| n this work we shall study the problem of time domain errors in signals. This thesis is divided into four parts, each of which addresses various aspects of the problem.;The concept of effective bandwidth as defined by Giardina and Chirlian is extended to signals of compact support with an arbitrary but finite number of finite discontinuities.;It is well known that the unit step function and all functions with discontinuities exhibit the Gibbs' phenomenon when their frequency spectra are truncated, wherein the amount of overshoot is independent of the frequency at which the function is truncated. Simple truncation is equivalent to passing the function through an ideal low-pass filter. On the other hand a filter with a triangular amplitude characteristic does not produce overshoot in the same function. A trapezoidal filter is intermediate between the ideal and triangular filters. Such a trapezoidal filter with slope k will be examined. Its impulse and step response will be determined and the relationship between the slope k and the amount of overshoot will be discussed. Furthermore, error bounds between a signal and its truncated form, using trapezoidal truncation, will be developed. These results are compared to previous results obtained by Giardina and Chirlian using ideal truncation.;Two theorems are introduced and proven which examine the time convolution of signals. If two functions satisfy a Lipschitz condition it is shown that their convolution also satisfies the Lipschitz condition and the Lipschitz constant is determined. Also, a simple bound on the convolution of two signals is developed.;The effect of distortion of the output signals caused by deviations in the amplitude and phase characteristics of non-ideal filters is examined. Bounds on the... |
