Ethan R. Duni

Analysis of Multiple-Antenna Systems With Finite-Rate Feedback Using High-Resolution Quantization Theory

This paper considers the development of a general framework for the analysis of transmit beamforming methods in multiple-antenna systems with finite-rate feedback. Inspired by the results of classical high-resolution quantization theory, the problem of finite-rate quantized communication system is formulated as a general fixed-rate vector quantization problem with side information available at the encoder (or the quantizer) but unavailable at the decoder. The framework of the quantization problem is sufficiently general to include quantization schemes with general non-mean-squared distortion functions and constrained source vectors. Asymptotic distortion analysis of the proposed general quantization problem is provided by extending the vector version of the Bennetts integral. Specifically, tight lower and upper bounds of the average asymptotic distortion are proposed. Sufficient conditions for the achievability of the distortion bounds are also provided and related to corresponding classical fixed-rate quantization problems. The proposed general methodology provides a powerful analytical tool to study a wide range of finite-rate feedback systems. To illustrate the utility of the framework, we consider the analysis of a finite-rate feedback multiple-input single-output (MISO) beamforming system over independent and identically distributed (i.i.d.) Rayleigh flat-fading channels. Numerical and simulation results are presented that further confirm the accuracy of the analytical results

Analysis of multiple antenna systems with finite-rate feedback using high resolution quantization theory

This paper considers the development of a general framework for the analysis of transmit beamforming methods in multiple-antenna systems with finite-rate feedback. Inspired by the results of classical high-resolution quantization theory, the problem of finite-rate quantized communication system is formulated as a general fixed-rate vector quantization problem with side information available at the encoder (or the quantizer) but unavailable at the decoder. The framework of the quantization problem is sufficiently general to include quantization schemes with general non-mean-squared distortion functions and constrained source vectors. Asymptotic distortion analysis of the proposed general quantization problem is provided by extending the vector version of the Bennetts integral. Specifically, tight lower and upper bounds of the average asymptotic distortion are proposed. Sufficient conditions for the achievability of the distortion bounds are also provided and related to corresponding classical fixed-rate quantization problems. The proposed general methodology provides a powerful analytical tool to study a wide range of finite-rate feedback systems. To illustrate the utility of the framework, we consider the analysis of a finite-rate feedback multiple-input single-output (MISO) beamforming system over independent and identically distributed (i.i.d.) Rayleigh flat-fading channels. Numerical and simulation results are presented that further confirm the accuracy of the analytical results

High-Rate Optimized Recursive Vector Quantization Structures Using Hidden Markov Models

This paper examines the design of recursive vector quantization systems built around Gaussian mixture vector quantizers. The problem of designing such systems for minimum high-rate distortion, under input-weighted squared error, is discussed. It is shown that, in high dimensions, the design problem becomes equivalent to a weighted maximum likelihood problem. A variety of recursive coding schemes, based on hidden Markov models are presented. The proposed systems are applied to the problem of wideband speech line spectral frequency (LSF) quantization under the log spectral distortion (LSD) measure. By combining recursive quantization and random coding techniques, the systems are able to attain transparent quality at rates as low as 36 bits per frame

A High-Rate Optimal Transform Coder With Gaussian Mixture Companders

This paper examines the problem of designing fixed-rate transform coders for sources whose distributions are unknown and presumably non-Gaussian, under input-weighted squared error distortion measures. As a component of this system, a flexible scalar compander based on Gaussian mixtures is proposed. The high-rate analysis of transform coders is reviewed, and extended to the case of input-weighted squared error. An algorithm is developed to set the parameters of the system using a data-driven technique that automatically balances the source statistics, distortion measure, and structure of the transform coder to minimize the high-rate distortion. The implementation of Gaussian mixture companders is explored, resulting in a flexible, low-complexity scalar quantizer. Additionally, modifications to the system for operation at moderate rates, using unstructured scalar quantizers, are presented. The operation of the system for the problem of wideband speech line spectral frequencies (LSF) quantization with log spectral distortion is illustrated, and shown to provide good performance with very low complexity

High-Rate Vector Quantization for Noisy Channels With Applications to Wideband Speech Spectrum Compression

This paper considers the high-rate performance of source coding for noisy discrete symmetric channels with random index assignment (IA). Accurate analytical models are developed to characterize the expected distortion performance of vector quantization (VQ) for a large class of distortion measures. It is shown that when the point density is continuous, the distortion can be approximated as the sum of the source quantization distortion and the channel-error induced distortion. Expressions are also derived for the continuous point density that minimizes the expected distortion. Next, for the case of mean squared error distortion, a more accurate analytical model for the distortion is derived by allowing the point density to have a singular component. The extent of the singularity is also characterized. These results provide analytical models for the expected distortion performance of both conventional VQ as well as for channel-optimized VQ. As a practical example, compression of the linear predictive coding parameters in the wideband speech spectrum is considered, with the log spectral distortion as performance metric. The theory is able to correctly predict the channel error rate that is permissible for operation at a particular level of distortion.

High-Rate Analysis of Vector Quantization for Noisy Channels

In this paper, the sensitivity of the high-rate performance of conventional source coding to symmetric channel errors (i.e., a channel where all index errors are equally likely) with arbitrary distortion measures is analyzed. It is shown that, in general, the overall distortion due to source quantization and channel errors cannot be expressed as the sum of the distortion due to the finite bit representation of the source and the distortion due to channel errors. An exception to this is when the distortion is measured as the mean-squared error. The binary symmetric channel with random index assignment is a special case of the analysis, and as the number of code-points gets large, the performance approaches a nonzero constant. Finally, the framework is applied to the wideband speech spectrum quantization problem, where it correctly predicts the channel error rate permissible for operation at a particular distortion level

Improved quantization structures using generalized HMM modelling with application to wideband speech coding

In this paper, a low-complexity, high-quality recursive vector quantizer based on a generalized hidden Markov model of the source is presented. Capitalizing on recent developments in vector quantization based on Gaussian mixture models, we extend previous work on HMM-based quantizers to the case of continuous vector-valued sources, and also formulate a generalization of the standard HMM. This leads us to a family of parametric source models with very flexible modelling capabilities, with which are associated low-complexity recursive quantization structures. The performance of these schemes is demonstrated for the problem of wideband speech spectrum quantization, and shown to compare favorably to existing state-of-the-art schemes.

Online training methods for Gaussian Mixture Vector Quantizers

This paper presents techniques relevant to the online training of Gaussian mixture vector quantizer (GMVQ) systems. techniques for learning from quantized data are considered, which enables online training configurations wherein the training is carried out remotely from the encoder. Next, methods for recursive training are presented, which eliminate the need to store large databases of example data, and also enable adaptive operation of the GMVQ system. These techniques are demonstrated on the problem of wideband speech spectrum quantization, and the performance losses due to the use of quantized training data are experimentally quantified as a function of the bit rate.

High-Rate Design of Transform Coders with Gaussian Mixture Companders

This paper examines the problem of designing fixed-rate transform coders for sources with arbitrary distributions, under input-weighted squared error distortion measures. As a component of this system, a flexible scalar compander using Gaussian mixtures is proposed. An algorithm is developed to set the parameters of the system using a data-driven technique that automatically balances the source statistics, distortion measure, and structure of the transform coder to minimize the high-rate distortion. The implementation of Gaussian mixture companders is explored, resulting in a flexible, low-complexity scalar quantizer. The operation of this system for the problem of wideband speech spectrum quantization with log spectral distortion is illustrated, and shown to provide good performance with very low, rate-independent complexity

High-rate training of Gaussian mixture vector quantizers

Summary form only given. This paper discusses the design of fixed-rate Gaussian mixture vector quantizers (GMVQs) under input-weighted squared error distortion measures. The goal is to select the system parameters so as to minimize the expected high-rate distortion. GMVQ systems produce low complexity by operating M Gaussian codebooks in parallel (typically with low-complexity structures) and then choosing amongst their outputs with an M-point vector quantizer. Thus, the total codebook is the union of the component Gaussian codebooks, and the total encoder regions are optimal provided the component encoders are optimal with respect to their individual codebooks