Zero-Delay Gaussian Joint Source-Channel Coding for the Interference Channel

1 Zero-Delay Gaussian Joint Source-Channel Coding for the Interfere nce Channel Xuechen Chen School of Electronics and Information T echnology Sun Y at-Sen Univ ersity , Guangzhou, China chenxch8@mail.sys u .edu.cn Abstract This paper studies zero-d e lay joint source chan nel codin g (JSCC) fo r transmission of cor related Gaussian sources over a Gaussian interfe r ence chann e l (GIC). W e pro pose to adopt de lay -free h ybrid digital and analog (HDA) scheme , whic h is, transmitting the superp osition of scaled source and its quan tized version after ap plying scalar quantizatio n to the sour ce at ea c h transmitter . At th e correspo n ding receiver , two kin d s of estimators a r e presented . It is shown that b o th the schemes, when optimized , beat the unco ded transmission if the channel signal-to-no ise ratio (CSNR) is h igher than a threshold value f or different cor relation coefficients and inte r ference values. Index T erms JSCC, Gau ssian IC, zero-delay , HD A I . I N T RO D U C T I O N In many emer ging app lica tions in v olving wireless sensor networks (WSN), low- cost se nsors with limited process ing capa bilities an d battery life are deployed wh ic h implies that enc oders with lo w complexity are nee ded. Su ch applications usually require real-time monitoring and control of un- derlying phys ical systems, which impos e strict delay constraints. As a result, novel coding method s instead o f traditional long block codes for se parate s o urce and channel coding (SSCC) which exhibit high complexity and high delay are needed. W e cons ider the extreme of zero-delay problem, i.e., the transmission is to be done in a sc alar fashion. It is w e ll kno wn that a zero-delay uncod e d (i.e., scale-and-transmit) sc heme c an achie ve the minimum sq uared distortion for a Gaussian source transmitted over an add iti ve white Gauss ian noise (A WGN) channel with an input power c onstraint. Howe ver , this is not the c a se for multi-terminal problems, in general. Also, in multi-t erminal sc enarios, the optimality of SS CC breaks down. Note that in WSN, measu rements collected by the sens ors close to each other exhibit s tatistical co rrelation. 2 As the correlation of sou rces can be adopted to gen erate correlated chann el inpu ts by JSCC, it is attracti ve to turn to JSCC. As a matter o f fact, uncoded sc heme is a special ca se of JSCC. In recent yea rs, many works have been don e to unde rstand multi-user JSCC p roblems. For example, [1] d eri ved the distortion lo wer bo und whe n a bi v ariate Gauss ia n is transmitted over a Gaussian multiple access ch annel (GMA C). The neces s ary conditions for optimality o f unco d ed transmiss ion for multi-user communications over Gaus sian broadca st ch annel (BC) and GMA C were gen eralized in [2]. [3] p roposed two distributed delay-free JSCC sc hemes for a biv ariate Ga ussian o n a GMA C. [4], [5] in vestigated ze ro-delay JSCC problems for a Gauss ian s ource in the W yn e r -Zi v Setting and over dirty-tape chann el respec ti vely . In this pape r , we consider zero-delay transmission of a pair of correlated Ga ussian sou rc e s ( S 1 , S 2 ) over a two-user Gaussian interferenc e channe l (IC). Each of two separate transmitters observes a dif ferent compone nt of the s ource pair and describes the observations to the co rresponding destination over a Ga ussian IC. Rece i vers i tries to recover the source S i , whe re i ∈ { 1 , 2 } with the minimum av erage d istortion. [6] and [7] gave achiev able distortion region for JSCC IC prob lem in the los s less setting and los sy setup respectiv ely . Abou t JSCC Gauss ian IC problem, [8] deri ved an outer bound on the achiev a ble region when the interference is we ak a nd showed the condition for optimality of uncode d transmiss ion. For strong interference case , see [9]. Our g o al here is to des ign low delay and low complexity JSCC techniques motiv a ted b y hy brid digital and analog strategies proposed in [7]. After applying sca lar q uantizer to eac h s ource, the source itself and the qu antized value a re scaled and sup erimposed to be chan nel inputs. At ea ch receiv er , we propos e two reconstruction methods. W e present nume rical results to show that as long as CS NR is higher than a threshold value, the propos ed scheme s offer better performance than unc oded transmission for dif ferent correlation coefficients an d interference values. The remainder of the letter is organized as follows. Sec tion II introduc es the GIC p rob lem while our s c alar HD A encoding method a nd two kinds o f estimation s chemes are presented in S ection III. Simulation res ults are giv en in Section IV . T h e sup plementary file for this pap er can be foun d in [ ? ]. S 1 S 2 X 1 X 2 c 1 c 2 + + Y 2 Y 1 ˆ S 1 ˆ S 2 !"#$%&'(%#)*%"+ !"#$%&'(%#)*%"+ '(%#)*%"+&, "+& ! " '(%#)*%"+&, "+& ! # ( ˆ T 1 , ˆ T 2 ) ( ˆ T 1 , ˆ T 2 ) $ " $ # - - + + % # & " & # % " ' " ' # Fig. 1: Framework of Our Scheme s 3 I I . P R O B L E M S T AT E M E N T W e ass ume that a s e quenc e of zero mean biv ariate Ga u ssian source { S 1 ,j , S 2 ,j } ∞ j =1 is indepen dent and identically distrib uted (i.i.d.) alon g time j , a nd the c ovari ance matrix for ea ch ( S 1 , S 2 ) is   1 ρ ρ 1   with | ρ | < 1 . In other words, S i ∼ N (0 , 1 ) and S i c = ρS i + N i i = 1 , 2 , i c = { 1 , 2 }\ i with N i ∼ N (0 , σ 2 N ) , where σ 2 N = 1 − ρ 2 and N i is not o nly indepe ndent of S i , but also of N i c . At the i -th trans mitter , an n -block of the i -th source { S i,j } n j =1 is to be mapp ed to cha nnel input { X i,j } n j =1 which should satisfy individual average power constraints, 1 n n X j =1 E [ | X i,j | 2 ] ≤ P i , i = 1 , 2 . Each X i then goes through a n a d diti ve Gaussian IC with i.i.d. no ise W i ∼ N (0 , σ 2 W ) wh ose outpu t is Y i , i.e., Y i = X i + c i c X i c + W i i = 1 , 2 , The source is recovered to be { ˆ S i,j } n j =1 as a function of { Y i,j } n j =1 . The qu ality o f the reconstruction is measured by the mean -squared-error distortion, i.e., D i = 1 n n X j =1 E h ( S i,j − ˆ S i,j ) 2 i . Then D = 1 2 ( D 1 + D 2 ) denotes average e n d-to-end distortion. At the extreme of zero-delay , the bloc k leng th n e q uals to 1 an d the en coding func tion is a sa mple- by-sample one. In the n ext se ction, we introduc e ou r z ero-delay JSCC encoding func tion and two kinds of correspo n ding decoding sche mes. I I I . P RO P O S E D S C H E M E S Our propose d HD A encode r is depicted in Fig. 1. Th e d igital information T i = Q ( S i ) is produce d by a midtread scalar quantizer with reco n struction levels { t k } ∞ k = −∞ , k ∈ Z , where Z den otes the set of integers. W e use ∆ = t k − t k − 1 to denote qua ntization ste p . Then it holds that t k = k ∆ . In parallel, the analog part is used to s e nd the so urce itself. Th e sc aled comb ination of T i and S i , X i = f i ( S i ) = δ i T i + β i S i , is then transmitted through the channe l. The av erage transmit power from encode r i is P i = E h ( δ i T i + β i S i ) 2 i = δ 2 i E [ T 2 i ] + β 2 i + 2 δ i β i E [ T i S i ] i = 1 , 2 , (1) 4 where E [ T i S i ] = X k t k + ∆ 2 Z t k − ∆ 2 t k sP S i ( s i ) ds i = X k t k √ 2 π × h exp ( − ( t k − ∆ 2 ) 2 2 ) − exp ( − ( t k + ∆ 2 ) 2 2 ) i = X k ∆ √ 2 π × exp  − ( t k − ∆ 2 ) 2 2  , (2) E [ T 2 i ] = X k t 2 k × 1 2 h erf ( t k + ∆ 2 √ 2 ) − erf ( t k − ∆ 2 √ 2 ) i . (3) As shown in Fig. 1, at i -th receiv er , ( T i , T i c ) are firstly rec overed by a joint estimator , then S i is reco n structed by utilizing the a n alog channe l outpu t Y i and the e stimated digital information pair ( ˆ T i , ˆ T i c ) j ointly . W e propose two kinds of dec oding schemes an d our object is to find minimum average distortion D with the a verage power constraint P 1 + P 2 = 2 P . A. Scheme A Gi ven the correlation ρ , we set the maximum distanc e be tween the qua ntization o u tput of S i and the one of S i c as M ∆ =  3 p 1 − ρ 2 + h ( | k | max − 1 2 )∆ − ρ ( | k | max − 1 2 )∆ i ∆  × ∆ . The deriv ation of M ∆ can be found in Appe ndix A. k den otes integer quantiza tion index. Its abs olute value is limited by | k | max , which sa tisfies ( | k | max + 1 2 )∆ = κ , where P r ( S i ∈ [ − κ, κ ]) ≈ 1 , tha t is , the overload dis tortion a pproximately equals to 0 . W e apply maximum a po sterior (MAP) e s timator for rec overy of digital information, ( ˆ t i , ˆ t i c ) = arg max t i,k ,t i c ,k ′ t i c ,k ′ ∈ [ t i,k − M ∆ , t i,k + M ∆] P ( T i ,T i c ) ,Y i  ( t i,k , t i c ,k ′ ) , y i  . Herein, P ( T i ,T i c ) ,Y i  ( t i,k , t i c ,k ′ ) , y i  = Z ∞ −∞ Z ∞ −∞ P S i ,S i c , ( T i ,T i c ) ,Y i  s i , s i c , ( t i,k , t i c ,k ′ ) , y i  ds i ds i c = Z t i,k + ∆ 2 t i,k − ∆ 2 Z t i c ,k ′ + ∆ 2 t i c ,k ′ − ∆ 2 P S i ,S i c ( s i , s i c ) P W i ( u t i,k ,t i c ,k ′ ) ds i ds i c , where u t i,k ,t i c ,k ′ = y i − δ i t i,k − β i s i − c i c ( δ i c t i c ,k ′ + β i c s i c ) . 5 After obtaining ( ˆ t i , ˆ t i c ) , ˆ S i is es timated as b e low , ˆ s i = E [ S i | ( T i , T i c ) , Y i ] = ∞ Z −∞ s i P S i | T i ,T i c ,Y i  s i | ( ˆ t i , ˆ t i c ) , y i  ds i = ∞ R −∞ ∞ R −∞ s i P S i ,S i c , ( T i ,T i c ) ,Y i ( s i , s i c , ˆ t i , ˆ t i c , y i ) ds i ds i c ∞ R −∞ ∞ R −∞ P S i ,S i c , ( T i ,T i c ) ,Y i ( s i , s i c , ˆ t i , ˆ t i c , y i ) ds i ds i c = R ˆ t i + ∆ 2 ˆ t i − ∆ 2 R ˆ t i c + ∆ 2 ˆ t i c − ∆ 2 s i P S i ,S i c ( s i , s i c ) P W i ( u ˆ t i , ˆ t i c ) ds i ds i c R ˆ t i + ∆ 2 ˆ t i − ∆ 2 R ˆ t i c + ∆ 2 ˆ t i c − ∆ 2 P S i ,S i c ( s i , s i c ) P W i ( u ˆ t i , ˆ t i c ) ds i ds i c . For Scheme A, it is h ard to o btain a nalytical form for D . B. Scheme B Note that X i can be rewritt en a s the summation of qua ntized v alue and qua n tization error R i = S i − T i , X i = δ i T i + β i ( T i + R i ) = α i T i + β i R i , where α i denotes δ i + β i . As R i is con strained in [ − ∆ 2 , ∆ 2 ] , we prop ose a pse udo maximum likelihood (ML) estimator as follows, ( ˆ t i , ˆ t i c ) = arg min t i,k ,t i c ,k ′ t i c ,k ′ ∈ [ t i,k − M ∆ , t i,k + M ∆] y i − α i t i,k − c i c α i c t i c ,k ′ . As Y i = α i T i + β i R i + c i c  δ i c T i c + β i c  ρ ( T i + R i ) + N   + W i = ( α i + c i c β i c ρ ) T i + c i c ( α i c − β i c ) T i c + ( β i + c i c β i c ρ ) R i + c i c β i c N + W i , the quan tiza tion error R i is es timated linearly b y ˆ r i = Γ i h y i − ( α i + c i c β i c ρ ) ˆ t i − c i c ( α i c − β i c ) ˆ t i c i , where Γ i is linear coefficient. Finally , ˆ S i = ˆ T i + ˆ R i . 6 Distortion Analys is: For Scheme B, we c an express the overall distortion D i in ana lytical form. By definition, D i = E [( T i + R i − ˆ T i − ˆ R i ) 2 ] = E "  λ +  1 − Γ i ( β i + c i c β i c ρ )  R i − Γ i c i c β i c N − Γ i W i  2 # = E [ λ 2 ] +  1 − Γ i ( β i + c i c β i c ρ )  2 σ 2 R + (Γ i c i c β i c ) 2 σ 2 N + Γ 2 i σ 2 W + 2  1 − Γ i ( β i + c i c β i c ρ )  1 − Γ i ( α i + c i c β i c ρ )  E h R i ( T i − ˆ T i ) i − 2  1 − Γ i ( β i + c i c β i c ρ )  Γ i c i c ( α i c − β i c ) E h R i ( T i c − ˆ T i c ) i , (4) where λ =  1 − Γ i ( α i + c i c β i c ρ )  ( T i − ˆ T i ) − Γ i c i c ( α i c − β i c )( T i c − ˆ T i c ) . The justification of (4) can be foun d in Ap pendix B. Next, we would analyz e the c o mponents of (4) one by on e. E [ λ 2 ] = X k k + M X m = k − M X l l + M X n = l − M Φ P ˆ T i c , ˆ T i ,T i c ,T i ( t n , t l , t m , t k ) , (5) where Φ =  (1 − Γ i ( α i + c i c β i c ρ ))( t k − t l ) − Γ i c i c ( α i c − β i c )( t m − t n )  2 . E [ R i T i ] = E [ T i S i ] − E [ T 2 i ] . (6) E [ R i T i ] can be obtained by sub stituting (2) and (3) into (6). E [ R i T i c ] = E h E [ R i T i c | T i ] i = X k P T i ( t k ) E [ R i T i c | T i = t k ] = X k P T i ( t k ) X m ∆ 2 Z − ∆ 2 t m r i P R i ,T i c | T i ( r i , t m | t k ) dr i = X k X m ∆ 2 Z − ∆ 2 t m r i P R i ,T i c ,T i ( r i , t m , t k ) dr i = X k k + M X m = k − M t m t k + ∆ 2 Z t k − ∆ 2 t m + ∆ 2 Z t m − ∆ 2 ( s i − t k ) P S i c ,S i ( s i c , s i ) ds i c ds i . (7) 7 The deriv ation of (7) can be foun d in Ap pendix D. E [ R i ˆ T i ] = E h E [ R i ˆ T i | T i ] i = X k X l ∆ 2 Z − ∆ 2 t l r i P R i , ˆ T i ,T i ( r i , t l , t k ) dr i = X k k + M X m = k − M X l l + M X n = l − M t l ∆ 2 Z − ∆ 2 ∆ 2 Z − ∆ 2 r i Ψ dr i c dr i , (8) where Ψ = P ˆ T i c , ˆ T i ,T i c ,T i ,R i c ,R i ( t n , t l , t m , t k , r i c , r i ) . Similarly , E [ R i ˆ T i c ] = E h E [ R i ˆ T i c | T i ] i = X k k + M X m = k − M X l l + M X n = l − M t n ∆ 2 Z − ∆ 2 ∆ 2 Z − ∆ 2 r i Ψ dr i c dr i . (9) T o calculate (5), (8), (9), we ne ed to find the value of joint p robability P ˆ T i c , ˆ T i ,T i c ,T i ( t n , t l , t m , t k ) . Gi ven the digital information pair ( T i , T i c ) = ( t k , t m ) , the distance d betwe e n the rec overed digital messag e α i t l + c i c α i c t n and the original digital me ssage α i t k + c i c α i c t m must be on e of the v alues in the set 1 as below , { ( pα i + q c i c α i c )∆ } , where p, q ∈ Z , m + q ∈ [ k + p − M , k + p + M ] Sort t he set in a scending orde r and check the position of d in it. W e c an obtain the one just before d and the one right after d . These two d istances are den o ted respe cti vely by d l and d u . The n P ˆ T i c , ˆ T i ,T i ,T i ( t n , t l , t m , t k ) can be expressed a s follows, P ˆ T i c , ˆ T i ,T i c ,T i ( t n , t l , t m , t k ) = t k + ∆ 2 Z t k − ∆ 2 t m + ∆ 2 Z t m − ∆ 2 P ˆ T i c , ˆ T i ,S i c ,S i ( t n , t l , s i c , s i ) ds i c ds i = t k + ∆ 2 Z t k − ∆ 2 t m + ∆ 2 Z t m − ∆ 2 ub Z lb P W i ( w i ) P S i c ,S i ( s i c , s i ) dw i ds i c ds i , (10) 1 The size of the set can be shrunk through limit i ng the absolute v alues of the elements by 2 ×  5 σ W + ( β i + c i c β i c ) ∆ 2  . 8 where ub = d + d u 2 − µ, l b = d + d l 2 − µ , and µ = β i ( s i − t k ) + c i c β i c ( s i c − t m ) . The deriv a tion of (10) can be found in Appe n dix C. According to (10), ∆ 2 Z − ∆ 2 ∆ 2 Z − ∆ 2 r i Ψ dr i c dr i = t k + ∆ 2 Z t k − ∆ 2 t m + ∆ 2 Z t m − ∆ 2 ub Z lb ( s i − t k ) P W i ( w i ) P S i c ,S i ( s i c , s i ) dw i ds i c ds i . (11) The de ri vati on of (11) can be found in Append ix D. Substituting (10) into (5), and (11) into (8), (9), we o btain all the comp onents nee ded to c alculate (4). 15 20 25 30 35 Channel SNR(dB) 5 10 15 20 25 30 SDR(dB) c = 2 SDR bound when  =0.95 SDR bound when  =0.75 Scheme A when  =0.95 Scheme B by simulation when  =0.95 Scheme B by calculation when  =0.95 Scheme A when  =0.75 Scheme B by simulation when  =0.75 Scheme B by calculation when  =0.75 uncoded transmission when  =0.95 uncoded transmission when  =0.75 (a) 15 20 25 30 35 Channel SNR(dB) 5 10 15 20 25 30 SDR(dB) c = 1.5 SDR bound when  =0.95 SDR bound when  =0.75 Scheme A when  =0.95 Scheme B by simulation when  =0.95 Scheme B by calculation when  =0.95 Scheme A when  =0.75 Scheme B by simulation when  =0.75 Scheme B by calculation when  =0.75 uncoded transmission when  =0.95 uncoded transmission when  =0.75 (b) 20 25 30 35 Channel SNR(dB) 8 10 12 14 16 18 20 22 24 26 28 30 SDR(dB) c = 0.8 SDR bound when  =0.95 SDR bound when  =0.75 Scheme A when  =0.95 Scheme B by simulation when  =0.95 Scheme B by calculation when  =0.95 Scheme A when  =0.75 Scheme B by simulation when  =0.75 Scheme B by calculation when  =0.75 uncoded transmission when  =0.95 uncoded transmission when  =0.75 (c) 20 25 30 35 Channel SNR(dB) 10 12 14 16 18 20 22 24 26 28 30 SDR(dB) c = 0.6 SDR bound when  =0.95 SDR bound when  =0.75 Scheme A when  =0.95 Scheme B by simulation when  =0.95 Scheme B by calculation when  =0.95 Scheme A when  =0.75 Scheme B by simulation when  =0.75 Scheme B by calculation when  =0.75 uncoded transmission when  =0.95 uncoded transmission when  =0.75 (d) 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 c 14 16 18 20 22 24 26 SDR(dB)  = 0.95 Scheme A when CSNR = 33.98 dB Scheme B when CSNR = 33.98 dB Scheme A when CSNR = 26.98 dB Scheme B when CSNR = 26.98 dB Scheme A when CSNR = 21.99 dB Scheme B wehn CSNR = 21.99dB uncoded transmission when CSNR = 33.98 dB uncoded transmission when CSNR = 26.98 dB uncoded transmission when CSNR = 21.99 dB (e) 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 c 8 10 12 14 16 18 20 SDR(dB)  = 0.75 Scheme A when CSNR = 33.98 dB Scheme B when CSNR = 33.98 dB Scheme A when CSNR = 26.98 dB Scheme B when CSNR = 26.98 dB Scheme A when CSNR = 21.99 dB Scheme B wehn CSNR = 21.99dB uncoded transmission when CSNR = 33.98 dB uncoded transmission when CSNR = 26.98 dB uncoded transmission when CSNR = 21.99dB (f) Fig. 2: (a)-(d) Performance co mparisons of relev ant s chemes , (e)-(f) Pe rformance trends of relev a nt scheme s with v arying c . I V . S I M U L A T I O N R E S U LT S W e c ompare our s chemes with unc oded transmission an d uppe r b ounds. Though the scheme s are proposed for general case, here we only present the symme tric interference cas e, tha t is, c 1 = c 2 = c . The performanc e is meas ured by sign al-to-distortion ratio S D R = σ 2 1 D . In our experiments, | k | max , the maximum value of | k | is chosen to be | k | max = ⌈ 6 ∆ − 1 2 ⌉ . | k | denotes the abso lute value of integer quantization index k . Fig. 2 (a), (b) show the c o mparison results when the interference is s trong. Herein, the upper bound is obtained by app lying the lo wer bo u nd resulting from Lemma 3 in [9]. The results while the interference is we ak ca n be found in Fig. 2 (c), (d). The upper b ound he re is 9 deriv ed through Lemma 1 in [8]. The ma rkers ’o’ represe nt the simulation resu lts b y Sc heme B using optimal parameters obtained from minimizing the average end-to-end distortion after (4) is s ubstituted, while the lines without marker sh ow the c alculation resu lts by Sc heme B. The e f fecti veness of (4) is demonstrated as markers ′ o ′ basically stick to correspond ing lines. From a ll these figures , it can be told that both Sche me A and Sc heme B we propose d are supe rior to the u ncoded trans mission when CSNR is larger than a thresho ld value for dif feren t interference values and correlation coefficients. Fixing interference factor c , the superiority of our schemes tow ards uncoded transmission becomes more obvious whe n ρ decrea ses, which is reflected by the fact that the threshold value of CS N R ge ts smaller . Comparing the Fig. 2 (a), (b) with the figures (c), (d), the a dvantage of our sche me s on unco ded transmission is more prominent for the ca ses with strong interference. In the scen arios with wea k interference, the a dvantage d e cays with the de creasing of c . This is rea sonab le as when c → 0 , the interference channe l degrades to point to point channel, and as well known, u ncoded transmission achieves optimal distortion in suc h sc enario. In Fig. 2 (e), (f), we fix the c orrelation ρ and plot S D R as a func tion of c with varying CSNR. Note that the performances of proposed scheme s en hance with the increas ing of c in the prese nce o f strong interference an d w ith the decrea sing of c for weak interference ch a nnel while the performance of uncode d transmission always dete riorates wit h the increasing o f c . A P P E N D I X A D E R I V A T I O N O F M ∆ As illustrated in the Section II, ( S 1 , S 2 ) are correlated gauss ian sources with zero mean and covari ance matrix   1 ρ ρ 1   . Then E [ S i c | S i = s i ] = E [ S i c ] + ρ ( σ S i c σ S i )( s i − E [ S i ]) = ρs i . VAR [ S i c | S i ] = σ 2 S i c (1 − ρ 2 ) = (1 − ρ 2 ) . In other words, S i c | ( S i = s i ) ∼ N ( ρs i , 1 − ρ 2 ) . Then with probab ility p , wh ere p ≈ 1 , S i c falls into the interval [ ρs i − 3 p 1 − ρ 2 , ρs i + 3 p 1 − ρ 2 ] . 2 2 W e hav e tried 4 p 1 − ρ 2 and 5 p 1 − ρ 2 and there were no adv anced performance improv ements. 10 For a midtread qu antizer , k ∆ deno tes reconstruction levels while ( k + 1 2 )∆ denotes decision lev els. In the follo wing, w e a ssume that k < 0 . Fig. 3 illustrates how the max imum distance is calculated when k < 0 . T i = k ∆ supposing that S i falls in the interval [( k − 1 2 )∆ , ( k + 1 2 )∆] . Fig. 3 (a ) shows the maximum distance be tween T i and T i c if S i equals to the right bo undary value ( k + 1 2 )∆ while (b) shows the maximum distance whe n S i equals to the left bound ary value ( k − 1 2 )∆ . Obviously , the max imum distance in (a) is larger tha n the one in (b). As a res ult, we only need to analyze the expression of the max imum distan ce in Fig. 3 (a). As de picted in (a), d max ( k ) = M ∆ =  3 p 1 − ρ 2 + h ρ ( k + 1 2 )∆ − ( k + 1 2 )∆ i ∆  × ∆ =  3 p 1 − ρ 2 − h ( k + 1 2 )∆ − ρ ( k + 1 2 )∆ i ∆  × ∆ (12) The proce dure to fin d d max( k ) for the c ase k > = 0 is similar du e to the symmetry . Howe ver , the ( k − 1 + 1 2 ) ∆ ( k + 1 2 ) ∆ ( k + 1 + 1 2 ) ∆ ρ ( k + 1 2 ) ∆ 3 1 − ρ 2 ! " ( k − 1 + 1 2 ) ∆ ( k + 1 2 ) ∆ ( k + 1 + 1 2 ) ∆ 3 1 − ρ 2 #$% & ' ( )* !" #$%&'# + " , $% - . / )0 ("#$%&'# + " , $% & . / )0 ! " 1 ! " ! " 1 Fig. 3: Illustration of max imum distanc e expression beco mes d max ( k ) =  3 p 1 − ρ 2 + h ( k − 1 2 )∆ − ρ ( k − 1 2 )∆ i ∆  × ∆ . So d max ( k ) is a increa s ing function of abs olute value of k as below d max ( | k | ) =  3 p 1 − ρ 2 + h ( | k |− 1 2 )∆ − ρ ( | k |− 1 2 )∆ i ∆  × ∆ . If | k | goe s to infinity , d max would go to infin ity as well. As matter of fact, the a b solute value of k is limited. W e limite | k | by | k | max which satisfies ( | k | max ∆ + 1 2 ∆) = κ , whe re P r ( S i ∈ [ − κ, κ ]) ≈ 1 , 11 that is, the overload d is tortion approximately equals to 0 . As a result, the maximum distan ce is se t to be M ∆ =  3 p 1 − ρ 2 + h ( | k | max − 1 2 )∆ − ρ ( | k | max − 1 2 )∆ i ∆  × ∆ . as shown in the pape r . In our experiments, we choose | k | max = ⌈ 6 ∆ − 1 2 ⌉ s o tha t κ > 6 . A P P E N D I X B J U S T I FI C A T I O N O F ( 4 ) Substituting ˆ R i = Γ i h Y i − ( α i + c i c β i c ρ ) ˆ T i − c i c ( α i c − β i c ) ˆ T i c i into D i = E [( T i + R i − ˆ T i − ˆ R i ) 2 ] , we h av e D i = E "  T i + R i − ˆ T i − Γ i  Y i − ( α i + c i c β i c ρ ) ˆ T i − c i c ( α i c − β i c ) ˆ T i c   2 # = E "  T i + R i − ˆ T i − Γ i  α i T i + β i R i + c i c α i c T i c + c i c β i c ( ρ ( T i + R i ) + N − T i c ) + W i − ψ   2 # (13) = E "  (1 − Γ i ( α i + c i c β i c ρ ))( T i − ˆ T i ) − Γ i c i c ( α i c − β i c )( T i c − ˆ T i c ) +  1 − Γ i ( β i + c i c β i c ρ )  R i − Γ i c i c β i c N − Γ i W i  2 # = E "  λ +  1 − Γ i ( β i + c i c β i c ρ )  R i − Γ i c i c β i c N − Γ i W i  2 # = E [ λ 2 ] +  1 − Γ i ( β i + c i c β i c ρ )  2 σ 2 R + (Γ i c i c β i c ) 2 σ 2 N + Γ 2 i σ 2 W + 2  1 − Γ i ( β i + c i c β i c ρ )  1 − Γ i ( α i + c i c β i c ρ )  E h R i ( T i − ˆ T i ) i − 2  1 − Γ i ( β i + c i c β i c ρ )  Γ i c i c ( α i c − β i c ) E h R i ( T i c − ˆ T i c ) i , where λ =  1 − Γ i ( α i + c i c β i c ρ )  ( T i − ˆ T i ) − Γ i c i c ( α i c − β i c )( T i c − ˆ T i c ) . In (13), we use ψ to d e note ( α i + c i c β i c ρ ) ˆ T i + c i c ( α i c − β i c ) ˆ T i c due to the space limit. 12 A P P E N D I X C D E R I V A T I O N O F ( 1 0 ) Suppose that ( T i , T i c ) = ( t k , t m ) , ( ˆ T i , ˆ T i c ) = ( t l , t n ) , then by the desc ription o f the midtread quantizer in the first paragraph of Section III, ( T i , T i c ) = ( k ∆ , m ∆) , ( ˆ T i , ˆ T i c ) = ( l ∆ , n ∆) . As a result, α i ( t l − t k ) + c i c α i c ( t n − t m ) , which is distance d between the recovered digital message α i t l + c i c α i c t n and the original digital mess age α i t k + c i c α i c t m , eq uals to α i ( l − k )∆ + c i c α i c ( n − m )∆ . According to the de finition of p seudo ML estimator , ( ˆ t i , ˆ t i c ) = arg min t i,k ,t i c ,k ′ t i c ,k ′ ∈ [ t i,k − M ∆ , t i,k + M ∆] y i − α i t i,k − c i c α i c t i c ,k ′ , (14) t n should b e in the interval [ t l − M ∆ , t l + M ∆] . In other words, n ∈ [ l − M , l + M ] . Assuming that l − k = p, n − m = q and p , q ∈ Z , conseque ntly , m + q ∈ [ k + p − M , k + p + M ] . Let us sort all the possible distances and label them o n the rea l axis, as shown in Fig. 4 3 . The origin d 0 denotes the cas e that d = 0 , i.e., T i , T i c are c orrectly recovered. d i , i > 0 denotes the i-th distance that is larger than 0 wh ile d i , i < 0 d enotes the i-th one that is smaller than 0 . When would the event that S i = s i , ( s i ∈ [ t k − ∆ 2 , t k + ∆ 2 ] ), S i c = s i c , ( s i c ∈ [ t m − ∆ 2 , t m + ∆ 2 ]) , ˆ T i c = t n , ˆ T i = t l happen ? Let us make an example for illustration. If d = d 2 , then d l = d 1 , d u = d 3 . According to (14), the event occu rs when α i t k + c i c α i c t m + β i ( s i − t k ) + c i c β i c ( s i c − t m ) + w i − α i t l − c i c α i c t n is minimum, i.e., β i ( s i − t k ) + c i c β i c ( s i c − t m ) + w i − d 1 is smaller than the d if ference between β i ( s i − t k ) + c i c β i c ( s i c − t m ) + w i and all other d . In other words, β i ( s i − t k ) + c i c β i c ( s i c − t m ) + w i 3 Though the number of possible distances can be infinite, we shrink the size by limit ing d by | d | ≤ 2 × ( 5 σ W + ∆ 2 ( β i + c i β i c )) 13 falls in the interval [ d 1 + d 2 − d 1 2 , d 2 + d 3 − d 2 2 ]. It conclud es that P ˆ T i c , ˆ T i ,T i c ,T i ( t n , t l , t m , t k ) = ∆ 2 Z − ∆ 2 ∆ 2 Z − ∆ 2 P ˆ T i c , ˆ T i ,T i c ,T i ,R i c ,R i ( t n , t l , t m , t k , r i c , r i ) dr i c dr i = t k + ∆ 2 Z t k − ∆ 2 t m + ∆ 2 Z t m − ∆ 2 P ˆ T i c , ˆ T i ,S i c ,S i ( t n , t l , s i c , s i ) ds i c ds i = t k + ∆ 2 Z t k − ∆ 2 t m + ∆ 2 Z t m − ∆ 2 p W i  d 1 + d 2 − d 1 2 ≤ w i + β i ( s i − t k ) + c i c β i c ( s i c − t m ) ≤ d 2 + d 3 − d 2 2  p S i c ,S i ( s i c , s i ) ds i c ds i = t k + ∆ 2 Z t k − ∆ 2 t m + ∆ 2 Z t m − ∆ 2 ub Z lb P W i ( w i ) P S i c ,S i ( s i c , s i ) dw i ds i c ds i , where ub = d 2 + d 3 − d 2 2 −  β i ( s i − t k ) + c i c β i c ( s i c − t m )  , l b = d 1 + d 2 − d 1 2 −  β i ( s i − t k ) + c i c β i c ( s i c − t m )  . The s ame proce dure app lies for the case s d = 0 and d < 0 . Fig. 4 (a), (b), (c) depict the three conditions respec ti vely . As a result, ub =          d + d u − d 2 − µ, if d > 0 d u + d − d u 2 − µ, if d < 0 d u 2 − µ if d = 0 lb =          d l + d − d l 2 − µ, if d > 0 d + d l − d 2 − µ if d < 0 d l 2 − µ if d = 0 The express ions o f ub a nd l b c a n be un ified into on e form instea d of three indivi dual forms for three case s. So we modified the express ion b y ub = d + d u 2 − µ, l b = d + d l 2 − µ 14 ! " # $ ! %& ! ' ! %' ! %( ! ( ! & ! ! ) ! * ! " # $ ! %& ! ' ! %' ! %( ! ( ! & ! ! ) ! * ! " # $ ! %& ! ' ! %' ! %( ! ( ! & ! ! ) ! * !"#$$%&$ ' !( #$ $%$)$ ' !*#$$%$ +$' Fig. 4: Illustration on how to ob ta in the u pper bound a nd lower b ound for three cas es. A P P E N D I X D D E R I V A T I O N S O F ( 7 ) A N D ( 1 1 ) Derivations of (7): E [ R i T i c ] = E h E [ R i T i c | T i ] i = X k P T i ( t k ) E [ R i T i c | T i = t k ] = X k P T i ( t k ) X m ∆ 2 Z − ∆ 2 t m r i P R i ,T i c | T i ( r , t m | t k ) dr i = X k X m ∆ 2 Z − ∆ 2 t m r i P R i ,T i c ,T i ( r i , t m , t k ) dr i = X k X m ∆ 2 Z − ∆ 2 ∆ 2 Z − ∆ 2 t m r i P R i c ,R i ,T i c ,T i ( r i c , r i , t m , t k ) dr i c dr i a = X k k + M X m = k − M t m t k + ∆ 2 Z t k − ∆ 2 t m + ∆ 2 Z t m − ∆ 2 ( s i − t k ) P S i c ,S i ( s i c , s i ) ds i c ds i , where (a) follo ws beca use P R i c ,R i ,T i c ,T i ( r i c , r i , t m , t k ) = P S i c ,S i ,T i c ,T i ( r i c + t m , r i + t k , t m , t k ) = P S i c ,S i ( r i c + t m , r i + t k ) P T i c ,T i | S i c ,S i ( t m , t k | r i c + t m , r i + t k ) = P S i c ,S i ( r i c + t m , r i + t k ) 15 Derivations of (11): ∆ 2 Z − ∆ 2 ∆ 2 Z − ∆ 2 r i P ˆ T i c , ˆ T i ,T c i ,T i ,R i c ,R i ( t n , t l , t m , t k , r i c , r i ) dr i c dr i b = t k + ∆ 2 Z t k − ∆ 2 t m + ∆ 2 Z t m − ∆ 2 ( s i − t k ) P ˆ T i c , ˆ T i ,S i c ,S i ( t n , t l , s i c , s i ) ds i c ds i = t k + ∆ 2 Z t k − ∆ 2 t m + ∆ 2 Z t m − ∆ 2 ub Z lb ( s i − t k ) P W i ( w i ) P S i c ,S i ( s i c , s i ) dw i ds i c ds i , where (b) follows from the same fact with (a). 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