Analysis and applications of some practical source coding systems

Abstract

This dissertation analyzes several practical data compression systems using an asymptotic approximation for quantizers with small step sizes. By using this high resolution quantization theory we present a simple formulation to analyze many source coding systems in the same framework and give practical applications of such systems. The first part of the research focuses on the analysis of a digital transmission system when the input to the system is a continuous time Gauss-Markov process of any order. By using fine quantization approximations we derive expressions for the time-average smoothed error for different quantization systems. We formulate our problem in a state space framework. The quantization systems that we study are: i) vector quantization of the original process, ii) state component and state vector quantization, iii) differential state quantization, iv) a scheme of quantizing the complex envelope of a narrowband process, and v) a sigma-delta modulator. Reconstruction filters are derived to estimate the vector process from its quantized value. The use of time average mean squared estimation error allows us to compare systems with different sampling rates. The tradeoff between the resolution of the quantizers and the sampling rate is shown both analytically and experimentally. The issue of the optimum choice of the reconstruction filter is addressed. Finally, we study the improvements, if any, brought by using optimized instead of uniform vector quantizers in our analysis. The state space approach allows us to consider Gauss-Markov processes of any order, or equivalently, processes with any arbitrary rational spectrum. It is shown that for most processes differential quantization of the state, an augmented process consisting of process and its derivatives, outperforms a simple state quantization and the vector quantization of the original process. In particular, for a second order lowpass process it is shown that when the overall rate R is high, the optimal smoothed error is proportional to 1/R3 for the differential scheme. This is better than the performance of DPCM and a modified vector DPCM, analyzed under the same framework. For both these schemes the asymptotic variation of the smoothed error is proportional to 1/R2. For differential state quantization, the resulting optimal size of the vector quantizers are small and can be used in practice. For a bandpass process the performance of all the schemes improves as the bandwidth decreases but the differential scheme still performs the best. We show that as the process becomes very narrowband the best quantization scheme, at low rates, is to differentially quantize the baseband complex envelope of the narrowband process. We analyze the case when the narrowband process is input to a sigma-delta modulator and derive simple asymptotic expressions for the quantization noise spectra. Next we study a universal source coding scheme with a vector quantizer codebook transmission. Again by using high resolution quantization theory we derive the optimal tradeoff between the quantizer resolution and the information used to transmit codebooks. We derive a formula that tightly bounds the signal-to-noise ratio of the universal coding system as a function of the interval between codebook transmissions. Another scheme of vector quantizing the transmitted codebooks is also studied and it is shown that under some reasonable conditions, uniform scalar quantization of the transmitted codebooks performs as well as vector quantizing them. We verify our results experimentally with stochastic and image data simulations.