Digital Filter Designs for Recursive Frequency Analysis

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.

Perspective Chapter: Cascaded-Resonator-Based Recursive Harmonic Analysis

“…In accordance with the prevailing trends in works dealing with this issue, in the initial works [13,16,22,23,25,26] the resulting filters of CR structures were of the FIR type. Later, in [27], IIR filters were used, which represent a computationally more efficient solution [28,29]. The unit characteristic polynomial of the transfer function is replaced by the optimized one.…”

Parallel software implementation of recursive multidimensional digital filters for point-target detection in cluttered infrared scenes

“…A somewhat similar approach is adopted here; however to better accommodate local variation, a 3-D (spatiotemporal) spectrum is estimated recursively for each pixel using the surrounding data within a 3-D analysis window. The sliding DFT (SDFT) is used in the spatial dimensions, which are of finite extent; whereas a deadbeat observer is used in the temporal dimension, which is effectively of infinite extent [12]. Both methods are implemented using filter banks with finite impulse responses (FIRs).…”

“…The SDFT was used for the spatial dimensions because it has a simple implementationusing a comb-filter cascaded with a bank of resonators [13]. Accumulated rounding errors due to pole-zero cancellation on the unit circle are negligible over the dimensions of a single frame [12].…”

“…The deadbeat observer implementation is slightly more complex and memory intensive as it uses a feedback loop with integrators in the forward path to drive steady-state errors to zero, which ensures that rounding errors do not accumulate over long periods of operation [12], [14].…”