G.F. Boudreaux-Bartels

Linear and quadratic time-frequency signal representations

A tutorial review of both linear and quadratic representations is given. The linear representations discussed are the short-time Fourier transform and the wavelet transform. The discussion of quadratic representations concentrates on the Wigner distribution, the ambiguity function, smoothed versions of the Wigner distribution, and various classes of quadratic time-frequency representations. Examples of the application of these representations to typical problems encountered in time-varying signal processing are provided.<<ETX>>

Application of the wavelet transform for pitch detection of speech signals

An event-detection pitch detector based on the dyadic wavelet transform is described. The proposed pitch detector is suitable for both low-pitched and high-pitched speakers and is robust to noise. Examples are provided that demonstrate the superior performance of the pitch detector in comparison with classical pitch detectors that use the autocorrelation and the cepstrum methods to estimate the pitch period. >

Time-varying filtering and signal estimation using Wigner distribution synthesis techniques

The short-time Fourier transform (STFT), the ambiguity function (AF), and the Wigner distribution (WD) are mixed time-frequency signal representations that use Fourier transform techniques to map a one-dimensional function of time into a two-dimensional function of time and frequency. These mixed time-frequency mappings have been used to analyze the local frequency characteristics of a variety of signals and systems. Although much work has also been done to develop STFT and AF synthesis algorithms that can be used to implement a variety of time-varying signal processing operations, no such synthesis techniques have thus far been developed for the WD. In this paper, a signal synthesis algorithm that works directly with the real-valued high-resolution WD will be derived. Examples of how this WD synthesis procedure can be used to perform time-varying filtering operations or signal separation will be given.

Fractional convolution and correlation via operator methods and an application to detection of linear FM signals

Using operator theory methods together with our previously introduced unitary fractional operator, we derive explicit definitions of fractional convolution and correlation operations in a systematic and comprehensive manner. Via operator manipulations, we also provide alternative formulations of those fractional operations that suggest efficient algorithms for discrete implementation. Through simulation examples, we demonstrate how well the proposed efficient method approximates the continuous formulation of fractional autocorrelation. It is also shown that the proposed fractional autocorrelation corresponds to radial slices of the ambiguity function (AF). We also suggest an application of the fast fractional autocorrelation for detection and parameter estimation of linear FM signals.

A comparison of the existence of 'cross terms' in the Wigner distribution and the squared magnitude of the wavelet transform and the short-time Fourier transform

It is shown that cross terms comparable to those found in the Wigner distribution (WD) exist for the energy distributions of the wavelet transform (WT) and the short-time Fourier transform (STFT). The geometry of the cross terms is described by deriving mathematical expressions for the energy distributions of the STFT and the WT of a multicomponent signal. From those mathematical expressions it is inferred that the STFT and the WT cross terms: (1) occur at the intersection of the respective transforms of the two signals under consideration, whereas the WD cross terms occur at mid-time-frequency of the two signals; (2) are oscillatory in nature, as are the WD cross terms, and are modulated by a cosine whose argument is a function of the difference in center times and center frequencies of the signals under consideration; and (3) can have a maximum amplitude as large as twice the product of the magnitude of the transforms of the two signals in question, like WD cross terms. It is shown that the presence of these cross terms could lead to problems in analyzing a multicomponent signal. The consequences of this effect with respect to speech applications are discussed. >

Group delay shift covariant quadratic time-frequency representations

We propose classes of quadratic time-frequency representations (QTFRs) that are covariant to group delay shifts (GDSs). The GDS covariance QTFR property is important for analyzing signals propagating through dispersive systems with frequency-dependent characteristics. This is because a QTFR satisfying this property provides a succinct representation whenever the time shift is selected to match the frequency-dependent changes in the signals group delay that may occur in dispersive systems. We obtain the GDS covariant classes from known QTFR classes (such as Cohens (1995) class, the affine class, the hyperbolic class, and the power classes) using warping transformations that depend on the relevant group delay change. We provide the formulation of the GDS covariant classes using two-dimensional (2-D) kernel functions, and we list desirable QTFR properties and kernel constraints, as well as specific class members. We present known examples of the GDS covariant classes, and we provide a new class: the power exponential QTFR class. We study the localized-kernel subclasses of the GDS covariant classes that simplify the theoretical development as the kernels reduce from 2-D to one-dimensional (1-D) functions, and we develop various intersections between the QTFR classes. Finally, we present simulation results to demonstrate the advantage of using QTFRs that are matched to changes in the group delay of a signal.

Broadband interference excision in spread spectrum communication systems via fractional Fourier transform

We demonstrate the use of the fractional Fourier transform (FRFT) in excising broadband, linear FM (chirp) type interferences in spread spectrum communication systems. This method is predicated on the fact that the FRFT perfectly localizes a linear FM signal as an impulse when the angle parameter of the transform matches the sweep rate (chirp rate) of the linear chirp signal. Therefore, a transform domain thresholding can often eliminate linear-FM type interferences without severely affecting the desired part of the received signal. Thus we propose a preprocessing of the received signal by an FRFT-based excision scheme prior to demodulation. The simulations demonstrate that this technique often improves the bit error performance of the receiver when compared to the case where there is no preprocessing.

On the use of cyclotomic polynomial prefilters for efficient FIR filter design

The authors present an efficient FIR (finite impulse response) filter design algorithm that generalizes existing cascaded FIR prefilter-equalizer methods. They propose using cyclotomic polynomial building blocks to form a multiplierless prefilter with impressive stopband performance, and they provide a straightforward strategy for choosing the polynomials to match filter specification. Two options for design of the equalizer are provided. A uniformly spaced equalizer can be optimally (L/sub infinity /) designed via a modified Parks-McClellan algorithm. A new algorithm, based on complex basis function subset selection methods, is also proposed for optimal design of a more efficient, nonuniformly spaced equalizer. The techniques, which can be applied to a broad class of filter design problems, typically provide a 35%-85% reduction in the number of additions and multiplications required, with a cost of 10%-45% additional delays. The methods also provide reduced coefficient quantization sensitivity and reduced roundoff noise. >

The power classes-quadratic time-frequency representations with scale covariance and dispersive time-shift covariance

We consider scale-covariant quadratic time-frequency representations (QTFRs) specifically suited for the analysis of signals passing through dispersive systems. These QTFRs satisfy a scale covariance property that is equal to the scale covariance property satisfied by the continuous wavelet transform and a covariance property with respect to generalized time shifts. We derive an existence/representation theorem that shows the exceptional role of time shifts corresponding to group delay functions that are proportional to powers of frequency. This motivates the definition of the power classes (PCs) of QTFRs. The PCs contain the affine QTFR class as a special case, and thus, they extend the affine class. We show that the PCs can be defined axiomatically by the two covariance properties they satisfy, or they can be obtained from the affine class through a warping transformation. We discuss signal transformations related to the PCs, the description of the PCs by kernel functions, desirable properties and kernel constraints, and specific PC members. Furthermore, we consider three important PC subclasses, one of which contains the Bertrand (1992) P/sub k/ distributions. Finally, we comment on the discrete-time implementation of PC QTFRs, and we present simulation results that demonstrate the potential advantage of PC QTFRs.

An overview of aliasing errors in discrete-time formulations of time-frequency representations

Discrete-time time-frequency representation (TFR) algorithms claim to provide alias-free approximations to their continuous-time TFR counterparts without requiring oversampling of the signal By counterexamples, we demonstrate that some of these claims are invalid. We give new necessary conditions for reducing aliasing errors in these discrete-time TFR algorithms.