Digital Filter Designs for Recursive Frequency Analysis

TL;DR

This paper reviews and enhances recursive digital filter techniques for frequency analysis, proposing new methods for stability and response optimization in DFT computations.

Contribution

It introduces novel stabilization techniques for IIR sliding DFT analyzers and compares various recursive methods within a unified framework.

Findings

FIR deadbeat observer outperforms SDFT methods in stability

New stabilization method ensures all poles are inside the unit circle

Windowing techniques improve frequency response quality

Abstract

Digital filters for recursively computing the discrete Fourier transform (DFT) and estimating the frequency spectrum of sampled signals are examined, with an emphasis on magnitude-response and numerical stability. In this tutorial-style treatment, existing recursive techniques are reviewed, explained and compared within a coherent framework; some fresh insights are provided and new enhancements/modifications are proposed. It is shown that the replacement of resonators by (non-recursive) modulators in sliding DFT (SDFT) analyzers with either a finite impulse response (FIR), or an infinite impulse response (IIR), does improve performance somewhat; however stability is not guaranteed, as the cancellation of marginally stable poles by zeros is still involved. The FIR deadbeat observer is shown to be more reliable than the SDFT methods, an IIR variant is presented, and ways of fine-tuning its response are discussed. A novel technique for stabilizing IIR SDFT analyzers with a fading memory, so that all poles are inside the unit circle, is also derived. Slepian and sum-of-cosine windows are adapted to improve the frequency responses for the various FIR and IIR DFT methods.