Binary Lattice Vector Quantization with Linear Block Codes and Affine Index Assignments

We determine analytic expressions for the performance of some low-complexity combined source-channel coding systems. The main tool used is the Hadamard transform. In particular, we obtain formulas for the average distortion of binary lattice vector quantization with affine index assignments, linear block channel coding, and a binary-symmetric channel. The distortion formulas are specialized to nonredundant channel codes for a binary-symmetric channel, and then extended to affine index assignments on a binary-asymmetric channel. Various structured index assignments are compared. Our analytic formulas provide a computationally efficient method for determining the performance of various coding schemes. One interesting result shown is that for a uniform source and uniform quantizer, the natural binary code is never optimal for a nonsymmetric channel, even though it is known to be optimal for a symmetric channel.

[1]  R. Gallager.  Information Theory and Reliable Communication , 1968 .

[2]  Kenneth Rose,et al.  Combined source-channel vector quantization using deterministic annealing , 1994, IEEE Trans. Commun..

[3]  I. Dostis,et al.  The effects of digital errors on PCM transmission of compandored speech , 1965 .

[4]  Qing Chen,et al.  Image coding using robust quantization for noisy digital transmission , 1998, IEEE Trans. Image Process..

[5]  Mikael Skoglund,et al.  Theory for transmission of vector quantization data , 1995 .

[6]  Thomas R. Crimmins.  On encoding and decoding maps for group codes (Corresp.) , 1976, IEEE Trans. Inf. Theory.

[7]  Nariman Farvardin,et al.  A study of vector quantization for noisy channels , 1990, IEEE Trans. Inf. Theory.

[8]  K. Zeger,et al.  Asymptotic Bounds on Optimal Noisy Channel QuantizationVia Random Coding , 1994 .

[9]  Amy R. Reibman,et al.  Asymptotic quantization for signal estimation with noisy channels , 1991, [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing.

[10]  Qing Chen,et al.  Robust quantization for image coding and noisy digital transmission , 1996, Proceedings of Data Compression Conference - DCC '96.

[11]  Allen Gersho,et al.  Vector quantization and signal compression , 1991, The Kluwer international series in engineering and computer science.

[12]  Masao Kasahara,et al.  A construction of vector quantizers for noisy channels , 1984 .

[13]  O. J. Tretiak,et al.  Design considerations in PCM transmission of low-resolution monochrome still pictures , 1967 .

[14]  David G. Daut,et al.  Combined Source-Channel Coding of Images Using the Block Cosine Transform , 1981, IEEE Trans. Commun..

[15]  G. Robert Redinbo,et al.  The optimum mean-square estimate for decoding binary block codes , 1974, IEEE Trans. Inf. Theory.

[16]  David L. Neuhoff,et al.  Optimal binary index assignments for a class of equiprobable scalar and vector quantizers , 1995, IEEE Trans. Inf. Theory.

[17]  Jerry D. Gibson,et al.  Hamming Coding of DCT-Compressed Images Over Noisy Channels , 1984, IEEE Trans. Commun..

[18]  Ke-Yen Chang,et al.  Analysis, Optimization, and Sensitivity Study of Differential PCM Systems Operating on Noisy Communication Channels , 1972, IEEE Trans. Commun..

[19]  Thomas R. Crimmins,et al.  Minimization of mean-square error for data transmitted via group codes , 1969, IEEE Trans. Inf. Theory.

[20]  D. J. Goodman,et al.  Using simulated annealing to design digital transmission codes for analogue sources , 1988 .

[21]  Thomas R. Crimmins,et al.  Mean-square-error optimum coset leaders for group codes , 1970, IEEE Trans. Inf. Theory.

[22]  K. Zeger,et al.  Tradeoff between source and channel coding , 1997, Proceedings of IEEE International Symposium on Information Theory.

[23]  Robert M. Gray,et al.  Joint source and noisy channel trellis encoding , 1981, IEEE Trans. Inf. Theory.

[24]  P. Noll,et al.  Effects of channel errors on the signal-to-noise performance of speech-encoding systems , 1975, The Bell System Technical Journal.

[25]  Hai-Shan Wu,et al.  Index allocation in vector quantisation for noisy channels , 1993 .

[26]  Allen Gersho,et al.  Pseudo-Gray coding , 1990, IEEE Trans. Commun..

[27]  G. Robert Redinbo.  On Minimum Mean-Square Error Linear Block Codes When the Data have q-adic Weighting , 1974, Inf. Control..

[28]  Allen Gersho,et al.  Vector quantizer design for memoryless noisy channels , 1988, IEEE International Conference on Communications, - Spanning the Universe..

[29]  R. Hagen,et al.  Robust Vector Quantization by Linear Mappings of Block-codes , 1993, Proceedings. IEEE International Symposium on Information Theory.

[30]  Richard E. Ladner,et al.  Codebook organization to enhance maximum a posteriori detection of progressive transmission of vector quantized images over noisy channels , 1993, 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[31]  Robert M. Gray,et al.  The design of joint source and channel trellis waveform coders , 1987, IEEE Trans. Inf. Theory.

[32]  R. Zelinski.  Effects of transmission errors on the mean-squared error performance of transform coding systems , 1979 .

[33]  K. Zeger,et al.  On the performance of affine index assignments for redundancy free source-channel coding , 1995, Proceedings DCC '95 Data Compression Conference.

[34]  N. Jayant,et al.  Digital Coding of Waveforms: Principles and Applications to Speech and Video , 1990 .

[35]  King Ngi Ngan,et al.  Error detection and correction of vector quantised digital images , 1990 .

[36]  P. Wintz,et al.  Quantizing for Noisy Channels , 1969 .

[37]  Erik Agrell,et al.  The Hadamard transform-a tool for index assignment , 1996, IEEE Trans. Inf. Theory.

[38]  K. Zeger,et al.  Affine index assignments for binary lattice quantization with channel noise , 1995, Proceedings of 1995 IEEE International Symposium on Information Theory.

[39]  R. Hagen,et al.  Design methods for VQ by linear mappings of block codes , 1994, Proceedings of 1994 IEEE International Symposium on Information Theory.

[40]  J. W. Modestino,et al.  Combined Source-Channel Coding of Images , 1978, IEEE Trans. Commun..

[41]  Roar Hagen,et al.  Robust Vector Quantization by a Linear Mapping of a Block Code , 1999, IEEE Trans. Inf. Theory.

[42]  NARIMAN FARVARDIN,et al.  Optimal quantizer design for noisy channels: An approach to combined source - channel coding , 1987, IEEE Trans. Inf. Theory.

[43]  Jerry D. Gibson,et al.  A constrained joint source/channel coder design , 1994, IEEE J. Sel. Areas Commun..

[44]  Nariman Farvardin,et al.  On the performance and complexity of channel-optimized vector quantizers , 1991, IEEE Trans. Inf. Theory.