Energy Efficient Estimation of Gaussian Sources Over Inhomogeneous Gaussian MAC Channels

Energy Eien t Estimation of Gaussian Soures Ov er Inhomogeneous Gaussian MA C Channels Sh uangqing W ei, Ra jgopal Kannan, Sitharama Iy engar and Nagesw ara S. Rao Abstrat It has b een sho wn lately the optimalit y of uno ded transmission in estimating Gaussian soures o v er homogeneous/symmetri Gaussian m ultiple aess  hannels (MA C) using m ultiple sensors. It remains, ho w ev er, unlear whether it still holds for an y arbitrary net w orks and/or with high  hannel signal-to-noise ratio (SNR) and high signal-to-measuremen t-noise ratio (SMNR). In this pap er, w e rst pro vide a join t soure and  hannel o ding approa h in estimating Gaussian soures o v er Gaussian MA C  hannels, as w ell as its suien t and neessary ondition in restor- ing Gaussian soures with a presrib ed distortion v alue. An in teresting relationship b et w een our prop osed join t approa h with a more straigh tforw ard separate soure and  hannel o ding s heme is then established. F urther omparison of these t w o s hemes with the uno ded approa h rev eals a la king of a onsisten t ordering of these three strategies in terms of the total transmission p o w er onsumption under a distortion onstrain t for arbitrary in-homogeneous net w orks. W e then form ulate onstrained p o w er minimization problems and transform them to relaxed on v ex geometri programming problems, whose n umerial results exhibit that either separate or un- o ded s heme b eomes dominan t o v er a linear top ology net w ork. In addition, w e pro v e that the optimal deo ding order to minimize the total transmission p o w ers for b oth soure and  hannel o ding parts is solely sub jet to the ranking of MA C  hannel qualities, and has nothing to do with the ranking of measuremen t qualities. Finally , asymptoti results for homogeneous net w orks are obtained whi h not only onrm the existing optimalit y of the uno ded approa h, but also sho w that the asymptoti SNR exp onen ts of these three approa hes are all the same. Moreo v er, the prop osed join t approa h share the same asymptoti ratio with resp et to high SNR and high SMNR as the uno ded s heme. 1 In tro dution Reen t y ears ha v e witnessed a tremendous gro wth of in terests in wireless ad ho  and sensor net w orks from b oth aademia and industry , due to their ease of implemen tation, infrastruture-less nature, 1 S. W ei is with the Departmen t of ECE, Louisiana State Univ ersit y , Baton Rouge 2 R. Kannan and S. Iy engar are with the Departmen t of CS, Louisiana State Univ ersit y , Baton Rouge 3 N. Rao is with the Computer Siene and Mathematis Division, Oak Ridge National Lab oratory 4 This w ork is funded in part from DOE-ORNL (Sensornets Program Sept. 2006- 2008) 1 as w ell as the h uge p oten tials in ivil and military appliations. In man y instanes, sensor no des are deplo y ed to serv e a ommon purp ose su h as surv eillane and monitoring en vironmen ts. One of the ma jor design metris is to maximize the lifetime of a sensor net while meeting the onstrain ts imp osed b y the qualit y of reonstrution, su h as the resulting ultimate distortion measure when data olleted b y sensors are fused to onstrut an estimate of the monitored soure. A ritial fator aeting the lifetime of a sensor net is the amoun t of total p o w er exp enditure that senor no des sp end on transmitting their measuremen ts to a fusion en ter. The p o w er onsumptions are losely related with the w a y sensors olleting and pro essing measuremen ts, as w ell as the omm uniation link qualit y b et w een sensor no des and the fusion en ter. In this pap er, assuming L sensor no des send measuremen ts of a Gaussian soure to a fusion en ter via a one-hop in terferene limited wireless link, w e in v estigate the issue of p o w er allo ations aross sensors with and without lo al ompression and  hannel o ding. A similar system mo del for Gaussian sensor net w orks has also b een adopted reen tly b y [1, 2, 3℄ and [4℄. In [1, 2℄, the authors in v estigated join t soure- hannel o ding paradigm and analyzed ho w distortion sales with resp et to the size of the net w ork. They sho w ed that uno ded transmission a hiev es the lo w er b ound of the mean squared error distortion as the n um b er of sensors gro w to innit y in symmetri net w orks. Ho w ev er, no exat soure and  hannel o ding s hemes are pro vided for general system settings other than the uno ded s heme. In [3 ℄, the exat" optimalit y of uno ded transmission is pro v ed ev en for the homogeneous Gaussian net w orks with nite n um b er of sensor no des. As p oin ted out in [ 3℄, it remains unlear though what approa h is more fa v orable when a system b eomes non-symmetri with a nite n um b er of sensors. The ob jetiv es of this pap er are t w o folds. First, w e will prop ose a join t soure- hannel o ding approa h and then establish its relationship with the separate soure and  hannel o ding strategy . Seond, w e will in v estigate the optimal rate and p o w er allo ation strategy in order to minimize the total transmission p o w er under the onstrain t that the mean squared error v alue in estimating the Gaussian soure remotely is no greater than a presrib ed threshold. In partiular, w e will ompare the resulting total p o w er onsumptions of three distint pro essing s hemes, namely , join t soure and  hannel o ding, separate soure and  hannel o ding and uno ded amplify-and-forw ard approa hes for in-homogeneous net w orks, and demonstrate the w ell kno wn result of the optimalit y of uno ded approa h for estimating Gaussian soures in homogeneous net w orks do es not alw a ys hold in inhomogeneous net w orks. Our on tributions in this pap er an b e summarized as follo ws: • A join t soure and  hannel o ding approa h is prop osed, whose a hiev able rate region is ob- tained. An in teresting relationship b et w een this approa h and separate soure and  hannel o ding approa h is then established. • Optimal deo ding order for b oth join t and separate soure  hannel o ding is found whi h is 2 only a funtion of MA C  hannel ranking order, and has nothing to do with the p o w er lev el of soure measuremen t noise, when w e in tend to minimize the total transmission p o w er. • Relaxed geometri programming problems are form ulated in order to ompare three s hemes. Numerial results demonstrate the uno ded transmission is not alw a ys the b est option. The ordering of the three s hemes is highly dep enden t on relativ e  hannel qualities, measuremen t noise lev els, as w ell as the distortion threshold. • Asymptoti results for large size homogeneous net w orks with nite SNR and SMNR are ob- tained whi h sho w the optimalit y of uno ded transmission from another p ersp etiv e. More imp ortan tly , a ondition is found under whi h the saling fator of reeiv ed  hannel SNR v er- sus signal-to-distortion-noise-ratio (SDNR), as SMNR gro ws to innit y , of join t approa h is equal to that of the uno ded s heme. In addition, w e pro v e the SNR exp onen ts of all three s hemes are the same. The pap er is organized as follo ws. System mo del is set up in Setion 2 . A join t soure and  hannel o ding s heme is prop osed in Setion 3, in whi h w e establish its a hiev able rate region, as w ell as its relationship with the separate soure and  hannel o ding s heme. In order to ompare join t and separate approa hes, the form ulated total p o w er minimization problems are solv ed using geometri programming approa h in Setion 4, where w e also obtain the optimal deo ding order for non-homogeneous net w orks. Uno ded approa h is revisited in Setion 5 from the p ersp etiv e of making omparisons with the former t w o approa hes. In Setion 6, w e ompare the aforemen tioned three s hemes in asymptoti region for homogeneous net w orks. Finally , n umerial results of our omparisons for arbitrary t w o-no de net w orks are presen ted in Setion 7. 2 System Mo del N 1 N L N 2 √ g 1 √ g 2 Y 1 X 2 X K Y 2 Y K X 1 Soure X 0 Sensor 2 Sensor 1 Sensor L Z √ g L ˆ X 0 W F usion Cen ter Figure 1: System mo del 3 Assume L sensor no des observ e a ommon Gaussian soure X 0 [ i ] , i = 1 , · · · , n , where X 0 [ i ] ∼ N (0 , σ 2 S ) are iden tially and indep enden tly distributed Gaussian random v ariables with mean zero and v ariane σ 2 S . The measuremen ts X j [ i ] = X 0 [ i ] + N j [ i ] , j = 1 , · · · , L from L sensors exp eriene in- dep enden t additiv e Gaussian measuremen t noise N j [ i ] ∼ N (0 , σ 2 N j ) , where indep endene is assumed to hold aross b oth spae and time. Let Y j [ i ] denote the transmitted signal from sensor j at time i , whi h satises an a v erage p o w er onstrain t: 1 n n X i =1 | Y j [ i ] | 2 ≤ P j , j = 1 · · · , L. (1) . The pro essed signals { Y j [ i ] } then go through a Gaussian m ultiple aess  hannel and are sup er- p osed at a fusion en ter resulting in Z [ i ] = P L j =1 √ g j Y j [ i ] + W [ i ] , where W [ i ] ∼ N (0 , σ 2 W ) are white Gaussian noise in tro dued at the fusion en ter and assumed indep enden t with N j [ i ] . Co eien ts g j , j = 1 , · · · , L apture the underlying  hannel pathloss and fading from sensors to the fusion en ter. In this pap er, w e assume oheren t fusion is onduted in the sense that g j are assumed p erfetly kno wn b y the fusion en ter. Up on reeiving { Z [ i ] } , the fusion en ter onstruts an estimate { ˆ X 0 [ i ] } of { X 0 [ i ] } su h that the a v erage mean squared error D E ∆ = lim n →∞ 1 n P n i =1 E    X 0 [ i ] − ˆ X 0 [ i ]    2 of the estimation satises D E ≤ D , where D is a presrib ed upp er b ound for estimation error. What in terests us in this pap er is p o w er eien t s hemes to estimate the Gaussian soure remotely with a presrib ed mean squared error. Three approa hes, namely , join t soure and  hannel o ding, separate soure and  hannel o ding, and uno ded amplify-and-forw ard s hemes, will b e in v estigated in the sequel. 3 Join t Soure-Channel Based F usion and Its Relationship with Sep- arate Soure and Channel Co ding In [5℄, a join t soure and  hannel o ding s heme is prop osed to estimate t w o orrelated Gaussian soures remotely at a fusion en ter where measuremen ts from t w o sensors are reeiv ed through a Gaussian MA C  hannel. A  hiev able rate region w as obtained as a funtion of the required distortion tuple in restoring t w o orrelated soures. Inspired b y their w ork, w e, in this setion, will rst dev elop an a hiev able rate region for our prop osed join t soure- hannel o ding (JSCC) approa h for an y arbitrary net w ork with L > 1 sensor no des and then demonstrate an in teresting relationship of JSCC with a separate soure and  hannel o ding s heme (SSCC) whi h is a straigh tforw ard om bination of the reen t ndings on CEO problem [6℄ and traditional MA C  hannel o ding [7℄ with indep enden t soures. 4 3.1 A  hiev able Rate Region for Distributed Join t Soure-Channel Co ding in Estimation of Gaussian Soures Let ˜ R j , j = 1 , · · · , L denote the ompression rate at the j -th sensor. There are total 2 n ˜ R j soure o dew ords U j = { U ( k ) j , k = 1 , · · · , 2 n ˜ R j } from whi h sensor j selets U ( m j ) j = { U ( m j ) j [ i ] , i = 1 , · · · , n } to represen t X j = { X j [ i ] , i = 1 , · · · , n } . The join t approa h w e prop ose here is to let ea h sensor diretly transmit a saled v ersion of a soure o dew ord U ( m j ) j . The saling fator in tro dued herein is to main tain the a v erage transmission p o w er P j b y sensor j , j = 1 , · · · , L . Sine L sensors see the same Gaussian soure with indep enden t measuremen t noise, L quan tization v etors { U ( m j ) j , j = 1 , · · · , L } are orrelated. As a result, the deo ding at fusion en ter needs to tak e in to aoun t of su h orrelation when it p erforms join t deo ding of these L o dew ords. The deo ded soure/ hannel o dew ord ˆ U ( m j ) j are then linearly om bined to obtain an MMSE estimate { ˆ X 0 [ i ] , i = 1 , · · · , n } of the Gaussian soure { X 0 [ i ] , i = 1 , · · · , n } . W e are in terested in deriving the a hiev able region of rate tuples { ˜ R j , j = 1 , · · · , L } su h that 2 n ˜ R j , j = 1 , · · · , L soure/ hannel o dew ords an b e deo ded with asymptoti zero error and the mean squared error D E satises D E ≤ D . Theorem 1. T o make D E ≤ D , ˜ R i satisfy ˜ R i = I ( X i ; U i ) = r i + 1 2 log " 1 + σ 2 S σ 2 N i  1 − 2 − 2 r i  # , i = 1 , · · · , L (2) wher e r i ≥ 0 , i = 1 , · · · , L ar e hosen b ase d on 1 D E = 1 σ 2 S + L X k =1 1 − 2 − 2 r k σ 2 N k ≥ 1 D (3) and I ( X j ; U j ) denotes the mutual information b etwe en X j and a Gaussian r andom variable U j , whih is asso iate d with X j by U j = X j + V j , j = 1 , · · · , L (4) wher e V j , indep endent of X j , ar e indep endent Gaussian r andom variables with me an 0 and varian e σ 2 V i = σ 2 N i / (2 r i − 1) . Pr o of. The pro of is a straigh tforw ard appliation of the te hniques used in pro ving Lemma 10 in [ 6 ℄. F or brevit y , w e only pro vide an outline here. W e quan tize { X j [ i ] } with 2 n ˜ R j Gaussian v etors { ˆ U j [ i ] } su h that the soure sym b ol X j [ i ] an b e onstruted from the quan tized sym b ol through a test  hannel [7℄: X j [ i ] = ˆ U j [ i ] + ˆ V j [ i ] , where ˆ V j [ i ] is a Gaussian random v ariable with mean zero and v ariane 2 − 2 ˜ R j σ 2 X j , whi h is indep enden t of ˆ U j [ i ] ∼ N  0 , (1 − 2 − 2 ˜ R j ) σ 2 X j  . Equiv alen tly , w e an also represen t ˆ U j [ i ] as ˆ U j [ i ] = αX j [ i ] + ˜ V j [ i ] , 5 where α is the linear-MMSE estimate o eien t and ˜ V j [ i ] is the resultan t estimation error. By orthogonal priniple, w e ha v e α = σ 2 ˆ U j σ 2 X j =  1 − 2 − 2 ˜ R j  and ˜ V j [ i ] is a Gaussian v ariable indep enden t of X j [ i ] with mean zero and v ariane 2 − 2 ˜ R j  1 − 2 − 2 ˜ R j  σ 2 X j . Therefore, after normalization, w e obtain U j = 1 α ˆ U j = X j + 1 α ˜ V j = X j + V j (5) where V j ∼ N  0 , σ 2 X j / (2 2 ˜ R j − 1)  . W e in tro due v ariables r j su h that 2 2 ˜ R j − 1 =  2 2 r j − 1  σ 2 X j σ 2 N j (6) whi h pro v es (4 ). W e an also see that r j is atually the onditional m utual information b et w een X j and U j giv en X 0 , i.e. r j = I ( X j ; U j | X 0 ) . Sine I ( X j ; U j ) = H ( U j ) − H ( V j ) , it is then straigh tforw ard to sho w that (2 ) holds. Giv en U j = X j + V j and X j = X 0 + N j , where N j and V j are indep enden t, w e an onstrut the LMMSE estimate of X 0 b y ˆ X 0 = P L j =1 β j U j , where o eien ts β j an b e determined again using Orthogonal Priniple. Based on Equations (95) and (96) in [6 ℄, w e obtain the desired result for the mean squared error in (3). F rom the pro of of Theorem 1, it an b e seen that U i and U j are orrelated due to the orrelation b et w een X i and X j , whose orrelation an b e aptured b y ρ i,j , the o v ariane o eien t b et w een X i and X j , whi h an b e omputed as ρ i,j = E [ X i X j ] p E | X i | 2 E | X j | 2 = σ 2 S q ( σ 2 S + σ 2 N i )( σ 2 S + σ 2 N j ) (7) The o v ariane o eien t ˜ ρ i,j b et w een U i and U j an b e obtained aordingly as: ˜ ρ i,j = ρ i,j q (1 − 2 − 2 ˜ R i )(1 − 2 − 2 ˜ R j ) . (8) After substituting ˜ R i determined in Theorem 1 in to it, w e obtain ˜ ρ i,j = r q i q j (1 + q i )(1 + q j ) , (9) 6 where q i = σ 2 S σ 2 N i (1 − 2 − 2 r i ) . F or an y giv en subset S ⊆ { 1 , 2 , · · · , L } , dene v etors U ( S ) = h U π 1 , · · · , U π | S | i and U ( S c ) = h U π | S | +1 , · · · , U π L i , where π is an arbitrary ordering of the L indexes. The o v ariane matrix of U = [ U ( S ) , U ( S c )] T an th us b e deomp osed as Σ U = E  UU T  = " Σ S Σ S , S c Σ S c , S Σ S c # , (10) where Σ S , Σ S c , Σ S , S c denote the auto- and ross-o v ariane matries of U ( S ) and U ( S c ) . The en tries of Σ U are ( Σ U ) i,j = ˜ ρ i,j p P i P j for i 6 = j and ( Σ U ) i,i = P i , i, j ∈ { 1 , · · · , L } , where ˜ ρ i,j is obtained in (9). After ea h sensor maps the observ ation v etor to U j , an additional saling fator γ j = r P j σ 2 U j is imp osed on U j , where σ 2 U j = σ 2 X j /  1 − 2 − 2 ˜ R j  in order to k eep the a v erage transmission p o w er of Y j [ i ] = γ j U j [ i ] as P j . The reeiv ed signal at the fusion en ter an th us b e written as Z [ i ] = L X j =1 γ j U j [ i ] √ g j + W [ i ] (11) Theorem 2. Given the r e  eive d signal Z [ i ] , i = 1 , · · · , n in (11), the quantization r ate ˜ R i , i = 1 , · · · , L obtaine d in The or em 1 satises the fol lowing ine qualities in or der to r estor e X 0 at the fusion  enter with distortion no less than D : ˜ R ( S ) ≤ | S |− 1 X i =1 I  U π i ; U π | S | π i +1  + I ( U ( S ); U ( S c )) + I ( U ( S ); Z | U ( S c )) , ∀ S ⊆ { 1 , · · · , L } , (12) wher e ˜ R ( S ) = P i ∈ S ˜ R i , U ( S ) = { U i , i ∈ S } , S c is the  omplementary set of S , { π 1 , · · · , π | S | } is an arbitr ary p ermutation of S , and U π | S | π i +1 = h U π i +1 , · · · , U π | S | i . Pr o of. The pro of follo ws the fo otsteps of the one pro ving a hiev abilit y of the apait y region for regular MA C  hannel with indep enden t  hannel o dew ords. The dierene here is that w e need to tak e in to aoun t the orrelations of the  hannel inputs from ea h user when the join t t ypial sequene te hnique is used to ompute the upp er b ound of the probabilit y of v arious error ev en ts. The details are deferred to the App endix A. It an b e easily seen that when inputs to the  hannel are indep enden t, the rst and seond terms in (12 ) v anish and onsequen tly the inequalit y redues to the one  haraterizing the apait y region for MA C  hannels with indep enden t inputs [7, Chap.15.3℄. 7 Next, w e pro v e a sequene of lemmas in order to establish a onnetion b et w een the JSCC and SSCC approa hes. Lemma 1. Given U j = X j + V j , j = 1 , 2 as in (4), U 2 → X 2 → X 1 → U 1 forms a Markov hain. As a r esult, we have I ( U 2 ; X 2 | U 1 ) = I ( U 2 ; X 2 ) − I ( U 1 ; U 2 ) (13) I ( U 1 ; X 1 | U 2 ) = I ( U 1 ; X 1 ) − I ( U 2 ; U 1 ) (14) I ( U 1 , U 2 ; X 1 , X 2 ) = I ( X 1 ; U 1 ) + I ( X 2 ; U 2 ) − I ( U 1 ; U 2 ) (15) Pr o of. See App endix B. Lemma 2. F or U j = X j + V j , j = 1 , · · · , L , the fol lowing r elation of mutual information holds I ( U ( S ); X ( S )) = X i ∈ S I ( U i ; X i ) − | S |− 1 X i =1 I  U π i ; U π | S | π i +1  , ∀ S ⊆ { 1 , · · · , L } . (16) Pr o of. WLOG, onsider S = { 1 , 2 , · · · , s } . Dene ˜ U 2 = [ U 2 , U 3 , · · · , U s ] and ˜ X 2 = [ X 2 , X 3 , · · · , X s ] . Apparen tly , ˜ U 2 → ˜ X 2 → X 1 → U 1 forms a Mark o v  hain. F rom (15 ), w e immediately obtain: I [ U ( S ); X ( S )] = I ( X 1 ; U 1 ) + I ( ˜ U 2 ; ˜ X 2 ) − I ( U 1 ; ˜ U 2 ) (17) Using same idea, it an b e sho wn that I ( ˜ U 2 ; ˜ X 2 ) = I ( U 2 ; X 2 ) + I ( U 3 · · · , U s ; X 3 , · · · , X s ) − I ( U 2 ; U s 3 ) , (18) where I ( U 3 · · · , U s ; X 3 , · · · , X s ) an b e deomp osed in a similar manner. Su h deomp osition an b e onduted iterativ ely un til w e rea h I ( X s − 1 , X s ; U s − 1 , U s ) = I ( X s − 1 ; U s − 1 ) + I ( U s ; X s ) − I ( U s ; U s − 1 ) (19) Com bining all iterations yields the desired result: I ( U ( S ); X ( S )) = s X i =1 I ( U i ; X i ) − s − 1 X i =1 I  U i ; U s i +1  (20) As the whole deriv ation do es not rely on the exat order of { 1 , · · · , s } , w e th us omplete the pro of of Lemma 2. 8 Lemma 3. F or the same U ( S ) and X ( S ) as in L emma 1, we have I ( U ( S ); X ( S )) − I ( U ( S ); U ( S c )) = I [ U ( S ); X ( S ) | U ( S c )] = I [ U ( S ); X 0 | U ( S c )] + X i ∈ S I [ U i ; X i | X 0 ] (21) Pr o of. See app endix Theorem 3. When e ah sensor p erforms indep endent ve tor quantization and subse quently tr ansmits the r esulting s ale d quantization ve tor thr ough a Gaussian MA C hannel, to r e  onstrut the Gaussian sour  e at fusion  enter with distortion no gr e ater than D , the ne  essary and suient  ondition is for any subset S ⊆ { 1 , 2 , · · · , L } , the fol lowing ine quality holds I [ U ( S ); X 0 | U ( S c )] + X i ∈ S I [ U i ; X i | X 0 ] ≤ I [ U ( S ); Z | U ( S c )] (22) wher e LHS = − 1 2 log " D E σ 2 S + X i ∈ S c D E σ 2 N i  1 − 2 − 2 r i  # + X i ∈ S r i (23) and RHS = 1 2 log  1 + 1 σ 2 W √ g ( S ) T Q Σ S √ g ( S )  (24) with √ g ( S ) T =  √ g i , i ∈ S  and Q Σ S = Σ S − Σ S , S c Σ S c − 1 Σ S c , S . The auto- and r oss- ovarian e matri es Σ S , Σ S c , Σ S , S c and Σ S c , S ar e dene d as in (10). Pr o of. T o onstrut an estimate of X 0 at a fusion en ter with distortion no greater than D is equiv alen t to requiring that the minim um ompression rate ˜ R i satises ˜ R i = I ( X i ; U i ) , as required b y lo al v etor quan tization, and that ˜ R i , i = 1 , · · · , L are in the region determined in Theorem 2. Consequen tly , the onditions are translated to X i ∈ S I ( X i ; U i ) ≤ | S |− 1 X i =1 I  U π i ; U π | S | π i +1  + I ( U ( S ); U ( S c )) + I ( U ( S ); Z | U ( S c )) (25) F rom Lemma 2 and Lemma 3, this ondition is equiv alen t to I [ U ( S ); X 0 | U ( S c )] + X i ∈ S I [ U i ; X i | X 0 ] ≤ I [ U ( S ); Z | U ( S c )] . (26) 9 Dene r i = I ( X i ; U i | X 0 ) . Then it is straigh tforw ard to sho w that the LHS of (26) is equal to that in (23 ) b y omputing the mean squared error of estimating X 0 using U ( S ) or [ U ( S ) , U ( S c )] [6℄, whi h is E  | X 0 | 2 | U ( S )  = " 1 σ 2 S + X i ∈ S 1 σ 2 N i  1 − 2 − 2 r i  # − 1 (27) Giv en I [ U ( S ); Z | U ( S c )] = H [ Z | U ( S c )] − H [ Z | U ( S ) , U ( S c )] and U and Z are Gaussian ran- dom v etor/v ariables, it is suien t to get the onditional v ariane of Z giv en the v etor U ( S c ) . This an b e b oiled do wn to nding the onditional v ariane of P i ∈ S √ g i γ i U i giv en U ( S c ) as Z = P L i =1 √ g i γ i U i + W . Based on Theorem 3 in [8℄, w e ha v e Co v [ U ( S ) | U ( S c )] = Σ S − Σ S , S c Σ S c − 1 Σ S c , S (28) Therefore, V ar X i ∈ S √ g i γ i U i ! = √ g ( S ) T Q Σ S √ g ( S ) . (29) The en trop y an th us b e omputed aordingly yielding H [ Z | U ( S c )] = 1 2 log  2 π e  σ 2 W + √ g ( S ) T Q Σ S √ g ( S )  H [ Z | U ( S c ) , U ( S )] = 1 2 log  2 π eσ 2 W  (30) whi h leads to (24), and hene ompletes the pro of. 3.2 Relationship With Separate Soure-Channel Co ding Approa h If w e lo ok losely at (22) and (23 ), w e an easily see that the LHS of the a hiev able rate region for the JSCC approa h atually  haraterizes the rate-distortion region for Gaussian soures with onditionally indep enden t (CI) ondition [6 ℄. Under the CI assumption, distributed soure o ding at sensors inludes t w o steps. The rst step is the same as in JSCC, in whi h an indep enden t v etor quan tization for Gaussian soure at ea h sensor is onduted with resp et to the observ ed signal X j = { X j [ i ] , i = 1 , · · · , n } , whi h generates a v etor U k j = { U k j [ i ] , i = 1 , · · · , n } , k ∈ { 1 , · · · , 2 n ˜ R j } . In the seond step, those indexes of k j are further ompressed using Slepian-W olf 's random binning approa h [6 , 9℄. Consequen tly , there are 2 nR j bins for sensor j , whi h on tain all represen tation v etors U k j of measuremen ts X j . It w as sho wn in [6 ℄ that R j satisfy: P j ∈ S R j ≥ I [ U ( S ); X 0 | U ( S c )] + P i ∈ S I [ U i ; X i | X 0 ] , for all S ⊆ { 1 , 2 , · · · , L } in order to restore X remotely with distortion no greater than D . 10 F or SSCC, to send indexes of bins orretly to the fusion en ter, indep enden t Gaussian o dew ords { Y j [ i ] ∼ N (0 , P j ) , i = 1 , · · · , n } for j = 1 , · · · , L for ea h bin index are generated at L sensors. T o ensure indexes are orretly deo ded at the fusion en ter, the rate tuple { R i , i = 1 , · · · , L } should also b e on tained in the apait y region of Gaussian MA C  hannel with indep enden t  hannel inputs under p o w er onstrain ts { P j , j = 1 , · · · , L } . The region is  haraterized b y P i ∈ S R i ≤ 1 2 log h 1 + P j ∈ S P j g j σ 2 W i , for all S ⊆ { 1 , 2 , · · · , L } . The data pro essing at the fusion en ter onsists of three phases. In the rst phase,  hannel deo ding is p erformed to reo v er the indexes of bins on taining { U k j , j = 1 , · · · , L } . In the seond phase, join t t ypial sequenes { U k j } are obtained from L bins whose indexes are restored. In the last phase, { U k j } are linearly om bined to estimate the soure v etor { X 0 [ i ] } under the minim um mean squared error (MMSE) riterion. Under SSCC, w e an therefore obtain the suien t and neessary ondition for restoring X 0 with MSE no greater than D : I [ U ( S ); X 0 | U ( S c )] + X i ∈ S I [ U i ; X i | X 0 ] ≤ 1 2 log   1 + X j ∈ S P j g j σ 2 W   , ∀ S ⊆ { 1 , 2 , · · · , L } . (31) In general, w e annot sa y whi h approa h, JSCC or SSCC, is b etter in terms of the size of rate region. This an b e seen more learly when w e lo ok at a partiular ase for L = 2 . When there are only t w o sensors, to reonstrut { X 0 [ i ] } with a distortion no greater than D using JSCC or SSCC prop osed as ab o v e, the transmission p o w ers P 1 and P 2 , as w ell as r 1 and r 2 satisfy: r 1 − 1 2 log ( D E σ 2 S + D E σ 2 N 2  1 − 2 − 2 r 2  ) ≤ 1 2 log 1 + P 1 g 1 (1 − ˜ ρ 2 1 , 2 ) σ 2 W ! (32) r 2 − 1 2 log ( D E σ 2 S + D E σ 2 N 1  1 − 2 − 2 r 1  ) ≤ 1 2 log 1 + P 2 g 2 (1 − ˜ ρ 2 1 , 2 ) σ 2 W ! (33) r 1 + r 2 + 1 2 log  σ 2 S D E  ≤ 1 2 log  1 + P 2 g 2 + P 1 g 1 + 2 ˜ ρ 1 , 2 √ P 1 g 1 P 2 g 2 σ 2 W  (34) 11 where ˜ ρ 1 , 2 denotes the o v ariane o eien t b et w een U i and U j , whi h is zero for SSCC and ˜ ρ 1 , 2 = r q 1 q 2 (1 + q 1 )(1 + q 2 ) , (35) for JSCC, as obtained in (9) with r i satisfying 1 /D E = 1 σ 2 S + 2 X k =1 1 − 2 − 2 r k σ 2 N k ≥ 1 D . (36) It an b e easily seen from (32 )-(34 ) that inequalities of (32) and (33 ) under JSCC are domi- nated b y those under SSCC, i.e. { P j , r j } satisfying (32) and (33 ) under JSCC also satises the orresp onding inequalities under SSCC, while the inequalit y ( 34 ) under JSCC dominates that under SSCC. T o ompare SSCC and JSCC, w e next form ulate a onstrained optimization problem in whi h the ob jetiv e is to minimize the total transmission p o w er of L sensors with a onstrain t that the distortion in restoring X is no greater than D . F or L = 2 , the problem an b e stated as min P i ,r i ,i =1 , 2 P 1 + P 2 , sub jet to (32 )-(34 ) and (36). (37) whi h b eomes p o w er/rate allo ations for SSCC and JSCC, resp etiv ely , for dieren t orrelation o eien ts ˜ ρ . The optimization results for SSCC and JSCC under dieren t  hannel and measuremen t parameters will rev eal to us the relativ e eieny of SSCC and JSCC, whi h will b e further ompared with that for an uno ded s heme, as in v estigated in the next few setions. 4 Optimal P o w er and Rate Allo ations to Minimize the T otal T rans- mission P o w er 4.1 Geometri Programming Solution to P o w er/Rate Allo ations The onstrained optimization problems in (37) are non-on v ex. They an, ho w ev er, b e solv ed e- ien tly using standard te hniques in on v ex optimization b y transforming the original problems in to relaxed on v ex geometri programming problems [10℄. In this setion, w e tak e SSCC as an example to demonstrate ho w it w orks. F or SSCC with ˜ ρ = 0 , the rate tuple ( R 1 , R 2 ) should b e tak en from the b oundary of the apait y region for t w o-user 12 Gaussian MA C  hannels to minimize P 1 + P 2 . Consequen tly , R 1 = α 2 log  σ 2 W + P 1 g 1 σ 2 W  + 1 − α 2 log  1 + P 1 g 1 g 2 P 2 + σ 2 W  R 2 = α 2 log  1 + P 2 g 2 g 1 P 1 + σ 2 W  + 1 − α 2 log  σ 2 W + P 2 g 2 σ 2 W  (38) where α ∈ [0 , 1] is a time sharing fator. Dene y j = 2 2 R j , z j = 2 2 r j for j = 1 , 2 . W e an transform this total p o w er minimization problem for SSCC with L = 2 to an equiv alen t generalized Signomial Programming problem [ 11 ℄: min P j ,y j ,z j ,j =1 , 2 P 1 + P 2 , sub jet to: (39)  g 2 P 2 + σ 2 W  (1 − α ) y 1 ≤  1 + g 1 P 1 σ 2 W  α  g 1 P 1 + g 2 P 2 + σ 2 W  (1 − α ) (40)  g 1 P 1 + σ 2 W  α y 2 ≤  1 + g 2 P 2 σ 2 W  (1 − α )  g 1 P 1 + g 2 P 2 + σ 2 W  α (41) σ 2 s σ 2 N 2 + σ 2 s z − 1 2 + D − 1 σ 2 s σ 2 N 2 σ 2 N 2 + σ 2 s y − 1 1 z 1 ≤ 1 (42) σ 2 s σ 2 N 1 + σ 2 s z − 1 1 + D − 1 σ 2 s σ 2 N 1 σ 2 N 1 + σ 2 s y − 1 2 z 2 ≤ 1 (43) y − 1 1 y − 1 2 z 1 z 2 σ 2 s /D ≤ 1 (44) D − 1 + σ − 2 N 1 z − 1 1 + σ − 2 N 2 z − 1 2 ≤ σ − 2 s + σ − 2 N 1 + σ − 2 N 2 (45) where onstrain ts (40 ) and (41 ) are obtained b y relaxing equalit y onstrain ts in (38 ), and onstrain ts (42 )-(45) result from the transformation of (32 )-(34 ), whi h are in the form of f ( x ) ≤ 1 , where f ( x ) is a p osynomial funtion of n v ariables [ 10 ℄: f ( x ) = P K k =1 c k x a 1 k 1 x a 2 k 2 · · · x a nk n , where c k ≥ 0 and x j > 0 for j = 1 , · · · , n and a ij ∈ R . In addition, onstrain ts (40 ) and (41) are in the form of generalized signomial funtions [11 , 12℄ with frational p o w ers. Single ondensation te hnique [11, 12℄ an then b e applied to on v ert this Signomial programming problem to a standard geometri programming (GP) problem. In this metho d, w e replae (1 + P 1 g 1 /σ 2 W ) in the RHS of (40 ) b y its geometri mean β β 11 11  g 1 P 1 σ 2 W β 12  β 12 , and similarly (1 + P 2 g 2 /σ 2 W ) in the RHS of (41) b y β β 21 21  g 2 P 2 σ 2 W β 22  β 22 , where β i,j ≥ 0 and β i, 1 + β i, 2 = 1 for i = 1 , 2 . In addition, w e also replae ( g 1 P 1 + g 2 P 2 + σ 2 W ) b y its geometri mean:  σ 2 γ 1  γ 1  g 1 P 1 γ 2  γ 2  g 2 P 2 γ 3  γ 3 , where γ i ≥ 0 and P 3 i =1 γ i = 1 . Finally , to handle frational p o w ers in the LHS of (40 ) and (41 ), w e in tro due t w o auxiliary v ariables t 1 and t 2 to replae g 1 P 1 + σ 2 W and g 2 P 2 + σ 2 W , resp etiv ely , in the LHS of 13 (40 ) and (41). A ordingly , t w o additional p osynomials are in tro dued on the list of onstrain ts: g i P i + σ 2 W ≤ t i for i = 1 , 2 . The resulting standard geometri programming problem an th us b e solv ed in an iterativ e manner b y rep eatedly up dating normalization o eien ts β ij and γ i , and applying in terior p oin t metho d for a giv en v etor of these o eien ts [11, 12 ℄. Using the similar metho d, w e an also transform the optimization problem for the JSCC approa h to a on v ex Geometri programming problem. The details are skipp ed here [ 13℄. 4.2 Optimal Soure/Channel Deo ding Order for Non-Symmetri Channels Although the optimization problems form ulated in (37) an only b e solv ed algorithmially , w e an still manage to obtain some insigh ts b y srutinizing the problem strutures. In this setion, w e will rev eal some relationships b et w een the optimal deo ding order and  hannel atten uation fators for non-symmetri net w orks. 4.2.1 Separate Soure and Channel Co ding W e rst sho w the optimal soure eno ding/deo ding order, as w ell as  hannel deo der order for SSCC is uniquely determined b y the ordering of  hannel atten uation fators { g i , i = 1 , · · · , L } , and has nothing to do with the ranking of sensor measuremen t noise p o w er { σ 2 N i , i = 1 , · · · , L } . Theorem 4. F or SSCC, let π ∗ denote any p ermutation of { 1 , · · · , L } suh that g π ∗ (1) ≤ g π ∗ (2) ≤ · · · ≤ g π ∗ ( L ) . T o minimize the total tr ansmission p ower, the optimal de  o ding or der for hannel  o des at r e  eiver is in the r everse d or der of π ∗ , i.e. interfer en e  an el lation is in the or der π ∗ ( L ) , π ∗ ( L − 1) , · · · , π ∗ (1) , whih is also the de  o ding or der of distribute d sour  e  o dewor ds. Pr o of. The pro of onsists of t w o steps. First, w e will determine the  hannel deo ding order for a giv en v etor of soure eno ding rates { R i , i = 1 , · · · , L } . F or SSCC, the rate tuple { R i } satises P i ∈ S R i ≤ 1 2 log  1 + P i ∈ S P i g i /σ 2 W  , whi h is equiv alen t to X i ∈ S X i ≥ f ( S ) ∆ = Y i ∈ S 2 2 R i − 1 , ∀ S ⊆ { 1 , · · · , L } . (46) where X i = P i g i /σ 2 W and f : 2 E → R + is a set funtion with E ∆ = { 1 , · · · , L } . Giv en { R i } , the optimization problem then b eomes min P L i =1 σ 2 W X i /g i , sub jet to (46). Based on the Corollary 3.13 in [14 ℄, the set of p o w er v etors { X i } satisfying (46) is a on tra- p olymatroid G ( f ) , as f satises (1) f ( φ ) = 0 (2) f ( S ) ≤ f ( T ) if S ⊂ T (3) f ( S ) + f ( T ) ≤ f ( S ∪ T )+ f ( S ∩ T ) . F rom Lemma 3.3 in [ 14℄, the minimizing v etor { X i , i ∈ E } for min P L i =1 σ 2 W X i /g i is a v ertex p oin t { X π ∗ ( i ) } of G ( f ) , where π ∗ is a p erm utation on the set E su h that 1 /g π ∗ (1) ≥ · · · ≥ 14 1 /g π ∗ ( L ) and X π ∗ ( i ) = f ( { π ∗ (1) , · · · , π ∗ ( i ) } ) − f ( { π ∗ (1) , · · · , π ∗ ( i − 1) } ) = i Y j =1 2 2 R π ∗ ( j ) − i − 1 Y j =1 2 2 R π ∗ ( j ) (47) whi h th us pro v es the rst part of this Theorem. W e next use (47 ) to transform the ob jetiv e funtion to L X i =1 σ 2 W X i /g i = L − 1 X i =1  1 g π ∗ ( i ) − 1 g π ∗ ( i +1)  i Y j =1 2 2 ∗ R π ∗ ( j ) (48) F or SSCC, rate tuples { R i , i = 1 , · · · , L } satisfy P i ∈ S R i ≥ I ( X ( S ); U ( S ) | U ( S c )) . It is therefore quite straigh tforw ard to sho w that in order to minimize the total transmission p o w er in ( 48), w e need to ha v e P i j =1 R π ∗ ( j ) a hiev e the lo w er b ound, i.e. i X j =1 R π ∗ ( j ) = I  X π ∗ ( i ) π ∗ (1) ; U π ∗ ( i ) π ∗ (1) | U π ∗ ( L ) π ∗ ( i +1)  (49) whi h implies that the deo ding order for soure o dew ords is π ∗ ( L ) , · · · , π ∗ (1) , indep enden t of the ordering of v arianes { σ N i , i = 1 , · · · , L } of measuremen t noise. Theorem 4 implies that sensor π ∗ ( L ) do es not ondut random binning and its quan tization v etor { U π ∗ ( L ) [ n ] } is restored rst. { U π ∗ ( L ) [ n ] } is then used as side information to restore sensor π ∗ ( L − 1) 's soure o dew ord { U π ∗ ( L − 1) [ n ] } from this no de's rst stage Gaussian soure v etor quan tization, whi h resides in the bin whose index is deo ded from the  hannel deo ding step. This pro ess on tin ues un til sensor π ∗ (1) 's rst stage quan tization v etor { U π ∗ (1) [ n ] } is restored b y using all other sensors' quan tization as side information. In the next setion, w e will see a similar onlusion an b e rea hed for JSCC. 4.2.2 Join t Soure and Channel Co ding Unlik e in the SSCC ase where w e ha v e a nie geometri (on tra-p olymatroid) struture whi h enables us to rea h a onlusion v alid for an y arbitrary asymmetri net w orks, JSCC in general la ks su h a feature for us to exploit. W e will instead , in this setion, fo us on a ase with only L = 2 sensor no des and establish a similar result as in Setion 4.2.1 for optimal  hannel deo ding orders. WLOG, w e assume g 1 > g 2 in the subsequen t analysis. 15 Theorem 5. Given a p air of quantization r ates ˜ R 1 and ˜ R 2 , when g 1 > g 2 , the optimal de  o ding or der to minimize the total tr ansmission p ower P 1 + P 2 is to de  o der no de 1 's signal rst, and then no de 2 's information after r emoving the de  o de d no de 1 's signal fr om the r e  eive d signal, i.e. ˜ R 1 = I ( U 1 ; Z ) = 1 2 log 1 +  √ P 1 g 1 + ˜ ρ √ P 2 g 2  2 σ 2 W + P 2 g 2 (1 − ˜ ρ 2 ) ! ˜ R 2 = I ( U 2 ; Z, U 1 ) = 1 2 log  P 1 (1 − ˜ ρ 2 ) + σ 2 W σ 2 W (1 − ˜ ρ 2 )  (50) wher e ˜ ρ is the same as in 35. Pr o of. See app endix D. 5 Uno ded Sensor T ransmission in F usion F or Gaussian sensor net w orks as mo deled in Setion 2, it has b een sho wn reen tly [1 , 2 ℄ that uno ded transmission, i.e. ea h sensor only forw ards a saled v ersion of its measuremen ts to the fusion en ter, asymptotially a hiev es the lo w er b ound on distortion when the n um b er of sensors gro w to innit y and system is symmetri. In the on text of the theme of this pap er, w e, in this setion, in v estigate the optimal p o w er allo ation strategy when a nite n um b er of sensors deplo y the uno ded s heme under more general  hannel onditions. F or uno ded transmission, the transmitted signal b y no de j is Y j [ i ] = α j X j [ i ] , where α j = r P j σ 2 S + σ 2 N j is a saling fator to mak e the transmission p o w er E | Y j [ i ] | 2 = P j . The reeiv ed signal at the fusion en ter is therefore Z [ i ] = P L j =1 Y j [ i ] √ g j + W [ i ] . The linear MMSE estimate of X 0 [ i ] is: ˆ X 0 [ i ] = γ Z [ i ] , where the o eien t γ an b e obtained using Orthogonal priniple: E h X 0 [ i ] − ˆ X 0 [ i ]  Z [ i ] i = 0 . The resultan t MSE is E    X 0 [ i ] − ˆ X 0 [ i ]    2 = σ 2 S P L j =1 P j g j σ 2 N j σ 2 S + σ 2 N j + σ 2 W P L j =1 P j g j + B + σ 2 W (51) where B = P L i =1 P L i 6 = j,j =1 ρ i,j √ P i g i p P j g j and the o v ariane o eien ts ρ i,j is the same as in (7). When L = 2 , the p o w er on trol problem under a distortion onstrain t E    X 0 [ i ] − ˆ X 0 [ i ]    2 ≤ D for 16 the uno ded s heme an b e form ulated as: min P 1 + P 2 sub jet to: 1 2 log " σ 2 S D 1 + σ 2 N 1 σ 2 S + σ 2 N 1 P 1 g 1 σ 2 W + σ 2 N 2 σ 2 S + σ 2 N 2 P 2 g 2 σ 2 W !# ≤ 1 2 log  1 + P 1 g 1 + P 2 g 2 + 2 ρ 1 , 2 √ P 1 g 1 P 2 g 2 σ 2 W  (52) This problem an again b e transformed to a GP problem using the ondensation te hnique applied in Setion 4.1. W e skip the details here. What deserv es our atten tion is that when w e ompare the onstrain t in (52 ) with (34), there is a striking similarit y when w e substitute r i with ˜ R i , whose relationship w as in tro dued in ( 6 ). After substitution, (34 ) b eomes 1 2 log  σ 2 S D [1 + A 1 ] [1 + A 2 ]  ≤ 1 2 log  1 + P 1 g 1 + P 2 g 2 + 2 ˜ ρ 1 , 2 √ P 1 g 1 P 2 g 2 σ 2 W  (53) where A i = σ 2 N i σ 2 S + σ 2 N i (2 2 ˜ R i − 1) for i = 1 , 2 , and ˜ ρ 1 , 2 and ρ 1 , 2 are asso iated as in (8 ). F or JSCC, w e ha v e ˜ R i ≤ I ( U i ; Y | U j ) + I ( U i ; U j ) , whi h is equiv alen t to 2 ˜ R i − 1 1 − ˜ ρ 2 1 , 2 ≤ P i g i σ 2 W (54) W e an infer from (32)-(34 ), as w ell as (52) that it is in general hard to argue whi h approa h is the most energy eien t in terms of the total p o w er onsumption under a ommon distortion onstrain t D , whi h will b e further exemplied in our sim ulation results in Setion 7. There, w e will see the most energy eien t approa h dep ends on exat v alues of σ 2 N j , as w ell as g i and D . Ho w ev er, when a system b eomes homogeneous and symmetri in the sense that σ 2 N i = σ 2 N j and g i = g j for all i, j ∈ { 1 , · · · , L } , w e ha v e onsisten t results for b oth nite n um b er of L and asymptotially large L , as rev ealed in the next setion, when w e ompare these three approa hes. 6 Energy Consumption Comparison for Homogeneous Net w orks: Finite and Asymptoti Results In this setion, w e pro vide analytial results on omparisons b et w een dieren t transmission strategies prop osed th us far, inluding JSCC, SSCC and uno ded s hemes in terms of their total transmission p o w er onsumptions when system b eomes homogeneous. 17 6.1 Comparison under nite L and nite SNR Theorem 6. When a system of L < ∞ sensors b e  omes symmetri, the total p ower  onsumption for the sep ar ate, joint and un o de d shemes pr op ose d pr eviously fol low:  P g σ 2 W  LoB <  P g σ 2 W  A <  P g σ 2 W  J <  P g σ 2 W  S (55) wher e  P g σ 2 W  LoB is the lower-b ound on tr ansmission p ower, and g is the  ommon hannel gain fr om e ah sensor to the fusion  enter. Indexes A, J and S r epr esent the un o de d, joint and sep ar ate en o ding shemes, r esp e tively. Pr o of. The pro of hinges up on the analysis of rate and p o w er allo ations for symmetri net w orks for dieren t s hemes. Separate Co ding: W e rst lo ok at the separate soure and  hannel o ding approa h. In symmetri net w orks, for soure o ding part, ea h no de emplo ys iden tial ompression rate R , whi h satises LR = Lr + 1 2 log σ 2 S D , where r is the solution to 1 σ 2 S + L σ 2 N 1 (1 − 2 − 2 r ) = 1 D . (56) As a result [6℄, LR = − L 2 log 1 − σ 2 N 1 L  1 D − 1 σ 2 S  ! + 1 2 log σ 2 S D (57) F or  hannel o ding part, to minimize the total transmission p o w er, it is optimal to let ea h sensor transmit at the same p o w er P and same rate R whi h an b e a hiev ed b y join tly deo ding all no de's information at the fusion en ter. Therefore, the total ompression rate also satises LR = 1 2 log  1 + LP g σ 2 W  (58) Com bining (57 ) and (58 ) yields  P g σ 2 W  S = − 1 L + σ 2 S LD  2 2 r  L = − 1 L + σ 2 S LD " 1 −  1 D − 1 σ 2 S  σ 2 N 1 L # − L (59) Join t Co ding: F or the join t soure- hannel o ding s heme, the v etor quan tization rate for ea h sensor is equal to ˜ R = I ( X 1 ; U 1 ) , whi h is asso iated with r as sho wn b y (2 ), where r an b e further obtained using 18 (56 ). By applying the te hniques used in driving the onstrain ts in (32 )-(34) to the L > 2 ase when  hannel is symmetri, the quan tization rate and the resultan t L-user m ultiple aess  hannel with L orrelated inputs { U 1 , · · · , U L } are asso iated b y Lr + 1 2 log σ 2 S D = I ( U 1 , · · · , U L ; Z ) = H ( Z ) − H ( Z | U 1 , · · · , U L ) , (60) where the onditional en trop y H ( Z | U 1 , · · · , U L ) = 1 2 log  2 π eσ 2 W  , and the en trop y of Z = P L j =1 √ g γ U j + W is H ( Z ) = 1 2 log h 2 π e  σ 2 W + g V ar ( γ P j U j ) i . Giv en the o v ariane o eien t b et w een U i and U j : ˜ ρ = σ 2 S σ 2 S + σ 2 N 1 (1 − 2 − 2 ˜ R ) , w e ha v e E | γ L X j =1 U j | 2 = LP + ( L 2 − L ) ˜ ρP (61) F rom (56 ) and (2), w e obtain ˜ ρ =  σ 2 S D − 1   L + σ 2 S D − 1  − 1 . (62) The transmission p o w er P for join t o ded s heme in symmetri net w orks an th us b e omputed using (60) and (61 ):  P g σ 2 W  J =  − 1 L + σ 2 S LD  2 2 r  L   1 +  L − 1 L  ˜ ρ  − 1 (63) Comparing (63 ) with (59), it is apparen t that  P g σ 2 W  J <  P g σ 2 W  S . Uno ded S heme When sensor transmits saled measuremen ts in a symmetri net w ork, w e an obtain the minim um transmission p o w er P b y making the mean squared error obtained in (51) equal to D and substituting P j , g j and σ 2 N j b y P , g and σ 2 N 1 , resp etiv ely . As a result, w e obtain  P g σ 2 W  A = " L 2 σ 2 S σ 2 S + σ 2 N 1 − L  σ 2 S D − 1  σ 2 N 1 σ 2 S + σ 2 N 1 # − 1  σ 2 S D − 1  (64) T o ompare  P g σ 2 W  A with  P g σ 2 W  J , w e need to in tro due an auxiliary v ariable. Dene ˜ Q L = 1 − σ 2 S + σ 2 N 1 σ 2 S ˜ ρ < 1 − ˜ ρ. (65) 19 W e an therefore re-deriv e the minim um p o w er for uno ded s heme:  P g σ 2 W  A = 1 L  1 ˜ Q L − 1  . (66) In addition, w e an express 2 2 r =  1 −  1 D − 1 σ 2 S  σ 2 N 1 L  as 2 2 r = " ˜ Q L L  L + σ 2 S D − 1  # − 1 = 1 − ˜ ρ ˜ Q L > 1 (67) whi h is used to transform  P g σ 2 W  J as  P g σ 2 W  J = 1 L (  1 − ˜ ρ ˜ Q L  L − 1 1 ˜ Q L − L − 1 + σ 2 S /D Lσ 2 S /D ) (68) where the seond term L − 1+ σ 2 S /D Lσ 2 S /D < 1 due to L > 1 and σ 2 S > D . Sine 1 − ˜ ρ > ˜ Q L , after omparing (68 ) and (66), it is straigh tforw ard to sho w  P g σ 2 W  J >  P g σ 2 W  A . Lo w er Bound of T ransmission P o w er Sine X 0 → { X 1 , · · · , X L } → Z → ˆ X 0 forms a Mark o v  hain, b y Data Pro essing Inequalit y [7 ℄, w e ha v e I ( X 0 ; ˆ X 0 ) ≤ I ( X 1 , · · · , X L ; Z ) . On one hand, to ensure E | X 0 − ˆ X 0 | 2 ≤ D , it an b e sho wn I ( X 0 ; ˆ X 0 ) ≥ 1 2 log σ 2 S D using rate distortion results [ 7℄. On the other hand, the m utual information I ( X 1 , · · · , X L ; Z ) is upp er-b ounded b y the m utual information of an additiv e noise Gaussian  hannel with  hannel gain √ g and total transmission p o w er upp er-b ounded b y E | P L j =1 X j | 2 = LP + ( L 2 − L ) ρP , where ρ = σ 2 S σ 2 S + σ 2 N 1 is the o v ariane o eien t b et w een X i and X j for i 6 = j . Consequen tly , I ( X 1 , · · · , X L ; Z ) ≤ 1 2 log  1 + P g σ 2 W  L + ( L 2 − L ) ρ   (69) F rom 1 2 log σ 2 S D ≤ I ( X 0 ; ˆ X 0 ) ≤ I ( X 1 , · · · , X L ; Z ) , w e obtain the lo w er-b ound of transmission p o w er:  P g σ 2 W  LoB = σ 2 S D − 1 L + ( L 2 − L ) ρ (70) Compare this lo w er b ound with ( 64), w e ha v e  P g σ 2 W  LoB <  P g σ 2 W  A . Therefore, w e ha v e sho wn that the order in ( 55 ) holds and th us ompleted the pro of. 20 6.2 Comparison under L → ∞ and nite SNR In this setion, w e pro vide the saling b eha viors of the total transmission p o w er of v arious s hemes studied so far. In partiular, w e are in terested in ho w P g σ 2 W sales with resp et to the n um b er of sensors L in a symmetri system under a ommon onstrain t on distortion no greater than D . The analysis is quite straigh tforw ard based on the results for a nite L n um b er of sensors that w e ha v e obtained in (59), (68 ), (66) and (70 ), for separate, join t, uno ded s hemes and the lo w er b ound, resp etiv ely . Theorem 7. lim L →∞  Lg P σ 2 W  S = σ 2 S D exp  σ 2 N 1  1 D − 1 σ 2 S  − 1 lim L →∞  Lg P σ 2 W  J = exp  σ 2 N 1  1 D − 1 σ 2 S  − D σ 2 S lim L →∞  L 2 g P σ 2 W  A = lim L →∞  L 2 g P σ 2 W  LoB =  1 D − 1 σ 2 S   σ 2 S + σ 2 N 1  (71) Pr o of. The pro of of these on v ergene results is quite straigh tforw ard based up on the results for nite L as ab o v e, and is skipp ed here. It is ob vious that the transmission p o w er of the uno ded s heme shares the same saling fator as the lo w er-b ound, whi h is in the order of 1 /L 2 . The asymptoti optimalit y of the uno ded s heme in symmetri Gaussian sensor net w orks is not a new result, whi h has b een attained previously in [1℄ [15℄. In [1, 15℄, the authors assumed a xed transmission p o w er and sho w ed that the distortion a hiev ed using uno ded approa h has the same asymptoti saling la w as that obtained via a lo w er b ound. Here, w e pro vide a dieren t p ersp etiv e in assessing its optimalit y for the uno ded s heme in terms of the total transmission p o w er while meeting a xed distortion onstrain t. Both join t and separate o ding s hemes ha v e the saling fator in the order of 1 /L . Asymptoti- ally , join t o ding s heme sa v es in total transmission p o w er b y a fator of σ 2 S /D as ompared with the separated approa h. 6.3 Comparison under nite L and SNR → ∞ Giv en the n um b er of sensors L < ∞ , w e are in terested in the saling fators asso iated with trans- mission SNR P /σ 2 W , mean squared error D and measuremen t noise v ariane σ 2 N in homogeneous net w orks. In partiular, w e need to in v estigate ho w the follo wing asymptoti fators are related, σ 2 N → 0 , D → 0 and P /σ 2 W → ∞ . 21 W e rst need to iden tify the limit imp osed up on the saling fator related with σ 2 N and D . As an b e seen from b oth (56) and (64 ) under SSCC, JSCC and uno ded approa hes, it is required that λ  σ 2 N  ∆ = σ 2 N L  1 D − 1 σ 2 S  ≤ 1 . (72) Denote γ ∗ = lim σ 2 N → 0 ln D ln σ 2 N . It an b e seen that γ ∗ has to satisfy γ ∗ ∈ [0 , 1] in order to ha v e the inequalit y in (72) hold. Consequen tly , λ ∗ = lim σ 2 N → 0 λ  σ 2 N  = ( 0 , 0 ≤ γ ∗ < 1 1 L γ ∗ = 1 (73) Theorem 8. The asymptoti r atios asso iate d with D , σ 2 N and P /σ 2 W have the fol lowing r elationship: lim σ 2 N → 0  P g/σ 2 W  S σ 2 S /D = 1 L (1 − λ ∗ ) L (74) lim σ 2 N → 0  P g/σ 2 W  J σ 2 S /D = 1 L 2 (1 − λ ∗ ) L (75) lim σ 2 N → 0  P g/σ 2 W  A σ 2 S /D = 1 L 2 (1 − λ ∗ ) (76) Pr o of. The pro of follo ws straigh tforw ardly with (59), (63) and (64 ) for SSCC, JSCC and uno ded s hemes, resp etiv ely , as w e let σ 2 N → 0 and σ 2 N D ≈ Lλ ∗ . W e an see from (74 )-(76 ) that when γ ∗ ∈ [0 , 1) , i.e. λ ∗ = 0 , the JSCC and uno ded s hemes ha v e the same a ysmptoti ratio b et w een the reeiv ed SNR and S-MSE-Ratio, whi h is smaller than that for the SSCC approa h b y a fator of 1 /L . If γ ∗ = 1 , i.e. λ ∗ = 1 /L , uno ded approa h has the smallest ratio among all s hemes. Ho w ev er, if w e in tro due the SNR exp onen t as dened in [ 16℄, η ∆ = − lim σ 2 N → 0 ln D ln  P g/σ 2 W  , (77) All three approa hes share the same ratio η = 1 for nite L . Theorem 8 therefore pro vides us another p ersp etiv e to ompare these remote estimation ap- proa hes. It demonstrates the prop osed join t soure and  hannel o ding s heme has p oten tially the same asymptoti p erformane as the uno ded one in high  hannel SNR and high measuremen t SNR regions for all ratio exp onen ts γ ∗ ∈ [0 , 1) . In addition, sp eaking of SNR exp onen t of distortion mea- sure in the high SNR region, all three s hemes in v estigated in this pap er share the same asymptoti ratio η = 1 . An additional remark w e next mak e is ab out the limitation as to adopting SNR exp onen t η as a metri to  haraterize the asymptoti p erformane in large SNR regions [ 16 ℄. It an b e seen 22 learly from the ab o v e analysis that the SNR exp onen t obsures the asymptoti dierene b et w een SSCC and JSCC, as w ell as the uno ded approa hes, whi h ha v e dieren t linear ratios as sho wn in Theorem 8. These dierenes are gone one log-sale is imp osed, ho w ev er. Note also that w e ha v e impliitly assumed that the sp etral eieny of the system mo del in this pap er, whi h is the ratio the soure bandwidth o v er  hannel bandwidth [ 16℄, is one. 7 Numerial Results d 0 d 1 d 2 F usion Cen ter Soure d L Figure 2: 1-D lo ation Mo del In this setion, the three approa hes prop osed in this pap er are examined and ompared with ea h other b y lo oking at ea h of their optimal total transmission p o w ers under the onstrain t of restoring X 0 with MSE no greater than D , some presrib ed threshold. P artiularly , w e onsider a linear net w ork top ology where a soure, fusion en ter and L = 2 sensor no des are lo ated on a same line as illustrated in Figure 2 . T o asso iate p ositions of sensor no des with  hannel gains and measuremen t noise, w e assume a path-loss mo del with o eien t β s and β c for g i and σ 2 N i , resp etiv ely: g i ∝ 1 / ( d 0 − d i ) β c and σ 2 N i ∝ d β s j , for i = 1 , · · · , L , where d 0 is the distane b et w een soure and fusion en ter, and d i is the distane b et w een the i -th sensor and soure. Giv en a distortion upp er-b ound D < σ 2 S and β c = β s = β , the distane b et w een the soure and fusion en ter has to satisfy the follo wing inequalit y , d 0 < ( L ) 1 /β h 1 D − 1 σ 2 S i − 1 /β , whi h is obtained b y making the MSE using { X i } to estimate X 0 no greater than D . W e then run the geometrial programming based optimization algorithm to determine the min- im um total transmission p o w ers for v arious approa hes. W e onsider 9 sp ots uniformly distributed b et w een the soure and fusion en ter for p ossible lo ations of t w o sensors, whi h are indexed b y in tegers 1 through 9 . The smaller the index v alue is, the loser the sensor is lo ated to the soure. Figure 3 (a), Figure 3(b) and Figure 3() demonstrate the total minim um p o w er onsumption P 1 + P 2 as a funtion of no des' lo ations for three sensor pro essing s hemes, from whi h w e ha v e follo wing observ ations: • As pro v ed in Theorem 6, when net w ork is symmetri, P total,A < P total,J < P total,S . • Under a relativ ely large distortion onstrain t (e.g. D = 0 . 5 ), uno ded s heme is the most energy eien t among the three andidates for all sensor lo ations, as sho wn b y Figure 3(a). 23 • Under relativ ely small distortion onstrain ts (e.g. D = 0 . 1 and D = 0 . 01 ), separate o ding approa h b eomes the most energy eien t when the relativ e p osition of t w o sensors b eomes more asymmetri. F or example, in b oth Figure 3(b) and Figure 3(), at a lo ation with an index pair (1 , 9) , i.e. the rst sensor is losest to the soure and the seond sensor is losest to the fusion en ter, w e ha v e P total,A > P total,J > P total,S . • Ov erall, to minimize the total p o w er exp enditure, w e should  ho ose either uno ded transmission or separate o ding s heme for a giv en pair of lo ations. This is a bit surprising as join t o ded approa h is often adv o ated more eien t (rate wise) than the separate one. It th us exemplies that exat v alues of  hannel onditions and the lev el of measuremen t noise are ruial to onluding whi h s heme is the most p o w er eien t in non-symmetri Gaussian net w orks with a nite n um b er of sensors. A Pro of of Theorem 2 Pr o of. F rom Theorem 1, w e kno w that ea h sensor nds from 2 n ˜ R j o dew ords the losest one U ( k ) j = { U ( k ) j [ i ] } to the observ ation v etor { X j [ i ] } and then amplify-and-forw ards { Y j [ i ] } to the fusion en ter. The deo der applies join tly t ypial sequene deo ding [7 ℄ to seek { U ( k ) j , j = 1 , · · · , L } from L o deb o oks whi h are join tly t ypial with the reeiv ed v etor { Z [ i ] } . WLOG, re-sh ue 2 n ˜ R j v etors su h that U (1) j is the v etor seleted b y sensor j ∈ { 1 , · · · , L } . W e assume that a subset U (1) ( S c ) = { U (1) j , j ∈ S c } has b een deo ded orretly , while its omplemen tary set U (1) ( S ) = { U (1) j , j ∈ S } is in error, whi h implies that the  hannel deo der at fusion en ter is in fa v or of a set of v etors U ( k ) ( S ) = { U ( k j ) j , k j 6 = 1 , j ∈ S } , instead. Next, W e will nd the upp er b ound of the probabilit y that  U (1) ( S c ) , Z  and U ( k ) ( S ) are join tly t ypial. The te hnique to upp er-b ond this probabilit y is quite similar as the one for MA C  hannels with indep enden t  hannel inputs [7, Chap 15.3℄. The ma jor dierene here is that the  hannel inputs from L sensors are orrelated b eause of the testing  hannel mo del used in indep enden t soure o ding, i.e. U j = X j + V j = X 0 + N j + V j , for j = 1 , · · · , L . The upp er b ound of the probabilit y that  U (1) ( S c ) , Z  and U ( k ) ( S ) are join tly t ypial is therefore 2 n ( H ( U ( S ) ,U ( S c ) ,Z )+ ǫ ) 2 − n ( H ( U ( S c ) ,Z ) − ǫ ) 2 − n P i ∈ S ( H ( U i ) − ǫ ) (78) = exp 2 − n H ( U ( S c ) , Z ) + X i ∈ S H ( U i ) − H ( U ( S ) , U ( S c ) , Z ) − ( | S | + 2) ǫ )) (79) where the rst term in (78) is the upp er b ound for the n um b er of join tly t ypial sequenes of ( U ( S ) , U ( S c ) , Z ) , the seond term in (78) is the upp er b ound of the probabilit y P  U (1) ( S c ) , Z  and 24 the last term in (78 ) is the upp er b ound of the probabilit y P  U ( k ) ( S )  . The summation in the last term in (78 ) is due to the indep endene of o deb o oks generated b y ea h sensor and the assumption that deo der is in fa v or of some U ( k j ) j for k j 6 = 1 and j ∈ S , whi h are indep enden t of U (1) ( S ) . Sine w e ha v e at most 2 n P j ∈ S ˜ R j n um b er of sequenes to b e onfused with U (1) j , j ∈ S , w e need X j ∈ S ˜ R j < H ( U ( S c ) , Z ) + X i ∈ S H ( U i ) − H ( U ( S ) , U ( S c ) , Z ) − ( | S | + 2) ǫ = | S |− 1 X i =1 I  U π i ; U π | S | π i +1  + I ( U ( S ); U ( S c ) , Z ) − ( | S | + 2) ǫ (80) for all S ⊆ { 1 , · · · , L } and an y arbitrarily small ǫ in order to a hiev e the asymptoti zero error probabilit y as n → ∞ , whi h th us ompletes the pro of. B Pro of of Lemma 1 Pr o of. As U 2 → X 2 → X 1 → U 1 forms a Mark o v  hain, w e ha v e I ( U 2 ; X 2 , X 1 | U 1 ) ( a ) = I ( U 2 ; X 1 , X 2 , U 1 ) − I ( U 2 ; U 1 ) ( b ) = I ( U 2 ; X 2 ) + I ( U 2 ; X 1 , U 1 | X 2 ) − I ( U 2 ; U 1 ) ( c ) = I ( U 2 ; X 2 ) − I ( U 2 ; U 1 ) (81) where equations ( a ) and ( b ) are due to the  hain rule on onditional m utual information [ 7℄. Giv en Mark o v  hain of U 2 → X 2 → X 1 → U 1 , U 2 and ( X 1 , U 1 ) are onditionally indep enden t giv en X 2 and onsequen tly I ( U 2 ; X 1 , U 1 | X 2 ) = 0 leading to equation ( c ) . On the other hand, follo wing equations also hold under similar argumen ts: I ( U 2 ; X 2 , X 1 | U 1 ) (1) = I ( U 2 ; X 2 | U 1 ) + I ( U 2 ; X 1 | X 2 , U 1 ) (2) = I ( U 2 ; X 2 | U 1 ) + I ( U 2 ; X 1 , U 1 | X 2 ) − I ( U 2 ; U 1 | X 2 ) (3) = I ( U 2 ; X 2 | U 1 ) (82) Therefore, om bining (81) and (82 ) yields: I ( U 2 ; X 2 | U 1 ) = I ( U 2 ; X 2 ) − I ( U 1 ; U 2 ) (83) 25 and similarly , I ( U 1 ; X 1 | U 2 ) = I ( U 1 ; X 1 ) − I ( U 1 ; U 2 ) (84) whi h th us pro v es ( 13) and (14 ). W e an pro v e (15 ) b y rstly sho wing that I ( U 1 , U 2 ; X 1 , X 2 ) = I ( X 1 , X 2 ; U 1 ) + I ( U 2 ; X 2 , X 1 | U 1 ) (85) under the  hain rule, where I ( X 1 , X 2 ; U 1 ) = I ( X 1 ; U 1 ) + I ( X 2 ; U 1 | X 1 ) = I ( X 1 ; U 1 ) (86) b eause of U 2 → X 2 → X 1 → U 1 . Sine w e ha v e already pro v ed (81), it is straigh tforw ard to sho w that (15) holds. C Pro of of Lemma 3 Pr o of. Due to the indep endene of measuremen t noise, U ( S ) → X ( S ) → X ( S c ) → U ( S c ) forms a Mark o v  hain. The rst equation in (21 ) is a diret appliation of (13). The pro of of the seond equation is based up on another Mark o v  hain b y adding the soure random v ariable X 0 in to the former one: U ( S ) → X ( S ) → X 0 → X ( S c ) → U ( S c ) . F rom this Mark o v  hain, w e an dedue I [ U ( S ); X 0 | U ( S c )] = I [ U ( S c ) , X 0 ; U ( S )] − I [ U ( S c ); U ( S )] = I [ X 0 ; U ( S )] + I [ U ( S c ); U ( S ) | X 0 ] − I [ U ( S ); U ( S c )] = I [ X 0 ; U ( S )] − I [ U ( S ); U ( S c )] (87) and X i ∈ S I [ U i ; X i | X 0 ] = I [ U ( S ); X ( S ) | X 0 ] = I [ U ( S ); X ( S )] + I [ U ( S ); X 0 | X ( S )] − I [ X 0 ; U ( S )] = I [ U ( S ); X ( S )] − I [ X 0 ; U ( S )] (88) where the rst equalit y is b eause of the onditional indep endene of ( X i , U i ) giv en X 0 . 26 It an b e seen that (87 ) and (88) yields I [ U ( S ); X 0 | U ( S c )] + X i ∈ S I [ U i ; X i | X 0 ] = I ( U ( S ); X ( S )) − I ( U ( S ); U ( S c )) , (89) whi h ompletes the pro of for Lemma 3 . D Pro of of Theorem 5 Pr o of. Dene Z i = 2 2 ˜ R i  1 − ˜ ρ 2  , for i = 1 , 2 . It an b e sho wn that Z i > 1 b y using ˜ ρ 2 = ρ 2 (1 − 2 − 2 ˜ R 1 )(1 − 2 − 2 ˜ R 2 ) and 0 < ρ < 1 as sho wn in ( 7). Giv en a pair of quan tization rates ( ˜ R 1 , ˜ R 2 ) , the original optimization problem b eomes min P 1 + P 2 , sub jet to: P 1 ≥ ( Z 1 − 1) σ 2 W (1 − ˜ ρ 2 ) g 1 ∆ = b 1 , P 2 ≥ ( Z 2 − 1) σ 2 W (1 − ˜ ρ 2 ) g 2 ∆ = b 2 P 1 g 1 + P 2 g 2 + 2 ˜ ρ p P 1 g 1 P 2 g 2 ≥ σ 2 W  2 2 ˜ R 1 +2 ˜ R 2 (1 − ˜ ρ 2 ) − 1  ∆ = b 3 (90) Dene a matrix A = " g 1 ˜ ρ √ g 1 g 2 ˜ ρ √ g 1 g 2 g 2 # (91) whose eigen v alue deomp osition is: A = Q Λ Q T , where the diagonal matrix Λ = diag { λ 1 , λ 2 } has eigen v alues λ i of A and the olumn v etors of the matrix Q = " q 1 , 1 q 1 , 2 q 2 , 1 q 2 , 2 # (92) are normalized eigen v etors asso iated with λ 1 and λ 2 resp etiv ely , whi h satisfy: Q T Q = " 1 0 0 1 # ∆ = I 2 (93) Notie that the last onstrain t in (90 ) is a quadrati form of v ariables √ P 1 and √ P 2 , whi h essen tially determines an ellipse, w e an p erform an unitary transformation b y in tro duing t w o new v ariables Y 1 and Y 2 : Y = [ Y 1 , Y 2 ] T = Q T [ p P 1 , p P 2 ] T (94) 27 su h that the original optimization problem in (90 ) is transformed to an equiv alen t one: min || Y || 2 = Y 2 1 + Y 2 2 , sub jet to: Q [ Y 1 , Y 2 ] T ≥ [ b 1 , b 2 ] λ 1 Y 2 1 + λ 2 Y 2 2 ≥ b 3 . (95) W e sho w next that Y 2 1 + Y 2 2 is minimized at the p oin t where the seond line q 2 , 1 Y 1 + q 2 , 2 Y 2 = b 2 in tersets with the ellipse λ 1 Y 2 1 + λ 2 Y 2 2 = b 3 as sho wn in Figure 4. In order to pro v e this, w e need to rst pro v e that the in tersetion of t w o lines q i, 1 Y 1 + q i, 2 Y 2 = b i , i = 1 , 2 is inside the ellipse. Let [ Y ∗ 1 , Y ∗ 2 ] denote the rossing p oin t of the t w o lines, whi h an b e determined as [ Y ∗ 1 , Y ∗ 2 ] = Q T [ b 1 , b 2 ] T . It is suien t to pro v e that λ 1 ( Y ∗ 1 ) 2 + λ 2 ( Y ∗ 2 ) 2 < b 3 , whi h is equiv alen t to ha ving [ b 1 , b 2 ] A [ b 1 , b 2 ] T < b 3 . (96) This holds as [ b 1 , b 2 ] A [ b 1 , b 2 ] T − b 3 = − σ 2 W 1 − ˜ ρ 2 h p ( Z 1 − 1)( Z 2 − 1) − ˜ ρ i 2 < 0 (97) where Z i w ere dened righ t b elo w ( 50), and hene obtain the desired result. Assume λ 1 > λ 2 . Solving quadrati funtion of | λ I 2 − A | = 0 , w e obtain eigen v alues λ 1 and λ 2 : λ 1 , 2 = 1 2 ( g 1 + g 2 ) h 1 ± √ 1 − ∆ i (98) where ∆ = 4 g 1 g 2 (1 − ˜ ρ 2 ) ( g 1 + g 2 ) 2 . The en tries of eigen v etors an b e omputed aordingly: q 1 , 1 = s λ 1 − g 2 2 λ 1 − g 1 − g 2 , q 1 , 2 = − s λ 2 − g 2 2 λ 2 − g 1 − g 2 q 2 , 1 = s λ 1 − g 1 2 λ 1 − g 1 − g 2 , q 2 , 2 = s λ 2 − g 1 2 λ 1 − g 1 − g 2 . (99) Based on (98 ), w e ha v e 2 λ 1 − g 1 − g 2 = − (2 λ 2 − g 1 − g 2 ) = ( g 1 + g 2 ) √ 1 − ∆ . Due to the non-negativ eness of the ratios in v olv ed in (99 ), it an b e sho wn that λ 1 > g 1 > g 2 > λ 2 . In addition, b eause of λ 1 + λ 2 = g 1 + g 2 , q i,j 's satisfy: q 1 , 1 > | q 1 , 2 | , q 2 , 1 < q 2 , 2 . (100) W e an therefore onlude that the lengths of the semi-axis 1 / √ λ 1 and 1 / √ λ 2 of the ellipse in the 28 diretion of Y 1 and Y 2 , resp etiv ely , satisfy 1 / √ λ 1 < 1 / √ λ 2 . Also, sine the slop es of the lines q 2 , 1 Y 1 + q 2 , 2 Y 2 = b 2 and q 1 , 1 Y 1 + q 1 , 2 Y 2 = b 1 ha v e the relationship of q 2 , 1 /q 2 , 2 < 1 < q 1 , 1 / | q 1 , 2 | , in addition, [ Y ∗ 1 , Y ∗ 2 ] is inside the ellipse, the minim um distane in ( 95 ) is attained at the p oin t where the line with smaller slop e in tersets with the ellipse, as illustrated b y Figure 4, whi h implies to minimize the total transmission p o w er P 1 + P 2 , the seond onstrain t on P 2 in (90 ), as w ell as the third one, should b e ativ e. This is equiv alen t to ha ving: ˜ R 2 = I ( U 2 ; Z, U 1 ) and ˜ R 1 + ˜ R 2 = I ( U 1 , U 2 ; Z ) + I ( U 2 ; U 1 ) , and ˜ R 1 = I ( U 1 ; Z ) . Consequen tly , when g 1 > g 2 , the deo ding at the fusion en ter follo ws exatly as that desrib ed in Theorem 5. Referenes [1℄ M .Gastpar and M. 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Nara y anan, On the distortion snr exp onen t of h ybrid digital-analog spae-time o ding, IEEE T r ansations on Information The ory , v ol. 53, no. 8, pp. 28672878, Aug. 2007. 30 1 2 3 4 5 6 7 8 9 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Node 1 position Node 2 position, D=0.5 Total Energy Uncoded Joint Separate (a) Result 1: D = 0 . 5 , σ 2 S = σ 2 W = 1 , β c = β s = 2 . 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 0 0.5 1 1.5 Node 1 position Node 2 position, D=0.1 Total Energy Uncoded Joint Separate (b) Results 2: D = 0 . 1 , σ 2 S = σ 2 W = 1 , β c = β s = 2 . 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Node 2 position,D=0.01 Node 1 position Total Energy Uncoded Joint Separate () Results 3: D = 0 . 01 , σ 2 S = σ 2 W = 1 , β c = β s = 2 . Figure 3: T otal p o w er onsumption for separate soure- hannel o ding (Red), join t soure- hannel o ding (Blue) and uno ded (Bla k) s hemes 31 Y 1 Y 2 q 1 , 1 Y 1 + q 1 , 2 Y 2 = b 1 ( Y ∗ 1 , Y ∗ 2 ) T otal p o w er minimizing p oin t q 2 , 1 Y 1 + q 2 , 2 Y 2 = b 2 Ellipse: λ 1 Y 2 1 + λ 2 Y 2 2 = b 3 Figure 4: T otal p o w er minimization 32 $d_0$ $d_1$ $d_2$ $d_K$ Fusion Center Source