Fast Fourier Optimization: Sparsity Matters

Many interesting and fundamentally practical optimization problems, ranging from optics, to signal processing, to radar and acoustics, involve constraints on the Fourier transform of a function. It is well-known that the {\em fast Fourier transform} (fft) is a recursive algorithm that can dramatically improve the efficiency for computing the discrete Fourier transform. However, because it is recursive, it is difficult to embed into a linear optimization problem. In this paper, we explain the main idea behind the fast Fourier transform and show how to adapt it in such a manner as to make it encodable as constraints in an optimization problem. We demonstrate a real-world problem from the field of high-contrast imaging. On this problem, dramatic improvements are translated to an ability to solve problems with a much finer grid of discretized points. As we shall show, in general, the "fast Fourier" version of the optimization constraints produces a larger but sparser constraint matrix and therefore one can think of the fast Fourier transform as a method of sparsifying the constraints in an optimization problem, which is usually a good thing.Comment: 16 pages, 8 figure

Hankel-norm approximation of FIR filters: a descriptor-systems based approach

We propose a new method for approximating a matrix finite impulse response (FIR) filter by an infinite impulse response (IIR) filter of lower McMillan degree. This is based on a technique for approximating discrete-time descriptor systems and requires only standard linear algebraic routines, while avoiding altogether the solution of two matrix Lyapunov equations which is computationally expensive. Both the optimal and the suboptimal cases are addressed using a unified treatment. A detailed solution is developed in state-space or polynomial form, using only the Markov parameters of the FIR filter which is approximated. The method is finally applied to the design of scalar IIR filters with specified magnitude frequency-response tolerances and approximately linear-phase characteristics. A priori bounds on the magnitude and phase errors are obtained which may be used to select the reduced-order IIR filter order which satisfies the specified design tolerances. The effectiveness of the method is illustrated with a numerical example. Additional applications of the method are also briefly discussed

Lecture notes on the design of low-pass digital filters with wireless-communication applications

The low-pass filter is a fundamental building block from which digital signal-processing systems (e.g. radio and radar) are built. Signals in the electromagnetic spectrum extend over all timescales/frequencies and are used to transmit and receive very long or very short pulses of very narrow or very wide bandwidth. Time/Frequency agility is the key for optimal spectrum utilization (i.e. to suppress interference and enhance propagation) and low-pass filtering is the low-level digital mechanism for manoeuvre in this domain. By increasing and decreasing the bandwidth of a low-pass filter, thus decreasing and increasing its pulse duration, the engineer may shift energy concentration between frequency and time. Simple processes for engineering such components are described and explained below. These lecture notes are part of a short course that is intended to help recent engineering graduates design low-pass digital filters for this purpose, who have had some exposure to the topic during their studies, and who are now interested in the sending and receiving signals over the electromagnetic spectrum, in wireless communication (i.e. radio) and remote sensing (e.g. radar) applications, for instance. The best way to understand the material is to interact with the spectrum using receivers and or transmitters and software-defined radio development-kits provide a convenient platform for experimentation. Fortunately, wireless communication in the radio-frequency spectrum is an ideal application for the illustration of waveform agility in the electromagnetic spectrum. In Parts I and II, the theoretical foundations of digital low-pass filters are presented, i.e. signals-and-systems theory, then in Part III they are applied to the problem of radio communication and used to concentrate energy in time or frequency.Comment: Added Slepian ref. Added arXiv ID to heade

MIMO decision feedback equalization from an H∞ perspective

We approach the multiple input multiple output (MIMO) decision feedback equalization (DFE) problem in digital communications from an H∞ estimation point of view. Using the standard (and simplifying) assumption that all previous decisions are correct, we obtain an explicit parameterization of all H∞ optimal DFEs. In particular, we show that, under the above assumption, minimum mean square error (MMSE) DFEs are H∞ optimal. The H∞ approach also suggests a method for dealing with errors in previous decisions

Adaptive polyphase subband decomposition structures for image compression

Cataloged from PDF version of article.Subband decomposition techniques have been extensively used for data coding and analysis. In most filter banks, the goal is to obtain subsampled signals corresponding to different spectral regions of the original data. However, this approach leads to various artifacts in images having spatially varying characteristics, such as images containing text, subtitles, or sharp edges. In this paper, adaptive filter banks with perfect reconstruction property are presented for such images. The filters of the decomposition structure which can be either linear or nonlinear vary according to the nature of the signal. This leads to improved image compression ratios. Simulation examples are presented