New non-uniform transmission and ADPCM coding system for improving both signal to noise ratio and bit rate

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1 New non-uniform ransmission and ADPCM coding sysem for improving boh signal o noise raio and bi rae Elisabeh Lahalle, Gilles Fleury, Rawad Zgheib To cie his version: Elisabeh Lahalle, Gilles Fleury, Rawad Zgheib. New non-uniform ransmission and AD- PCM coding sysem for improving boh signal o noise raio and bi rae. IEEE Signal Processing Leers, Insiue of Elecrical and Elecronics Engineers, 2011, 18 (3), pp < /LSP >. <hal > HAL Id: hal hps://hal-supelec.archives-ouveres.fr/hal Submied on 4 Mar 2011 HAL is a muli-disciplinary open access archive for he deposi and disseminaion of scienific research documens, wheher hey are published or no. The documens may come from eaching and research insiuions in France or abroad, or from public or privae research ceners. L archive ouvere pluridisciplinaire HAL, es desinée au dépô e à la diffusion de documens scienifiques de niveau recherche, publiés ou non, émanan des éablissemens d enseignemen e de recherche français ou érangers, des laboraoires publics ou privés.

2 1 New non-uniform ransmission and ADPCM coding sysem for improving boh signal o noise raio and bi rae Elisabeh Lahalle*, Gilles Fleury, Rawad Zgheib Absrac Here we address he problem of adapive digialransmission sysems. New sysems based on a nonuniform ransmission (NUT) principle are proposed, uilizing a recenly proposed algorihm for adapive idenificaion and reconsrucion of AR processes subjec o missing daa. We propose a new adapive sampling (nonuniform ransmission) mehod combined wih he adapive reconsrucion algorihm. A new NUT-ADPCM coding-decoding sysem is designed. The proposed sysem is demonsraed for audio-signal compression and compared o he ADPCM G.726 sandard. The new sysem yields improvemens in boh signal-o-noise raio and average bi rae. Index Terms Adapive ransmission, adapive reconsrucion, audio-signal compression. I. INTRODUCTION IN he design of digial ransmission sysems, i is imporan o find a good radeoff beween a low bi rae and a high signal-o quanizaion-noise raio [7], [9]. In a previous sudy [5], a new concep for he design of digial coding sysems was inroduced, based on nonuniform ransmission of signal samples. The idea is o avoid ransmiing a sample if i can be efficienly prediced. The use of he nonuniform ransmission principle in an ADPCM coding sysem was firs suggesed in [6]. An LMS-like algorihm [4] was suggesed for he predicion of samples ha were no ransmied. However, his algorihm converges oward biased esimaions of he model parameers and does no use an opimal predicor in he leasmean-squares sense [10]. Recenly, we proposed wo new adapive algorihms for he idenificaion and reconsrucion of nonsaionary AR processes subjec o missing daa using a Kalman filer for he predicion. The firs algorihm is based on a pseudolinear RLS algorihm for he idenificaion [11], and he second [12] is based on a laice filer srucure, hus guaraneeing sabiliy and robusness [2], [3]. Boh algorihms are fas and offer an opimal reconsrucion error in he leas-meansquares sense. They showed good performance in erms of quadraic reconsrucion error when applied o speech-signal reconsrucion wih a high probabiliy of missing samples. Indeed, nonsaionary AR processes can model a large number of signals in pracical siuaions such as speech signals [7], [3]. We here propose o use hese algorihms in a nonuniform ransmission (NUT) sysem, previously inroduced in [5], o E. Lahalle, G. Fleury and R. Zgheib are wih E3S - SUPELEC Sysems Sciences / Dep. of Signal Processing and Elecronic Sysems, Gif-sur-Yvee, France (firsname.lasname@supelec.fr), el: 33(0) , fax:33(0) improve is performance. We hen combine hem wih adapive reconsrucion algorihms o design a new NUT-ADPCM coding-decoding sysem. In he following, we begin by recalling he nonuniform ransmission idea inroduced in [5], hen he adapive algorihm for he reconsrucion of AR processes wih missing observaions [11]. The performance of NUT ransmission sysems using [4] and [11] idenificaion algorihms are compared. Finally, a new NUT-ADPCM sysem is designed. I is compared in he las secion o he ADPCM [1] coding sysem hrough simulaions of speech signals. A. NUT Coding 1) Adapive sampling: In a simplified digial-ransmission sysem, assuming ha he channel is perfec, he reconsrucion error is equal o he quanizaion error. Mirsaidi e al. [5] proposed a novel nonuniform ransmission sysem ha reduced quanizaion error by using a parameric signal-modeling approach for he predicion. In he receiver, a sample a ime n is prediced using he esimaed model parameers a ime n 1 and he available quanized samples. The key idea of he sysem hey proposed is ha a ime n an efficien predicion may provide a smaller error han a quanizaion. In his case, i is beer o replace he ransmied quanized sample wih is predicion in he receiver. Thus, using an efficien predicion mehod for signals subjec o missing samples, he number of ransmied samples is considerably reduced and he reconsrucion performance is also improved. 2) Kalman Pseudo-linear RLS Algorihm: Le {x n } be an AR process of order L wih parameers {a k }, and {ɛ n } he corresponding innovaion process of variance σɛ 2. The loss process is modeled by an i.i.d. binary random variable {c n }, where c n = 1 if x n is available; oherwise, c n = 0. Le {z n } be he reconsrucion of he process {x n }. If x n is available z n = x n ; oherwise, z n = ˆx n, he predicion of x n. To idenify he AR process subjec o missing daa in real ime, we use he pseudolinear RLS algorihm [8]. In he case of missing observaions, he regression vecor Ψ n = [x n 1... x n L ] canno be consruced wih only available daa. Missing daa are replaced by heir predicions, i.e., ˆΨ n = [z n 1... z n L ]. Thus, ˆΨ n depends on he available parameers. We show [11] ha he leas-squares esimae of he parameers is unbiased if he predicor used is opimal in he leas-mean-squares sense. A Kalman filer is used for recursive opimal predicion. Le x n = [x n... x n L1 ] be he sae vecor. The prediced

3 2 and filered esimaes are denoed by ˆx n1 n and ˆx n1 n1, respecively. P n1 n and P n1 n1 are, respecively, he prior and poserior predicion-error covariance marices. K n1 is he Kalman filer gain. The Kalman pseudolinear RLS algorihm is summarized, a ime n1, as follows : A n1 = â n, â n,l (1) P n1 n = AP n n A R ɛ, (2) ˆx n1 n = Aˆx n n, (3) ˆΨ n = ˆx n n, (4) If x n1 is available, i.e. c n1 = 1, he prediced sae vecor ˆx n1 n is filered by he Kalman filer ((5),(6),(7)) and he AR parameers are updaed using he RLS-like algorihm ((8),(9),(10)): K n1 = P n1 n c n1 (c n1p n1 n c n1 ) 1, (5) P n1 n1 = (I d K n1 c n1)p n1 n, (6) ˆx n1 n1 = ˆx n1 n K n1 (y n1 c n1ˆx n1 n), (7) γ n1 = c n G n ˆΨn1 λ ˆΨ n1 G n ˆΨ n1, (8) â n1 = â n γ n (y n1 c n1ˆx n1 n), (9) G n1 = 1 λ (I d γ n1 ˆΨ n1 )G n (10) If x n1 is missing, i.e. c n1 = 0, he prediced sae vecor is no filered by he Kalman filer and he AR parameers are no updaed using he RLS-like algorihm : K n1 = 0, (11) P n1 n1 = P n1 n, (12) ˆx n1 n1 = ˆx n1 n, (13) γ n1 = 0, (14) â n1 = â n, (15) G n1 = 1 λ G n (16) 3) Comparison: In his secion, he periodic ransmission sysem, he nonuniform ransmission sysem using he LMSlike algorihm proposed in [4], and he Kalman pseudolinear RLS Algorihm are compared; he mehods are referred o hereafer as Mehods 1, 2 and 3, respecively. The es signal used was Mary had a lile lamb, is fleece was whie as snow sampled a F s = 8kHz. The forgeing facor used in he pseudolinear RLS algorihm [11] is λ = and he sepsize used in he LMS-like algorihm [4] is µ = In boh parameric mehods, he speech signal is modeled by an AR process of order 10. The hree mehods are evaluaed in erms of mean bi rae and he signal-o-reconsrucion ( error raio N ) i=1 given by: (SNR) = 10log x2 i 10. In he case of N i=1 (x i ˆx i ) 2 a ransmied sample, he reconsrucion error is he same as he quanizaion error; in he case of a missing sample, i is he same as he predicion error. For he periodic ransmission sysem, he bi rae is given by r = BF s, where B is he number of bis per sample. For oher wo mehods, a onebi flag is ransmied for each nonransmied sample. We consider he average bi rae, given by r = (T B 1)F s, where T sands for he proporion of samples ransmied. TABLE I COMPARISON OF THE THREE METHODS Mehod B T (SNR) db Bi rae ADPCM For comparison purposes, Table I shows he resuls of he hree mehods. They are compared wih respec o he signal-o-noise raio (SNR) and he bi-rae performance of he G.726 ADPCM codec a 32 kbis. For almos he same SNR, Mehod 2 shows a higher average bi rae han for periodic ransmission wih B = 5, which is of course due o he onebi flag. In conras, Mehod 3, using an opimal predicor [11], ouperforms all oher mehods in boh SNR and average bi rae. Consequenly, a new nonuniform ransmission ADPCM coding sysem using an opimal predicor [11] [12] is proposed below. B. NUT-ADPCM Coding The ADPCM codec, defined by he ITU-T Recommendaions G.726 [1], is based on he differenial coding principle (DPCM). Because he difference beween he signal and is predicion has a lower variance han he original samples, i can be accuraely quanized wih fewer bis han he original samples. The performance of he codec is aided by using adapive predicion and quanizaion, so ha he predicor and difference quanizer adap o he changing characerisics of he signal being coded. The adapive predicor used is based on an ARMA(2,6) model of he signal. 1) sysem design: When he predicion mehod used is efficien, he predicion error of some samples migh be of a magniude less han he minimal quanizaion sep provided by he adapive quanizer. We propose avoiding ransmiing

4 3 he quanized predicion errors in ha case and so expec o reduce he number of ransmied bis wihou sacrificing reconsrucion qualiy. Therefore, he adapive predicor used in he ransmission and recepion mus solve he problem of he reconsrucion of signals wih missing samples. In addiion, he ransmission of a one-bi flag o indicae he ransmission mode is required. The proposed sysem differs from he ADPCM codec mainly in he adapive predicor used. Addiionally, i conains a ransmission-decision elemen for he quanized predicion error and for he one-bi flag. Adapive predicor: We hen use he Kalman Recursive Leas-Squares Laice algorihm (KRLSL) [12] algorihm as an adapive predicor in he NUT-ADPCM coding sysem insead of he Kalman RLS [11] o ensure he sabiliy of he sysem. The signal is modeled by a nonsaionary AR(L) process of order 10 insead of an ARMA(2,6). Recall here ha he KRLSL algorihm is based on an adapive Burg algorihm for he idenificaion of he AR parameers, hus guaraneeing a each ime poin he sabiliy of he idenified filer [12]. Addiionally, he KRLSL algorihm uses a Kalman filer for he predicion of he samples. When a sample is available, he sae, consising of he las L samples, is updaed by he Kalman filer proporionally o he predicion error of he sample [12]. However, in he case of NUT-coding sysems, he receiver has only quanized versions of he predicion errors. Therefore, even if he rue predicion error of a sample is known a he ransmier, he quanized predicion error is used wih he Kalman filer o updae he sae. This reproduces he same sample predicion as is done in he receiver. Hence, when a quanized predicion error is received, he sample reconsrucion is compued as in equaion ˆx n n = ˆx n n 1 e Q n,p. The sample hus reconsruced is used o iniialize he laiceidenificaion algorihm ((17)). In he KRLSL algorihm, he recursive equaions of he RLSL algorihm ((17),(18),(19),(20),(21),(22),(23)) are used insead of he equaions of he RLS algorihm ((8),(9),(10)) a each available sample, x n1. They are used o esimae ˆk he reflecion coefficiens a each ime since he las available sample. The AR parameers, a ime n 1, are hen deduced from he reflecion coefficiens ˆk n1 using he Durbin-Levinson recursions. (17) Ĉ ˆD ˆf ˆb ˆf (0) (0) = ˆb = ˆx n1,, = λĉ 1 = λ ˆD 1 = ˆk ˆk (0) = 1, (18) (l 1) 2 ˆf ˆb(l 1) 1, (19) (l 1)2 ˆf ˆb(l 1)2 1, (20) = Ĉ ˆf (l 1) ˆD, (21) ˆk ˆb(l 1) 1, (22) (l 1) = ˆb ˆk ˆf (l 1) 1 (23) Transmission decision: Once he sample is prediced via he Kalman filer, he predicion error is quanized using he same adapive quanizer as described in he recommendaions of he G.726 specificaions [1]. If he sample is perfecly prediced (i.e., he predicion error is quanized o zero), he code corresponding o he quanized predicion error is no ransmied. A one-bi flag mus also be ransmied o he receiver. In he ransmier, he adapive predicor mus be informed abou he ransmission decision of he sample code o execue he appropriae seps, which are also execued in he receiver. In his scheme, o reduce he number of bis ransmied, he number of codes ransmied should compensae for he exra bi flag. Thus, le N be he size of a signal coded using an ADPCM codec a B bis. The oal number of bis ransmied is hen equal o B ADP CM = NB bis. The nonuniform ransmission ADPCM sysem also uses B bis (he same number as he ADPCM codec o which i is compared) o code a quanized predicion error. The oal number of bis ransmied by he NUT-ADPCM sysem is hen equal o B NUT = pnb N f, where p is he raio of ransmied samples and N f is he number of he ransmied flags. A flag D n = 1 is ransmied a each new ransmission following a sequence of non-ransmied samples. A flag D n = 0 is ransmied when he ransmission sops. The condiion for a profiable use of he NUT-ADPCM sysem is hus pnb N f < NB. A simplified diagram of he NUT- ADPCM codec described above is presened in Figure 1. xn _ xes enp NU ADPCM Coder In Inverse quanizer NU ADPCM Decoder Adapive predicor enpq Adapive quanizer Transmission decision xrec xes Adapive predicor Dn xrec In enpq PCM conversion ADPCM oupu Dn Inverse quanizer synchronous coding adjusmen 64kbps oupu Fig. 1. Simplified diagram of a non-uniform ransmission ADPCM coder decoder. 2) Simulaions: The nonuniform ransmission ADPCM sysem described above is compared here o he ADPCM hrough simulaions on he same speech signal as above. Differen values of he number of bis are used o code he predicion error B. The differen mehods are coded using MATLAB and execued on a Penium 4 PC wih a 3GHz

5 4 processor. The es resuls are presened in Table II. The erm CPU in he able indicaes he compuaion ime in seconds required by each of he mehods. TABLE II COMPARISON OF ADPCM AND NUT-ADPCM Mehod B p% Bi rae (kbps) (SNR) db CPU (s) ADPCM NUT-ADPCM ADPCM NUT-ADPCM ADPCM NUT-ADPCM ADPCM NUT-ADPCM Referring o Table II, he NUT-ADPCM yields in all cases an improvemen in boh average ransmission bi rae and SNR compared o ADPCM. Moreover, lisening ess show a beer qualiy using he NUT-ADPCM mehod han using he ADPCM for he same number of bis. The performance improvemen is neverheless obained a he expense of a sligh increase in compuaion ime. [5] S. Mirsaidi, G. Fleury and J. Oksman, Reducing quanizaion error using predicion/non uniform ransmission, Proceedings of he Inernaional Workshop on Sampling Theory and Applicaions, Aveiro, Porugal, [6] S. Mirsaidi, G. Fleury and J. Oksman, An ADPCM-like sysem based on non uniform signal ransmission, Proceedings of he Inernaional Workshop on Sampling Theory and Applicaions, Aveiro, Porugal, [7] L. Rabiner and R. Schafer, Digial processing of speech signals, Prenice Hall, [8] R. Sanchis and P. Alberos, Recursive idenificaion under scarce measuremens-convergence analysis, Auomaica, vol. 38, pp , [9] J. G. Wade, Codage e raiemen du signal, Masson, [10] R. Zgheib, G. Fleury and E. Lahalle, New fas recursive algorihms for simulaneous reconsrucion and idenificaion of AR processes wih missing observaions, Proceedings of he foureenh European Signal Processing Conference, (EUSIPCO), Florence, Ialy, Sepember [11] R. Zgheib, G. Fleury and E. Lahalle, New fas algorihm for simulaneous idenificaion and opimal reconsrucion of non saionary AR processes wih missing observaions, Proceedings of he IEEE welveh Digial Signal Processing workshop, (DSP), Wyoming, USA, Sepember [12] R. Zgheib, G. Fleury e E. Lahalle, Laice algorihm for adapive sable idenificaion and robus reconsrucion of non saionary AR processes wih missing observaions, IEEE Trans. on Signal Processing., vol. 56, pp , II. CONCLUSION New nonuniform ransmission coding sysems were proposed based on nonuniform ransmission mehods using adapive parameric predicion mehods [11], [12]. Nonuniform ransmission was sudied in he differenial coding case. A nonuniform ransmission mehod for quanized error predicion was proposed. As he predicion error is lower han he quanizaion error, his mehod can be considered o be a nearlossless compression mehod (or even a lossless compression mehod disregarding he quanizaion effec. A NUT-ADPCM coding sysem based on his nonuniform ransmission mehod was described using he same adapive quanizaion mehod as he ADPCM described in he G.726 [1] recommendaions. In conras o he ADPCM, he adapive-predicion mehod used in he NUT-ADPCM is he one proposed in [12], hus he signal was modeled by an AR process. In addiion, ransmission decisions for he quanized predicion errors and he flags were inroduced. The NUT-ADPCM coding sysem was compared o he ADPCM as described in he G.726 recommendaions hrough simulaions wih speech signals. Improvemens in boh SNR and average ransmission bi rae were observed. Thus, when he predicion errors are coded wih four bis (he ADPCM working a 32kbis/s), he NUT- ADPCM coding offers an improvemen of 10% in he average bi rae and an improvemen of 6dB in he SNR compared o he ADPCM coding sysem. REFERENCES [1] 40, 32, 24, 16 kbi/s Adapive differenial pulse code modulaion (ADPCM), CCITT : Recommendaion G.726, Geneve, [2] J. Makhoul, Linear predicion: A uorial review, Proceedings of he IEEE, vol. 63, No. 4, pp , [3] J. Makhoul and L. Cosell, Adapive laice analysis of speech, IEEE Trans. on Circuis and Sysems, vol. 28, pp , [4] S. Mirsaidi, G. Fleury and J. Oksman, LMS Like AR modeling in he case of missing observaions, IEEE Trans. on Signal Processing, vol. 45, No. 6, 1997.