two paths with the same first and last states in their sequences, the path with th e smaller accumulati ve dista nce is kept and the others are discarded. Thus, by co mputing th e minimum distance path in the virtual cod ing trell is, we can then get [0] [ 1 ] [2]... z z z . After that, the original packets A X and B X can then be obtained using (7). Since Viterbi decoding is a kind of optimal ML decoding method for convolutio nal code, we have the following proposition for o ur Viterbi-like decodin g. Proposition 1 : Viterbi-like decod ing is an ML decoding method among all po ssible CRESM decoding meth ods. The proof of this theo rem is similar to that of [11 ]. IV. F URTHER D ISCUSSION In this sectio n we discuss two extensions to CRESM. The first is the resolution of n -packet collision where n can be more than two. The second is CRESM without the assumption of carrier phase synchroniz ation. A. CRESM with Collisions of More than Two Packets For co llisions of more th an two pack ets, the b asic idea remains the sam e except that we n eed to increase over-sampling. Proposition 2 : The co llision of n packets with symbol misalignment with ( 1 ) n − different time s hifts requires n samplings within on e symbol. For illu stration, consider a 3-packet collision, say of nod es A , B and C . Let X A , X B and X C be the cor responding packet-vectors 1 1/2 -1 -1/-2 -1/0 1/0 Input bit 1 Input bit -1 i / o means Input/Output 1 -1 1 -1 1 -1 z [1]=2 2 2 -2 0 0 0 1 -1 2 0 0 1 -1 2 0 0 1 -1 2 0 0 1 -1 2 0 0 z [2]=0 z [3]=0 z [4]=2 z [5]=0 z [6]=-2 1 -1 2 0 0 z [7]=0 -2 -2 -2 -2 -2 5 containing information symbols. Then we hav e a virtual input streams [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] . .. [ 0 ] [ 0 ] [ 0 ] [ 1 ] [ 1 ] [ 1 ] . . . A B C A B C v v v v v v x x x x x x = to a virtual convolutional encoder as shown in Fig. 9. And the output of the virtual convolutional encoder is [ 1 ] [ 0 ] [ 1 ] . .. [ 0 ] ( [0 ] [ 1 ] ) ( [ 0 ] [ 1 ] [2 ] ) . .. z z z v v v v v v − = + + + which is used as the source for a Viterbi-li ke decoding method with 4 states. This virtual output s tream Z can be obtained by three sa mples within one symb ol. Fig. 9. The virtual encoder fo r CRESM with 3 collisions The decoding pro cess is similar to t hat of 2-packet collision except that the virtual convolutional encod ing trellis has 4 states (‘ 1,1’; ‘1,-1 ’; ‘-1,1’; and ‘-1,-1’). Du e to the limit ed spac e here we omit the d etailed descript ion. B. CRESM without Car rier-Phase Syn chronization In this subse ction we present a gen eralized v ersion o f CRESM (G-CRESM) that does not ass ume carrier-pha se synchronizat ion. We combine the ef fects of non-synchro nization into th e c omplex channel coefficients h A and h B . The performance of G-CRESM is sli ghtly better than CRESM with carrier- phase synchroni zation, as will be seen from our simula tion results in Se ction V. The coeff icients h A and h B do not have any effect on the successive d ecoding algo rithm in Se ction IIC. We can see this from the modifie d successive dec oding outcome s 1 [ ] ( [2 ] [2 1 ] [ 1 ]) 1 [ ] ( [ 2 1 ] [2 ] [ 1 ]) A A A B B B x k z k z k x k h x k z k z k x k h = − − + − = + − + − (10) For Viterbi-like decoding in Secti on III, from (2) we ca n get 0 ( ) 1 ( ) [ 2 ] 2 [ 2 ] ( ) cos( ) [ ] [ ] | | [ 1 ] [2 1 ] 2 [ 2 1 ] ( ) cos( ) ( 1 ) [ ] [ ] | | [ ] B A B A G c A A i B A B A A G c A A i B A B A A y k y k y t t dt h h h w k x k e x k h h y k y k y t t dt h h h w k x k e x k h h ϕ ϕ ϕ ϕ ω ω ∆ − ∆ − = = ∆ ⋅ = + − + + + = = − ∆ ′ = + + ∫ ∫ (11) where B A ϕ ϕ − is the relative p hase of the X A and X B. We could rewrite th e equations (11 ) as ( ) ( ) [ 2 ] [ ] [ 1 ] [ ] [ 2 1 ] [ ] [ ] [ ] B A B A i G A B i G A B y k x k Hx k e n k y k x k Hx k e n k ϕ ϕ ϕ ϕ − − = + − + ′ + = + + (12) where | | B A H h h = , an d [ ] n k and [ ] n k ′ are t he sca led Gaussian noises with variances of 0 2 | | A N h ∆ and 0 2( 1 ) | | A N h − ∆ respectively. The constellation map of G-CRESM in genera l has four points due to the phase difference of two transmissions, as illustra ted in Fig . 10. The Euclidean distance of a received symbol is well defined based o n this figure. The decodin g of G-CRESM is simi lar to that of CRESM described in Section IIIB ; the only difference is the v alues of transition outputs in the virtual e ncoding trellis as shown in Fig. 11. Fig. 10. Constellation map of G-CRESM with BPSK modulation and / 4 B A ϕ ϕ π − = Fig. 11. Encoding trellis of G-CRESM V. S IMULATION R ESULTS We have performed simulation s to investigate the performance of CRESM and G-CRESM. W e assumed BPSK modulation with no channel coding. We used Vi terbi-like decoding t o resolve pac ket collision s, limiting our attention, however, to coll isions involv ing two packets . 5 6 7 8 9 10 11 12 13 14 15 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR(dB) BER The performance of CRESM compared with BPSK and SIC BPSK without collisions SIC CRESM with ∆ = 0.01 CRESM with ∆ = 0.1 CRESM with ∆ = 0.2 CRESM with ∆ = 0.3 CRESM with ∆ = 0.4 CRESM with ∆ = 0.5 Fig. 12. The simulation results o f CRESM with BPSK modulation Fig. 12 shows the BER resu lts. For benchmarking, we al so present the results of non-sim ultaneous transmi ssion with BPSK m odulation, and SIC [3 , 7, Ch.6] with the s ame transmission setup as in CRESM but which simply treats interference as noise during the decoding proces s. For CRESM, we can see that as ∆ varies fro m 0 to 0.5 symbol leng th (and by symmetry, 1.0 to 0.5), the aver age BER decre ases rapidly. For SIC, symbol misalignment has no e ffect because the power received from the other transmitter is simply treated as noise ; furthermore, the BER does not improve with the increase of image real -1 -j -2 2 1 j 1/-1 received signal constellation point d i Euclidian distance a/b X A sent bit / X B sent bit 1/1 -1/1 -1/-1 d 1 d 2 d 3 d 4 1 -1 1 -1 1 -1 1.7+0.7j 0.3- 0 .7j 1 -1 1.7+0.7j 1.7+0.7j -0.3 + 0.7j 0 . 3 - 0 .7 j -1.7-0.7j -1.7-0.7j 0 . 3 - 0 . 7 j - 0 . 3 + 0 . 7 j 1 -1 1.7+0.7j -0 . 3+ 0 .7 j -1.7-0.7j 0 . 3 - 0 . 7 j 1 -1 1.7+0.7j 0.3- 0.7j -1.7-0.7j - 0 . 3 + 0 . 7 j D v z D 6 SNR s ince in ou r set -up the po wers used b y b oth transmitter are the same. 5 6 7 8 9 10 11 12 13 14 15 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR(dB) BER The performance of G−CRESM (with φ B − φ A = π /4) BPSK without collisions SIC G−CRESM with ∆ = 0.01 G−CRESM with ∆ = 0.1 G−CRESM with ∆ = 0.2 G−CRESM with ∆ = 0.3 G−CRESM with ∆ = 0.4 G−CRESM with ∆ = 0.5 Fig. 13. The simulation results of G-CRESM with different ∆ From Fig. 12 we also se e that CRESM with half sy mbol misalignment ( 0.5 ∆ = ) is 3.7 dB wor se tha n the singl e-source BPSK case. This performance penalty from co llision resolution is fundamen tally due to two reas ons. The fi rst reason is over-sampling , which broadens the band width of the noise by a factor of max{ 1 / , 1 /( 1 )} ∆ − ∆ ; as a result, the “ effective” SNR is less t han BPSK by at le ast 3dB. The second rea son is the dependence of the decodi ng of succe ssive bits; i.e., the decoding o f the curr ent bit depends no t only on the c urrent received symbo l, but also on th e previous decod ed bit. 5 6 7 8 9 10 11 12 13 14 15 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR(dB) BER The performance of G−CRESM (with ∆ = 0.5) BPSK without collisions SIC G−CRESM with φ B − φ A = 0.01 π G−CRESM with φ B − φ A = 0.1 π G−CRESM with φ B − φ A = 0.2 π G−CRESM with φ B − φ A = 0.3 π G−CRESM with φ B − φ A = 0.4 π G−CRESM with φ B − φ A = 0.5 π Fig. 14. The simulation results of G-CRESM with different relative phases Fig. 13 shows the results of G-CRESM when the phases of the carri ers of the two sources a re not synchronized. We note that the BER res ults in Fig. 1 3 are actually better than th ose in Fig. 12, where the phases of the two sources are perfec tly synchronized . Fig. 14 ex plores the phase affection on G-CRESM an d the worst case is just the CRESM (G-CRESM with 0 B A ϕ ϕ − =  ). We notice t hat phase ha s smaller impac t on the G-CRESM th an ∆ . Thi s me ans w e do no t nee d t o deliberately sync hronize the phase difference of the two sources: i.e., usin g G-CR ESM we co uld deal with the phase asynchrony at t he receiver. VI. C ONCLUSION AND F UTURE W ORK This paper has proposed an d investig ated CRESM, a n ovel packet col lision re solution scheme that treats c ollided s ignals as of the outpu t of a virtual convolutional encoder. The main essence of CR ESM is that b y over-sampl ing the o verlapping signals at the receiv er, one co uld extract the in dividual packets from the transmitters. Within this ge neral con struct, we propose a specific Viterbi-like decoding scheme that minimizes the BER. As far as we know, this is the fi rst paper t hat propos es to treat collisions as a kind of convolutional code, to which simple digital signal proces sing (DSP) techniques could t hen be applied for dec oding purpos es. Although we have described CRESM in the context of 802.11, the idea be hind CRESM is in fact quite general and is applicable to other MAC protocols (e.g ., Aloha). Within a larger cont ext, CRESM can b e viewed as a technique fo r multiple-pac ket reception (MPR) [1 2]. An attractive fea ture of CRESM is th at no s ymbol-level syn chronization is requ ired of the simultaneo usly transmitting stations – in fact CRESM exploits the naturally occurring symbol misalignment to perform MPR. Given that collisions can be resolved in a si mple manner at the physica l layer by CRESM, an implicat ion is that the MAC protocol should be r edesigned in such a way as to encourage multiple packet tran smissions. That is, the stati ons should be more aggressive in the ir transmissions. For ex ample, in [4], it was shown th at t he n etwork throughput can be in creased by n times if one cou ld resolve n -pa cket collisions . In th is p aper we have only investi gated the coll isions of BPSK packets. The idea of CRESM, howev er, is independent of modulation. As extension work, it will be i nteresting to investigate C RESM using an info rmation-theoret ic approach. R EFERENCES [1] S. Gollakota and D. Kattabi, “ZigZag Decoding: Combating Hidden Terminals in Wireless N etworks,” MIT-CSAIL-TR-2008-018 April 8, 2008. [2] D. Halperin, T. Anderson, and D. We therall, “Interference Cancellation: Better Receivers for a New Wirele ss MAC,” Hotnets 2007. [3] D. Halperin, T. A nderson, and D. Wetherall, “Taking the String out of Carrier Sense: Inte rference Cancellation for Wireless LANs,” ACM MOBICOM 2008, San Francisco, USA. [4] P. X. Zheng, Y. J . Zha ng, an d S. C. 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