This paper considers compensation of anticipated erasures in a discrete-time (DT) signal such that the desired interpolation can still be accomplished, with minimum error, through a linear time-invariant (LTI) filter. The algorithms presented may potentially be useful in the compensation of a fault in a digital-to-analog converter where samples are dropped at known locations prior to reconstruction. Four algorithms are developed. The first is a general solution that, in the presence of erasures, minimizes the squared error for arbitrary LTI interpolation filters. In certain cases, e.g., oversampling and a sinc-interpolating filter, this solution is specialized so it perfectly compensates for erasures. The second solution is an approximation to the general solution that computes the optimal, finite-length compensation for arbitrary LTI interpolation filters. The third is a finite-length windowed version of the oversampled, sinc-interpolating solution using discrete prolate spheroidal sequences. The last is an iterative algorithm in the class of projection onto convex sets. Analysis and results from numerical simulations are presented.

In some contexts, DACs fail in such a way that specific samples are dropped. For example, in flat-panel video displays, some of the pixel LEDs can malfunction and get permanently set to particular values. We refer to this as the "missing pixel" problem. Under certain conditions, it may be possible to compensate for the dropped samples by preprocessing the digital signal. The paper describes a number of such compensation strategies. Each strategy is analyzed and results from numerical simulations are presented. Of particular interest is the relationship between compensation and the class of discrete prolate spheroidal sequences.

In this paper, we present a novel algorithm for sampling rate conversion by an arbitrary factor. Theoretically, sampling rate conversion of a discrete-time (DT) sequence can be performed by converting the sequence to a series of continuous-time (CT) impulses. This series of impulses is filtered with a CT lowpass filter, and the output is then sampled at the desired rate. If the CT filter is chosen to have a rational transfer function, then this system can be simulated using a DT algorithm for which both computation and memory requirements are low. The DT implementation is comprised of a parallel structure, where each branch consists of a time-varying filter with one or two taps, followed by a fixed recursive filter operating at the output sampling rate. The coefficients of the time-varying filters are calculated recursively. This eliminates the need to store a large table of coefficients, as is commonly done.

In this thesis, we consider three main resampling problems. The first is the sampling rate conversion problem in which the input and output grids are both regularly spaced. It is known that the output signal is obtained by applying a time-varying filter to the input signal. The existing methods for finding the coefficients of this filter inherently tradeoff computational and memory requirements. Instead, we present a recursive scheme for which the computational and memory requirements are both low. In the second problem which we consider, we are given the instantaneous samples of a continuous-time (CT) signal taken on an irregular grid from which we wish to obtain samples on a regular grid. This is referred to as the nonuniform sampling problem. We present a noniterative algorithm for solving this problem, which, in contrast to the known iterative algorithms, can easily be implemented in real time. We show that each output point may be calculated by using only a finite number of input points, with an error which falls exponentially in the number of points used. Finally we look at the nonuniform lowpass reconstruction problem. In this case, we are given regular samples of a CT signal from which we wish to obtain amplitudes for a sequence of irregularly spaced impulses. These amplitudes are chosen so that the original CT signal may be recovered by lowpass filtering this sequence of impulses. We present a general solution which exhibits the same exponential localization obtained for the nonuniform sampling problem. We also consider a special case in which the irregular grid is obtained by deleting a single point from an otherwise regular grid. We refer to this as the missing pixel problem, since it may be used to model cases in which a single defective element is present in a regularly spaced array such as the pixel arrays used in flat-panel video displays. We present an optimal solution which minimizes the energy of the reconstruction error, subject to the constraint that only a given number of pixels may be adjusted. Thesis Supervisor: Alan V. Oppenheim Title: Ford Professor of Engineering

The problem of changing the sampling rate of a signal by a rational factor of L/M is discussed. It is shown that infinite impulse response (IIR) filters can be efficiently implemented using the polyphase decomposition. The computational cost of the recursive and nonrecursive parts of the interpolation filter are considered separately, and a gain in efficiency of a factor of LM/(L+M-1) is achieved for the recursive part. This gain is only significant when both L and M are larger than one.