Digital resampling is a process that converts a digital signal from one sampling rate to another. This process is performed by means of interpolating between the input samples to produce output samples at an output sampling rate. The digital interpolation process is accomplished with an interpolation filter.<p> The problem of resampling digital signals at an output sampling rate that is incommensurate with the input sampling rate is the first topic of this thesis. This problem is often encountered in practice, for example in multiplexing video signals from different sources for the purpose of distribution. There are basically two approaches to resample the signals. Both approaches are thoroughly described and practical circuits for hardware implementation are provided. A comparison of the two circuits shows that one circuit requires a division to compute the new sampling times. This time scaling operation adds complexity to the implementation with no performance advantage over the other circuit, and makes the 'division free' circuit the preferred one for resampling.<p> The second topic of this thesis is performance analysis of interpolation filters for Quadrature Amplitude Modulation (QAM) signals in the context of timing recovery. The performance criterion of interest is Modulation Error Ratio (MER), which is considered to be a very useful indicator of the quality of modulated signals in QAM systems. The methodology of digital resampling in hardware is employed to describe timing recovery circuits and propose an approach to evaluate the performance of interpolation filters. A MER performance analysis circuit is then devised. The circuit is simulated with MATLAB/Simulink as well as implemented in Field Programmable Gate Array (FPGA). Excellent agreement between results obtained from simulation and hardware implementation proves the validity of the methodology and practical application of the research works.

Digital resampling is a process that converts a digital signal from one sampling rate to another. This process is performed by means of interpolating between the input samples to produce output samples at an output sampling rate. The digital interpolation process is accomplished with an interpolation filter.<p> The problem of resampling digital signals at an output sampling rate that is incommensurate with the input sampling rate is the first topic of this thesis. This problem is often encountered in practice, for example in multiplexing video signals from different sources for the purpose of distribution. There are basically two approaches to resample the signals. Both approaches are thoroughly described and practical circuits for hardware implementation are provided. A comparison of the two circuits shows that one circuit requires a division to compute the new sampling times. This time scaling operation adds complexity to the implementation with no performance advantage over the other circuit, and makes the 'division free' circuit the preferred one for resampling.<p> The second topic of this thesis is performance analysis of interpolation filters for Quadrature Amplitude Modulation (QAM) signals in the context of timing recovery. The performance criterion of interest is Modulation Error Ratio (MER), which is considered to be a very useful indicator of the quality of modulated signals in QAM systems. The methodology of digital resampling in hardware is employed to describe timing recovery circuits and propose an approach to evaluate the performance of interpolation filters. A MER performance analysis circuit is then devised. The circuit is simulated with MATLAB/Simulink as well as implemented in Field Programmable Gate Array (FPGA). Excellent agreement between results obtained from simulation and hardware implementation proves the validity of the methodology and practical application of the research works.

This research develops efficient solution methods for linear systems with scalar and multi-level Toeplitz structure. Toeplitz systems are common in one-dimensional signal-processing applications, and typically correspond to temporal- or spatial-invariance in the underlying physical phenomenon. Over time, a number of algorithms have been developed to solve these systems economically by exploiting their structure. These developments began with the Levinson-Durbin recursion, a classical fast method for solving Toeplitz systems that has become a standard algorithm in signal processing. Over time, more advanced routines known as superfast algorithms were introduced that are capable of solving Toeplitz systems with even lower asymptotic complexity. For multi-dimensional signals, temporally- and spatially-invariant systems have linear-algebraic descriptions characterized by multi-level Toeplitz matrices, which exhibit Toeplitz structure on multiple levels. These matrices lack the same algebraic properties and structural simplicity of their scalar analogs. As a result, it has proven exceedingly difficult to extend the existing scalar Toeplitz algorithms for their treatment. This research presents algorithms to solve scalar and two-level Toeplitz systems through a constructive approach, using methods devised for specialized cases to build more general solution methods. These methods extend known scalar Toeplitz inversion results to more general scalar least-squares problems and to multi-level Toeplitz problems. The resulting algorithms have the potential to provide substantial computational gains for a large class of problems in signal processing, such as image deconvolution, non-uniform resampling, and the reconstruction of spatial volumes from non-uniform Fourier samples.