Antonia Papandreou-Suppappola

Quadratic Time-Frequency Representations with Scale Covariance and Generalized Time-Shift Covariance: A Unified Framework for the Affine, Hyperbolic, and Power Classes

Abstract We propose the generalized class of quadratic time-frequency representations (QTFRs) that satisfy the scale covariance property, which is important in multiresolution analysis, and the generalized time-shift covariance property, which is important in the analysis of signals propagating through systems with specific dispersive characteristics. We discuss a formulation of the generalized class QTFRs in terms of two-dimensional kernel functions, a generalized signal expansion related to the generalized class time-frequency geometry, an important member of the generalized class, a set of desirable QTFR properties and their corresponding kernel constraints, and a “localized-kernel” generalized subclass that is characterized by one-dimensional kernels. Special cases of the generalized QTFR class include the affine class and the new hyperbolic class and power classes. All these QTFR classes satisfy the scale covariance property. In addition, the affine QTFRs are covariant to constant time shifts, the hyperbolic QTFRs are covariant to hyperbolic time shifts, and the power QTFRs are covariant to power time shifts. We present a detailed study of these classes that includes their definition and formulation, an associated generalized signal expansion, important class members, desirable QTFR properties and corresponding kernel constraints, and localized-kernel subclasses. Also, we investigate the subclasses formed by the intersection between the affine and hyperbolic classes, the affine and power classes, and the hyperbolic and power classes. These subclasses are important since their members satisfy additional desirable properties. We show that the hyperbolic class is obtained from Cohens QTFR class using a “hyperbolic time-frequency warping” and that the power classes are obtained similarly by applying a “power time-frequency warping” to the affine class. The affine class is a special case of the power classes. Furthermore, we generalize the time-frequency warping so that when applied either to Cohens class or to the affine class, it yields QTFRs that are always generalized time-shift covariant but not necessarily scale covariant.

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.

Generalized time-shift covariant quadratic time-frequency representations with arbitrary group delays

We propose new classes of quadratic time-frequency representations (QTFRs) that satisfy the generalized time-shift covariance property important for analyzing signals propagating through systems with group delay dependent characteristics. We obtain these classes from Cohens (1966) class and the affine class of constant time-shift covariant QTFRs using a generalized warping that depends on the desirable group delay time-shift covariance. We develop formulations for the new classes, desirable properties with kernel constraints, new QTFR members, and intersection subclasses. We also propose the new exponential class of frequency-shift covariant and exponential time-shift covariant QTFRs.

Power class time-frequency representations: interference geometry, smoothing, and implementation

The nth power class (PC/sub n/) of quadratic time-frequency representations (QTFRs) is specifically suited for the multiresolution analysis of signals passing through systems whose dispersion characteristic is approximately f/sup /spl kappa//. This paper considers several aspects of PC analysis important in applications. We discuss the geometry of PC auto terms and cross terms, and we propose new PC QTFRs that attenuate cross terms via smoothing. We describe the implementation of PC QTFRs via warping techniques. Finally, simulation results demonstrate the advantages of PC analysis.

The effect of mismatching analysis signals and time-frequency representations

We study the time-frequency geometry underlying quadratic time-frequency representations (QTFRs) defined based on a generalized time-shift covariance property. These QTFRs include the generalized warped Wigner distribution (WD) and its smoothed versions that are useful for reducing cross terms in multicomponent signal analysis applications. The generalized warped WD is ideal for analyzing nonstationary signals whose group delay matches the specified time-shift covariance. Its smoothed versions may also be well suited for various signals provided that their smoothing characteristics match the signals time-frequency structure. Thus, we examine the effects of a mismatch between the analysis signal and the chosen QTFR. We provide examples to demonstrate the advantage of matching the signals group delay with the generalized time-shift covariance property of a given class of QTFRs, and to demonstrate that mismatch can cause significant distortion.

New concepts in narrowband and wideband Weyl correspondence time-frequency techniques

We propose the new P/sub 0/-Weyl symbol to analyze system induced time shifts and scale changes on the input signal. This new Weyl symbol (WS) is useful in wideband signal analysis. We also propose new WS as tools for analyzing systems which produce dispersive frequency shifts on the signal. We obtain these generalized, frequency-shift covariant WS by warping conventional, narrowband WS. Using the new, generalized WS, we provide a formulation for the Weyl correspondence for linear systems with instantaneous frequency characteristics matched to user specified characteristics. We also propose a new interpretation of linear signal transformations as weighted superpositions of nonlinear frequency shifts on the signal. Application examples in signal analysis and detection demonstrate the advantages of our new results.

New higher order spectra and time-frequency representations for dispersive signal analysis

For an analysis of signals with arbitrary dispersive phase laws, we extend the concept of higher order moment functions and define their associated higher order spectra. We propose a new higher order time-frequency representation (TFR), the higher order generalized warped Wigner distribution (HOG-WD). The HOG-WD is obtained by warping the previously proposed higher order Wigner distribution, and is important for analyzing signals with arbitrary time-dependent instantaneous frequency. We discuss links to prior higher order techniques and investigate properties of the HOG-WD. We extend the HOG-WD to a class of higher order, alternating sign, frequency-shift covariant TFRs. Finally, we demonstrate the advantage of using the generalized higher order spectra to detect phase coupled signals with dispersive instantaneous frequency characteristics.

A wideband time-frequency Weyl symbol and its generalization

We extend the work of Shenoy and Parks (1994) on the wideband Weyl correspondence. We define a wideband Weyl symbol (P/sub 0/WS) in the time-frequency plane based on the Bertrand (1988) P/sub 0/-distribution, and we study its properties, examples and possible applications. Using warping relations, we generalize the P/sub 0/WS and the wideband spreading function (WSF) to analyze systems producing dispersive time shifts. We provide properties and special cases (e.g. power and exponential) to demonstrate the importance of our generalization. The new generalized WSF provides a new interpretation of a system output as a weighted superposition of dispersive time-shifted versions of the signal. We provide application examples in analysis and detection to demonstrate the advantages of our new results for linear systems with group delay characteristics matched to the specific warping used.

Localized subclasses of quadratic time-frequency representations

We discuss the existence of classes of quadratic time-frequency representations (QTFRs), e.g. Cohen, power, and generalized time-shift covariant, that satisfy a time-frequency (TF) concentration property. This important property yields perfect QTFR concentration along group delay curves. It also (1) simplifies the QTFR formulation and property kernel constraints as the kernel reduces from 2-D to 1-D, (2) reduces the QTFR computational complexity, and (3) yields simplified design algorithms. We derive the intersection of Cohens class with the new power exponential class, and show that it belongs to Cohens localized-kernel subclass. In addition to the TF shift covariance and concentration properties, these intersection QTFRs preserve power exponential time shifts, important for analyzing signals passing through exponentially dispersive systems.

New time-frequency symbol classification

We propose new time-frequency (TF) symbols as the narrowband Weyl symbol (WS) smoothed by an appropriate kernel. These new symbols preserve time and frequency shifts on a random process. Choosing specific smoothing kernels, we can obtain various new symbols (e.g. Levin symbol and Page symbol). We link a quadratic form of the signal to the new symbols and Cohens (1992) class of quadratic time-frequency representations, and we derive a simple kernel constraint for unitary symbols. We also propose an affine class of symbols in terms of the wideband Weyl symbol (P/sub 0/WS). These symbols preserve scale changes and time shifts. Furthermore, we generalize the smoothed versions of the WS and P/sub 0/WS to analyze random processes undergoing generalized frequency shifts or generalized time shifts.