Channel Code Design with Causal Side Information at the Encoder

1 Channel Code Design with Causal Side Information at the Encoder Hamid Farmanbar , Shahab Ov e is Gharan, and Amir K. Khandani Coding and Signal T ra nsmissi on Laboratory Department of Electrical and Computer Engineering Uni v ersity of W aterloo W aterloo, Ontario, N2L 3G1 Email: { hamid,shahab,khandani } @cst.uwaterloo.ca Abstract The problem of channe l code design for the M -ary input A WGN chan nel with additive Q -ary interferen ce where the sequence o f i.i.d. interference symb ols is k nown causally at the encoder is co nsidered. The code d esign c riterion at high SNR is der i ved by definin g a new distance measure between the input symbols of the Sh annon’ s associa ted chann el. F or the case of binary- input ch annel, i.e., M = 2 , it is shown that it is sufficient to use only two (ou t of 2 Q ) input symb ols of the associated chann el in the encoding as far as the d istance spectrum of code is c oncerned . This redu ces the p roblem of chan nel code d esign for the binar y-inpu t A WGN channel with k nown interferenc e at the encoder to design of bin ary cod es for the binary symmetric channel where th e Hamming d istance among codew ords is the major factor in the perfor mance of the code. Index T erms Causal side in formation , Shann on’ s associa ted channel, chan nel coding , p airwise error probab ility . This work was presented in part at the IE EE Canadian W orkshop on Information Theory , Edmonton, Alberta, Canada, June 6-8, 2007. 2 I . I N T R O D U C T I O N Information transmissio n over channels with known interference at the transmit- ter has recently found applications in various communi cation problems such as digit al watermarking [1] and broadcast schemes [2]. A remarkable result on such channels was obtained by Costa, wh o showed that the capacity of the addi tiv e white Gauss ian noise (A WGN) channel with additive Gaussian i.i.d. interference where th e sequence of interference symbols is k nown non-causally at the t ransmitter is the s ame as t he capacity of t he A WGN channel [3]. Therefore, the Gaussian interference does not incur any loss in the capacity . This result was extended to arbitrary (random or deterministic) interference in [4] by using a precoding scheme based on multi-dimensional lattice quantization. Follo wing Costa’ s “Writ ing on Di rty P aper” famous title [3], coding for the channel with non-causally known i nterference at t he transm itter is referr ed to as “dirty paper coding” (DPC). By analogy , coding for the chann el with causall y-known interference at the transmitter is sometim es referred to as “dirty tape coding” (DTC). The result obtain ed by Costa does no t hold for th e case that the s equence of interference sy mbols is k nown causally at the transm itter . Recently , dirty paper cod ing has emerged as a building block in mult iuser communi- cation. In parti cular , there has been considerable research studyin g the application of dirty paper coding to broadcast ov er multi ple-input mult iple-output (MIMO) channels. In such systems, for a given user , the sign als sent to other users are consi dered as interference. Since all signals are kn own to the transmit ter , successiv e “dirty paper” cancelation can be used in transmis sion after some l inear preprocessing [2]. It was shown that DPC in fact achie ves the sum capacity of the MIMO broadcast channel [5], [6], [7]. Most recently , it has been shown th at t he same is true for the entire capacity region of the MIMO broadcast channel [8]. These de velopments motiv ate fi nding r ealizable dirty paper coding techniques. Build- ing upon [4], Erez and ten Brink [9] proposed a practical code design based on vector 3 quantization via trell is shaping and using po werful channel codes. Due to the complexity of impl ementation, their scheme uses the knowledge of interference up to six future symbols rather t han the whole interference sequence. Bennatan et al. [10] gave another design based on superposit ion coding and successiv e cancelation decoding. Their design uses a trellis coded quant izer with m emory length n ine and a low density parity check (LDPC) code as channel code. W ei Y u et al. [11] gav e a design based on con volutional shaping and chann el codes. The schemes that u se the i nterference sequence up t o the current symbo l can be used as lo w-complexity solut ions for the dirty paper problem. For example, in [1], scalar lattice quantization is prop osed for data-hidin g e ve n though in that cont ext, the host signal in clearly known non-causally . In thi s paper , we consider the prob lem of channel code design for the M -ary input A WGN channel with additiv e causally-known di screte i nterference. The discrete interference model is mo re appropriate for many practical applications . For example, in the MIMO broadcast channel where the transmitter uses a finite constellatio n, the interference caused b y other users is discrete rather than conti nuous. Our design does not rely on the subo ptimal (in terms of capacity ) precoding schem e based on scalar lattice quantization for the dirty tape channel [4], [12]. Instead, we consider a new approach based on code design for the Shannon’ s associated channel over all p ossible i nput sym bols. Another distin ction between our work and t he related research in the field i s that we consi der a finite channel input alphabet rather than a continuous on e. This paper is organized as foll ows. In the next section, we su mmarize Shannon’ s work on channels wit h causal sid e information at th e transmitter . In s ection III, we introduce the channel model. In section IV, we derive the code d esign criterion for the A WGN channel with causall y-known discrete interference at the encoder . In section V, we consider channels with binary input for wh ich we show that the design criterion deriv ed in section IV reduces to maximizing the Hammi ng di stance. In section VI, we 4 consider a special case for which the result for the binary channel also ho lds for the M -ary channel. In section VII, we consider a more general channel m odel for which the main resul ts of this work hold . W e conclude this paper in s ection VIII. I I . C H A N N E L S W I T H S I D E I N F O R M A T I O N A T T H E T R A N S M I T T E R Channels with known i nterference at the transm itter are special case of channels with si de inform ation at the transmitter which were considered by Shannon [13] in the causal knowledge setting and by Gel’fand and Pinsker [14] i n the no n-causal knowledge setting. Shannon considered a discrete m emoryless channel (DMC) whose transition matrix depends on the channel state. A s tate-dependent discrete memoryl ess channel (SD-DMC) is defined by a finite input alphabet X , a finite output alphabet Y , and transiti on prob- abilities p ( y | x, s ) , where the s tate s t akes on values in a finite alp habet S . The block diagram of a state-dependent channel w ith state inform ation at the encoder is shown in fig. 1 . In the causal knowledge setting , the encoder m aps a message w into X n as x i = f i ( w , s 1 , . . . , s i ) , 1 ≤ i ≤ n. (1) Shannon showed that it is suffi cient to con sider the coding schemes t hat use only the current state s ymbol in t he encoding p rocess t o achieve the capacity o f an SD-DMC with i .i.d. st ate sequence known causally at t he encoder [13]. The SD-DMC can be used in th e way sho wn i n fig. 2 to transmit information . A precoder is added in front of the SD-DMC. A message w is mapp ed into T n , where T is a new alphabet. The output of t he precoder ranges over X and depends on the current interference symb ol. The regular (witho ut state) channel from T to Y is defined by the transition prob abilities q ( y | t ) = X s ∈S p ( s ) p ( y | x = t ( s ) , s ) , (2) 5 Encoder State Channel Generator Decoder p ( y | x, s ) w ˆ w s y ∈ Y x ∈ X s ∈ S Fig. 1. SD-DMC with state information at the encoder . State Channel Generator Decoder Encoder Precoder p ( y | x, s ) ˆ w w t ∈ T s ∈ S s x ∈ X y ∈ Y Fig. 2. The associated re gular DMC. where p ( s ) is the probabili ty of the state s . The DMC defined in (2) i s called the associated channel. Th e codes for the as sociated channel describe the codes for the SD-DMC that use only the current s tate s ymbols in the encoding operation. In order to describe all coding schemes for the SD-DMC that use only the current state sy mbol in the encoding process, T must include all functions from the st ate alphabet to the i nput alphabet of the st ate-dependent channel. T here are a total of |X | |S | of such functions, where | . | d enotes the cardinality of a set. Any of the functions can b e represented by a |S | -tup le ( x 1 , x 2 , . . . , x |S | ) composed of elements of X , imp lying th at th e value of the function at s tate s is x s , s = 1 , 2 , . . . , |S | . I I I . T H E C H A N N E L M O D E L W e consider data transmi ssion over the channel Y = X + S + N , (3) 6 where X is t he channel input, which takes on values in a real finite set X , Y is t he channel output, N is additive white Gaussian noise wit h power σ 2 , and the interference S i s a discrete random variable that takes on values in a real finit e set S . The sequence of i.i.d. i nterference symbo ls is known causall y at the encoder . The above channel can be considered as a special case of the state-dependent channel considered by Shannon with one exception, that the channel output alphabet is conti nuous. In our case, the likelihood function f Y | X, S ( y | x, s ) is used instead of the transition probabiliti es. W e denote the input to the ass ociated channel by T , wh ich can be considered as a functi on from S to X . W e denot e the cardinality of X and S by M and Q , respecti vely . Then th e cardinali ty of T will be M Q , which is the number all functions from S to X . The likelihood function for the ass ociated channel is given by f Y | T ( y | t ) = X s ∈S p ( s ) f Y | X, S ( y | t ( s ) , s ) = X s ∈S p ( s ) f N ( y − t ( s ) − s ) , (4) where p ( s ) is the probability of the interference symbol s and f N denotes t he pdf of the Gaussian no ise N . Although in this work, we consi der a fixed channel input alp habet X , t he transmi tted power is not fixed in general. In fact, for probabili ty d istribution p ( s ) on S and for a giv en coding scheme for the a ssociated channel which induces probabil ity dis tribution p ( t ) on the sym bols of T , the transmitt ed p owe r is given by E [ X 2 ] = X t ∈T X s ∈S p ( t ) p ( s ) E [ X 2 | t, s ] = X t ∈T X s ∈S p ( t ) p ( s ) t 2 ( s ) . (5) Thus, in general, the transm itted power depends on th e probability distri bution on the interference alphabet. The binary-input channel with X = {− x, x } is an exception, howe ver , for which we have t 2 ( s ) = x 2 for all s ∈ S . Therefore, for any cod ing scheme 7 and any prob ability dis tribution on the interference alphabet, t he transmitted power is equal to x 2 . In this work, we do not impose any constraint on the po wer of the transmitted signal. Howe ver , in t he p erformance comparisons giv en in section s V and VI for different scenarios, we ens ure that the transmitted power i s the same in all s cenarios. I V . T H E C O D E D E S I G N C R I T E R I O N Any coding scheme for t he associated channel defined by (4) translates to a coding scheme for the actual channel defined by f Y | X,S ( y | x, s ) . W e us e t he pairwise error probability (PEP) approach to derive the code design criterion at h igh SNR. Since in this work, we consider fixed channel inp ut and int erference alphabets , t he hi gh SNR scenario is realized b y making the noise power σ 2 suffi ciently small. Thi s i s equiv alent to s cale up the transmitt ed signal and t he i nterference by the same factor for a given noise power . Suppose that the mess ages w 1 and w 2 are encoded into codewords t n 1 ≡ t 1 t 2 . . . t n and r n 1 ≡ r 1 r 2 . . . r n , respecti vely , where t i and r i belong to the alphabet T , i = 1 , . . . , n . In the abs ence of noise, transmi ssion of the codew ord t n 1 can result in many diff erent recei ved sequences at the channel output depending on the interference sequence s n 1 ≡ s 1 s 2 . . . s n . In specific, all sequences in { ( t 1 ( s 1 ) + s 1 , t 2 ( s 2 ) + s 2 , . . . , t n ( s n ) + s n ) : s n 1 ∈ S n } represent the transm itted codew ord t n 1 at the channel out put. On t he other hand, all sequences in { ( r 1 ( s 1 ) + s 1 , r 2 ( s 2 ) + s 2 , . . . , r n ( s n ) + s n ) : s n 1 ∈ S n } represent the codeword r n 1 . Using maximum li kelihood decoding, the probability of the event t hat message w 2 is 8 decoded giv en m essage w 1 was sent is giv en by Pr { w 1 → w 2 | w 1 } = X s n 1 p ( s n 1 ) Pr { w 1 → w 2 | w 1 , s n 1 } = X s n 1 p ( s n 1 ) Pr  f Y | T ( y n 1 | t n 1 ) ≤ f Y | T ( y n 1 | r n 1 ) | w 1 , s n 1  = X s n 1 p ( s n 1 ) Pr ( n Y i =1 f Y | T ( y i | t i ) ≤ n Y i =1 f Y | T ( y i | r i ) | w 1 , s n 1 ) = X s n 1 p ( s n 1 ) Pr ( n Y i =1 X s ∈S p ( s ) f N ( y i − t i ( s ) − s ) ≤ n Y i =1 X s ∈S p ( s ) f N ( y i − r i ( s ) − s ) | w 1 , s n 1 ) . (6) In appendix I, we have shown that the above error probabili ty at high SNR is given by Pr { w 1 → w 2 | w 1 } = O Q p P n i =1 d 2 SI ( t i , r i ) 2 σ !! , (7) where Q ( x ) = Z ∞ x 1 √ 2 π exp  − y 2 2  dy , (8) and d SI ( t, r ) ( SI stands for side i nformation), the distance between t wo i nput symb ols of the associated channel t and r , is defined as d SI ( t, r ) = min s 1 ,s 2 ∈S | t ( s 1 ) + s 1 − r ( s 2 ) − s 2 | . (9) According to (7), at high SNR, t he code design criterion is t o maximize the minimum distance between the codewords with t he distance measure defined in (9). A. No Side Info rmation at the Encoder - A Compari son In order to see how the knowledge of int erference at the encoder can result in lar ger dist ances between codew ords, consider the channel model int roduced in section III wi th the exception that the i nterference sequence is no t known at th e encoder . In this case, the discrete interference is con sidered as nois e. In order to obtain the PEP for this 9 channel, sup pose that mess ages v 1 and v 2 are encoded into x n 1 ≡ x 1 · · · x n ∈ X n and z n 1 ≡ z 1 · · · z n ∈ X n , respectively . Sim ilarly , it can be shown th at the PEP at hig h SNR is given by Pr { v 1 → v 2 | v 1 } = O Q p P n i =1 d 2 ( x i , z i ) 2 σ !! , (10) where d ( x, z ) , t he di stance between two symbols x and z o f X i s defined as d ( x, z ) = min s 1 ,s 2 ∈S | x + s 1 − z − s 2 | . (11) Comparing (9) and (11), it becomes clear that l ar ger distances among code words are possible for the channel with side inform ation at the encoder . In fact, the distance d ( x, z ) is equal to d SI ( t, r ) for t = ( x, . . . , x ) and r = ( z , . . . , z ) . Howev er , T has m any o ther symbols, which may yi eld lar ger distances. For example, consider the channel with X = S = {− 1 , +1 } . For the case wi thout side i nformation at t he encoder , we can compute the distances between sym bols of X according to (11) as d (1 , 1) = d ( − 1 , − 1) = d (1 , − 1) = 0 . Hence, according to (10), it is imposs ible to transmit data over this channel with low error probabil ity e ven at high SNR. For the case with s ide informati on at the encoder , the fou r symbols o f the ass ociated channel can be represented as u 1 = ( − 1 , +1) , u 2 = (+1 , − 1) , u 3 = (+1 , +1 ) , u 4 = ( − 1 , − 1) . Using (9), it is easy to check that the dist ances between all p airs of the symbols are zero except for d SI ( u 1 , u 2 ) wh ich i s 2 . As will be seen in s ection V, u 1 and u 2 can be used in the encodi ng to achieve arbit rarily low error probabilities as SNR increases. It is worth ment ioning that the dist ance measures defined in (9) or (1 1) do not s atisfy the triangl e inequality . For example, again consi der the channel with X = S = {− 1 , +1 } . The dist ances between all pairs of the input symbols o f the associated channel are zero except for d SI ( u 1 , u 2 ) which i s 2 . Therefore, t he triangle in equality does not hold for d SI ( u 1 , u 3 ) , d SI ( u 3 , u 2 ) , and d SI ( u 1 , u 2 ) . 10 V . T H E B I N A RY C H A N N E L W e call t he channel introduced in (3) a bina ry channel wh en the channel accepts binary input, i.e., M = 2 . There is no constraints on t he cardinalit y of th e int erference alphabet. For th e b inary channel, th e s ize o f T i s 2 Q . Howe ver , we may not need to use all the symbols of the alphabet in the encodin g. In this section, we show that it i s suffi cient to u se only two symbols of T in the encodi ng as far as t he distance spectrum of the code is concerned. W e begin with the following lem ma for the binary channel. Lemma 1: For the binary channel, there e xist at least tw o symbols in T with nonzero distance. Pr oo f: W e m ay explicitly denote the channel input and interference alphabets by X = { x 1 , x 2 } and S = { s 1 , . . . , s Q } , where x 1 < x 2 and s 1 < s 2 < · · · < s Q . From th e definition of di stance in (9), it is sufficient to show that there exist two elements t and r in T such that the correspondi ng multi-sets 1 (of size Q ) { t ( s 1 ) + s 1 , . . . , t ( s Q ) + s Q } and { r ( s 1 ) + s 1 , . . . , r ( s Q ) + s Q } are disjoint . W e prove this by induction on Q . The statement of the lemma hol ds for Q = 1 since we may take t = ( x 1 ) and r = ( x 2 ) . Then th e s ets { x 1 + s 1 } and { x 2 + s 1 } are disjoi nt. Now sup pose that the statement of the lem ma is true for s ome Q . Therefore, the exist two Q -tuples compo sed of elements of X (two input sy mbols of the a ssociated channel) such that the corresponding multi-sets are disj oint. W e prove that th e st atement of the lemma hold for Q + 1 . The element x 2 + s Q +1 is larger than any element of the two multi-sets (of size Q ). Hence, it does not belon g to any of the m ulti-sets. If x 1 + s Q +1 does not belong to any of the multi -sets t oo, then we can includ e the new elements x 1 + s Q +1 and x 2 + s Q +1 in the multi-sets of size Q arbitrarily (one elements in each mul ti-set). The resulting multi-set s of size Q + 1 will be d isjoint. If x 1 + s Q +1 belongs to one of the m ulti-set of size Q , we in clude it i n t hat mult i-set and i nclude x 2 + s Q +1 in the other mul ti-set to form the new disjoint m ulti-sets of size Q + 1 . The two ( Q + 1) -tu ples (the two i nput sym bols 1 A multi-set dif fers from a set in that eac h member may hav e a multiplicity greater than one. For example, { 1 , 3 , 3 , 7 } is a multi-set of size four where 3 has multiplicity two. 11 of the a ssociated channel) are then obtained from t he two multi-sets of si ze Q + 1 by subtracting the int erference sym bols from their elements. Lemma 1 is in fact a special case of theorem 2 in [15 ], which was s tated i n t he context of capacity . Let u 1 and u 2 be two input symbo ls of the associa ted channel with the maxi mum distance among all pairs o f input symbols of the associ ated channel. Since d SI ( u 1 , u 2 ) > 0 (according to Lemma 1), we have u 1 ( s ) 6 = u 2 ( s ) , ∀ s ∈ S , otherwise, from (9), d SI ( u 1 , u 2 ) = 0 . W e choose an arbit rary i nterference symbol s ∈ S to partition T as follows. W e put t ∈ T i n T 1 if t ( s ) = u 1 ( s ) , otherwise (i.e., t ( s ) = u 2 ( s ) ) we put t in T 2 . Note that the distance between any two sym bols in T j is zero, j = 1 , 2 . Suppose that a codebook is designed for the binary channel with code words com- posed of elements of T . W e construct a ne w codebook from the origi nal one by replacing the element s of t he codewords that belong to T 1 by u 1 and replacing the elements of the codewords that belong to T 2 by u 2 . Since the codewords of the new codebook are composed of j ust two elements, we may call the new cod e a binary code. Theor em 1: The distance spectrum of the binary code constructed by the procedure described above is at least as go od as the distance spectrum of t he original code. Pr oo f: Consider any two codewords ( t 1 , . . . , t n ) and ( r 1 , . . . , r n ) from the orig inal codebook, where t i , r i ∈ T . The squared distance between the two code words is equal to P n i =1 d 2 SI ( t i , r i ) . For any i ∈ { 1 , 2 , . . . , n } , we cons ider t wo cases: Case 1 : t i and r i belong to the same partit ion. Then d SI ( t i , r i ) = 0 , so th e replace- ment wi ll not change the distance. Case 2 : t i and r i belong t o differ ent partiti ons. Then s ince d SI ( t i , r i ) ≤ d SI ( u 1 , u 2 ) , the replacement will not decrease the distance. According to theorem 1, as far as the dis tance sp ectrum of t he cod e in concerned, it is suffic ient to use two s ymbols of T wit h the maximum dist ance, namely u 1 and u 2 , i n t he encoding for a bin ary channel. Since T has size 2 Q for the binary channel, a brute-force search for finding two symbols in T with the maxim um dis tance will 12 hav e exponential comp lexity with respect to Q . W e hav e p roposed an algorit hm with polynomial compl exity for finding two symb ols with the maxi mum distance in appendi x II. Since it i s sufficient t o use u 1 and u 2 in the encoding for the binary channel, we can define the Hammi ng distance between any two code words, wh ich is the num ber of positions at whi ch the two codewords are different. Consider two codew ords c 1 = ( t 1 , . . . , t n ) and c 2 = ( r 1 , . . . , r n ) with elements from the binary set { u 1 , u 2 } . The squared distance between these codewords is given by n X i =1 d 2 SI ( t i , r i ) = d 2 SI ( u 1 , u 2 ) d H ( c 1 , c 2 ) , (12) where d H ( c 1 , c 2 ) is t he Hamming di stance between c 1 and c 2 . Therefore, t he problem of designing codes for the binary channel where the i nterference sequence is known causally at t he encoder reduces t o t he desi gn of codes for the binary symmetric channel. The only diffe rence is that the coding i s over the set { u 1 , u 2 } rather th an { 0 , 1 } . A. Comparison wit h t he Interfer ence-F r ee Channel If we were to use a binary code for the interference-free binary channel wi th the input alphabet X = { x 1 , x 2 } , then the E uclidean distance between any two code words c 1 and c 2 of length n for the interference-free channel would be d 2 E ( c 1 , c 2 ) = ( x 1 − x 2 ) 2 d H ( c 1 , c 2 ) , (13) where d E denotes the Eucl idean d istance. Using (12) and (13), we can compare th e performance of a zero-one binary code for the binary channel with causal side information at the encoder wi th the same zero-one binary code for the interference-free binary channel. In the case of channel wit h side information, zero and one are mapped to u 1 and u 2 , and in the case of the interference- free channel, zer o and one are mapped to x 1 and x 2 , respectively . N ote that u 1 and u 2 are functions from the int erference alphabet S to t he channel input alphabet X = { x 1 , x 2 } . 13 It is clear from (9) that d SI ( u 1 , u 2 ) ≤ | x 1 − x 2 | . (14) Therefore, usin g (12) and (13), the distance s pectrum of t he code for the in terference- free channel is at least as good as the distance-spectrum o f the code for the channel with known interference at the encoder . Of course, thi s is not surprising. Howe ver , it is interesting to search for the conditio ns that (14 ) is satisfied with equality . If (14) is satisfied with equalit y , the distance s pectrum of the two codes will be the same. In other words, if (14) is satisfied with equality , the knowledge of interference at the encoder enabl es us to achie ve the same performance (in terms of order of p robability of error) as the interference-free case at high SNR. W e may explicitly denote th e in terference alphabet by S = { s 1 , . . . , s Q } , where s 1 < s 2 < · · · < s Q . Then the following theorem holds. Theor em 2: d SI ( u 1 , u 2 ) = | x 1 − x 2 | if and only if min i 6 = j | s i − s j | ≥ | x 1 − x 2 | . Pr oo f: If min | s i − s j | ≥ | x 1 − x 2 | , we may t ake u 1 = ( x 1 , x 2 , x 1 , . . . ) and u 2 = ( x 2 , x 1 , x 2 , . . . ) . Then we have d SI ( u 1 , u 2 ) = min i,j | u 1 ( s i ) + s i − u 2 ( s j ) − s j | = min {| x 1 + s k − x 2 − s k | , | x 1 + s 2 k 1 +1 − x 2 − s 2 k 2 +1 | k 1 6 = k 2 | x 1 + s 2 k 1 +1 − x 1 − s 2 k 2 | k 1 ,k 2 , | x 2 + s 2 k 1 − x 2 − s 2 k 2 +1 | k 1 ,k 2 } = min {| x 1 − x 2 | , | x 1 + s 2 k 1 +1 − x 2 − s 2 k 2 +1 | k 1 6 = k 2 , | s 2 k 1 +1 − s 2 k 2 | k 1 ,k 2 } . (15) W e also hav e | x 1 + s 2 k 1 +1 − x 2 − s 2 k 2 +1 | ≥ | s 2 k 1 +1 − s 2 k 2 +1 | − | x 1 − x 2 | ≥ 2 min | s i − s j | − | x 1 − x 2 | for k 1 6 = k 2 ≥ | x 1 − x 2 | (16) 14 and | s 2 k 1 +1 − s 2 k 2 | ≥ min | s i − s j | ∀ k 1 , k 2 ≥ | x 1 − x 2 | . (17) Therefore, d SI ( u 1 , u 2 ) = | x 1 − x 2 | . For the other direction, suppose that min | s i − s j | < | x 1 − x 2 | . W e will show that d SI ( u 1 , u 2 ) < | x 1 − x 2 | . Suppose that s k , s k +1 ∈ S achiev e th e m inimum of | s i − s j | and t 1 and t 2 are arbitrary elements of T . W e consider two non-trivial cases: Case 1 : t 1 ( s k ) = t 1 ( s k +1 ) = x 1 and t 2 ( s k ) = t 2 ( s k +1 ) = x 2 . Then d SI ( t 1 , t 2 ) ≤ | t 1 ( s k +1 ) + s k +1 − t 2 ( s k ) − s k | < | x 1 − x 2 | . Case 2 : t 1 ( s k ) = x 1 , t 1 ( s k +1 ) = x 2 and t 2 ( s k ) = x 2 , t 2 ( s k +1 ) = x 1 . Then d SI ( t 1 , t 2 ) ≤ | t 1 ( s k ) + s k − t 2 ( s k +1 ) − s k +1 | < | x 1 − x 2 | . As an example, consi der a binary channel wit h X = S = {− 1 , +1 } and equiprob- able interference symbols. The two symbols with the maxim um d istance in the input alphabet of the associated channel are u 1 = ( − 1 , +1) , u 2 = (+1 , − 1) . W e hav e si mulated the error probabil ity performance of th e above un coded system with maxi mum l ikelihood decoding. T he error p robability vs. SNR  = 1 σ 2  for t he above channel is plotted in fig. 3. The error prob ability curve for the i nterference-free channel wit h X = {− 1 , +1 } is plotted for comparison. For the interference-free channel, P e = Q ( 1 σ ) . It is easy t o check that i n this example, d SI ( u 1 , u 2 ) = | x 1 − x 2 | = 2 . As it can be seen, the error probabilit y curves decay at the same rate with increasing SNR as expected. The error p robability curve for th e scenario t hat t he interference is not known at the encoder , is plotted for comparison. In this scenario, the error probability curve reaches an error floor of 1 4 . Another example is illustrated i n fig. 4. For t his example, X = {− 1 , +1 } , S = {− 1 , 0 , +1 } . W e can find by ins pection two symbols of the ass ociated channel in put alphabet with the maximu m dist ance as u 1 = ( − 1 , − 1 , +1) , u 2 = (+1 , +1 , − 1) . Here, we h a ve d SI ( u 1 , u 2 ) = 1 < | x 1 − x 2 | = 2 . Th erefore, the error probabi lity curve for the channel with known interference at the encoder does not decay as fast as the error 15 2 4 6 8 10 12 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) P e Channel with unknown interference Channel with known interference Interference−free channel Fig. 3. Error prob ability vs. S NR for the binary input A W GN channel with/without kno wn/unkno wn interference. X = S = {− 1 , +1 } . probability curve for the interference-free channel. For the scenario that the interference is not kno wn at the encoder , the error probabi lity curve reaches an error floor of 1 6 . V I . T H E M - A RY C H A N N E L In general, the statement of theorem 1 is not extendable to the case with M > 2 channel input symbols. In fac t, b y using more than M input symbols of t he associated channel, we can obtain a better codebook in terms of dist ance spectrum than any ot her codebook composed of just M input s ymbols of t he associated channel. An example showing this is given in appendix III. Howe ver , under some condition on the channel input and int erference alphabets, the statement o f theorem 1 can be generalized to the case wit h M > 2 . Theor em 3: As far as t he distance spectrum of code i s concerned, it is su f ficient to 16 2 4 6 8 10 12 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) P e Channel with unknown interference Channel with known interference Interference−free channel Fig. 4. Error prob ability vs. S NR for the binary input A W GN channel with/without kno wn/unkno wn interference. X = {− 1 , +1 } , S = {− 1 , 0 , +1 } . use M (out o f M Q ) i nput sy mbols of the associated channel in the encoding if min s i ,s j ∈S | s i − s j | ≥ 2 max x i ,x j ∈X | x i − x j | . Pr oo f: Consider the M input sym bols of the as sociated channel u 1 = ( x 1 , . . . , x 1 ) , u 2 = ( x 2 , . . . , x 2 ) , . . . , u M = ( x M , . . . , x M ) . W e use these symbols to partition the associated channel in put alphabet T as follows. Put t ∈ T in T i if the first elem ent of t is x i , i = 1 , 2 , . . . , M . Not e that T i has size M Q − 1 and the d istance between any t wo symbols i n T i is zero, i = 1 , 2 , . . . , M . For any p, q = 1 , . . . , M , we have d SI ( u p , u q ) = min k 1 ,k 2 | x p + s k 1 − x q − s k 2 | = min {| x p − x q | , | x p + s k 1 − x q − s k 2 | k 1 6 = k 2 } . (18) 17 W e also hav e | x p + s k 1 − x q − s k 2 | ≥ | s k 1 − s k 2 | − | x p − x q | ≥ 2 max | x i − x j | − | x p − x q | for k 1 6 = k 2 ≥ | x p − x q | , (19) Therefore, d SI ( u p , u q ) = | x p − x q | . Note that the distance between any two symbols from T p and T q is at mo st | x p − x q | = d SI ( u p , u q ) . Suppose that a codebo ok is designed wit h codewords composed of possibly all elements of T . W e construct a new codebook from t he original one by replacing the elements of the codewords t hat belong to T i by u i , i = 1 , 2 , . . . , M . It is easy to check that the distance spectrum of t he new code is at least as goo d as the distance spectrum of the origin al code. According to theorem 3, it is suffi cient t o use only the symbols u 1 , . . . , u M in the encoding. But any of these symbo ls is a const ant function from S to X . Therefore, the s ame symbol ent ers the channel regardless of the current i nterference sym bol. Thi s suggests that the knowledge of interference symbol s at the encoder is not helpful in terms of distance spectrum i mprovement provided that the conditio n of theorem 3 is sati sfied. In fact, with the condition o f th eorem 3, we hav e d SI ( u i , u j ) = d ( x i , x j ) = d E ( x i , x j ) , i, j = 1 , . . . , M . (20) where d ( ., . ) , defined in (11), is the distance measure when the interfer ence is not kno wn at the encoder and d E ( ., . ) is the Euclidean distance measure. Therefore, the error probability performance of a code for the channel with known/unknown interference at the encoder will be the same as the performance of the same code for the interference-free chann el at high SNR. It is worth mention ing that for the above-mentioned three scenarios the codes for the interference-free channel, t he channel with known in terference at the encoder , and the channel with unknown interference use the same transm itted power . 18 V I I . A M O R E G E N E R A L C H A N N E L M O D E L Although we have considered the A WGN channel with additive interference so far , our treatment appl ies to more general channels characterized by Y = f ( X , S ) + N , (21) where f is an arbitrary function of two variables, S is t he channel state which is known causally at the encoder , X is t he channel inpu t, and N is white Gaussi an noise. Another special case of thi s more general channel is the fast fading channel Y = S X + N , (22) where S is the fa ding coef ficient. For the general channel model (21), the distance between two symbol s t and r o f T is defined as d SI ( t, r ) = min s 1 ,s 2 ∈S | f ( t ( s 1 ) , s 1 ) − f ( t ( s 2 ) , s 2 ) | . (23) Theorem 1 on the binary channel also holds for the general channel model. Howe ver , the maxim um distance among pairs of s ymbols of T may be zero; i.e., lemm a 1 does not hold true in general. Theorems 2 and 3 do not hold for the more general channel model i n (21) and are specific to t he A WGN with additive interference channel model. V I I I . C O N C L U S I O N In this paper , we deri ved the code design criterion at high SNR for t he M -ary input A WGN channel wi th additive Q -le vel interference, where the sequence of interference symbols is kn own causally at the encoder . The code design is over an input alphabet T of size M Q . The performance of a code for our channel at high SNR is governed by the m inimum di stance between the codew ords with elements from T . W e may not need to us e all sym bols of T in the encoding. In particul ar , we showed that for th e case M = 2 , as far as the distance sp ectrum of th e code is concerned, we just need to use two symbols of T with the m aximum distance amon g all pairs of sy mbols. This reduces the code design problem for our chann el to code design for binary symmetric channel which has been well researched in the l iterature. 19 A P P E N D I X I D E R I V A T I O N O F C O D E D E S I G N C R I T E R I O N A T H I G H S N R Define A i = { t i ( s ) + s : s ∈ S } , i = 1 , . . . , n, (24) B i = { r i ( s ) + s : s ∈ S } , i = 1 , . . . , n. (25) It is worth mentioni ng that the cardinality of A i (or B i ) can be l ess than Q , i = 1 , . . . , n, since different interference sym bols m ay yield the s ame el ement in A i (or B i ). For any i = 1 , . . . , n , we hav e X s ∈S p ( s ) f N ( y − t i ( s ) − s ) = X a ∈A i p ( a ) f N ( y − a ) , (26) X s ∈S p ( s ) f N ( y − r i ( s ) − s ) = X b ∈B i p ( b ) f N ( y − b ) , (27) where p ( a ) and p ( b ) are obtai ned from p ( s ) according to p ( a ) = X s ∈S : t i ( s )+ s = a p ( s ) , (28) p ( b ) = X s ∈S : r i ( s )+ s = b p ( s ) . (29) For any sequence a n 1 ≡ a 1 · · · a n ∈ A 1 × · · · × A n and b n 1 ≡ b 1 · · · b n ∈ B 1 × · · · × B n , we d efine the eve nts E 1 ( a n 1 ) = n \ i =1  a i = arg min a ∈A i | y i − a |  , (30) E 2 ( b n 1 ) = n \ i =1  b i = arg min b ∈B i | y i − b |  , (31) giv en that w 1 has been sent and the interference sequence s n 1 has o ccurred. The e vent E 1 ( a n 1 ) sim ply m eans that a i is th e clos est point to the received signal y i (giv en w 1 has been s ent and the int erference sequence s n 1 has occurred) among all poi nts of A i for all i = 1 , . . . , n . 20 Any term i n the error probabil ity in (6) can be written as Pr ( n Y i =1 X a ∈A i p ( a ) f N ( y i − a ) ≤ n Y i =1 X b ∈B i p ( b ) f N ( y i − b ) | w 1 , s n 1 ) = X a n 1 X b n 1 Pr ( n Y i =1 X a ∈A i p ( a ) f N ( y i − a ) ≤ n Y i =1 X b ∈B i p ( b ) f N ( y i − b ) , E 1 ( a n 1 ) , E 2 ( b n 1 ) | w 1 , s n 1 ) = X a n 1 X b n 1 Pr      n Y i =1 f N ( y i − a i )    p ( a i ) + X a ∈A i a 6 = a i p ( a ) f N ( y i − a ) f N ( y i − a i )    ≤ n Y i =1 f N ( y i − b i )     p ( b i ) + X b ∈B i b 6 = b i p ( b ) f N ( y i − b ) f N ( y i − b i )     , E 1 ( a n 1 ) , E 2 ( b n 1 ) | w 1 , s n 1        = X a n 1 X b n 1 Pr ( n X i =1 ( y i − a i ) 2 ≥ n X i =1 ( y i − b i ) 2 + K σ 2 , E 1 ( a n 1 ) , E 2 ( b n 1 ) | w 1 , s n 1 ) , (32) where K = K ( y n 1 , a n 1 , b n 1 ) is given by K ( y n 1 , a n 1 , b n 1 ) = 2 n X i =1 log p ( a i ) + P a ∈A i a 6 = a i p ( a ) f N ( y i − a ) f N ( y i − a i ) p ( b i ) + P b ∈B i b 6 = b i p ( b ) f N ( y i − b ) f N ( y i − b i ) . (33) Giv en the e vents E 1 ( a n 1 ) and E 2 ( b n 1 ) , it is easy to check that K ( y n 1 , a n 1 , b n 1 ) is bound ed as K 1 ( a n 1 ) = 2 n X i =1 log p ( a i ) < K ( y n 1 , a n 1 , b n 1 ) < K 2 ( b n 1 ) = 2 n X i =1 log 1 p ( b i ) . (34) As we cons ider the hi gh SNR regime, we may assume that the noise power i s suffi ciently small so that the error probabili ty (6) can be well approximated by X s n 1 p ( s n 1 ) X a n 1 X b n 1 Pr ( n X i =1 ( y i − a i ) 2 ≥ n X i =1 ( y i − b i ) 2 , E 1 ( a n 1 ) , E 2 ( b n 1 ) | w 1 , s n 1 ) . (35) 21 Any term i n the sum mation (35) can be upper bounded as Pr ( n X i =1 ( y i − a i ) 2 ≥ n X i =1 ( y i − b i ) 2 , E 1 ( a n 1 ) , E 2 ( b n 1 ) | w 1 , s n 1 ) ≤ Pr ( n X i =1 ( y i − c i ) 2 ≥ n X i =1 ( y i − b i ) 2 , E 1 ( a n 1 ) , E 2 ( b n 1 ) | w 1 , s n 1 ) ≤ Pr ( n X i =1 ( y i − c i ) 2 ≥ n X i =1 ( y i − b i ) 2 | w 1 , s n 1 ) = Q p P n i =1 | c i − b i | 2 2 σ ! ≤ Q p P n i =1 d 2 SI ( t i , r i ) 2 σ ! , (36) where c i = t i ( s i ) + s i , i = 1 , . . . , n. (37) The first inequ ality is due to the fact that g iv en E 1 ( a n 1 ) , we have | y i − a i | ≤ | y i − c i | , i = 1 , . . . , n . In the following, we show that th e upp er boun d (36) is tigh t for the term(s) in the summation (35) s atisfying { a i , b i } = arg min a ∈A i b ∈B i | a − b | , i = 1 , . . . , n, (38) and a i = c i , i = 1 , . . . , n. (39) Any term i n (35) equals the int egral of the joi nt probability distribution of y n 1 ≡ y 1 · · · y n (giv en w 1 , s n 1 ) over the region in t he n -dim ensional Euclidean space defined by ( y n 1 : n X i =1 ( y i − a i ) 2 ≥ n X i =1 ( y i − b i ) 2 , E 1 ( a n 1 ) , E 2 ( b n 1 ) ) . (40) This region is illus trated by the shaded area ABCD in fig. 5 for n = 2 . The horizontal and vertical bound aries of ABCD correspond to t he events E 1 ( a 2 1 ) and E 2 ( b 2 1 ) . The elements of A i and B i are shown by ◦ and × , respectiv ely . The other boundary 22 a 1 b 1 b 2 a 2 D A B C δ y 1 y 2 Fig. 5. Illustrating the regions of integration for dimension n = 2 . of ABCD which correspond s to P 2 i =1 ( y i − a i ) 2 ≥ P 2 i =1 ( y i − b i ) 2 is the perpendicular bisector of t he l ine segment connecting a 2 1 to b 2 1 . W e may con sider an n -cube i nside t his region with sides equ al to som e δ > 0 as shown in fig. 5 and perform the integration over this smaller region t o ob tain a lower b ound for the t erm(s) in the summation (35 ) satisfying (38) and (39). 23 In summary , for the terms i n (35) which satisfy (38) and (39 ), we have Pr ( n X i =1 ( y i − a i ) 2 ≥ n X i =1 ( y i − b i ) 2 , E 1 ( a n 1 ) , E 2 ( b n 1 ) | w 1 , s n 1 ) ≥  1 − Q  δ 2 σ  n − 1  Q  k b n 1 − a n 1 k 2 σ  − Q  k b n 1 − a n 1 k + δ 2 σ  ≃ Q  k b n 1 − a n 1 k 2 σ  as σ → 0 = Q p P n i =1 d 2 SI ( t i , r i ) 2 σ ! , (41) where t he right h and side o f t he inequality in (41) equals t he integral of the joint probability distri bution o f y n 1 ≡ y 1 · · · y n (giv en w 1 , s n 1 ) over the smaller region, which is obtained by us ing the fac t that y n 1 is Gaussian centered at c n 1 = a n 1 and by app lying the necessary rot ation. A P P E N D I X I I A P O L Y N O M I A L C O M P L E X I T Y A L G O R I T H M F O R FI N D I N G T W O S Y M B O L S O F T W I T H T H E M A X I M U M D I S T A N C E W e propose an al gorithm for findi ng two sy mbols of T wi th distance greater than or equal to som e d 0 > 0 . Then we explain how to find two symbo ls in T with the m aximum distance. Consider the bi partite graph G ( U, V , E ) shown in fig. 6 with 2 Q vertices at each part. E ach of the non-intersecting sets U 1 , · · · , U Q contains two vertices of the upper part U and each of t he nonintersecting s ets V 1 , · · · , V Q contains two vertices of the lower part V . The vertices of the sets U i = { u i 1 , u i 2 } and V i = { v i 1 , v i 2 } are labeled by the elements of t he set X + s i = { x 1 + s i , x 2 + s i } , i = 1 , . . . , Q . A verte x in U i is connected to a verte x in V j if the absolute value of t he difference of their labels is greater than or equal to d 0 , i, j = 1 , . . . , Q . From t he definition of dist ance in (9), there exist two s ymbols in T wit h distance d ≥ d 0 if and only if G has a complete bipartite subgraph K Q,Q with exactly one vertex in each U i and each V j . If such a su bgraph exists, we label the edges of th e subgraph 24 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 0 1 0 1 0 1 0 1 . . . . . . x 1 + s 1 x 2 + s 1 x 1 + s 1 x 2 + s 1 V 1 U 1 V 2 x 2 + s 2 U 2 U Q x 2 + s Q x 2 + s Q V Q x 1 + s Q x 1 + s Q x 2 + s 2 x 1 + s 2 x 1 + s 2 Fig. 6. Graph representation for the problem of finding two symbols of T with the maximum distance. by 1 and we l abel t he rest of the edges of G by 0 . W e denote the label of edge e by y e ∈ { 0 , 1 } . Such a labeling satisfies t he fol lowing set of constraints X e : e ∩ U i 6 = φ y e = Q, i = 1 , . . . , Q, (42) X e : e ∩ V i 6 = φ y e = Q, i = 1 , . . . , Q, (43) y e ∈ { 0 , 1 } . (44) Note that by d efinition, an edge of a graph is a set of two vertices. Therefore, the notation e ∩ U i in (42) i s meaningful. The equation s (42) and (43) state that the s um of the labels of the edges going out of any U i and V i is Q . W e devise an objectiv e function for the cons traints (42), (43), and (44) such that the objective function takes a given maximum value on ly for a labeling wi th label 1 for the edges of the subgraph K Q,Q and label 0 for the rest of the edges. Consi der the following 25 optimizatio n problem max y e Q X i =1 2 X j =1   X e : u ij ∈ e y e   2 + Q X i =1 2 X j =1   X e : v ij ∈ e y e   2 subject t o X e : e ∩ U i 6 = φ y e = Q, i = 1 , . . . , Q, X e : e ∩ V i 6 = φ y e = Q, i = 1 , . . . , Q, y e ∈ { 0 , 1 } . (45) In the following, we find t he m aximum of t he above optimization problem for the foregoing labeling. Given the cons traints o f (45), we have 2 X j =1   X e : u ij ∈ e y e   = X e : e ∩ U i 6 = φ y e = Q, i = 1 , . . . , Q, (46) 2 X j =1   X e : v ij ∈ e y e   = X e : e ∩ V i 6 = φ y e = Q, i = 1 , . . . , Q. (47) If th e su m of two no nnegati ve variables is constant, then the sum of their squares takes it s maximum if one of the variables is zero. Therefore, for any i = 1 , . . . , Q , th e m aximum of 2 X j =1   X e : u ij ∈ e y e   2 and 2 X j =1   X e : v ij ∈ e y e   2 will be Q 2 and this maxi mum occurs if and onl y if one vertex in any of U 1 , . . . , U Q and V 1 , . . . , V Q is connected to Q edges wi th label 1 and t he other verte x in any of U 1 , . . . , U Q and V 1 , . . . , V Q is not connected to any edge with label 1 . This is equiv alent to t he existence of a s ubgraph K Q,Q . Then the maximum of the objective function in (45) will be Q × Q 2 + Q × Q 2 = 2 Q 3 . 26 W e may relax the in tegrality cons traint (44) and change equality signs in (42) and (43) to inequali ty si gns to obtain the foll owing opti mization program max y e Q X i =1 2 X j =1   X e : u ij ∈ e y e   2 + Q X i =1 2 X j =1   X e : v ij ∈ e y e   2 subject t o X e : e ∩ U i 6 = φ y e ≤ Q, i = 1 , . . . , Q, X e : e ∩ V i 6 = φ y e ≤ Q, i = 1 , . . . , Q, 0 ≤ y e ≤ 1 . (48) Using the same argument as in the previous paragraph, the value 2 Q 3 is also achiev able for t he above maximization problem if and only if a subgraph K Q,Q of the graph G exists. The above optim ization problem is a quadrati c pr ogr amming probl em [16] with con vex objectiv e function and can be solved in pol ynomial t ime [17] in t erms of the number of edges of G , which is at m ost 4 Q 2 . In su mmary , we turned the prob lem of findi ng two sym bols in T with distance at least d 0 > 0 into the quadratic programmi ng problem (48). If the maximum value of (48) is 2 Q 3 , then two such sym bols are obtained from the opti mal sol ution of (48). Otherwise, two such symbo ls do no t exist. T o find two sym bols i n T wi th the maxim um distance, we need t o run the d escribed algorithm for a few values for d 0 . W e can ob tain an upper bound on th e num ber o f possible di stances between symbols of T . From the definition of distance in (9), a loose upper b ound is M 2 Q 2 = 4 Q 2 . By using the binary search algorithm [18], the search over possible dist ances can be done with logarit hmic comp lexity wi th respect t o the nu mber of possible dist ances. It is worth menti oning that our propo sed algorithm can be extended to find K ≥ 2 symbols of T wit h t he m aximum minim um distance among K sym bols for th e general case M ≥ 2 . 27 A P P E N D I X I I I A N E X A M P L E T H A T S H OW S U S I N G M O R E T H A N M S Y M B O L S O F T R E S U L T S I N L A R G E R M I N I M U M D I S TA N C E ( M > 2 ) Consider the channel with X = { 1 , 4 , 5 , 7 } and S = { 0 , 4 } . Consider the following codebook wit h six codew ords of lengt h two that uses s e ven sym bols of the ass ociated channel. Code word 1 : ((4 , 1 ) , (5 , 1)) Code word 2 : ((4 , 1 ) , (1 , 5)) Code word 3 : ((5 , 4 ) , (5 , 4)) Code word 4 : ((5 , 4 ) , (4 , 5)) Code word 5 : ((1 , 5 ) , (4 , 1)) Code word 6 : ((1 , 5 ) , (1 , 4)) The m inimum distance of the above code is 3 . 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