Hybrid Digital-Analog Joint Source Channel Coding for Broadcast Multiresolution Communications

Transcription

1 217 25th European Signa Processing Conference (EUSIPCO) Hybrid Digita-Anaog Joint Source Channe Coding for Broadcast Mutiresoution Communications O. Fresnedo, P. Suárez-Casa, L. Castedo Department of Eectronics and Systems University of A Coruña, SPAIN {oscar.fresnedo, pedro.scasa, uis}@udc.es J. García-Frías Department of Eectrica & Computer Engineering University of Deaware, USA jgf@ude.edu Abstract Mutiayer coding represents an appeaing soution for the broadcasting of common data when we want to ensure the receivers decode the signa with different distortion eves depending on the quaity of the received signa. In addition, the combination of digita and anaog techniques to encode the source data aows to sove the imitation of these approaches when they are appied separatey. In this work, we consider a Hybrid Digita-Anaog (HDA) mutiayer system where the digita ayers are empoyed to satisfy different Quaity of Service (QoS) requirements and an anaog ayer refines the estimates of the source symbos. The resuting scheme provides good performance for different epansion factors and number of ayers, and it aso presents good scaing for a SNR vaues. Area #1 Area #2 T I. INTRODUCTION Communication systems designed according to the sourcechanne separation principe are in genera suboptima for broadcast channes. In this case, when the receivers eperience a different quaity of the received signa, the superimposition of severa data ayers transmitted with different eves of protection provides better performance [1]. In such mutiayer schemes, source coding based on successive refinement is empoyed to iterativey generate new information ayers that successivey reduce the distortion and, at the same time, it is possibe to reconstruct the source data from any ayer for a given distortion target [2]. Different approaches have been considered to decode the source data at the receivers with different resoutions or distortion eves, such as hierarchica moduations and embedded channe codes [3]. Another strategy is mutiayer Hybrid Digita-Anaog (HDA) where coarse information is encoded using digita schemes to ensure a the receivers are abe to decode at east that information, whie the quantizer errors are directy transmitted to refine the source estimates in the best receivers. These schemes are particuary hepfu in broadcast settings where users at different distances must be abe to decode a common message [4]. Mutiayer HDA schemes usuay send the information corresponding to the anaog part using inear encoding or uncoded transmission. Non-inear strategies can aternativey be considered to propery adjust the anaog rate to the digita one. Anaog Joint Source Channe Coding (JSCC) mappings based on space-fiing parametric curves have been shown to achieve near optima performance in different scenarios Fig. 1. Eampe of the broadcast scenario with different two coverage regions. [5] [7], speciay for bandwidth compression. In the case of bandwidth epansion, other non-inear mappings such as spherica codes [8], non-differentiabe anaog mappings [9], orthogona poynomias [1] or anaog codes based on chaotic dynamica systems [11] have been proposed. In this work, we focus on the broadcasting of common messages from one transmitter to severa receivers at different distances. A mutiayer HDA scheme is considered where the digita ayers satisfy different QoS requirements and the ast ayer encodes the information using some adequate anaog JSCC mapping. Compared to [4], the matched tandem codes used to transmit the quantization errors are repaced by anaog JSCC mappings. II. SYSTEM MODEL Let us consider the transmission of a common message from one transmitter to severa receivers paced at different distances. We are interested in ensuring that the furthest receivers be abe to decode at east part of the message, whie the source information is decoded with better quaity at ocations coser to the transmitter. Let us divide a coverage area into L circuar broadcast regions with their corresponding QoS requirements, i.e. ξ r ɛ i r : d r d i, i = 1,..., L, (1) ISBN EURASIP

2 217 25th European Signa Processing Conference (EUSIPCO) Quantization Anaog Mapping Moduator Moduator Transmitter Receiver Demod Moduator Demod Moduator Anaog Power Aocation Fig. 2. Bock-diagram of the proposed mutiayer architecture. where ɛ i is the distortion eve required for the i-th region, d i is the distance from the transmitter to the furthest points in the i-th region, ξ r represents the actua distortion between the source information and the data decoded by the receiver r, and d r is the distance between r and the transmitter. Figure 1 shows an eampe for L = 2 with two circuar broadcast regions around the transmitter, and their corresponding distortion requirements. In this contet, a mutiayer transmitter with L digita ayers is considered to satisfy the distortion requirements in (1). Figure 2 shows the bock-diagram of the proposed mutiayer system. As observed, a vector of anaog source symbos s is first quantized and then encoded with a particuar digita scheme at each ayer i = 1,..., L. The digita components and the aocated power must convenienty be chosen at each ayer i to ensure that the signa distortion for the receivers in the i-th region is ower than ɛ i. In Section III-A, we wi provide the detais for the design of the digita ayers. Once the transmit power is propery aocated to the L digita ayers, the remaining power can be epoited to improve the signa quaity of those receivers coser to the transmitter. In this case, two aternatives can be considered: 1) using one or severa digita ayers in a way simiar to previous ayers, or 2) using a purey anaog ayer where the quantization errors are directy mapped to anaog channe symbos. As aready commented, we choose the second approach since the utiization of an anaog ayer avoids the foor effect of the digita schemes, and the signa distortion continuousy decreases as the distance to the transmitter owers. The symbo vectors corresponding to the L + 1 ayers are Dequantization superimposed into a singe vector to be transmitted. Thus, L wi wa = i + a, (2) γ i γ a where i and a represent the symbo vector of the i-th digita ayer and the anaog mapped symbos, respectivey, and γ i and γ a are normaization factors to guarantee the power of the encoded symbos at each ayer is equa to 1. Finay, w i and w a are the power aocated to the i-th digita ayer and to the anaog one, respectivey. The resuting vector is assumed to be broadcast over an Additive White Gaussian Noise (AWGN) channe. Hence, the received signa can be epressed as y = + n, where n N C (, σn) 2 represents the AWGN. Without oss of generaity, the variance of the channe noise is assumed to be σn 2 = 1. Hence, the Signa-to-Noise Ratio (SNR) at the receivers is directy measured as SNR r = P r, where P r is the power of the received signa. Assuming free space propagation, the path oss eponent is n = 2 and the received power is given by P r = P t d 2 r, with P t the transmit power. At the receivers, an estimate of the symbos is computed from the received vector y. The different ayers are iterativey decoded using a Succesive Interference Canceation (SIC) strategy. In this case, each receiver starts decoding the first ayer. If successfu, it subtracts the information corresponding to the first ayer from the received signa and proceeds to decode the net ayer. It repeats these operations unti it is not abe to decode a ayer. If a receiver successfuy decodes the L digita ayers, it can epoit the anaog data to refine its estimate of the source symbos. In genera, the receivers in the -th region are abe to decode ony the first ayers, whie those which fa into the cosest region can aso improve their estimation with the anaog data of the ast ayer as ong as the received power is above a certain vaue. Otherwise, the anaog estimates are disregarded. Finay, the distortion between the source and estimated symbos is measured according to the Mean Square Error (MSE) criterion as ξ r = s ŝ r 2, (3) where ŝ r is the vector estimated for the receiver r. III. MULTILAYER SCHEME In the ensuing subsections, we describe the transmit ayers and the SIC decoder corresponding to the digita part of the system, and the mapping and the decoder empoyed in the anaog scheme. Finay, the power aocated to each ayer is determined depending on the number of ayers, the quaity requirements and the avaiabe transmit power. A. Digita Layers As observed in Figure 2, the digita part of the mutiayer transmitter consists of a quantizer and L ayers where a channe encoder is concatenated to a digita moduator. The discrete-time continuous-ampitude source symbos are quantized to obtain a discrete representation of the origina ISBN EURASIP

3 217 25th European Signa Processing Conference (EUSIPCO) anaog message. The quantization is based on source coding with successive refinement [2]. Thus, the quantizer provides L bit sequences such that the first sequence corresponds to a coarse representation of the input. The net sequences can be empoyed to refine this representation iterativey. For simpicity, in this work we focus on scaar quantization, but other type of quantizers can be considered instead. Let b 1, b 2,... b L be the number of bits at the quantizer output for each ayer. The mutiayer quantizer can be interpreted as a coection of L scaar quantizers Q H() ( ) with H() = 2 bi the number of quantization eves for the -th ayer. Each quantizer spits the source space into H() equa ength intervas (or bins), T r = (t r 1, t r ], r = 1,..., H(), each one represented by its midde point, v r. Notice that the bins for the + 1-th ayer are determined from the bins of the previous ayer by spitting each bin into severa smaer bins depending on b +1. Given a source vector s = [s 1, s 2,..., s N ], the quantized vaues for the -th ayer are hence obtained as Q H() (s i ) = v r i = 1,..., N, (4) when s i fas into the r-th bin, i.e., t r 1 < s i t r. Net, the binary representation of the source vector is determined by encoding the quantized vaues to the corresponding b -bit words. It is important to assign propery the words to the quantizer bins to guarantee the successive refinement when the received signa is decoded. The quantization error for the -th ayer is computed as the distortion between the origina and quantized symbos, measured according to the MSE metric, i.e. [ s H() D = E Q (s) 2] H() tr = s v r 2 p(s) ds, (5) t r 1 where p(s) is the source pdf. Notice that when the encodermoduator tandem is propery designed to ensure that no errors occur during the transmission, the distortion after decoding the received signa is given by the quantization error in (5). In this situation, the distortion observed by a receiver r that is abe to decode the first ayers is ξ r = D and, hence, the number of quantization bits for each ayer can be determined from the distortion requirements of each area to satisfy that D ɛ. Net, the bits of each ayer are individuay encoded using a channe encoder and a digita moduator. Assuming the -th channe encoder has rate R c () and the -th moduator uses M() eves, the number of source symbos transmitted per channe use is given by R = R c() og 2 (M) b. (6) The parameters of the channe encoders and moduators are chosen in such way that R is the same for a the digita ayers, and it shoud be seected to ensure that the distortion requirements are feasibe. In genera, we focus on scenarios with R() 1, where digita communications ceary outperform anaog JSCC techniques. As epained earier, the observed symbos are decoded at the receivers using the SIC strategy. B. Anaog Layer As introduced in Section II, an anaog ayer is incorporated into the mutiayer scheme to improve the signa quaity of the receivers in the nearest region. In this ayer, the quantization errors are mapped using a particuar anaog JSCC scheme. Given a source vector s = [s 1, s 2,..., s N ], the eements of the error vector e q = [e q1, e q2,..., e qn ] are given by e qi = s i Q H (s i ) i = 1,..., N, (7) where H = 2 L bi is the tota number of bins considering the L digita ayers. In this work, we consider spherica codes based on the eponentiay chirped moduation to map the quantization errors. These codes have been shown to provide good performance and good scaing with the SNR for bandwidth epansion [8]. Using this type of spherica codes, the encoded data corresponding to the anaog ayer is obtained as [ a = cos(2πe q ), sin(2πe q ), cos(2παe q ), sin(2παe q ), ]..., cos(2πα ρ/2 1 e q ), sin(2πα ρ/2 1 e q ), (8) where α is the parameter that determines the mapping shape, is the normaization factor to ensure the power of the mapping output is equa to 1, and ρ represents the epansion factor. Since the quantization errors are rea-vaued, whie the digita symbos and the channe response are compe-vaued, the epansion factor of the anaog mapping is ρ = 2R. Note that athough ρ is a non integer vaue, we can achieve any epansion factor by using an adequate anaog K:S scheme, where part of the information is encoded with an 1:S spherica epansion code, whie the remaining is sent uncoded. After the quantization errors are coded, the resuting symbos are intereaved to generate the corresponding compe-vaued symbos and then superimposed to the data of the digita ayers. At the receiver, if the L digita ayers are successfuy decoded, an estimate of the quantization errors can be computed by using the Minimum Mean Square Error (MMSE) estimator. Let ˆ a be the vector of anaog symbos after decoding the L digita ayers. Then, the MMSE estimates for the quantization errors are cacuated from ê q = E[e ˆ a ] = 1 p(ˆ a ) + e p(ˆ a e)p(e) de. (9) The estimates are obtained by soving the integra numericay. C. Power Aocation An important issue is the aocation of the avaiabe transmit power among the L + 1 ayers. As commented, each digita ayer is empoyed to satisfy the quaity requirements for a given region. Thus, the first ayer shoud ensure that the receivers paced at a distance d 1 from the transmitter achieve the distortion target ɛ 1, and the same for the other ayers. In genera, this is accompished when the power aocated to the -th ayer guarantees an error free transmission of the quantized bits or with negigibe error probabiity to those receivers whose distance to the transmitter is d. Hence, we ISBN EURASIP

4 217 25th European Signa Processing Conference (EUSIPCO) design an iterative aocation scheme such that the digita encoder-moduator tandem is abe to provide a ow error probabiity at each ayer in the worst case, i.e., considering the argest distance in the -th region, d. In that point, the received power is Pr = P t /d 2 and the actua SNR for the -th ayer is w P r η = 1 + L+1 i=+1 w i P r, (1) where w is the power for the -th ayer and L+1 i=+1 w i represents the interference caused by the net ayers. Assuming that the encoder-moduator dupe in the -th ayer achieves a sufficienty ow error probabiity (e.g., p e 1 5 ) for SNRs arger than a given threshod η th, the minimum power which must be aocated to the -th ayer is determined from (1) by equating η to η th. From the transmit power constraint L+1 =1 w = P t, the term L+1 i=+1 w i can be rewritten as L+1 i=+1 w i = P t ( 1 ) w i + w and, therefore, (1) is transformed into η th = 1 + P r ( P t w P r ( 1 (11) )). (12) w i + w Since the above equation ony depends on the power assigned to previous ayers, we can determine the power aocation for each ayer iterativey as w = η th ( 1 + P r P r ( )) P t 1 w i ( ) 1 + η th = 1,..., L. (13) Once the power is aocated among the first L ayers, the remaining power is assigned to the ast ayer to improve the decoding of the cosest receivers, i.e., w L+1 = P t L w i. In the case of channe encoders based on iterative decoding and the echange of og-ikeihood ratio (LLR) messages, the SNR threshod from which the error bit probabiity drasticay decays can be determined using EXtrinsic Information Transfer (EXIT) Chart [12] or Density Evoution (DE) [13]. IV. RESULTS We net present the resuts of severa computer eperiments to iustrate the performance of the proposed HDA mutiayer scheme. The source symbos are generated from a zero-mean unit-variance Gaussian distribution and quantized using the mutiayer quantizer in Section III-A. The L digita ayers use the Low Density Parity Check () codes of the IEEE (WiMAX) standard, and PSK or QAM moduations. The number of quantizer bits, the code rates and the number of moduation eves are chosen to satisfy the distortion requirements and to ensure R is equa for the L digita ayers. The quantization errors are encoded using the anaog mappings described in Section III-B with a proper epansion factor. SDR (db) Anaog Uncoded Anaog Spira Anaog Ep Chirped Digita 4QAM 16QAM 64QAM 256QAM Fig. 3. Performance comparison between the HDA scheme with different mappings and the digita approaches with 2 ayers and R 1 = R 2 = 1/ Fig. 4. Performance of anaog chirped mappings for different vaues of α. Since the source information is anaog, the signa quaity is measured in terms of the Signa-to-Distortion Rate (SDR) SDR r = 1 og 1 (1/ξ r ), (14) where ξ r represents the MSE between the source and decoded symbos given by (3). In this work, system performance is measured in terms of the SDR with respect to the SNR at reception, which depends of the distance to the transmitter. In the first eperiment, we consider a broadcast transmission where receivers at distance d 18 m must decode the information with SDR r 4.5 db. If the transmit power is P t = 3 db, that distance corresponds to a SNR r 5 db. Since we have a singe requirement, the transmitter uses one digita ayer and one anaog ayer to refine the symbo estimates. In particuar, we chose as digita parameters b 1 = 2, R 1 c = 1/2 and M 1 = 4 (i.e., R 1 =.5). The anaog mapping epansion factor is hence 1:4 and the parameter α is set to 3. Figure 3 pots the SDRs obtained for the proposed scheme when the SNR at reception ranges from db to 3 db. This performance is aso compared to severa schemes that use other we-known strategies in the anaog ayer. Red and green curves correspond to the use of the 1:2 Archimedean spira ISBN EURASIP

5 217 25th European Signa Processing Conference (EUSIPCO) SDR (db) Anaog Uncoded Anaog Spira Anaog Ep Chirped Digita QPSK 8PSK 16QAM bandwidth epansion where the data is protected against the channe distortion by adding redundancy. As the epansion factor owers, the improvement of digita transmissions vanishes unti they even achieve inferior performance than anaog communications for bandwidth compression. Simiar resuts are epected for arger number of ayers since the gap between purey digita systems and the proposed HDA scheme is more reated to the rate R Fig. 5. Comparison of the the proposed HDA scheme to severa digita approaches when considering three ayers and an epansion factor R = 1. to map the quantization errors and the uncoded transmission of those, respectivey. Bue curves represent the performance when the refinement ayer is aso digita with a rate 1/2 code and different QAM moduation schemes depending on the number of bits in the quantization step. On one hand, the SDR vaues obtained for SNR r > 5 db are arger than the threshod vaue, SDR r 4.5 db. On the other hand, the anaog mapping based on the chirped moduation provides better performance than the other anaog schemes, athough it performs worse than the digita schemes. However, this comparison is not fair in practice because it is not possibe to impement severa digita schemes in a singe ayer and, consequenty, we shoud choose one of them. Hence, if we restrict to a particuar digita scheme, ony some receivers wi show better performance, whie a users in the inner region ehibit good behaviour with the anaog scheme. Figure 4 compares the performance of the proposed mutiayer scheme with the eponentiay chirped mappings when considering different vaues of α. As observed, increasing α improves the system performance in the high SNR region (cosest receivers) but penaizes the furthest receivers. The opposite behaviour is observed when α owers. It is hence important to estabish a trade-off by seecting α vaues which provide good performance for the whoe range of SNRs. In the second eperiment, the requirements are SDR 1 r 4.5 db and SDR 2 r 1 db for SNR 1 r > 1 db and SNR 2 r > 14 db, respectivey. The mutiayer scheme hence comprises two digita ayers and the anaog one. Since the requirements are ess restrictive in this case, the digita components are chosen such that R = 1, i.e., one source symbo per channe use is now transmitted. Figure 5 shows the performance curves obtained with the same schemes as in the previous eperiment for the refinement ayer: uncoded transmission, 1:2 Archimedean spira, chirped moduation mappings and purey digita. Simiar behaviour is observed for the different schemes. However, the mutiayer scheme with chirped mappings cosey approaches the fuy digita mutiayer schemes. This is because digita transmissions are more suitabe for V. CONCLUSION A HDA mutiayer scheme for the transmission of common data from one transmitter to severa receivers considering different QoS requirements has been proposed. Such scheme shows good behaviour and scaing for a the range of SNRs, as we as for different epansion factors and number of ayers, with respect to previous anaog and digita strategies. The system stands out for its feibiity since both the digita ayers and the anaog mappings can be configured depending on the considered scenario and the distortion constraints. ACKNOWLEDGMENT This work has been funded by Office of Nava Research Goba of United States, the Xunta de Gaicia (ED431C , ED341D R216/12, ED431G/1), the Agencia Estata de Investigación of Spain (TEC C4-1-R, TEC REDC, TEC C4-1-R) and ERDF funds of the EU (AEI/FEDER, UE). REFERENCES [1] T.Cover, Broadcast channes, IEEE Transactions on Information Theory, vo. 18, no. 1, pp. 2 14, January [2] W. H. R. Equitz and T. M. 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