Volterra Models for Digital PWM and Their Inverses

Abstract

In an all-digital class-D audio amplifier each signal sample is mapped into a pulse using digital pulse-width-modulation (PWM), and this intrinsically generates nonlinear distortion. This article develops discrete-time Volterra models for digital PWM. The analysis considers two types of demodulation filter, an ideal filter which provides insight into the behavior of the PWM itself, and an analog low-order demodulator filter which models the interaction of the PWM mapping with a real demodulator filter. Symmetric, trailing edge, and leading edge PWM are considered. Using the Volterra models, formulae for the inverse systems are developed. This facilitates digital precompensation of the above mentioned distortion by digitally pre-filtering the modulating signal prior to the PWM mapping. This method is first simulated and compared to pseudo-natural PWM, a well-known method for PWM distortion reduction. Moreover, the method is verified by real measurements from a physical realization of the prefilter setup using real-time digital signal processing.

Definition of Simplified Frequency-Domain Volterra Models With Quasi-Sinusoidal Input

“…The reason is that the constraint about the real-valued output signal has not been fully imposed. Only the right (positive) part of the spectra has been considered as in (9), which corresponds to impose that the ac part of the output signal has to be real-valued. However, also the dc component at the output must be real-valued.…”

“…A broad variety of techniques based on very different approaches can be found in the literature [6]. Among them, the Volterra approach to the modeling of nonlinear time invari-ant (NTI) systems [7], [8] has been employed for a long time, and it is still widely employed in many applications [9]- [13]. The main advantage lies in its conceptual simplicity, since it represents a blend between the usual LTI system theory (both in time and frequency domain) and the Taylor series expansion.…”

“…The main drawback is that a (discrete) Volterra model is defined by a number coefficients that grows more than exponentially with the considered order of nonlinearity [14]. For this reason, only very low orders of nonlinearity (usually second and third) are employed in most practical applications [9], [10]- [13]: otherwise, the computational burden and the identification pro-cedure become troublesome. Similarly to a low-order Taylor expansion, such polynomial models may be not accurate for a broad input range, and in any case they are not the proper tools to represent strong nonlinearities.…”

Existing methods for improving the accuracy of digital-to-analog converters

“…These methods have the potential to improve the linearity of a conventional DAC that can switch between a multiple of fixed discrete voltage or current levels. As an alternative to producing multiple levels, time-domain averaged switching methods, such as pulse-width modulation (PWM) with pre-distortion 32,33 or 1-bit ∆-Σ modulation 34 , can be used for accurate time-varying signal reproduction. The latency introduced in switched conversion tends to make such techniques ill-suited to feedback control applications.…”