A family of chaotic pure analog coding schemes based on baker s map function

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1 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 DOI.86/ RESEARCH Open Acce A family of chaotic pure analog coding cheme baed on baker map function Yang Liu * JingLi Xuanxuan Lu Chau Yuen and Jun Wu 3 Abtract Thi paper conider a family of pure analog coding cheme contructed from dynamic ytem which are governed by chaotic function baker map function and it variant. Variou decoding method including maximum likelihood ML minimum mean quare error MMSE and mixed ML-MMSE decoding algorithm have been developed for thee novel encoding cheme. The propoed mirrored baker and ingle-input baker analog code perform a balanced protection againt the fold error large ditortion and weak ditortion and outperform the claical chaotic analog coding and analog joint ource-channel coding cheme in literature. Compared to the conventional digital communication ytem where quantization and digital error correction code are ued the propoed analog coding ytem ha graceful performance evolution low decoding latency and no quantization noie. Numerical reult how that under the ame bandwidth expanion the propoed analog ytem outperform the digital one over a wide ignal-to-noie SNR range. Keyword: Analog error correction code; Chaotic dynamic ytem; Mean quare error MSE; Maximum likelihood ML decoding; Minimum mean quare error MMSE decoding Introduction Currently pervaive communication ytem in practice are almot digital-baed. Shannon ource-channel eparation theorem ha long convinced people that information can be tranmitted without lo of optimality by a two-tep procedure: compreion and encoding. Thi fundamental reult ha laid the foundation for the typical tructure of modern digital communication ytem the tandem tructure of ource coding followed by channel coding. Although digital communication ytem have been well developed over the lat decade they ha inherent drawback. Firt to tranmit continuou-alphabet ource ignal are quantized which introduce permanent lo in information. Second to preciely repreent real-valued ignal via digit the bandwidth i uually expanded. Moreover the ubequent channel coding procedure make tranmiion further bandwidth demanding. Third the digital error correction code are highly ignal-to-noie-ratio SNR dependent. Take turbo and low-denity parity-check LDPC code *Correpondence: Electrical and Computer Engineering Lehigh Univerity Bethlehem PA USA Full lit of author information i available at the end of the article a example. When the receiving SNR i under ome threhold value the decoding performance i uually very poor. On the contrary once SNR exceed thi threhold their bit error ratio BER fall down dratically in a narrow SNR range waterfall region. Thi ungraceful degradation in performance can caue problem in application. A typical cenario i the broadcating ytem where the SNR for different receiver can vary over a large range. At the ame time the digital error correction code are not energy efficient ince more tranmiion power lightly increae performance a long a the receiving SNR i modetly above the threhold. Lat but not leat digital error correction code with atifying performance uually require a long block length which introduce high latency for decoding and proceing at the receiver. In addition to the claical ource-channel eparate digital ytem the analog tranmiion ytem can erve a an alternative olution to data tranmiion. The analog ytem ha advantage over it pure digital peer it doe not introduce the granularity noie and it performance evolve gracefully with SNR. Mot of the analog tranmiion ytem ever preented in literature are joint ourcechannel coding JSCC ytem where compreion and 5 Liu et al. Thi i an Open Acce article ditributed under the term of the Creative Common Attribution Licene creativecommon.org/licene/by/4. which permit unretricted ue ditribution and reproduction in any medium provided the original work i properly credited.

2 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page of 8 encoding are performed in one tep and ignal are in pure analog or hybrid-digital-analog HDA form. The tudy of analog communication can date back to the paper 4. Reference 4 how that direct tranmiion of a Gauian ource over an additive white Gauian noiy AWGN channel with no bandthwidth expanion or compreion i optimal. For the bandwidth expanion cae 5 obtain the reult that the fatet decay peed of the mean quare error MSE cannot be better than the quare invere of the SNR. Although until now no practical cheme have been found to achieve thi decaying peed. In 6 7 the optimal linear analog code are treated. The deign of practical nonlinear analog coding cheme ha alway been an open iue. Some intereting paradigm have been found. Fuldeth 8 and Chung 9 dicu numerical-baed analog ignal encoding cheme. Vaihampayan and Cota propoe a cla of analog dynamic ytem contructed by firt-order derivative equation which generate algebraic analog code on toru or phere. Cai and Modetino Hekland et al. and Floor and Ramtad 3 tudy the deign of the Shannon-Kotel nikov curve. The minimum mean quare error decoding cheme for the Shannon-Kotel nikov analog code and their modified verion combined with hybrid digital ignal are dicued in 4 5. Among the family of analog coding cheme one pecial cla i contructed through chaotic dynamic ytem. In dynamic ytem the ignal equence i generated by iteratively invoking ome predefined mapping function. To be pecific the next ignal tate i obtained by performing a mapping to the current ignal tate and the whole ignal tate equence i initialized by the input ignal. For a chaotic dynamic ytem the function governing the ignal generation tate tranition i choen a chaotic function. Chaotic function are characterized by their fat divergence which i more well known a the remarkable butterfly effect. Thipropertymeanthat even a very tiny difference in initial input will oon reult in ignificantly different ignal equence. From the ignal pace expanion viewpoint thi indicate that a pair of point in ource pace with mall ditance will have a large ditance in the code pace. So chaotic dynamic ytem can potentially entitle ignal with error reitance. The eminal work 6 propoe an analog ytem baed on the tent map dynamic ytem and it performance i extenively dicued in 7. A with the analyi performed in 8 9 the drawback of the tent map code i that it performance convergence Cramer-Rao lower bound CRLB require very high SNR. Roenhoue and Wei propoe an improvement cheme by protecting the itinerary of the tent map code with digital error correction code. However thi hybrid-digital-analog cheme till uffer from the drawback rooted in digital error correction code. In thi paper we focu on a new pure analog chaotic dynamic encoding cheme which i contructed via a two-dimenional chaotic function baker map. Thi tructure i cloely related to and more complicated than the one reported in 6. The pecific contribution of thi paper include the following: we develop variou decoding method for the baker coding ytem and analyze it MSE performance. Baed on that we proceed to propoe two improved coding tructure and extend variou decoding method to thee new tructure. Thee propoed improvement effectively balance the protection for all ource ignal and have more atifying MSE performance compared to the tent map code. We alo compare our propoed analog coding cheme with the claical ource-channel eparate digital coding cheme where turbo code i applied. By uing equal power and bandwidth our propoed coding cheme outperform the digital turbo cheme over a wide SNR range. Thi paper i organized a follow: in Section the original baker dynamic ytem i dicued including it encoding tructure decoding method and performance analyi. Two modified chaotic ytem baed on the baker ytem are dicued in Section 3 and 4 including their encoding and decoding cheme. In Section 5 numerical reult and dicuion are preented and performance i dicued. Section 6 conclude the paper. In thi paper we aume that the ource ignal are mutually independent and uniformly ditributed on the interval which i alo adopted in previou work 6 and 7. By thi aumption the MMSE decoding method ha cloed form olution and we can compare performance with previou work. However it hould be pointed out that the maximum likelihood ML decoding method doe not require thi condition and i applicable to ignal with arbitrary ditribution. We aume that the tranmiion channel i AWGN and the decoding method obtained can be eaily extended to block fading channel. The baker map analog coding cheme In thi ection we introduce the analog encoding cheme baed on the baker map function. The baker map function F: i a piecewie-linear chaotic function given a follow: x ignuu Fu v y ignu v u v. The above baker map ha a cloe connection with the ymmetric tent map function dicued in 6 and 7 which i defined a Gx x x.

3 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page 3 of 8 Although the ymmetric tent map in i non-invertible once the ignx i given it value can be determined. The invere ymmetric tent map function with the ign of x given i G y y. Comparing the baker map and the ymmetric tent map function the baker map can be alternatively defined via the ymmetric tent map function a follow: x y Fu v Gu G ignu v u v. 3 Baed on the baker map function above a dynamic analog encoding cheme can be performed. For a pair of independent ource y a chaotic ignal equence i generated by repeatedly invoking baker mapping i.e. xn+ Fx n y n n N 4 y n+ where N i the bandwidth expanion. Thi equence can be viewed a a rate-/n analog code with and y a continuou information bit. In the following we ue x x x N T and y y y y N T to denote the codeword of two input ignal repectively. An important concept about the baker dynamic encoding ytem i the itinerary which i defined a N ign ignx ignx N. In fact if the itinerary of the code equence i given x k and y k can all be expreed a affine function of and y. Specifically x k and y k can be repreented via y in the following form: xk y a k + b k k N. y k y c k y + d k. 5 The affine parameter in 5 are function of itinerary. Forapecific they can be obtained in the following recurive way: a k+ k a k b k+ k b k c k+ k N 6 kc k d k+ k d k with the tarting point a b c d. In fact the collection of N itinerarie one-to-one map onto a partition of the feaible pace of i.e. the egment +. The itinerary i a function of input. For any pecific itinerary the admiible value of fall in a egment of length / N which i called a cell 7 and denoted a C e l e u. The two endpoint e l and e u of the cell aociated with are determined a e l min e u max bn + a N bn + a N } b N a N } b N a N. Thi concept i illutrated in Fig.. In the left part of Fig. a when N the itinerary ha bit i.e. + }. The two correponding cell are repectively the left and right half of the egment +. Thi concept i extended to length of N n + in the right of Fig. a where G n x denote n-fold compoition of G.Infact once the itinerary j i given the endpoint e lj and e uj } of the cell and the affine parameter a kj b kj c kj d kj can all be determined a function of j a hown in Fig. b. Next we dicu decoding cheme for the above baker dynamic encoding ytem.. Maximum likelihood decoding Under the AWGN channel aumption the received ignal can be repreented a rxn x n + n xn n N 9 r yn y n + n yn i.i.d. where n xn n yn N σ n N. We denote r x T r x r x r xn and ry T. ry r y r yn The likelihood function of the obervation equence r x r y with given ource y i pr x r y y πσ N exp r } x x + r y y σ. 8 The maximum likelihood etimate of ource pair ˆx ML ŷ ML i ML ˆ ŷ ML } arg max pr x r y y y arg min y N k rxk x k y + r yk y k y. The lat equality emphaize the fact that all x k y k are function of and y. Baed on connection between itinerarie and cell dicued in 5 8 the original maximum likelihood ML etimation problem in can be further tranformed into

4 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page 4 of 8 ML ˆ ŷ ML arg min C arg min N k rxk a k + b k + ryk c k y + d k } min N e e y k rxk a k + b k + ryk c k y + d k }. Fig. Partition and itinerary. a When N itinerary ha jut bit +or left; for general N n+ itinerary ha n pattern right. Each pecific pattern j correpond to one egment cell C j of the feaible region. b The feaible region + i partitioned into N cell with each cell C j correponding to one pecific itinerary pattern j. The parameter in the affine repreentation of the codeword and the endpoint of the cell can be determined once the itinerary j i given

5 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page 5 of 8 For any given itinerary the inner minimization problem in Eq. i convex and quadratic. Without conidering the contraint it optimal olution x y i given in a cloed form x at r x b a T a y ct r 3 y d c T c where a T a a N b T b b N c Tandd c c N T.Taking into account that the feaible y aociated with d d N hould lie within admiible range a limiting procedure mut be performed to obtain the olution to the inner minimization with pecific i.e. e l if x x inner < e l e u if x > e u x otherwie. if y y inner < + if x > +. 4 y otherwie. Since there are totally a finite number of poible itinerary pattern by enumerating all poible itinerarie and electing the x inner y inner } which minimize the outer minimization the ML etimation of y i obtained a ML ˆ ŷ ML arg min N k rxk a k x inner + b k + r yk c k y inner + d k }} The ML decoding cheme doe not require a priori knowledge of the ource ditribution. So it i applicable regardle of the probability ditribution of the ource.. Minimum mean quare error decoding The ML decoding method i not optimal in the ene of mean quare error performance. In thi ubection we focu on the minimum mean quare error MMSE olution to the baker dynamic ytem. The MMSE etimator i given in a general form a ˆX MMSE y EX y} xf x ydx 5 where X i random parameter to be determined and y i a pecific realization of the noiy obervation Y. Iti worth noting that the above general olution uually cannot reult in a cloed form olution for concrete problem. Fortunately under the uniform ditribution aumption of theourceignalthecloedformmmseetimatorforthe baker map can be obtained. To provide the reult of the MMSE decoder here we introduce the following notation: A a ; B a T b r x ; C b r x ; A c ; B c T d r y ; C d r y ; 6 and } B E exp A C σ A A Q σ e u + B A ; D Q σ ; A } B E exp A C σ ; D Q A A Q σ ; + B σ A J exp σ A e l + B A J exp σ A + B A σ e l + B σ A A + B σ σ A } exp } exp σ A σ A e u + B A + B A } ; } 7 where the function Q i the well-known Gauian-Q function which i defined a Qx x πσ e t dt. 8 The MMSE etimator of and y igiveninacloed form a follow: π σ A E E D A J πb D A 3/ ˆx MMSE ŷ MMSE π 9 π A E A E D D σ A J πb D A 3/. π A E A E D D A E E D The detailed proof of the above reult i rather involved and relegated to the Appendix..3 Mixed ML-MMSE decoding cheme The MMSE etimator involve highly nonlinear numerical evaluation like the Q-function which are computation demanding and cotly for implementation. Next we introduce ome kind of mixed ML and MMSE etimator for the baker analog code. A previouly dicued once the itinerary i given the analog codeword can be written a an affine function in y. For pecific itinerary by packing the codeword

6 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page 6 of 8 x and y into one vector v and uing 5 we can rewrite the baker dynamic ytem a follow: v x a y c } } G T x y } } u b + d } } t G T u + t where parameter a b c andd are defined in 6. Recall that in ML decoding a detection of the itinerary can be obtained. By ubtituting in with the ML detection ŝ ML and packing the received ignal r x and r y T into one vector r rx T rt y 9 can be expreed in a compact form r ŝ ML r tŝml G Ṱ ML u. Thu the baker map code i equivalent to a N linear analog code with encoder G. Now the problem to determine the ource ignal u in the above equation become the tandard minimum MSE receiving problem whoe olution i the well-known Wiener filter and given a û MMSE ŝml GŝMLG Ṱ ML + 3σ I GŝMLr ŝ ML. 3 A licing operation then follow the above Wiener filtering to enure the final etimate ˆx ML MMSE and ŷ ML MMSE lie in e lŝ ML e uŝ ML and + repectively. For the mixed ML-MMSE method ML decoding i performed to obtain ŝ ML. Then the Wiener filtering and limiting procedure follow. The mixed ML-MMSE decoding method require a priori knowledge of the ource and involve only linear computation operation..4 Performance analyi In Fig. the different decoding algorithm performance for the baker analog code with different length i plotted. E u mean the average power for each ource ignal and N denote the unilateral power pectral denity i.e. N σ. The ML MMSE and ML-MMSE decoding algorithm have identical MSE performance for high SNR. In low SNR range the MMSE decoding method ha the bet performance. In the following we analyze the MSE performance of the baker dynamic coding ytem by conidering the Cramer-Rao lower bound. CRLB i a lower bound for the unbiaed etimator 3. It hould be pointed out that the ML decoding method dicued above are the biaed etimator due to the licing operation. However when the SNR i large the decoding error i ufficiently mall uch that the licing rarely impact the decoding reult. So MSE Performance of Different Decoding Method for Baker Analog Code 3 4 log MSE MLN5 MMSEN5 ML MMSEN5 MLN3 MMSEN3 ML MMSEN3 Uncoded 5 5 E /N u Fig. MSE performance of different decoding algorithm for the baker dynamic ytem

7 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page 7 of 8 CRLB can preciely predict the decoding error when the SNR i modetly large and i ueful a tool to undertand the ytem performance. Thi will alo be verified by the following numerical reult. The Cramer-Rao lower bound for i given a 3 } CRLB baker E log pr x r y y 4 E x x σ N rxk a k b k k +r yk c k y d k } 5 σ N 3σ k a 4 N. 6 k where pr x r y y i defined in and E x denote the expectation with repect to. The recurive relation in 6 and the fact k are ued to obtain 6. Similarly the CRLB for y obtained a CRLB baker y E y σ N k c k } log pr x r y y y 3σ 4 /4 N. 7 When N i modetly large CRLB x 3σ /4 N.Each increment in N can decreae the decoding ditortion of by 3/4. Comparatively increment in N lightly improve y determination which i nearly a contant a 3σ /4. The CRLB reveal that the two ource are under unequal protection and there i inufficient coding gain on y. Recall that the x-equence in codeword i obtained by continuouly tretching and hifting the ignal. Intuitively the ignal i locally magnified. In comparion the y- equence i obtained by compreing the ignal. That i whythetermof N and N appear in the denominator of CRLB for and y repectively. Thi inight i verified by Fig. 3 where eparate MSE decoding performance of and y are plotted with their CRLB illutrated a benchmark. Although ha an obviou coding gain y i poorly protected and it ditortion dominate the overall decoding performance. From the CRLB analyi we realize that the bottleneck of the baker analog code lie in the weak protection to y. Thu to improve the baker map code effective protection hould alo be performed to y. 3 Improvement I mirrored baker analog code A analyzed in the lat ection the unatifying performance of the original baker map lie in the poor protection of y.toenhancetheprotectionofy anatural idea i to perform a econd original baker map encoding MSE Performance for Two Input of Baker Analog Code log MSE MSE N5 X MSE Y N5 CRLB X N5 CRLB N5 Y MSE X N3 MSE N3 Y CRLB N3 X CRLB Y N3 Uncoded E u /N Fig. 3 MSE performance for and y of the baker ytem

8 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page 8 of 8 by witching the role of and y.thuboth and y obtain balanced and effective protection. Thi idea lead to the improvement cheme to be dicued in thi ection the mirrored baker dynamic coding ytem. The mirrored baker tructure comprie two branche with it firt branch being the original baker encoder and the econd branch exchanging the role of and y to perform the original baker encoding for a econd time. For a given N the mirrored baker ytem form a 4N analog code. Here we adjut our notation for the new ytem to make our following dicuion clear. The two codeword aociated with two branche are labeled with ubcript and repectively. In the firt branch i the tent map encoded and o doe y in the econd branch. The codeword aociated with and y of the two branche are denoted a x y } and x y }repectively with their correponding noiy obervation a r x r y } and r x r y } repectively. The encoding procedure i expreed a xn+ y n+ yn+ x n+ Fx n y n Fy n x n n N ; 8 with x x and y y y.the obervation are repreented a rjxn x jn + n jxn j ; n N. r jyn y jn + n jyn 9 The mirrored baker dynamic ytem ha two itinerarie and from the firt and econd branche repectively the two of which compoe the entire itinerary for the mirrored baker ytem. A previouly dicued indicate a partition of the feaible domain of.sodoe to y. The entire feaible domain for the ource pair y which i a quare centered at the origin on the plane i uniformly divided into N cell with each cell being a tiny quare having edge of length N. Auming that the ource y i known to live in ome pecific cell the itinerarie and can be determined and the codeword can be expreed a affine function: xk y a k + b k y k y c k y + d k xk y a k + b k 3 y k y c k y + d k with k N. The parameter a k b k c k d k } and a k b k c k d k } are for the firt and the econd branche repectively and can be determined recurively for k N a follow: a k+ k a k b k+ k b k c k+ kc k d k+ k d k with the tarting point a a b b c c d d. c k+ k c k d k+ k d k a k+ ka k b k+ k b k 3 3 We denote a jj a jj a jj a jn j T j and define b jj c jj and d jj in the ame way for j. For a pecific itinerary } we denote it indicated admiible cell ha projection C onto feaible domain and projection C onto y feaible domain i.e. C e l e u y C e l e u with bn + e l min a N b } N a N bn + e u max a N b } N a N dn + e l min c N d } N c N dn + e u max d } N c N c N 33 Next we dicu decoding method for the mirrored baker dynamic ytem. Thee decoding method are obtained by traightforwardly extending the reult for the original baker ytem. In the following main reult are provided with detail omitted. 3. ML decoding In thi ubection the ML decoding of the mirrored baker map code i preented. The etimate ˆx ML ŷ ML maximizing the likelihood function i equivalently given a ML ˆ ŷ ML arg min min N e l e u e l y e j u k r jxk a jkj + b jkj + } r jyk c jkj y + d jkj. 34

9 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page 9 of 8 Foragivenpairofequence } the optimal olution of the inner minimization of the above equation i given a x at rx b +a T rx b a T a +a T a y ct ry d +c T ry d c T c +c T c followed by the hard limiter x inner y inner 35 e l ifx < e l e u ifx > e u x otherwie. e l ify < e l e u ify > e u. 36 y otherwie. The ML etimation i given by electing the x inner y inner among different itinerarie } which minimize the outer minimization in MMSE decoding To introduce the MMSE decoding reult for the mirrored baker ytem we adopt the following notation: Ā a + a ; B a T b r x +a T b r x ; C b r x + b r x ; Ā c + c ; B c T d r y + c T d r y ; C d r y + d r y ; B } j Ā j C j Āj Ē j exp σ Ā ; D j Q j σ e jl j + B j σ Āj Āj Q σ e ju j + B j ; σ Āj J j exp σ Āj e jlj + B j Ā j exp σ Āj e juj + B j j. Ā j 37 The calculation of the MMSE etimation till follow imilar line a dicued for the ingle baker ytem. The major difference i that ince the ign equence of y contribute to the itinerary the integration of y hould be decompoed into part over different C. The MMSE etimation of can be given a ˆx MMSE E r x r y r x r y } + f r x r y r x r y dx f rx r y r x r y f x d } C f r x r y r x r y 38 4 f r x r y r x r y } C } C f r x r y r x r y y dy d π σ Ā Ē Ē D J Ā π B D Ā 3/. 39 π Ē Ē D D Ā Ā } } Similarly the MMSE etimation of y for the mirrored baker map code i given a follow: ŷ MMSE π σ J Ā π B } Ā 3/. 4 π Ē Ē D D } Ā Ā Ā Ē Ē D D 3.3 ML-MMSE decoding If the itinerary } i given the codeword of the mirrored baker map ytem can be repreented a an affine function of the original ource y. The correponding coefficient can be determined recurively by uing Eq. 3 and 3. Thu the mirrored baker dynamic ytem can be rewritten a follow: v x y x y a c a c x y b + d b GT u + t d 4 We can firt perform the ML etimation dicued in Section 3. and thu obtain the ML detection of the itinerary ŝ ML ŝ ML }. By taking the ML detection of the itinerary a true value the linear MMSE etimator i invoked to etimate the original value of y } a follow: û MMSE ŝml ŝ ML GŝML ŝ ML G Ṱ ML ŝ ML +3σ I GŝML r tŝml ŝ ML. 4 Then a limiting procedure i performed to obtain admiible decoding reult.

10 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page of 8 4 Improvement II ingle-input -D baker analog code Inpired by the performance analyi in Section.4 to enhance the original baker map performance effective protection mut be performed equally to all ource. Beide the mirrored tructure propoed in the lat ection here we propoe an alternative improving trategy that i to feed the y-equence with input which actually form a ingle-input -D baker analog code. By feeding the two input of the original baker map with one ource the problem of poor protection of y vanihe and protection of i enhanced. In other word the protection to all ource i equal and trengthened. Furthermore another unconpicuou yet profound apect of motivation of thi -D cheme i that it perform a hidden repetition code of the itinerary which i explained in full detail a follow. A pointed out in the paper 8 and 9 reliably determining the itinerary i a key factor impacting decoding performance. In the original baker analog coding ytem the y-equence doe not help to protect the itinerary ince each of it ignal i uncorrelated with.recall that the codeword of the y-equence of the baker ytem i generated by the invere tent map function uing ign equence from the x-equence. By feeding the y- equence with wehavey G ign. Equivalently G y. So actually y can be regarded a the tate immediately before in the tent dynamic ytem which we denote a x. Following thi manner we can regard y i atheimmediateprevioutateofy i in a tent map dynamic equence for i N. Thu by rewriting the y-equence ignal a y N y N y } x N x N } and concatenating it with the x-equence ignal we actually obtain a long tent map analog code except that there are two copie of here. Moreover thi obtained equivalent tent map equence ha it pecial pattern: the firt half itinerary i reverely identical with the econd half itinerary. In other word the -D baker analog code actually contruct a hidden repetition code for the itinerary equence. Both the x- and y- equence now become analog parity bit of the itinerary. Thi intereting alternative view of the -D baker dynamic ytem i illutrated in Fig. 4. Next ticking to the notation introduced above for the baker ytem we give out the decoding reult for thi one-dimenional baker analog code. 4. ML decoding cheme Similar to the previou dicuion for each given itinerary the optimal olution to inner minimization x i obtained by x inner e l ifx <e l e u ifx >e u x otherwie with x at r x b +c T r y d a T a +c T c. 43 The ML etimate i obtained by going over all poible itinerarie and electing the x inner which minimize the likelihood function. 4. MMSE decoding cheme Defining the following parameter A a + c ; B a T b r x + c T d r y ; C b r x + d r y ; B } AC A E exp σ ; D Q A σ e l + B σ A A Q σ e u + B σ ; A J exp σ A e l + B } exp A σ A e u + B } A 44 TheMMSEetimateifgivena ˆx MMSE π A E σa J π A ED πb D A 3/. 45 Fig. 4 -D baker dynamic encoding ytem

11 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page of ML-MMSE decoding cheme Aume that the ML detection of the itinerary i ŝ ML then the received ignal can be written in an affine form of a rx r y aŝml cŝml + bŝml dŝml. 46 The linear MMSE etimate i obtained by performing a limiting procedure to the following value: ˆx MMSE ŝ ML aṱ ML r x bŝml + c Ṱ ML r y dŝml aŝml + cŝml + 3σ Simulation reult and dicuion In thi ection numerical reult and dicuion are preented. The MSE performance of ML MMSE and ML-MMSE decoding algorithm for mirrored baker and ingle-input baker ytem i preented in Fig. 5 and 6 repectively where E u repreent the average power for each ource ignal and N denote the unilateral power pectral denity. In our experiment the ource ignal are independent and uniformly ditributed over +. For each coding ytem code with N 3andN 5are teted. The aociated CRLB determined explicitly in Eq. 48 and uncoded performance are plotted to erve a benchmark. Numerical reult verify the validity of the decoding algorithm developed in previou ection and how that both of the mirrored and ingle-input tructure have improved MSE performance of the original baker coding ytem. Figure 7 compare the performance of the mirrored and ingle-input baker map and the tent map analog code propoed in 6 7 where code rate of /6 and/ are conidered for each coding cheme. Although the tent map encoding cheme can be proved to have a lower CRLB it actual performance i diadvantageou to the improved baker cheme over a wide SNR range. Generally the ditortion of analog tranmiion ytem can be decompoed into two part : anomalou ditortion and weak ditortion. Weak ditortion temming from the channel noie can become very mall and cloe to zero a long a the channel noie i ufficiently mall. A analyzed in 3 to reduce the ditortion of etimation the tranmitted ignal mut be tretched a much a poible which can be intuitively een a amplifying the ignal. However due to tranmiion power contraint tranmitted ignal have to be bounded and thu the tretching cannot be arbitrarily extenive without folding. Thi mean the tretched ignal will have multiple fold. The ML decoding project the received ignal to a valid codeword with minimum Euclidean ditance. Projection onto an erroneou fold reult into an anomalou ditortion which introduce a rather notable etimation error. In practical code deign the weak ditortion and MSE Performance for Mirrored Baker Analog Code log MSE MLN5 MMSEN5 ML MMSEN5 CRLBN5 MLN3 MMSEN3 ML MMSEN3 CRLBN3 Uncoded E /N u Fig. 5 MSE of different decoding algorithm for the mirrored baker analog code

12 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page of 8 MSE Performance for D Baker Analog Code log MSE MLN5 MMSEN5 ML MMSEN5 CRLBN5 MLN3 MMSEN3 ML MMSEN3 CRLBN3 Uncoded E u /N Fig. 6 MSE performance of different decoding algorithm for the ingle-input baker analog code MSE Performance of Different Analog Code 5 log MSE 5 5 Mirrored Baker MLcoderate/ D Baker MLcoderate/ Tentcoderate/ Mirrored Baker MLcoderate/6 D Baker MLcoderate/6 Tentcoderate/6 Uncoded E /N u Fig. 7 MSE performance of the tent map code mirrored baker map code and ingle-input baker map code code rate /

13 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page 3 of 8 the anomalou ditortion are two competing apect lengthening the codeword curve will relieve the weak ditortion but will inevitably introduce more fold and a narrower pace between fold and hence a higher chance for anomalou ditortion; likewie hortening the codeword curve will reduce the chance for anomalou ditortion but increae the weak ditortion. The key i to trike a bet balance between thee competing factor. Specifically the weak error can be accurately characterized by the CRLB and the anomalou error can be roughly indicated by the BER. The CRLB for and y of the mirrored baker ytem i given in the following which i alo CRLB of the ingleinput baker code CRLB mirror E x σ N k a k + N k a k σ log pr x r y r x r y y N k k + N k k 3σ 4 N 4 N + 3 CRLBmirror y CRLB d y. 48 For comparionthe CRLB for the tent map code code rate /N i given a CRLB tent 3σ 4 N. 49 It i not hard to verify the fact that CRLB tent < CRLB mirror CRLB d N N +. 5 Thi mean under equal bandwidth expanion or code rate the tent map ytem will alway have a lower weak ditortion. For tent map and baker map coding ytem itinerary error caue anomalou ditortion. To compare the anomalou ditortion of different analog coding ytem we examine the bit error rate BER performance of the itinerary bit for each code. We tet the tent map code mirrored baker code and ingle-input code with N 5 each of which ha itinerary length of 4. The BER of each itinerary bit for different ytem i illutrated in the ubfigure in Fig. 8. It hould be noted that in Fig. 8 the tent map code ha a code rate of /5 while the mirrored baker and ingle-input baker ytem have a code rate of /. The BER performance for the firt four itinerary bit of the tent map ytem with rate / i even wore than that for the tent map code with rate /5. BER for S BER for S BER: S 3 tent map ingle input baker mirrored bakerbranch BER: S 5 5 E u /N db 5 5 E u /N db BER for S BER for S 3 BER: S BER: S E /N db u 5 5 E /N db u Fig. 8 BER of itinerary bit for the tent map code mirrored baker map code and ingle-input baker map code N 5

14 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page 4 of 8 From the figure in Fig. 8 the mirrored baker map code and ingle-input baker map code have obviou advantage in the itinerary BER performance. The mirrored tructure exhibit equal protection for different itinerary bit and the BER decay with teeper lope than that of the tent map code. Comparatively the ingle-input baker ytem preent an unequal protection of different itinerary bit. The BER for itinerary bit with maller indice decay much fater than that with larger indice. Since error in itinerary bit with maller index caue more eriou ditortion the ingle-input baker ytem perform a clever unequal protection to itinerary bit adaptive to their ignificance. Thi alo explain the ingle-input baker map code advantageou performance over the mirrored baker map code in the medium SNR range. From the above comparion it can be een that although the improved baker analog code have larger weak ditortion than the tent map code their anomalou ditortion ha been effectively uppreed. The modified baker map code achieve a better balance between the protection againt two kind of ditortion and conequently outperform the tent map code in a wide SNR range. Next we compare the baker map code with optimum performance theoretically attainable OPTA and exiting analog coding cheme in literature 4. OPTA can be obtained by equating the rate ditortion function with the channel capacity. From 4 we know that the rate ditortion function depend on the ource ditribution and uually doe not have a cloed-form expreion. One of the few exception i the Gauian ource whoe ditortion function can be obtained analytically Theorem 3.3. in 4. In the Gauian cae OPTA can be obtained in a cloed form and thi i part of the reaon why the exiting literature tend to chooe Gauian ource a the cae of tudy like 5 do. However Gauian ource cannot be fed directly to the family of baker map encoder whoe input are required to be bounded +. Neverthele to make our propoal comparable with OPTA and other previou work we perform the comparion in an approximated manner by uing a truncated Gauian ource. The ource ignal i firt generated from the Gauian ditribution N m σ N /3.We then truncate it uing a limiting range of 3σ uch that 99.7 % of the probability ma fall in the region of +. The ignal value i et a + if it exceed + and ifitdropbelow. We performed mirrored baker coding on thi truncated Gauian ource and the reult are hown in Fig. 9. It hould be noted that in the figure the OPTA bound i calculated with the true Gauian ource the only ource that i analytically tractable. Since the imulated coding cheme ue a truncated Gauian ource we therefore ee a mall Approximated OPTA Mirrored Baker SDR and Shannon Kotel nikov Spiral SDR 5 Approx. OPTA: coderate/ CRLB: coderate/ SDR of ML: coderate/ SDR of ML MMSE mixed: coderate/ Uncoded Shannon Kotel nikov Spiral coderate/ 4 SDRdB E /N db u Fig. 9 Approximated OPTA SDR of the mirrored baker map code and SDR of Shannon-Kotel nikov piral with different parameter

15 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page 5 of 8 dicrepancy and the baker code actually appear to lightly outperform the OPTA at the low SNR region. At the ame time we alo plot the erie of the Shannon- Kotel nikov piral with parameter optimized for different channel SNR Fig. 9 in. It hould be noted that the MSE performance of the mirrored baker code and Shannon-Kotel nikov piral in Fig. 9 are obtained by the ML method which can be improved by the MMSE method according to 4 and our previou dicuion. The advantage of the parameterized Shannon- Kotel nikov piral curve approach i that by optimizing the parameter with repect to the ource ditribution and the channel condition the performance of the code can be made within ome 5 db from the OPTA. The cot however i that one mut know the exact ource ditribution and the accurate SNR information. A hown in Fig. 9 each curve repreent a Shannon-Kotel nikov piral with it parameter optimized toward one pecific channel SNR. Every time the channel condition change i.e. a different SNR the parameter mut be adapted or the code will uffer from a quick performance deterioration due to channel mimatch. The propoed baker analog code do not require the knowledge of the ource ditribution nor the channel SNR in order to perform encoding and ML decoding. Intead of deigning a equence of code the one optimized for each channel SNR in in our approach a ingle code iuedforawiderangeofsnrrange.figure9reflect that our propoal SDR in db ha identical lope for high channel SNR or diverity a that of optimized Shannon- Kotel nikov piral. The improved baker analog code univerally outperform the Shannon-Kotel nikov piral optimized for low-channel SNR and have obviou advantage in the low SNR range for all Shannon-Kotel nikov piral. Additionally the ML decoding algorithm of our propoed chaotic analog code ha imple cloed-form expreion which i abent for piral code. Latwecomparethepropoedanalogencodingytem with the conventional digital encoding ytem for analog ignal tranmiion. In our experiment the ource ignal are uniformly ditributed between the range +. For digital ytem uniform quantization and turbo code with recurive ytematic convolutional code +D+D +D 3 areued.thebcjrlog-mapalgorithm +D+D 3 with eight decoding iteration i performed for decoding the turbo code. Uniform puncturing i utilized to appropriately adjut the code rate when applicable. Due to the different ignificance of bit obtained by quantization equal error protection EEP and unequal error protection UEP are conidered. The detail of the teted ytem are given a follow:. Analog: 6 analog code i ued by utilizing the mirrored baker code with N and puncturing the ytem ignal y for the econd branch. Auming that codeword are tranmitted uing in-phae and quadrature form which can be regardeda -QAM modulation the ytem ha bandwidth expanion of 3/.. Digital-EEP: 8-bit quantization /3 turbo code and 56-QAM are ued. Sytem bandwidth expanion i 3/. 3. Digital-UEP: 8-bit quantization i performed. The four leat ignificant bit LSB are left uncoded. The four mot ignificant bit MSB are encoded by / turbo code. Both the coded and uncoded bit are 56-QAM modulated. Sytem bandwidth i 3/. 4. Digital-UEP: 8-bit quantization i performed. The two LBS are uncoded. The ix MSB are encoded by /5 turbo code. All bit are 56-QAM modulated. Sytem bandwidth i 3/. 5. Digital-UEP3: 8-bit quantization i performed. The four LSB are uncoded. The four MSB are encoded by /5 turbo code. The coded and uncoded bit go through 64-QAM modulation. Sytem bandwidth i 3/. The performance of the propoed analog and four digital ytem i plotted in Fig.. The propoed analog code exhibit an obviou advantage to the digital competitor over a wide range when SNR ha low and medium value. The digital ytem enter their waterfall region at rather high SNR and exhibit error floor which i the reult of the quantization noie. In fact due to the bandwidth limitation quantization noie will alway exit for the digital tranmiion cheme and eventually form an error floor that limit the overall ytem performance even a the SNR increae to infinity. In Fig. the digital coding cheme outperform the analog cheme in a narrow E u /N o range which i due to the fact that the digital error correction code performance boot dratically in a very narrow SNR range the o-called waterfall region. The digital code reilience to noie although powerful i finally uppreed by the quantization noie. Comparatively analog coding cheme have a very graceful performance evolution and their ditortion can be made arbitrarily mall if the channel i ufficiently good. 6 Concluion Thi paper introduce a family of pure analog chaotic dynamic encoding cheme baed on the baker map function. We firt dicu the coding cheme uing the original baker map function including it encoding and decoding cheme. Mean quare error analyi indicate that the intrinic unbalanced protection of it input reult in an unatifying performance. Baed

16 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page 6 of 8 MSE Performance of Different Analog Code 4 6 log MSE EEP: turbo374856qam UEP: 4 MSB by turbo564864qam UEP: 4 MSB by turbo qam UEP: 6 MSB by turbo QAM Analog: 6 Mirrored Baker Fig. Analog ignal tranmiion ytem v digital ytem E /N u on that two improvement encoding cheme are propoed mirrored baker and ingle-input baker ytem. Thee two cheme provide ufficient protection to all encoded analog ource. The variou decoding method for the original baker coding ytem are extended to the modified ytem. Compared to the claical tent map analog code the improved baker map encoding cheme achieve a better balance between the anomalou and weak ditortion and have advantageou performance in a wide practical SNR range. Moreover our improved encoding cheme alo exhibit competition or even better performance than the claical analog joint ource-channel coding cheme epecially in the low SNR range while maintaining much lower complexity in the decoding procedure. We alo compare the analog and conventional digital ytem uing turbo code to tranmit analog ource ignal. The digital ytem uffer from granularity noie due to quantization large decoding latency and threhold effect. Comparatively the analog coding cheme ha a graceful performance degradation and outperform over a wide SNR region. Appendix In thi appendix we provide detailed proof of the cloedform olution of the MMSE decoder for the original baker map code in 9. Following notation in Section we tart from Eq. 5; the MMSE etimate of can be given a ˆx MMSE + E r x r y } f r x r y d 5 f r x r y d 5 C f r x r y f d 53 C f r x r y + f r x r y y f y dy d f r x r y 4 f r x r y C C + f r x r y y dy d + N 4 f r x r y C πσ exp N rxk σ a kn +b kn k r yk c kn y +d kn } } dy d. 56

17 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page 7 of 8 In the above equation we utilize the fact that and y are independently uniformly ditributed over the range +.Toproceedwiththeabovederivationweintroduce ome intermediate parameter a follow: A a ; B a T b r x ; C b r x ; A c ; B c T d r y ; C d r y ; 57 Thuthecalculationin5canbefurtherwrittena πσ ˆx MMSE N exp } A 4 f r x r y C n σ x +B +C d } } I + exp } A σ y + B y + C dy. } } I 58 Similarly the MMSE etimator of ŷ MMSE can be alo obtained tarting from 5 and i determined a ŷ MMSE Ey r x r y } + y + 4f r x r y + + y f y r x r y dy 59 f r x r y y f y f y d dy f r x r y 6 N y f r x r y y d dy C n n 6 πσ N + y exp } A 4f r x r y σ y + B y + C dy } } I 3 exp } A C n σ x + B + C d. } } I 4 6 Oberving Eq. 58 and 6 the term f r x r y till need to be determined which can be calculated a f r x r y + + f r x r y y f f y d dy 63 πσ N I I where I and I 4 are defined in 58 and 6 repectively. Here we further introduce the following notation: } B E exp A C σ A A Q σ e u + B A ; D Q σ ; A } B E exp A C σ ; D Q A A Q σ ; + B σ A J exp σ A e l + B } exp A J exp σ A + B } exp A σ e l + B σ A A + B σ σ A σ A σ A e u + B A + B A } ; } 65 where the function Q i the well-known Gauian-Q function which i defined a Qx e t dt. 66 πσ x After ome manipulation the integral I I I 3 and I 4 defined previouly can be given by ue of the notation in 7 a σ I E J A σ I 3 E J A πb σ D ; I A 3/ πb σ D ; I 4 A 3/ π σ E D ; A π σ E D ; A 67 Thu by ubtituting Eq and 67 into 58 and 6 we can finally obtain the MMSE etimator of and y a follow: π σ A E E D A J πb D A 3/ ˆx MMSE ŷ MMSE π 68 π A E A E D D σ A J πb D A 3/. 69 π A E A E D D A E E D The proof ha been completed. Endnote Here we ambiguouly ue the terminology partition ince every two adjacent cell overlap with their common endpoint.butthidoenotharmthedecoding procedure. Competing interet The author declare that they have no competing interet. Acknowledgement Thi work i upported by the National Science Foundation under Grant No and

18 Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 Page 8 of 8 Author detail Electrical and Computer Engineering Lehigh Univerity Bethlehem PA USA. Singapore Univerity of Technology and Deign SUTD Singapore Singapore. 3 Computer Science and Technology Department Tongji Univerity Shanghai China. Received: 6 January 5 Accepted: June 5 Reference. CE Shannon Communication in the preence of noie. Proceeding of IRE VA Kotel nikov The Theory of Optimum Noie Immunity. McGraw-HillNew York NY USA JM Wozencraft Principle of Communication Engineering. John Wiley & Son Hoboken New Jerey USA TJ Goblick Theoretical limitation on the tranmiion of data from analog ource. IEEE Tran. Inform. Theory J Ziv The behavior of analog communication ytem. IEEE Tran. Inform. Theory KH Lee DP Peteren Optimal linear coding for vector channel. IEEE Tran. Commun Y Liu J Li K Xie in 46th Annual Conference on Information Science and Sytem CISS. Analyi of linear channel code with continuou code pace Princeton USA 8. A Fuldeth Robut ubband video compreion for noiy channel with multilevel ignaling. Diertation Norwegian Univerity of Science and Technology S-Y Chung On the contruction of ome capacity-approaching coding cheme. Diertation Maachuett Intitute of Technology. V Vaihampayan SIR Cota Curve on a phere hift-map dynamic and error control for continuou alphabet ource. IEEE Tran. Inform. Theory X Cai JW Modetino in 4th Annual Conference on Information Science and Sytem CISS. Bandwidth expanion Shannon mapping for analog error-control coding Princeton USA 6. F Hekland PA Floor TA Ramtad Shannon-Kotel nikov mapping in joint ource-channel coding. IEEE Tran. Commun PA Floor TA Ramtad Shannon-Kotel nikov mapping for analog point-to-point communication. 4. Y Hu J Garcia-Fria M Lamarca Analog joint ource-channel coding uing non-linear curve and mme decoding. IEEE Tran. Commun G Brante RD Souza J Garcia-Fria Spatial diverity uing analog joint ource channel coding in wirele channel. IEEE Tran. Commun HC Papadopoulu GW Wornell Maximum likelihood etimation of a cla of chaotic ignal. IEEE Tran. Inform. Theory B Chen GW Wornell Analog error-correction code baed on chaotic dynamical ytem. IEEE Tran. Commun I Hen N Merha On the threhold effect in the etimation of chaotic equence. IEEE Tran. Inform. Theory SM Kay Aymptotic maximum likelihood etimator performance for chaotic ignal in noie. IEEE Tran. Signal Proce I Roenhoue AJ Wei Combined analog and digital error-correcting code for analog information ource. IEEE Tran. Commun SM Kay Fundamental of Statitical Signal Proceing Volume I Etimation Theory. Prentice Hall Upper Saddle River New Jerey USA 993. S Haykin Adaptive Filter Theory 4th Ed. Prentice Hall Berlin Germany 3. HV Poor An Introduction to Signal Detection and Etimation nd Ed. Springer Upper Saddle River New Jerey USA TM Cover JA Thoma Element of Information Theory. John Wiely & Son Hoboken New Jerey USA 99 Submit your manucript to a journal and benefit from: 7 Convenient online ubmiion 7 Rigorou peer review 7 Immediate publication on acceptance 7 Open acce: article freely available online 7 High viibility within the field 7 Retaining the copyright to your article Submit your next manucript at 7 pringeropen.com

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