A simplified approach to optimum quadrature sampling

Abstract

A deterministic bandpass signal x(t) bandlimited to ‖ ω ‖ ε I1≡(ω0−σ/2, ω0+σ/2) can be represented in the form x(t)=p1(t) cosω0′t−q1(t) sinω0′t, where p1, q1 are low-pass (the ’’in phase’’ and ’’quadrature’’) components bandlimited to ‖ ω ‖?σ/2+‖ ω0−ω0′ ‖ and ω0′ is an arbitrary frequency within the band I1. Quadrature sampling has as its aim the recovery of the low-pass components p1, q1 directly from samples of both the bandpass signal x(t) and its quarter-wavelength translation x(t−π/2ω0′), the samples being taken at a low-pass rate. Grace and Pitt [J. Acoust. Soc. Am. 44, 1453 (1968)] obtained a result for the case ω0′=ω0?σ requiring an overall or average sampling rate of (σ/π)(γ/[γ]) samples/s, where γ=ω0/σ and [⋅] denotes the greatest integer function. By reducing the problem to an application of the classical Shannon sampling theorem in a special case (2ω0/σ =integer) and then proceeding to the general case via an embedding technique, we arrive here at an improved average sampling rate of (2γ+1/[2γ+1])(σ/π) samples/s. The method, assuming only ω0?σ/2 and finite energy signals, provides a simpler and more accessible treatment of the quadrature sampling problem than the earlier techniques of Grace and Pitt, Kohlenberg [J. Appl. Phys. 24, 1432 (1953)], and the author [IEEE Trans. Aerosp. Electron. Syst. AES-15, 366 (1979)].