Blind Known Interference Cancellation

Blind Known Interfe r ence Cancellation Shengli Z hang*, Soung Chang Liew § , Hui W ang* *Communication Engineering Department, Shenzhen University , Shenzhen, China § Information Engineering Department, Chinese University of Hong K ong, Hong Kong, China Abstract: This paper investi gates i nterference-cancellation schemes at t he r eceiver , i n whi ch the ori ginal dat a of the interference is known a priori . Such a priori knowledge is common in wireless relay networks. For example, a transmitti ng rela y could be rela ying data t hat was previousl y transmitted by a node, in which cas e the int erference received b y the node now is a ctually self information . Besides the case of self information, the node cou ld also have overheard or received the interference data in a prior transmission by another node. Directly removing the known interference r equires accurate estimate of th e interference channel, w hich may be di f ficult in many situations. In this paper , we propose a novel scheme , Blind Known Interference Cancellation (BKIC), to cancel known int erference without interference chann el information. BKIC consists of two steps. The first step combines adjacent s ymbols to cancel the interference, exploiting the fact that the channel coeff icients ar e almost the s ame between successive symbols. After such int erference cancellation , however , the signal of interest is also distorted. The second step rec overs the signal of interest amidst t he distortion. W e propose t wo algorithms for the critical second s teps. The first algorithm (BKIC-S) is bas ed on th e principle of s moothing. It i s simple and has n ear optimal per formance in the slow fading scenari o. The second al gorithm (BKIC-RBP) is based on the principle of real-valued belief propagation. I t can achi eve MAP-optimal performance with fast convergence, and has near optimal performance even in the fast fading scenario. Both BKIC schemes out perform the traditional self-interference cancellation schemes with perfe ct initial channel information b y a larg e margin, while having lower complexities. I. Intr oduction The use of rel ay in wireless networks is att racting increasing attention [1, 2] because of the man y advantages it brings, such as improved connectivit y and reduced power consumption. Many multi-hop relay standards, including 802.16j, 802.11s, are being developed. In wireless relay n etworks, a node may rec eive a targ et si gnal superimp osed with int erferences. However , in many scena rios, the receiver actually knows the data contained in the i nterference [3], either because the interference was signal previously received b y the no de, or t he interference is self-information previousl y transmitted by the node and now transmitted by the rela y superimposed with signal from another node. One example of known interference is when ph ysical-layer network coding over infinite field [ 4] (e.g., analog network coding [5]) is used in a two-w ay r elay channel , as shown in Fig. 1a. Ano ther example is a linear -chain one-way rela y netw ork [6] , as shown in Fig. 1b. Many other scenarios of known interference can be found in [3 ]. The method to deal with known interference is straightforward in theory . The receiver first estimates the channel coefficient of the inter ference signal and then removes the known interference from the targe t signal [3]. W e refer to this scheme as traditional KIC (Known Interference Cancellation) in this paper . I n p ractice, however , the scheme doe s not pe rform w ell when the channel estimation is inaccurate. Accurate channel estimation is a non-trivial problem even i n t he absen ce o f int erference. This is the reason why non-coherent detection schemes that d o not require chann el information are still widely studied and used in wireless communication s ystems [7]. In the pr esence of interference, we face additional challenges. Fi rst, channel estim ation is more difficult because t he trainin g sequ ences are corrupted by the superposi tion of two signals. Estim ation of the channel coefficients of the targeted packet and the interferin g p acket can be compl ex. For example, when th e two training sequen ces are the same and the y overlap with each other , we can only obtain the summation of the two channel coef ficients but not their individual values. Second, when t he power of the i nterfering packet is much larger than that of the tar get packet, a tiny estimati on error on the interferenc e channel may cause t he interference c ancellation process to le ave behind a relativel y large residual interference with respect to t he power o f th e target packet. Third, it may be impossible to esti mate the channel accuratel y [8] in mobile environment with fast fading. The cha nnel estimated from the training sequence may have changed b y the time the da ta is received. W e note that the above difficulties appl y regardless of whether the in terference is known or unknown. This is because for bot h cases, the training sequ ences are presumed known in order that channel estimation can be enabled. Basicall y , we need to estimate the channels of two superimposed packets. There has been some work trying to tackle t his problem. For ex ample, in physical-layer network coding [9, 10] , the channels of t wo s uperimposed packets need to be estimated. T o deal with the channel estimation problem, the ph ysical-layer network coding implementation in [1 1] uses orthogonal sequences for t he two packets. The an alog network coding sch eme in [5] , on the ot her hand, attaches the training sequence to both the front end and back end of a packet, and time the transmissions of the two packets so that one of them has interference-free front end and the other one has interference-free back end. These schemes use new fr ame designs and are not comp atible with l egacy wireless systems. Ref. [12] uses an optimiz ation scheme to es timate t he channels with two speciall y designed t raining sequen ces. The estimation accuracy is m uch poorer than that in single-channel estimation. Non-coherent ANC schemes that avoid chan nel estimation at the end nodes have also been studied [13, 14]. However , non-coherent schemes suf fer from SNR degradation of about 3dB compared with the c oherent sch emes. B y co ntrast, t he bli nd kno wn interference cancellation (BK I C) schem es proposed in this paper can obtain near-perfect performance – specificall y , performance clos e to that of a point-to-point communication link without interference – while avoiding estimation of the interference chann el. BKIC has three advantages o ver the t raditional methods: 1) better pe rformance; 2) no need for interference channel estimation; and 3) compatibility with legacy s ystems. The principle on which BKIC op erates i s based on the observation that the wireless ch annel t ypically r emains almost unchanged between adjacent symbols [8] . BKIC uses the int erference i n one s ymbol to cancel the interference in its adjacent s ymbol. For example, if the interference channe l is h and the interference symbol is 1, then the interference i n the current symbol is h . If the inte rference data in the adjacen t symbol is -1, then the co rresponding known interference is approximately – h . The interference c an be cancelled with each other if we combine the two s ymbols. Such adjacent-symbol combination, however , ma y result in distortion of the target signal. Thus, a ke y is sue is how to remove such distortion as the nex t step. W e propose two schemes, smoot hing (BKIC-S) and real-valued b elief propagation (BKIC-RBP), to equalize the resulting distortion. This paper considers BKIC schemes for both flat fading channel and frequenc y s elective ch annel. W e show that the per formance of ou r schemes i s alm ost t he same as that of a pur e coh erent point-to-point channel w ithout interference (note: t his is a theoretical upp er bound for ou r s ystem). Besides it s ex cellent perf ormance, our s chemes ar e also attractive because of their low complexity and compatibili ty with legacy s y stems. Specifically , our schemes do not require s pecial chan ges to the frame structure or th e operation of the transmitter . A salient feature of our schemes is t hat the y could be realized b y an add-on module i nserted i nto the signal proces sing path of the r eceiver without requiring complicate modifications to the exis ting module. The remainder o f this pa per is organized as follo ws. In se ction II, w e pres ent the s ystem model and architecture o f BK IC. S ection III explains BKIC under the flat fading channel assumpti on, and Section IV extends the discussion to the frequency s elective fading channels. In Section V , w e analyze the performan ce. W e validate and supplement the analytical results with numerical simulation in Section VI. Finally , Section VII concludes this paper . II. System Model and Architectur e Known interference is com mon in many wireless networks. TWR C with analog network codin g [5] in Fig. 1a, and one-way rela y chain n etwork in Fi g. 1b, are two examples with known interference. In this section, we present the general mathematical formulation for known interference s ystems. For a focus, consider the chain network in Fig. 1 b. The source node S tr ansmits to the destination node D through two relay nodes, R 1 and R 2 , and there are no cross-hop tra nsmissions. For simplicit y , we assume one dimensional q -ar y ASK modulation at all the nodes; our method can be easil y extended to other m odulations, including two-dimensional modu lations. As shown in Fig. 1b, in ti me slot 1, S t ransmits a packet to relay R 1 ; i n ti me slot 2, R 1 forwards it to the second rela y R 2 ; and in time slot 3, R 2 forwards it to destination D , while node S sends a new packet to R 1 at the same time. The transmiss ions i n tim e slots 2 and 3 are r epeated for the delivery of successive packets from S to D. Note that node R 1 receives a superposition of the two packets, one from S a nd one from R 2 . W ith the assumption of symbol level synchronization [15], the k -th received s y mbol at R 1 can be expressed as 0 0 ( ) ( , ) ( ) ( , ) ( ) ( ) x I L L x l l r k h k l x k l h k l I k l n k = = = − + − + ∑ ∑ (1) where ( ), ( ) { 1 , 3 , 3 , 1 } x k I k A q q q q ∈ = − + − + − −  is the k -th target symbol sent from S and ( ) I k is the k -th interfering s ymbol sent from R 2 ; n ( k ) is the zero mean Gaussian no ise with variance 2 σ ; ( , ) h k l and ( , ) x h k l are respectivel y the l -th tap channel coefficients from R 2 and S to R 1 for their k -th s ymbols; L x and L I are t he respective max imum tap delays of the two signals. The transmit powers, and t he effects of transmit and receive puls e shapes, are co mbined into the channe l coeff icients, ( , ) h k l and ( , ) x h k l . According to the W SSUS model of B ello [ 16], all the channel taps are i ndependent of each other; for each tap, the channel variation satisfies { } 0 ma x ( , ) ( 1 , ) ( 2 ) E h k l h k l J f π τ − = , where 0 max ( 2 ) J f π τ denotes the z ero-th order Bessel function of the first kind, max f is t he max imum Doppler frequenc y , and τ is t he s ymbol duration. Hereafter , we use the bold x / I letter to denote the corresponding vector of the whole packet. The system formulation for an alog network coding in two-way rela y channel is the same as in (1) and the details are omitted here. System Arc hitecture: In this paper , we propose a bli nd known interference cancellation scheme which can cancel the interference i n (1) and t ransform ( ) r k to the signal of interest, 0 ( , ) ( ) ( ) x L x l h k l x k l n k = − + ∑ , plus a small noise as 0 ( ) ( , ) ( ) ( ) ( ) ' ( ) ( ) ( ) x L x l z k h k l x k l n k w k x k n k w k = = − + + = + + ∑ (2) where w ( k ) is th e residua l interference introduced during the interference cancellation processing of our BKIC scheme. According to (2), we have the fol lowing two definitions: Definition 1 : Desired Signal (DS): 0 ' ( ) ( , ) ( ) x L x l x k h k l x k l = = − ∑ , w hich does not contain any nois e or interference and it is the targe t of the whole s ystem. Definition 2 : Desired Signa l plus Noise (DSN): 0 ' ( ) ( ) ( , ) ( ) ( ) x L x l x k n k h k l x k l n k = + = − + ∑ , which is equivalent to the received si gnal from a pure poi nt-to-point transmission without any i nterference. The function of BKIC is to remove the known interference. In the ideal c ase, it s output should be exactly DSN. In reality , BKIC needs to estimate DSN as accuratel y as possible. The signal ( ) z k in (2) is DS N with a small extra noise, from which the tradit ional si gnal detection algorithm can then proceed to detect x as i n conve ntional receivers. As will be shown later , the smal l noise w ( k ) can be approxi mated as a Gaussian noise with negligible variance. Therefore, an y transmitting specifications employed b y source S or an y ch annel enviro nment experienced b y S do not af fect our BKIC scheme. A rela y enabled with o ur interference cancellation ma y b e built as in the system architecture shown in Fig. 2. When the signa l is received, th e rel ay firs t checks if the known interference is present with the block “int erference check”. Interference c heck may be impleme nted with data sequence correlation [18, 19]. T o limit the scope of this paper , we will not delve into the details of “interference check”. If the interference is not present, then t he packet is directly fed to th e conventional receiver for data det ection. If known interference is pres ent, then the pa cket is fed to the BK I C block to cancel the known interference. After that, the si gnal is fed to the conventional receiver for target data detection. III. Blin d Known Interfer ence Cancellation in Flat Fading channel In this section, we present our B K I C schemes assuming flat f ading channel. The nex t section extends the treatment to th e general multi path cha nnel. BKIC consists of two steps. In th e first step, the interference is cancel ed by combining adjacent symbols. In the second step, the DSN, i.e., the point-to-point form of the signal, distorted during the cancellation st ep, is recovered. Th e second step is the no n-trivial step, and we present two algorithms for it. The first algorithm, which serves more like a benchmark, recovers DSN b y means of smoothing; the second algorithm, by a nove l real-valued belief propagation framework specially designed for our purpose here. S tep 1: Interference cancellation W ith flat fading, L I =1 for the interference channel, we can rewrite (1) as ( ) ' ( ) ( ) ( ) ( ) I r k x k h k I k n k = + + . (3) In practice, the channel ma y be time var ying, and the variation depends on the m oving speed an d other environmental fact ors. However , for adjacent s y mbols, the channel variation is very small. Then we can approximate the channel variation as ( 1 ) ( ) ( ) I I h k h k k + ≈ + ∆ . (4) In (4), ( ) k ∆ is governed by the Doppler rate, and it is almost negligible in modern wire less communication systems [20, 24 ]. This is a key observation in this paper that enables us t o use adjacent s ymbols to cancel the known int erference without channel estimation. T o do so, we obtain a new signal ( ) t k by combining ( ) r k and ( 1 ) r k + as follows: ( ) ( ) ( ) ( 1 ) ( 1 ) ( ) ( ) ' ( ) ' ( 1 ) ( ) ( 1 ) ( ) ( ) ( 1 ) ( 1 ) I k t k r k r k I k I k I k x k x k n k n k I k k I k I k = − + +   = − + + − + − ∆   + +   { 1 , 2, 1 } k N ∈ −  (5) In (5), almost all the i nterference terms have been removed from t ( k ). However , the signal of interest to us is DSN, ' ( ) ( ) x k n k + , rather than t ( k ). In the nex t step, we show how to ext ract DSN from t ( k ). S tep 2: Interested signal r ecovery In (5), the target signal ' ( ) x k is distorted into the form of t ( k ) after the interference cancellation step. At first glance, t ( k ) may appear to be the signal ' ( ) x k passing through an I nter -Symbol Interference (ISI) channel, in which case traditional ISI equalization schemes such as filtering, V iterbi detection and Belief Propagation (BP) [ 21] detection, could be used to recover ' ( ) x k . However , a closer examination reveals an important difference between t he signals t ( k ) in (5) and that from a traditional ISI channel. Specifically , the difference is t he correlated noise i n (5) for adjacent symbols t ( k ) and t ( k+ 1). Although V iterbi/BP detection achieves optimal M AP performance for indepen dent noise in ISI equalization, i ts performance is far from optimal for the recovery of the target signal here bec ause of th e co rrelated nois e, as will b e shown in our nume rical simulation. Noise whitening is a standard te chnique for d ealing with correlated noi se. However , we cannot directly whit en the noise in (5) because it is im possible to transform t he N -1 equations i n (5) into N equations with independent noise terms while m aintaining the interference cancellation effe ct. Another no ise whitening s cheme is noise prediction and whitenin g pro cess in [ 22]. As will be shown in our simulation results later , this scheme is also far from optimal. W e now propose two schemes to recover DSN with near optimal performan ce. Recovery by Smoothing From (5), we could write ( ) 1 ( 1 ) ( 1 ) ( 1 ) ' ( 1 ) ( 1 ) ( ' ( 2) ( 2)) ( 1 ) ( 1 ) ( 2) ( 1 ) ( 1 ) ( ) ( 1 ) ( ) ' ( 1 ) ( 1 ) ' ( 1 ) ( 1 ) ( 1 ) ( ) ( ) ( 1 ) for {2 , - 2 } k m I u t x n x n I I I I u k u k t k x n x k n k I m I k I k k N = = = + − + − ∆ = − + = + − + + + − ∆ + ∈ ∑  (6) Then, we obtain the estimate of ' ( 1 ) ( 1 ) x n + as follows: 1 1 1 ( 1 ) ( ) ' ( 1 ) ( 1 ) ( 1 ) 1 N k z u k x n w N − = = = + + − ∑ (7) where the residual int erference ( ) 1 1 1 ( 1 ) ( 1 ) ' ( 1 ) ( 1 ) ( 1 )( ) ( ) 1 ( 1 ) N k I w x k n k I N k k N I k − =   = − + + + + − ∆   − +   ∑ is independent of x’ (1) and n (1). As will be shown in the next section, w ( k ) can be approximated by Gaussian dist ribution. For slow fading ( α almost equal 1), its variance is very small; for fast fading, it ma y become la rg er due to the accumulation of errors (note: this e ff ect i s i nevitable in an y channel estimation scheme). Then, we can remove ' ( 1 ) ( 1 ) x n + from each signal in (5) to o btain the estimate of ' ( ) ( ) x k n k + as: ( ) 1 ( ) ( ) ( 1 ) ( 1 ) ' ( ) ( ) ( )( ( 1 ) / ( 1 ) ( ) ) '( ) ( ) ( ) for 2 ( 1 ) k m I k z k z u k x k n k I k w I m x k n k w k k I = = − − = + + + ∆ = + + ≥ ∑ (8) W e can then fed z ( k ) to a conventional receiver for final data detection. Recovery by Real-valued BP Belief propagation is a powerful technique to infer information from a la rg e amount of correlated data. I n the conventional method of apply ing BP to equaliz e ISI as in (5), x’ and its I SI form ( ) ' ( ) ' ( 1 ) ( 1 ) I k x k x k I k − + + are associated with the corresponding variabl e nodes. Th eir estimates are refined with message pa ssing [ 21]. Direct BP ap plication as such assum es the noise terms in the 1 N − equations in (5) are independent. Strictly speaking, this is not true. For correlated nois e terms, [22] proposed to predict and whiten the corre lated nois e durin g the message passing pro cedure to improve the performance of the traditional BP algorithm. However , this method cannot m ake full use of the sp ecial noise correlation form in (5) and achieve onl y suboptimal performance. For better performance, we propose a novel BP scheme whe re DSN x’+n and the post-cancellation signal t , rather tha n DS as in traditional B P , are associated with the variable nod es. Since th ey are real-valued signals, we r efer to ou r BP detection algorithm as BKIC with Real-valued BP (BKIC-RBP). The T anner graph of our BP algorithm is shown in Fig. 3. An important subtl ety in the T anner graph is that x’+n i s treated as “signal s ymbols”. W ith respect to Fig. 2, x’+n is the target signal th at will be fed to the conventional receiver after t he interference cancelation process. With reference to (5), the o bservation t ( k ) ar e made up of adjace nt “signal symbols” plus noi se in t he cancellation process, w hich is ( ) ( ) I k k − ∆ and does not inc lude ( ) n k and ( 1 ) n k + . The noise ( ) n k will be dealt with by the conventional receiver later . Remark: W e stress th at associating DSN rather t han DS to the left variable node in Fi g. 3 is the ke y of our BKIC-RBP schem e. First, directly esti mating DS cannot im prove t he performance since th e noise is i ndependent of t he interference. Mo re im portantly , ( ) ( ) ( 1 ) ( ) ( ) ( 1 ) I k n k n k I k k I k − + − ∆ + becomes the general noi se in t an d the relation between a djacent noi se terms in ( 5) i s wiped off f rom t he figure when associating DS to the left variable node. W ith the above setting, the tar get of BKIC-RBP i s to find a vector x' + n to maximize ( | ) ( | ) ( ) ( ( ) | ' ( ) ( ), ' ( 1 ) ( 1 )) ( ) k P P P P t k x k n k x k n k P ∝ = + + + + ∏ x' + n t t x' + n x' + n x' + n . (9) Based on (9), a correspo nding T anne r Graph can be establis hed as in Fig. 3, where the messa ges being passed between the variable nod es and the check nodes are the probabilit y densit y fu nctions of the variable nodes at left hand side. The algorithm includes three criti cal steps: initializ ing the messages, updatin g the messages at the variable nodes and updati ng the messages at check nodes in an iterative way . Message initialization: The variable nodes x ’+n on the left si de of Fig. 3 are not associated with any chann el outputs, and we initializ e the PDF of x’+n with t he a prior i probabilities. The variable x’ adopts a dis crete value , and the discrete distribution is determined by the constellation set and the multipath c hannel. However , we cannot obtain the distribution because we assume channel inf ormation is not available in BKIC. Generally speaking, the upper bound of the interference power , max P , can be derived easily . For example, max P could be set to the max power of the received signal r 1 . W e assume x ’ is uniformly distributed between max P − and max P 2 . Si nce the noise is of Gaussian distribution, the messages (i.e., the a priori probabilities) associated with the leftmost edges in the T anner graph can be expr essed as ( ) ( ) ( ) ' ' ' ' 2 2 max ' ( ' ) ( ) 1 exp ( ) / 2 2 2 max max max max P x n x n x n x n P P P m p y p x n y p x s p n y s ds y s ds P σ π σ + + + − − = = + = = = = − = − − ∫ ∫ . (10) For each left variable node ' ( ) ( ) x k n k + , t here is a incoming edge from an adjacent check node whos e messages is denoted b y '( ) ( ) x k n k m +  and an outgoing edge to an adjacent check node whose message is denoted by '( ) ( ) x k n k m +  . The initial values of both '( ) ( ) x k n k m +  and '( ) ( ) x k n k m +  are set to ' x n m + . W ith the ini tial messages, we then iteratively update t hem to obtain the final esti mation. Since our T anner graph in Fig. 3 does not include an y circles, one it eration is enough to o btain t he opt imal MAP performanc e. Each iteration includes t wo parallel messa ge update pr ocesses. One suc cessively updates the messa ges, '( ) ( ) x k n k m +  and '( ) ( ) x k n k m +  , one b y on e from top to bottom as illustrated in Fig. 3(a). The other process successivel y updates them from bottom to top as illustrated in Fig. 3(b). F or example, for top-to-bottom messag e updates, the chec k-node update ru le and the va riable-node 1 P max obtained in thi s way also includes th e power of DSN and the power o f t he interference. So it is a loose upper bound o f the maximal interference power . 2 A more accurate di stribution o f x’ + n should improve the performance of BKIC-RBP . For tunately , BKIC-RBP with the approximate a prior distribution still performs very well, as demonstrated by our simulation result s later . update rule in the foll owing are applied i n an alte rnating manner because each check-node messa ge update depends on the previous variable-node message update, and vice ver sa. Message updates at the check nodes: First consider the top-to- bottom process where the messages associated with the edges betw een the left variable nodes and the check nodes are updated one b y one from top to bottom as in Fig. 3 (a). For a check node connect ed to the ri ght evidence node [ ] t k , the two left v ariable nodes connected to it are ' ( ) ( ) x k n k + and ' ( 1 ) ( 1 ) x k n k + + + . Given the input PDF as '( ) ( ) x k n k m +  , t hen the output PDF can be calculated according to (5): ( ) 2 2 '( ) ( ) '( 1) ( 1) '( ) ( ) '( ) ( ) ( ) / 2 '( ) ( ) ( 1 ) ( ) ( ) ( ) ( ) ( ) ( 1 ) ( 1 ) ( ) ( ) ( ) ( ( ) ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 ) x k n k x k n k x k n k x k n k k s x k n k I k I k m p y p y t k I k k I k I k I k I k p y t k I k s p k s ds I k I k I k p y t k I k s e ds I k σ ∆ + + + + + + ∆ − +   + = = + + ∆   +     + = + + ∆ =   +     ∝ + +   +   ∫ ∫  (1 1) where 2 σ ∆ is the upper bound of the variance of the interference term ( ) k ∆ 3 . For block fading, ( ) 0 k ∆ = with probability 1. The PDF of ' ( 1 ) ( 1 ) x k n k + + + in (1 1) ca n be si mplified to ( ) '( ) ( ) '( 1) ( 1) '( ) ( ) ( ) ( ) ( 1 ) x k n k x k n k x k n k I k m p y p y t k I k + + + + +   = ∝ +   +    . (12) Now consider the bottom-to-top proc ess where the messages associated with th e edges between the left variable nod es and th e check nod es are update d one b y one from the bottom to the top as in Fig. 3 (b). Analogous to (1 1), the PDF of ' ( ) ( ) x k n k + can be up dated from the PDF of ' ( 1 ) ( 1 ) x k n k + + + and the observation ( ) t k based on the following equation: ( ) 2 2 / 2 '( ) ( ) '( ) ( ) '( 1) ( 1) ( 1 ) ( 1 ) ( ) ( 1 ) ( ) ( ) s x k n k x k n k x k n k I k I k m p y p y t k I k s e ds I k I k σ ∆ − + + + + +   + + = ∝ − − +     ∫  (13) For block fading, (13) can be simplified to ( ) '( ) ( ) '( ) ( ) ' ( 1) ( 1) ( 1 ) ( 1 ) ( ) ( ) ( ) x k n k x k n k x k n k I k I k m p y p y t k I k I k + + + + +   + + = ∝ −      (14) Message updates at the variable nodes: 3 BKIC-RBP is robust to 2 σ ∆ as shown in the simulation. Therefore, we can fix it to a relative high value, such as 0.001 with almost no performance loss. Message upd ates at the left v ariable nodes is the same for both top-to -bottom process and th e bottom-to-top process. Each left variable node is connected to three edges, whose associate messages are output P DF '( ) ( ) x k n k p +  , input PDF '( ) ( ) x k n k p +  and the a priori PDF '( ) ( ) x k n k p + respectively . Each output PDF is updated with the input PDF and the a priori PDF as ( ) ( ) ( ) ' ( ) ( ) ' ( ) ( ) ' ( ) ( ) 1 x k n k x k n k x k n k p y p y p y T + + + = ⋅   . (15) where T is the normalization factor . At the end of the processing, we need to collect the information contained in all the messages an d make a final estim ate of DSN. For the k -th variable node with ' x n + , there is the a priori P DF ( ) ' ( ) ( ) x k n k p y + , t he input P DF ( ) ' ( ) ( ) x k n k p y +  after t he top-to-bott om process, and the input P DF ( ) * ' ( ) ( ) x k n k p y +  after t he bott om-to-top process (there is onl y one input PDF of th e first and the last l eft variable nod e). Th en the final prob ability di stribution of ' ( ) ( ) x k n k + can b e c alculated as t he following product: ( ) ( ) ( ) ( ) * ' ( ) ( ) ' ( ) ( ) ' ( ) ( ) '( ) ( ) 1 f x k n k x k n k x k n k x k n k p y p y p y p y T + + + + =   (16) where T is the normalization factor . ( ) ' ( ) ( ) f x k n k p y + contains all the i nformation about x’+n and it can be fed to the traditional detection block for further target data detection. In o rder to compare wit h the BKIC-S scheme, an estimate of ' x n + is given by ' ( ) ( ) ( ) ar g max ( ) f x k n k y z k p y + = (17) Discussion: we can regard t as an inner encoder output with input x’ . Then the decoder of it, i.e., the hard/soft decision based o n ( ) ' ( ) ( ) f x k n k p y + can be combined wit h the channel decoding procedure so that the T urbo like detection-decodin g can applied to achieve even better performance. IV. BKIC in Frequency Selective Fading channel The previous section proposes the BKIC scheme for flat fading channel. In this section, we extend the scheme to the frequ ency-selective channel. W hen BK IC is performed in a totall y blind manner , we have n o prior information about the multi-path characteristics. In this case, i t i s sti ll reasonable to assume knowledge of the m aximum delay of all the paths, L , as in many current broadband wireless s ystems. For example, the length of the predefined CP (cyclic prefix) in OFDM s ystem implies the maxim um dela y of all paths. The refore, we can perform interference cancellation and recover the equivalent DSN for each potential path in a successive wa y as follows. W e first rewrite the received signal in (1) as 0 1 0 ( ) ' ( ) ( , ) ( ) ( ) ' ( ) ( , ) ( ) ( , 0) ( ) ( ) ' ( ) ( , 0) ( ) ( ) L I d L I I d I r k x k h k d I k d n k x k h k d I k d h k I k n k x k h k I k n k = = = + − + = + − + + = + + ∑ ∑ . (18) In (18), we select the fi rst path of the interfering signal as th e interf erence to be cancelled and combine the other interfering paths to DS , ' ( ) x k . Then, the selected interfering path can be removed by applying the BKIC scheme as in the preceding section. After that, we can obtain 1 0 1 ( ) ' ( ) ( , ) ( ) ( ) ( ) L I d z k x k h k d I k d w k n k = = + − + + ∑ (19) where 0 ( ) w k is the residual interference after removing the first path. Comparing z 1 ( k ) and r ( k ) , we see th at the first path of interference has be en removed and generated a new residual interference term 0 w . In a simi lar w ay , we can remov e the second path o f interference to obtain z 2 ( k ). Repeat the BKIC scheme L +1 times, we finall y obtain that 1 0 ( ) ( ) ' ( ) ( ) L L d d z k z k x k w n k + = = = + + ∑ . (20) V. Performance Analysis In this section, we analyze the BER and SINR performance of the proposed BKIC schemes unde r flat fading channels and frequency selective channels. 1. BKIC under Flat Fading Channel Proposition 1: The performance of th e BKIC sch emes is upper bounded by the cl ean s ystem where there is no interference at all. This proposition is easy t o understand since the interfering dat a is independent to the target data and it can not help to detect the target d ata. Due to the exi stence of the resid ual interference w , BKIC schemes can never achieve this upper bound exact ly . Fortunatel y , we can approach it ver y closely . Proposition 2: For the BK IC-S scheme, the residual interference ( ) w k can b e well approximated b y a Gaussian noise (0, ( )) N k µ , where the variance i s upper bounded by 2 2 2 2 ( ) ( ) ( ) ( 1 ) { 1 / ( )} ( 1 ) 3 x I P NP k I k I j N σ µ α   + = + − Ε   −   . Proof: According to (8), we can show that the res idual interference h as t he largest variance for the first symbol, which can be expressed as ( ) 1 1 ( ) 1 ( 1 ) ' ( 1 ) ( 1 ) ( ) ( ) 1 ( 1 ) N j I k w x j n j N j j N I j − =   = − + + + − − ∆   − +   ∑ . When N is very l arge, the total residual int erference, w , can b e re garded as of nor mal distributi on based on larg e number theor y . It is straightforward to verif y that its mean value is zero and its variance is 2 2 2 2 2 2 2 2 2 2 ( ) ( ) ( 2 1 ) ( ) ( 1 ) {| ( ) | } { } ( ) ( 1 ) 6( 1 ) ( ) ( ) ( ) ( 1 ) { 1 / ( )} ( 1 ) 3 x j x I P I k N N k w k I k N N I j P NP I k I j N σ µ µ σ σ α ∆ + − ≤ = Ε = Ε + − −   + ≈ + − Ε   −   , (21) where P x is t he received power of the target signal x’ and P I is t he received power of the i nterfering signal. From the above proposition, we can obtain some important observations. Corollary 1: In BKIC-S, the residual interference is independent of the received power o f the interfering signal for block fading channel. In block fading channel, 1 α = and 2 0 σ ∆ = . Then the power of the interfering si gnal, embedded in h I , does not affect the performance of the BKIC-S scheme. In general slow fading channel, t he block fading assumption holds ver y well. This independent prop erty is desi red especially when the interfering signal power is much stronger . In contrast, the t raditional known interference cancellation [3] scheme, which is based on estim ated chann el information, performs poo rly in this case b ecause the residua l interference is proporti onal to the interfering signal power wi th a given channel estimation mean square error (MSE). In block fading, the residual interference is ver y small for large packet l ength. For exampl e, when N =1 00, the residual inter ference power is decreased by about -20dB. Assuming equal int erference power and DS power , t he residual interference is 20d B less t han the DS pow er , which is much smaller than the general 10d B S INR requirement for wireless receiver . Corollary 2: In BKIC-S, there is an opti mal pack et length 4 N to m inimize the residual i nterference µ for continuous fading channel. The first term in (2 1) decreases with N as in th e block f ading channel. The s econd term i n (21) increases with N . When N is l arg e, the channel v aries far from its average value and the residual interference comin g from channel v ariation accumul ates (fortunately , B KIC-RBP is insensitive t o channel va riation.). If th e interference data adopts a constant power modu lation (PSK modulation), the optimal N can be calculated as 2 2 3 ( ) / 0 1 x opt P N N σ µ σ ∆ + ∂ ∂ = ⇒ = + (22) In real communication systems, 2 σ ∆ is very small and opt N is large . Besides the performance, compl exity i s also an im portant issue. The complex ity of our BKIC-S scheme is quite low . O nly on e multiplication a nd two addition proce sses ar e needed for e ach symbol. Corollary 3 : W ith BKI C-S, the SNR los s compared to the clean system (only DSN signal exists) is 2 2 2 10 log( ) 10 log( ) 1 0 log( 1 ) x x TSN BKIC S TSN P P SNR SNR SNR µ µ σ σ µ σ − ∆ = − = − = + ≈ + . (23) In BKIC-S, the residual int erference is fix ed when the power of DS pl us noise is given. As a result, the SNR loss of BK I C-S depends on the SNR of DS N. For smaller residu al interference, the S NR loss is approximatel y proportional to the SNR of DSN. For the performance of BKIC-RBP , we have Proposition 3 : The performance of the BK IC-RBP scheme a chieves the MAP opt imal DSN recovery performance, which is lower bounded by the BKIC-S. As i s well known, the l oop free BP detection has MAP-optimal performance. In our BKIC-RBP , there are no circl es i n t he T anner graph in Fi g. 3, s o ex act MAP performance of si gnal r ecover y can be achieved. As a result, the performance of BK IC-RBP i s lower bounded by the BKIC-S scheme. Proposition 4: The complexity of BKIC-RBP is l inear in terms of packet length. 4 In our paper, packet length is just the processing length of the algorithms. Dividing a packet into several parts for processing is not considered. The loop-free propert y in Fig. 3 also guarantees fast converge nce of the BP algorithm. Onl y one iteration (one top-to-bott om process and one bott om-to-top process) in the RBP algorithm is enough to obtain t he MAP performance. The complexity of our RBP recover y is onl y 4 6 N − message update operations, which is linear in terms of the packet length. However , re al valued processing i s needed to achi eve the optim al MAP performance. In practice, we need to quantize the real valued PDF into discrete form with controll ed complexity . As shown in our sim ulation and man y other works [23] , there is t y picall y little performance loss associated with quantization errors in belief propagation algorithms. 2. Analysis of selective fadi ng BKIC In multipath channel, the flat fading BKIC is executed several ti mes to successively get rid of th e interference of each path. Therefore , the performance anal ysis is similar . W e have Proposition 5: For the BKIC-S i n multipath channel, the total residual interference w can b e approximated by a Gaussian noise 0 (0, ) L m m N µ = ∑ , where m µ is the variance of th e residual interference generated after cancellation of the m -th path. Its value can be obtained as in (21). According to Pr oposition 5 , we can obtain the c orresponding coroll aries as those in the fl at fading channel. W ith respect to Pr oposition 3 , we can obtain a similar proposition as follows: Proposition 6 : I n the multi path channel, BK IC-RBP achieves the MAP optimal performance for each path. Its performance is lower bounded by the performance of BKIC-S. VI. Numerical Si mulation This section presents numerical simulation results for the performance of BK IC. W ithout loss of generality , we assume the BPSK modulation, i.e., , { 1 , 1 } x I ∈ − , for bot h tar get signal and the interfering signal. As is clear from the ear lier disc ussion, the tar get signal does not affect the operation of BKIC. Thus, without loss of genera lit y in BKIC, we could assume flat fading with unit channel coefficie nt for the target-signal chan nel (note: with respect to Fi g. 2, i t is in th e conventional receive r tha t t he channel characteristics of the t arget signal that matters, and it is over there that the actual fading characteristics of the target signal channel come into play). For the interference channel, block fading, cont inuous fading channel and multi-path fading are simulated. The system SNR is defined as 2 1 / σ , where 1 is th e power of t he target si gnal. For BKIC-RBP , we need to quanti ze the messages (t he PDFs) int o discrete f orm to enable simulation with Matlab. In our sim ulation ex periments, the quantization interval for BK I C-RBP is 0.025 for 1-7dB and it is 0.0125 for 8-10dB. Quantiz ation of the messages results i n quantization errors, and smaller quantization step can further improve the performance at the cost of high complexit y . For comparison purposes, we also simulate the scheme of Noise Predicti ve Belief Propagation (NPBP) in [22] for the second step of BK I C. In our simulation, the length of the whitening filter in [22] is set to two. The com plexit y of BKIC-NPBP is 30( 4 6) N − message u pdate operations plus 3 whitening operations, which is much higher than BKIC-S and BKIC-RBP . Block fading: W e first consider single p ath block fading int erference channel. In this case, the channel coefficie nts are set to a constant unit within the whole block. In Fig.4 and Fi g. 5, we show the residual interference v ariance and BER of the proposed BKIC schemes respectively , with packet length N =100 (bits). W e first look at the s imulation performance of the BKIC-S. For BKIC-S, the theoretical variance of the residual i nterference is 2 ( 1 ) ( 1 ) N σ µ + = − according to (21 ). Based on it, we can calculate the theoretical B ER of BKIC-S as 2 ( 1 /( ) ) Q µ σ + as in [ 24]. I n both figures, the si mulation results match the theoretical values ver y w ell. This validates the Gaussian approxim ation i n Pr oposition 2 . In Fig. 4, we can see that t he residual i nterference of B KIC-S is almost independent of t he SNR. The near -constant residual interference becomes more significant compared to noise in hi gh SNR region. As a result, comp ared to the low er bound (i.e., the standard BPSK wi thout interference), th e SNR loss in Fig. 5 is larg er when SNR i ncreases. The S NR at BER of 2E-3 is around 10dB, corresponding to an SNR loss of than 0.5 dB. As predicted by Proposition 3 , the performan ce of BKIC-RBP , i ncluding residual interference and BER, is better than BKIC-S. Th e improvem ent becomes even l arge r in high S NR region. Both figures show that BKIC-RBP can benefit more from the SNR increase. The reason is that high SN R will sharpen a priori distribution of DSN, which improves the performance of BK IC-RBP . By contrast, the conventional NPBP scheme is worse than BKIC-RBP b y about 1.5dB. If we do not know the channel is block fading or continuous fading, and fixed 2 σ ∆ as 0.001, the performance of BKIC-RBP1 is obtained. The almost identical performance shows that BKIC-RBP is robust to 2 σ ∆ . Fig. 6 shows the BER performances of di fferent schemes for N =1000. As predicted in the analysis, the BER performanc e of BK IC-S is significantl y improved as N increases. It is almost the same as the theoretical lower bou nd. On the ot her hand, the BER performance im provement of BKIC-RBP is ne gligible compared to the case where N =100 due to the fixed quantization error in our simulation. Continuous Fading: In this part, we s et the interference channel to single path continuous fading with the first order Markov channel mode in as in [ 17] and the parameter α is s et to 3 1 10 − − , which corresponds to a fast fading channel. 2 σ ∆ is set to 0.001. Fig. 7, and Fig. 8 show the residual interference variance and BER o f the BKIC s chemes wh en N =100, respectively . For comparison, we also give the BER performance of the traditional known interference cancellation (T r aditional K I C) scheme where t he channel co efficient of the first symbol is perfectl y known. Therefore, th e given performance is an upp er bound of th e actual T raditional KIC with non-ideal channel estimation. From both figures, BKIC-RBP out performs all the other schemes by at le ast 1dB. BK IC-S is next to the best scheme when SNR is less than 10 dB. Compared to Fig. 5, all schemes de grade in c ontinuous fading ch annel. The d egradation for BKIC-RBP and BKI C- NPBP is about 0.1 dB, which is much smalle r than t hat of BK IC-S and traditional KIC. In Fig. 9, we present the BER perfo rmance of th e BKIC schemes wh en N =1000. W ith large N , t he channel varies more significant and the BKIC-S scheme performs even worse. The BKIC-RBP scheme performs best a mong all the schemes and there i s onl y 0.2 dB S NR loss compar ed to t he block fading case. BK IC-RBP i s about 1d B bett er than the BKIC-NPBP scheme. It indicates that the BP al gorithm is not significantly affected by channel variation. Frequency Selective Fading: W e also investigate the BER pe rformance of BK I C with fr equenc y sele ctive fading channel. Th e results are shown in Fig. 10. For illustration, a simple multi path interference channel scenario where there are two interference paths is considered. Th e amplitudes of the tw o paths are the s ame and unchanged within the packet; th e delays of the two paths are 0 and 2 respectively . Com pared to the flat fading c ase in Fig.5, both t he performances of B KIC-RBP and BKIC-S are de graded a s analyzed in the previous section. BK IC-S degrades more t han BKIC-RBP , which has onl y t in y degradation. VII. Conclusions This paper presents two known interference-cancellation schemes with good performance and low complexity . Although t here has been much th eoretical work in this area, d eployments of the previously proposed sch emes are difficult bec ause of t heir needs for accurate channel esti mation and their high comple xity . T o our knowledge, there has been no eff ective blind known interference-cancellation scheme that does not r equire estimation of the i nterference channel. Our work fills a gap in that regard. Specifically , this paper p roposes a framework for blind known interference cancellation ( BKIC), as embodied in Fi g. 2. The principle on which BK IC operates is based on the observation the channel coeff icient is almost constant for adjacent interference symbols. Thus, if t he interference symbols are known, b y combinin g adjacent s y mbols (i.e., combining the received signal and a wei ghted of f-shifted version of it), we can obtain a ne w signal that is almos t free of the interference. Thi s, however , causes distorti on to our t arg et signal. A ke y component of B KIC, t herefore, is how to compensate for this distortion. T o do so, we propose and investigate two schemes: BKIC-S, which is based on the principle of smoothing; and BKIC-RBP , which is based on the principle of real-valued be lief propagation. BKIC-RBP has MAP-optimal performance. The algorithmic complexities of both schemes are linear in the size of the packet. W e show that both BKIC-S and BKIC-RBP have superior performance than the traditional schemes and their performance is ve ry close to the theoretical performance bound, especially for block fading interference channel. The performance of BKIC-S improves with packet size, while the performance of BKIC-RBP is not sensitive to the packet s ize, but is dependent on the quantization step used in the alg orithm to approximate real values. I mportantly , BK IC-RBP is very robust against fast fading in which the channel coefficients may vary in a dynamic manner within a packet. Going forward, to full y expl oit the potential of BKIC in wireless netwo rks, new MAC la yer and network layer protocols need to be d esigned. Beside s relay networks, recently th ere ha s been increased interest in t he wireless networking com munity on the realization of wireless full-duplex communication [ 25]. In the full duplex m ode, a node transmits and r eceives at the same t ime. 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[23] S.-Y. Chung, J. G. D. Forney, T. Richardson, and R. Urbanke, “On the design of low-density p arity-check cod es within 0.0045 dB of the Shannon limit,” IEEE Commun. Lett. , vol. 5, pp. 58–60, Feb. 2001. [24] J. G. Proakis, Digital Communication s . New York: McGraw Hill, 1995. [25] J. Cho i, M. Jain, K. Srinivasan, P . Levis, and S. Katti, “Achieving single channel, full duplex wireless communication” in Proc. ACM MobiCom 2010. N 1 Time Slot 1 Time Slot 2 N 1 N 2 N 2 R R (a) T wo way relay with analog network coding, where the two s ources N1 and N2 transmit simultaneo usly to the rela y in the first ti me slot, and the re lay amplifies and broadc asts the received si gnal to b oth sources in t he second time slot. S S Time Slot 1 Time Slot 2 D D R 1 R 1 R 2 R 2 S Time Slot 3 D R 1 R 2 (b) One way relay channel in a chain. Fig. 1. T w o wireles s networks with known inter ference. BKIC Conventional Receiver Interference check Fig. 2. Sy stem architect ure with bli nd known interfer ence cancellat ion Variable Node Variable Node Check node Evidence Node t (1) t (2) t ( N -1 ) ' ( 1 ) ( 1 ) x n + ' (2) (2) x n + ' ( ) ( ) x N n N + Variable Node Variable Node Check node Evidence Node t (1) t (2) t ( N -1 ) ' ( 1 ) ( 1 ) x n + ' ( 2) ( 2) x n + ' ( ) ( ) x N n N + (a) Top-to-Bottom (b) Bottom-to-Top Fig. 3. T anner Gra ph for Continuous B P , where blank circle s denote the varia ble nod es and the filled cir cles denote the evidence nod es and the recta ngles denote t he check nod es Fig. 4. Residual interference variance with N =100. Fig. 5. BER performance for block fading with N =100. Fig. 6. BER performance for block fading with N =1000 Fig. 7. Residual interference for continuous fading channel with N =100 Fig. 8. BER performance for continuous fading channel with N =100. Fig. 9. BER performance for continuous fading channel with N =1000 Fig. 10. BER performance for frequenc y selective fading channel with N =100