The time-frequency (TF) version of the wavelet transform and the "affine" quadraticlbilinear TF representations can be used for a TF analysis with constant-Q characteristic. This paper considers a new approach to constant-Q TF analysis. A specific TF warping transform is applied to Cohen's class of quadratic TF representations, which results in a new class of quadratic TF representations with constant-Q characteristic. The new class is related to a "hyperbolic TF geometry" and is thus called the hyperbolic class (HC). Two prominent TF representations previously considered in the literature, the Bertrand Po distribution and the Altes-Marinovic Q-distribution, are members of the new HC. We show that any hyperbolic TF representation is related to both the wideband ambiguity function and a "hyperbolic ambiguity function." It is also shown that the HC is the class of all quadratic TF representations which are invariant to "hyperbolic time-shifts" and TF scalings, operations which are important in the analysis of Doppler-invariant signals and self-similar random processes. The paper discusses the definition of the HC via constant-Q warping, some signal-theoretic fundamentals of the "hyperbolic TF geometry,'' and the description of the HC by 2-D kernel functions. Several members of the HC are considered, and a list of desirable properties of hyperbolic TF representations is given together with the associated kernel constraints. I. INTRODUCTION UADRATIC time-frequency representations (QTFR's) are useful in the analysis of nonstationary signals [l]. This paper discusses a new class of "constant-Q" QTFR's. By way of introduction, and to establish the notation used, we first give a brief review of two basic QTFR classes. In the following, let x (t) be a signal with Fourier transform X (f) = jYm x(t)e-'2*fr dt.
The affine and hyperbolic classea of quadratic timefrequency representations (QTFRB) provide frameworks for multiresolution or constant-Q time-frequency analysis. This paper studies the QTFR propertiea of regularity (QTFR reversibility) and unitarity (preservation of inner products, Moyal's formula) in the context of affine and hyperbolic 9TFRa. We develop the calculus of inverse kernels and iscuss important implications of regularity and unitarity, such as signal recovery, the derivation of other quadratic signal repreeentations, optimum detection, leastsquares signal syntheais, the effect of linear signal tranaforms, and the construction of QTFR basis systems.
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