L. Ozarow, "On a source-coding problem with two channels and three receivers," Bell Syst. Tech. J. 59, pp. 1909.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the source with distortions d 1 and d 2 , whereas the central receiver reconstructs the source with a smaller distortion d 0 . A theoretic solution to the MD problem, based on Shannon s random coding approach, was published in [6] For specific cases, the converse proof was found in [1] [15], 13] and [23] yet, the complete solution for the MD problem is still unknown. The practical interest in this problem generated quantization schemes for compression with multiple descriptions. Early solutions were based on the natural correlation in the signal that remains even after ....

....the ECDQ to the multiple description schemes, we found that both the TSDQ and the dithered MDSQ schemes become equivalent to sets of additive noise channels. Using this equivalence, we compare the performance of the proposed encoders to the optimum solutions for the quadratic memoryless Gaussian [15] and the non Gaussian [23] cases. The total rate (R 1 R 2 ) achieved by the entropy coded dithered periodic MDSQ (PMDSQ) exceeds the theoretical bounds by log( 0:5 bit per sample (corresponding to 1.5dB increase in distortion per each description) at high resolution for any smooth ....

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L. H. Ozarow, On a source coding problem with two channels and three receivers, Bell Sys. Tech. Jour. 59 (1980.

.... find the set of achievable values for the quintuple (R 1 ; R 2 ; D 0 ; D 1 ; D 2 ) El Gamal and Cover determined an achievable rate region for general memoryless sources [10] while Ozarow showed that this region is tight for the case of memoryless Gaussian sources and squared error distortions [23]. Ahlswede [1] studied the case of no excess rate (when there is equality in (3) and Zhang and Berger [39] considered the no excess marginal rate case (when there is equality in (1,2) Zhang and Berger also showed by counterexample that in the excess rate case the achievable region of El Gamal ....

....this regime, performance of the systems considered will not be predicted well by the theory, and there is much to be learned by means of numerical simulations. We conclude in Section 5. II. Asymptotic Multiple Description Rate Region For a memoryless Gaussian source with variance oe , Ozarow [23] gave an explicit characterization of the set of achievable distortions (D 0 ; D 1 ; D 2 ) for a given pair of rates R 1 ; R 2 : 4) 5) D 0 1 Gamma ( Pi Gamma Delta) 6) where Pi = 1 Gamma D 1 =oe ) 1 Gamma D 2 =oe and Delta = D 1 D 2 =oe ) Gamma ....

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L. Ozarow. On a source-coding problem with two channels and three receivers. Bell Syst. Tech. J., 59(10):1909.

....for up to e=n k erasures but then drops very sharply when the actual number of erasures is greater than e. An alternative approach for reliable transmission of multimedia data that provides graceful degradation of performance in presence of channel noise is Multiple Description Coding (MDC) [6], 7] In MDC, two or more independently decodable descriptions of the source are sent to the receiver (Fig. 1) If only description S 1 (or S 2 ) is received, the signal can be reconstructed with acceptable side distortion D S1 (or D S2 ) If both descriptions are received, the distortion ....

....there are three different decoders, each corresponding to one of the possible loss scenarios. In an ideal MDC channel environment, the channels are independent and data on each channel is either completely lost or received intact. This environment has been studied extensively both theoretically, [6], 7] and practically, 8] 9] 10] 11] 12] etc. The paper by Goyal [5] provides a good overview of MDC systems. Figure 1 Generic MDC system. The source is encoded in two descriptions, S 1 and S 2 . They are transmitted through the network over different channels. At the decoder, if ....

L. Ozarow. "On a source-coding Problem with two Channels and three Receivers". Bell System Technical Journal, vol. 59, page(s) 1909.

....Performance of MDVQ with the initial and the MD BSA optimized index assignments for a Gaussian source signal. The rate is 2 bits per source sample per description. It is interesting to compare the performance of the optimized MDVQ scheme with the rate distortion bounds for multiple descriptions [8] and the MDSQ [1] for the Gaussian source with zero mean and unit variance. In Table I the side distortions of several schemes that all have the same central distortions are compared. As stated above, the vector quantizers had a dimension of two and indexes with 6 bits (central distortion of ....

....distortion side distortion (central SNR dB) side SNR dB) OPTA MDVQ (dim. 2) MDSQ 0.030 0.063 0.153 0.190 (15.2) 12.0) 8.2) 7.2) 0.015 0.083 0.256 0.482 (18.2) 10.8) 5.9) 3. 2) TABLE I SIDE DISTORTIONS OF THE OPTIMIZED MDVQ, THE RATE DISTORTION BOUND FOR MULTIPLE DESCRIPTIONS [8] (OPTA) AND THE MDSQ [1] FOR THE SAME CENTRAL DISTORTIONS. ALL SYSTEMS HAVE TWO DESCRIPTIONS, EACH WITH A RATE OF 2 BITS PER SOURCE SAMPLE. mapped to two 4 bit descriptions. The side distortions for the optimized MDVQ were measured and inserted into the table. At the same central distortions ....

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L. Ozarow, "On a source-coding problem with two channels and three receivers, " The Bell System Technical Journal, vol. 59, no. 10, pp. 1909.

....the Rayleigh amplitude, and therefore no soft information is used for decoding. II. MULTIPLE DESCRIPTION BLOCK CODES Following the information theoretical definition of the multiple description problem and the characterization of the region of achievable rates for the memoryless Gaussian source [5], Vaishampayan s Multiple Description Scalar Quantizers (MDSQ) 1] constituted the first constructive approach towards a practical MD scheme. The basic idea of MDSQ is to generate two coarse side descriptions of a scalar source sample using two independent scalar quantizers. The latter refine ....

L.Ozarow, "On a source-coding problem with two channels and three receivers, " Bell Systems Technical Journal, vol. 59, pp. 1909.

....the Rayleigh amplitude, and therefore no soft information is used for decoding. II. MULTIPLE DESCRIPTION BLOCK CODES Following the information theoretical definition of the multiple description problem and the characterization of the region of achievable rates for the memoryless Gaussian source [5], Vaishampayan s Multiple Description Scalar Quantizers (MDSQ) 1] constituted the first constructive approach towards a practical MD scheme. The basic idea of MDSQ is to generate two coarse side descriptions of a scalar source sample using two independent scalar quantizers. The latter refine ....

L.Ozarow, "On a source-coding problem with two channels and three receivers, " Bell Systems Technical Journal, vol. 59, pp. 1909.

.... the Properties of Multiple Description Codes Jo ao Barros, Joachim Hagenauer Institute for Communications Engineering (LNT) Munich University of Technology(TUM) Arcisstr.21, D 80333 M unchen joao.barros ei.tum.de Abstract The traditional formulation of the multiple description problem [1] is based on a two channel communications system like the one shown in Figure 1. Two distinct yet correlated descriptions of the source are sent over each channel with rates R 1 and R 2 . If both channels work, the receiver is able to reconstruct the original information with distortion D 0 . ....

....theoretical point of view, the multiple description problem still o ers many open questions. In fact, the achievable rate distortion region, i.e. the set of achievable rates (R 1 ; R 2 ; D 1 ; D 2 ; D 3 ) is only known for a memoryless Gaussian source with a squared error distortion measure [1], which is described by D 0 2 2(R 1 R 2 ) 1 ( 2 ; 1) with = D 1 D 2 2 2(R1 R2 ) and = 1 D 1 ) 1 D 2 ) Among the developments based on this theoretical result, two main coding techniques have emerged, namely Multiple Description Scalar Quantization (MDSQ, 2] and ....

L. Ozarow, \On a Source-Coding Problem with Two Channels and Three Receivers", Bell Systems Technical Journal, Vol.59, No.10,pp.1909.

.... description (MD) source coding asks the question of what is the minimum distortion ED(X; g 0 (f 1 (X) f 2 (X) under the constraints that ED(X; g i (f i (X) D i and H(g i (f i (X) R i (i = 1; 2) Work on information theoretic bounds on the performance of these codes has been reported in [4, 9, 13], and work on the design of codes and on their performance analysis have been reported in [3, 5, 12] among (many) others. Multiple description codes are of interest in practical scenarios because they admit a natural interpretation in terms of providing resilience against the type of errors ....

L. H. Ozarow. On a Source Coding Problem with Two Channels and Three Receivers. Bell Syst. Tech. J., 59(10):1909-1921, 1980.

....over two unreliable channels, so that the signal can be reproduced with a good quality even if only one of the two descriptions is available to the decoder. The multiple description coding paradigm was introduced in the early eighties in the information theory community (see for instance Ozarow [8] and El Gamal Cover [5] Recent applications of such coders include data transmission over packet switched networks or diversity systems. This paper describes a new method for the design of multiple description codes based on unconstrained vector quantizers. We assume the existence of a ....

L. Ozarow. On a source coding problem with two channels and three receivers. Bell Syst. Tech. J., 59:1909-1921, December 1980.

....coding scheme using a vector quantizer formulation. Although many multiple description schemes are not defined as vector quantizers, it is shown, as in many other coding paradigms, that vector quantizers asymptotically achieve the theoretically optimal performance. These bounds are presented in [79], 38] and [34] among others, and we describe one of them briefly. 5.1.1 Transmission model The multiple description coding paradigm has been introduced in the early eighties to model data transmission on a diversity system. A coder sends # 1 1 # # 0 Figure 5.1: Block diagram of a ....

L. Ozarow. On a source coding problem with two channels and three receivers. Bell Syst. Tech. J., 59:1909.

....interest in Multiple Description (MD) coding. Primary reason for this is that MD coding is a form of joint source channel coding that is well suited to transmission over channels such as lossy packet networks. MD coding itself is not new, however. First references on the topic include [3] and [4] and date back to early 1980 s. The original problem was posed as follows. Source coder produces two descriptions of the source at rates R 1 and R 2 ; as shown in Figure 1.1. These descriptions are sent over two dierent channels. At the receiver, each description is either received correctly or ....

L. Ozarow, "On a source coding problem with two channels and three receivers," Bell Syst. Tech. J., vol. 59, pp. 1909-1921, Dec. 1980.

....: Ek X GammaX k dg I(X; X) and I( Delta) denotes mutual information. A single letter characterization of the set of achievable distortions DX (R 1 ; R 2 ) for the case of a memoryless ( white ) Gaussian source of variance oe and squared error distortion measure was found by Ozarow [Oza80]. This set, denoted D ; R 1 ; R 2 ) and its inverse set ; d 0 ; d 1 ; d 2 ) are described in the next section. For a general source, however, the complete solution for RX (d 0 ; d 1 ; d 2 ) is still unknown. El Gamal and Cover [GC82] characterize an inner bound for RX (d 0 ; d 1 ; d 2 ....

....bound above becomes tight, and is realized by the optimum Gaussian test channel [GC82] This result parallels the well known asymptotic tightness of the Shannon lower bound (right hand side of (4) at high resolution; see [Ber71, LZ94] II. Ozarow s Solution of the White Gaussian Case Ozarow [Oza80] proved that the set of achievable squared distortions D ; R 1 ; R 2 ) for a memoryless (i.i.d. Gaussian source with variance oe is given by the union of all triplets d 0 ; d 1 ; d 2 satisfying 1 Gamma Gamma fi Gamma p Delta 2 (6c) where = 1 Gamma d 1 =oe ....

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L. H. Ozarow. On the source coding problem with two channels and three receivers. Bell Sys. Tech. Jour., 59:1909--1922, 1980.

....the concurrent limitations on qualities of these descriptions taken separately and jointly This problem would come to be known as the mttltple description problem. The primary theoretical results in this area were provided by the aforementioned researchers along with Ahlswede, 1We may infer from [3], 4] 5] that the problem originated with Gersho and Witsenhausen. Gersho proposed a scheme (unpublished) that utilized the redundancy in coded speech waveforms to mitigate the effects of a channel breakdown and Witsenhausen subsequently studied the more general, abstract problem. SOURCE ....

....for a binary source and Hamming distortion. At that very meeting, Cover and E1 Gamal determined and reported the achievable rate region later published in [2] Ozarow s contribution was to show the tightness of the E1 Gamal Cover region for memoryless Gaussian sources and squared error distortion [3]. Subsequently, Ahlswede [18] showed that the E1 Gamal Cover region is tight in the no excess rate sum case (where there is equality in (1) and Zhang and Berger [6] showed that this region is not tight when there is excess rate. The complementary situation, where (2) 3) hold with equality, is ....

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L. Ozarow. On a source-coding problem with two channels and three receivers. Bell Syst. Tech. J., 59(10):1909.

....performance if a minimal subset of descriptions is available and to achieve increasingly better performance if more descriptions are available. Early work on multiple descriptions considered the case of two descriptions. Theoretical results for the achievable rate region include results in [44, 79]. Currently the achievable rate distortion is completely known only for the two channel memoryless Gaussian source. Recent work in the area of multiple descriptions includes [80, 81, 82, 83, 84, 85, 86, 87] Previous work on MD coding includes methods which allow a wide range of redundancy levels ....

L. Ozarow, "On a source coding problem with two channels and three receivers," Bell System Technical Journal, vol. 59, Dec.

....frame erasure channels. The performance of MR DPCM over packet switched networks and mobile radio channels is obtained. A. Multiple Description Scalar Quantizer The multiple description problem was rst formulated in [2] and rate distortion bounds for memoryless gaussian sources was derived in [3], 2] The problem of multiple description quantizer design for independent source samples was described in [4] The block diagram of the multiple description scalar quantizer (MDSQ) system is shown in Figure 1. Assume the source has pdf p X ( where X is a random variable with zero mean and ....

L. Ozarow , \On a source coding problem with two channels and three receivers," The Bell System Technical Journal, vol. 59, pp.

....robust transmission. Source Coder Description 1 (R1) Description 2 (R2) Decoder 1 (D1) Decoder 0 (D0) Decoder 2 (D2) Fig. 1: The multiple description problem Early papers on MDC are information theoretic and try to find the set of achievable values for the quintuple (R 1 ; R 2 ; D 0 ; D 1 ; D 2 ) [1, 2]. More recent papers consider the problem of designing practical multiple description systems [3, 4, 5] In [4, 5] a blockwise transform is applied to the input vector to obtain the multiple description property. The input vector is usually a jointly Gaussian vector, the basic idea being to ....

L.Ozarow, "On a source coding problem with two channel and three receivers". Bell Syst. Tech. J., 59(10): pp.

....of bits generally results in a single channel reconstruction with distortion, D 1 , that is unacceptably high. Theoretical work in the early 1980 s developed rate distortion bounds governing the tradeoff between two channel (D 0 ) and single channel (D 1 ) distortions for various sources [1, 2, 3]. For the Gaussian source [2] the lower bound establishes that any partitioning of the SDC bitstream forces D 1 to be very high. To reduce D 1 below this bound, the total bitrate over both channels must exceed R . The extra bitrate (beyond R ) does not effect D 0 = D , but allows dependencies ....

....in a single channel reconstruction with distortion, D 1 , that is unacceptably high. Theoretical work in the early 1980 s developed rate distortion bounds governing the tradeoff between two channel (D 0 ) and single channel (D 1 ) distortions for various sources [1, 2, 3] For the Gaussian source [2], the lower bound establishes that any partitioning of the SDC bitstream forces D 1 to be very high. To reduce D 1 below this bound, the total bitrate over both channels must exceed R . The extra bitrate (beyond R ) does not effect D 0 = D , but allows dependencies to be incorporated into the ....

L. Ozarow, "On a source coding problem with two channels and three receivers," The Bell System Technical Journal, vol. 59, p.

....of MD coding consists then of minimizing the distortion in the presence of loss of data, given some allowed redundancy ae. Early papers on MD coding are information theoretic in nature, and deal with the problem of finding sets of achievable values for the quintuple (R 1 ; R 2 ; D 0 ; D 1 ; D 2 ) [3, 7]. More recent papers however consider the problem of designing practical MD systems. MD quantizers are designed in [8, 10] A suitable blockwise transform is applied to an input vector before coding to obtain the MD property in [4, 6] the input vector is usually a jointly Gaussian vector, the ....

....sources, there is a large gap between the performance of the the proposed filter banks and that of the MD rate region; however, this gap is reduced as the memory in the source increases. 2 The Multiple Description Rate Region For a memoryless Gaussian source with variance oe 2 , Ozarow [7] has found an explicit characterization of the set of achievable distortions (d 0 ; d 1 ; d 2 ) for a given pair of rates R 1 ; R 2 . The inverse of the functions presented in [7] are the following [5, 7] R 1 1 2 log oe 2 d 1 (1) R 2 1 2 log oe 2 d 2 (2) R 1 R 2 1 2 ....

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L.Ozarow, "On a Source Coding Problem with Two Channels and Three Receivers". Bell Syst. Tech. J., 59(10):

....First, mapping the information, generally a real number, to a set of integers (designing a quantizer) second, mapping each integer to the set of components to be transmitted over the channels (index assignment) This paper deals with the second part of the problem. The relevant works include [3] [4], 1] 12] 10] 11] 6] 8] 2] Applications of this problem arise in video and speech communication. To the best of our knowledge, this paper is rst to give lower bounds for the problem, and provide optimal algorithms that achieve the lower bound. In addition, the general case of k ....

Ozarow, L., \On a source coding problem with two channels and three receivers", Bell Syst. Tech. J., 59 (

....3. Maximum pairwise di erence of the bigs and smalls lists of an arrangement is a lower bound on the maximum spread of the arrangement, spread(A) That is, for all A max j fbig(A) j small(A) j g spread(A) This statement is known as the Ski Instructor problem and the proof can be found in [21]. Since for the bounding sequences hbi and hai we have b j big(A) j and a j small(A) j for all A and j, then pairing smaller b j with smaller a j gives a lower bound on the spread for all possible arrangements. For all A: max j (b j a j ) max j fbig(A) j small(A) j g spread(A) The ....

Ozarow, L., \On a source coding problem with two channels and three receivers", The Bell System Technical Journal, 59 (

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L. Ozarow, "On a source-coding problem with two channels and three receivers," Bell Syst. Tech. J. 59, pp. 1909.

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L. Ozarow. On a source-coding problem with two channels and three receivers. Bell Syst. Tech. J., 59(10):1909.

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L. Ozarow. On a source-coding problem with two channels and three receivers. The Bell System Technical Journal, 59(10), Dec. 1980.

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L. Ozarow. On a source-coding problem with two channels and three receivers. Bell Syst. Tech. J., 59(10):1909.

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L.Ozarow, "On a Source Coding Problem with Two Channels and Three Receivers". Bell Syst. Tech. J., 59(10):1909-1921, 1980.

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L.Ozarow, "On a source coding problem with two channel and three receivers". Bell Syst. Tech. J., 59(10): pp. 1909-1921, December 1980.

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L. Ozarow, "On a source coding problem with two channels and three receivers," The Bell Syst. Tech. J., vol. 59, p. 1921, Dec. 1980.

No context found.

L. H. Ozarow, "On a source coding problem with two channels and three receivers," Bell Syst. Tech. J., vol. 59, pp. 1909.

No context found.

L. Ozarow, "On a source coding problem with two channels and three receivers," Bell System Technical Journal, vol. 59, pp. 1909.

No context found.

L. Ozarow, "On a Source Coding Problem with Two Channels and Three Receivers," Bell Syst. Tech. J., vol. 59, pp. 1909.

No context found.

L. Ozarow, "On a source-coding problem with two channels and three receivers, " Bell Syst. Tech. J., vol. 59, no. 10, pp. 1909-1921, Dec. 1980.

No context found.

L. Ozarow, "On a source coding problem with two channels and three receivers," Bell System Technical Journal, vol. 59, pp. 1909.

No context found.

L. Ozarow, "On a source coding problem with two channels and three receivers," Bell Syst. Tech. J., vol. 59, pp. 1909.

No context found.

L. Ozarow. On a source-coding problem with two channels and three receivers. Bell Syst. Tech. J., 59(10):1909.

No context found.

L. Ozarow, "On a source-coding problem with two channels and three receivers," The Bell System Technical Journal, vol. 59, no. 10, pp. 1909-1921, Dec. 1980.

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