Truncation and approximation errors in digital pulse waveform generators†

Abstract

Abstract Truncation error in digital pulse waveform generators of the binary transversal filter type (implemented by shift registers) is formulated by calculating the spectrum of the truncated part of the desired time-domain pulse and then taking the inverse Fourier transform of its in-band portion back into the time domain. Error or distortion function corresponding to a base-band signal of sinusoidal roll-off spectrum is expressed in terms of certain definite integrals convenient for machine computation. Tap weight settings which compensate this truncation error in the first order are suggested. Analysis of the spectrum of the approximate pulse (untruncated) generated by a shift register of finite shift period reveals that it contains as a factor the Fourier series of the exact desired signal spectrum, from which fact a meaningful measure of the in-band approximation error is proposed. Tap weights which compensate this error are simplified in terms of sine and cosine integrals. Samples of computer plots of both error spectra which serve as useful design guides are included, and method of extending the calculation of these errors for a vestigial side-band signal of arbitrary amount of spectral roll-off is also indicated.