The fan-chirp transform for non-stationary harmonic signals

Abstract

This paper presents a novel transform related to the framework of warping operators when the continuous time warping mapping is a second-order polynomial. This case is proven in the paper to be the only one from the aforementioned group that marginalizes the Wigner distribution along line paths, in particular, with a fan geometry. The properties and attributes of the fan-chirp transform (FChT) along with the analytical characterization of harmonically related Gaussian chirplets bear especial relevance in the paper. This analysis shows that for chirp-periodic signals the FChT can reach the limit of the time–frequency (TF) uncertainty principle, while simultaneously keeping the cross-terms at minimum level. The formulation of the fast digital computation of the FChT is also provided in the paper. Two practical scenarios—the analysis of speech with natural intonation and bat ultrasound—validate the theoretical developments and shows manifestly the eloquent competitive performance of the new transform.

Introduction

Identifying frequency-modulated (FM) sinusoids or chirps in a signal is known to be a tough challenge for classical linear analysis. Most of the alternative solutions for this problem has come from the field of time–frequency (TF) analysis, mainly in the form of Cohen's class bilinear time–frequency distributions (TFD) [1]. However, the long debate on the relevance and meaning of the cross-terms [2] or the dilemma on the need of positive TFDs [3], [4] have moved the attention towards different approaches, such as matching pursuits [5] over redundant chirplet dictionaries [6], [7], [8], or chirp-based transforms that marginalize the Wigner–Ville distribution according to certain geometries [10], [11], [12], [13], [14]. Although the redundant dictionary remains the most popular method for chirplet decomposition so far, chirp-based transforms are especially interesting because they can provide a broader picture of the TF content of the signal.

Several signal processing transforms are related to the term “chirp”: the Chirp-Z [9] and the “Chirplet” transforms [10] contain explicitly the term, whilst the fractional Fourier transform [11], [12] and the warped-time operators [13], [14] are conceptually related to it. The Chirp-Z transform [9] is an efficient algorithm for calculating discrete Fourier transforms (DFTs) at frequencies not related to a power-of-two fraction of the sampling bandwidth. Although a discrete-time chirp signal is used in the mechanism, the usage of this algorithm is not actually related to the context of this paper.

The first relevant chirp-based transform, the chirplet transform (CT) [10], is described by the inner product between the signal and a chirplet as

$$X_{\beta }(f)={\int }_{-\infty }^{\infty }x(t)g_{\varrho ,\tau ,f,\beta }^{*}(t)\mathrm{d}t\text{,}$$

where t is time, $x(t)$ is the analysis signal, ${}^{*}$ denotes complex conjugate, and $g_{\varrho ,\tau ,f,\beta }(t)$ is a Gaussian chirplet of unit energy

$$g_{\varrho ,\tau ,f,\beta }(t)=\frac{{\mathrm{e}}^{-(1/2)((t-\tau )/\varrho {)}^2}}{\sqrt[4]{\pi {\varrho }^2}}{\mathrm{e}}^{\mathrm{j}2\pi (f(t-\tau )+(1/2)\beta (t-\tau {)}^2)}\text{.}$$

Here $\nu$ is the instantaneous frequency (IF) at $t=\tau ,\beta$ the frequency variation rate and $\varrho$ the time spread. By disregarding the Gaussian term and setting $\tau =0$ for simplicity, it is easy to deduce that the squared magnitude of the Chirp(let) transform is

$$|X_{\beta }(f){|}^2={\int }_{-\infty }^{\infty }{\mathrm{WD}}_x(t,f-\beta t)\mathrm{d}t\text{,}$$

where ${\mathrm{WD}}_x(t,f)$ is the Wigner–Ville distribution [2] of $x(t)$ (henceforth Wigner distribution, WD). Thus, the CT yields the “slanted” marginal of the WD.

Another well-established chirp-based transform is the fractional Fourier transform (FrFT) [11]

$$X_{\theta }(u)={\int }_{-\infty }^{\infty }x(v)K_{\theta }(v,u)\mathrm{d}v\text{,}$$

where $K_{\theta }(v,u)$ is the transformation kernel [12]. The FrFT involves products with linear-FM chirps in such a way that it yields the marginalization of the WD along the angular direction $\theta$, i.e.,

$$|X_{\theta }(u){|}^2={\int }_{-\infty }^{\infty }{\mathrm{WD}}_x(\mathrm{c}u-\mathrm{s}v,\mathrm{s}u+\mathrm{c}v)\mathrm{d}v\text{,}$$

where $\mathrm{c}=\cos \theta$ and $\mathrm{s}=\sin \theta$.

The last chirp-based transform considered here is the warping operator [13], [14], [15], defined as

$$X_{\psi (·)}(f)={\int }_{-\infty }^{\infty }x(\psi (t))\sqrt{|{\psi }^{\prime }(t)|}{\mathrm{e}}^{-\mathrm{j}2\pi \mathit{ft}}\mathrm{d}t\text{,}$$

where $\psi (t)$ is a continuous differentiable time mapping and ${\psi }^{\prime }(t)$ its derivative. An equivalent formulation to this warped-time Fourier transform is

$$X(f;\phi (·))={\int }_{-\infty }^{\infty }x(t)\sqrt{|{\phi }^{\prime }(t)|}{\mathrm{e}}^{-\mathrm{j}2\pi f\phi (t)}\mathrm{d}t\text{,}$$

where $\phi (t)={\psi }^{-1}(t)$.¹ Eq. (7) represents the inner product of signal $x(t)$ with a non-linear chirp, which is the basic mechanism in TF redundant dictionaries for chirp-based signal decomposition [8]. The warped-time framework has also given rise to new TF distributions, such as the generalized warped Cohen's class (GWCC) [16], which introduces new ways of interpreting the TF content.

This paper tackles the warping operator (7) constrained to the mapping $\phi (t)$ being a second-order polynomial. This case is proven here to marginalize the WD along straight line paths. Thus, it can be seamlessly compared against other linear chirp-based transforms, such as the CT and FrFT (see Fig. 1 for an introductory illustration), and studied with more detail apart from the general framework. The organization of the paper follows: Section 2 introduces the analysis and synthesis equations of the proposed fan-chirp transform (FChT); in Section 3 its main basic attributes, the marginalization geometry and representation of chirp-periodic signals are studied; Section 4 contains a discussion on previous works related to the FChT; Section 5 addresses the estimation of the only user-defined parameter of the FChT, the chirp rate, in order to better match the TF geometry of the analysis signal; Section 6 elaborates on the practical aspects of the digital implementation; Section 7 presents the performance evaluation of the FChT on synthetic and real scenarios, namely, the analysis of natural speech and sound of mammals; the conclusions close the paper.