Representation and Reconstruction of Finite-Energy Band-Limited Signals via Pulse-Width Modulation

Abstract

In circuit implementations, pulse-width modulation (PWM) is often used as a convenient quasi-periodic representation of finite-energy, band-limited signals. Due to power and circuit area constraints, an engineering practice is to implement modulation-to-reconstruction as a purely analog pipeline; for modulation , a continuous-time modulating signal is compared dynamically over time to a continuous-time, periodic reference signal and for reconstruction , an analog low-pass filter is applied directly to the continuous-time PWM signal. This practical pipeline creates challenges from theory and implementation perspectives: When generated by an arbitrary reference signal, PWM acts as a sample-and-encode mechanism yet does not necessarily represent the modulating signal without loss of information in the Landau sense. In order to avoid loss of information, high-order approximations of sawtooth reference signals are generated in practice, which requires the use of less power efficient signal generators. Low-pass filter reconstruction is a practical, yet suboptimal reconstruction mechanism, which has been analyzed theoretically only for sinusoidal modulating signals. This paper addresses these challenges: First, for finite-energy, band-limited modulating signals, we characterize reference signals that ensure the existence of a lossless reconstruction mechanism in terms of their amplitude and period, which is related to what is referred as the oversampling factor. As a corollary, we note that even highly non-linear reference signals can ensure a lossless representation. Furthermore, we analyze the frequency domain characteristics of PWM signals with different pulse orientations and show that an analog low-pass filter losslessly reconstructs the underlying finite-energy, band-limited signal asymptotically in oversampling factor.