4.1 Introduction

In this chapter we use the rules-and-pairs notation and technique to derive several sampling theorem results, which can be done very concisely in some cases. In fact, the wideband (or baseband) sampling theorem and the Hilbert sampling theorem for narrowband (or RF and IF) waveforms are obtained here following the derivations of Woodward [1]. Two other narrowband sampling techniques, uniform sampling and quadrature sampling, have been analyzed by Brown [2], but these results have been obtained here much more easily using Woodward's approach and have been extended to show what sampling rates are acceptable, rather than just giving the minimum sampling rates presented by Brown.

Woodward's technique is to express the spectrum U of the given waveform u in a repetitive form, then gate it to obtain the spectrum again. The Fourier transform of the resulting identity shows that the waveform can be expressed as a set of impulses of strength equal to samples of the waveform, suitably interpolated. This is the converse of repeating a waveform to obtain a line spectrum: if a waveform is repeated at intervals T, a spectrum is obtained consisting of lines ( ?-functions in the frequency domain) at intervals F = 1/ T with envelope U, the spectrum of u. Conversely, if a spectrum U is repeated at intervals F, we obtain a waveform of impulses ( ?-functions in the time domain) at intervals T = 1 /F with...