On the use of operators versus warpings versus axiomatic definitions of new time-frequency (Operator) representations

The purpose of this paper is to give an overview of three techniques for deriving new time-frequency-scale representations, emphasizing their relative advantages and disadvantages. The three techniques considered are Cohen\u27s generalized characteristic function method, the axiomatic approach, and unitary transformation methods

Mixed time-frequency signal transformations

Mixed time-frequency representations are transformations of time-varying signals that depict how the spectral content of a signal is changing with time. They are multidimensional, timevarying extensions to conventional Fourier transform spectral analysis of signals and systems. Most time-frequency representations (TFRs) transform a one-dimensional signal, x(t), into a twodimensional function of time and frequency, Tx(t, f). These transformations represent a surface above the time-frequency “plane” which gives an indication as to which spectral components of the signal are present at a given time and their relative amplitude. They are conceptually similar to a musical score with time running along one axis and frequency along the other.37,50 Just as the location and the shape of the notes on a musical score represent the pitch, time of occurrence, and duration of each sound in a piece of music, so too does the location of the local maximum and the shape of the surface of the TFR give an indication as to the frequency content, onset, and duration of various dominant signal components. Such representations are useful for the analysis, modification, synthesis, and detection of a variety of nonstationary signals with time-varying spectral content.1-4,30,91 The purpose of this chapter is to give an overview of many of the linear and quadratic time-frequency representations that have been developed over the past 60 years. The chapter first reviews various one-dimensional spectral representations, such as the Fourier transform, the instantaneous frequency, and the group delay. A brief discussion follows of a few commonly used TFRs such as the short-time Fourier transform (STFT), the Wigner distribution (WD), the Altes Q distribution, and the Bertrand unitary P0 distribution. These TFRs will be used as examples in the subsequent section, which describes many useful properties that an ideal TFR should satisfy. Unfortunately, no one TFR exists which satisfies all of these desirable properties. The relative merits of these TFRs can be understood by grouping them into “classes” of TFRs that share two or more properties. The remainder of the chapter is devoted to defining and understanding these classes. Important insights into these classes of TFRs can be gained by examining a set of five two-dimensional kernel functions that are unique to each TFR. These kernels greatly simplify the analysis and application of TFRs

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 this event based pitch detector in comparison with classical pitch detectors that use the autocorrelation and the cepstrum methods to estimate the pitch period. © 1992 IEE

Fractional autocorrelation and its application to detection and estimation of linear FM signals

We present an alternative, efficient implementation of fractional autocorrelation. The algorithm utilizes the fast discrete approximation of the fractional Fourier transform (FRFT) along with the FFT algorithm. We relate fractional autocorrelation with radial slices of the narrowband ambiguity function (AF) and demonstrate that relationship with simulation examples. We also show that fractional autocorrelation is related to a detection statistic recently proposed for detection of multicomponent linear FM signals. Since our fractional autocorrelation based detection algorithm does not require the calculation of either the AF or the Radon transform, we propose its use as a computationally fast alternative for detecting linear FM signals and estimating their chirp rates

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

The wavelet transform (WT), a time-scale representation, is linear by definition. However, the nonlinear energy distribution of this transform is often used to represent the signal; it contains “cross terms” which could cause problems while analyzing multicomponent signals. In this paper, we show that the cross terms that exist in the energy distribution of the WT are comparable with those found in the Wigner distribution (WD), a quadratic time-frequency representation, and the energy distribution of the short time Fourier transform (STFT), of closely spaced signals. The cross terms of the WT and the STFT energy distributions occur at the intersection of their respective WT and STFT spaces, while for the WD they occur midtime and midfrequency. The parameters of the cross terms are a function of the difference in center frequencies and center times of the perpended signals. The amplitude of these cross terms can be as large as twice the product of the magnitudes of the transforms of the two signals in question in all three cases. In this paper, we consider the significance of the effect of the cross terms on the analysis of a multicomponent signal in each of these three representations. We also compare the advantages and disadvantages of all of these methods in applications to signal processing. © 1992 IEE

Joint fractional representations

By using Cohen\u27s characteristic function operator method, we derive the formulation of joint fractional representations (JFRs). JFRs generalize conventional time-frequency representations into fractional domains which are defined by the fractional Fourier transform. We present some of the properties of the fractional Wigner distribution along with the properties of the fractional ambiguity function. We calculate fractional Wigner distributions of some signals and provide some examples

Generalization of the Choi-Williams Distribution and the Butterworth Distribution for Time-Frequency Analysis

We provide a generalization to the Choi-Williams exponential distribution and introduce the new Butterworth distribution. We compute design equations and curves for the optimal choices of their parameters and for the optimal value of σ of the exponential distribution. We also provide examples that demonstrate the superior performance of our method over the Choi-Williams. Our new distributions can achieve cross-term reduction and autoterm preservation simultaneously, whereas the Choi-Williams distribution suffers from an inherent tradeoff between the two. © 1993 IEE

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

Using operator theory methods together with our recently 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

ON THE OPTIMALITY OF THE WIGNER DISTRIBUTION FOR DETECTION.

A variety of methods have been proposed for the detection of a signal with unknown signal parameters in a noisy environment. Here the noise statistics are incorporated to reveal that certain processing of the Wigner distribution (WD) signal representation can lead to an optimal, and often easy to compute, detection scheme. For the special case of linear FM signals in complex white Gaussian noise, it is shown that the optimal detector is equivalent to integrating the WD along the line of instantaneous frequency. If the position and the sweep rate of the linear chirp are unknown, then a generalized likelihood ratio test leads one to integrate the WD along all possible lines in the time-frequency plane and choose the largest integrated value for comparison to a threshold. Simulation examples demonstrate the utility of the proposed method. Finally, some comments concerning the detection of the general phase modulated signal are offered

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. Our 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. Our simulations demonstrate that this technique often improves the bit error performance of the receiver when compared to the case where there is no preprocessing