The fan-chirp transform for non-stationary harmonic signals
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 aswhere t is time, is the analysis signal, denotes complex conjugate, and is a Gaussian chirplet of unit energyHere is the instantaneous frequency (IF) at the frequency variation rate and the time spread. By disregarding the Gaussian term and setting for simplicity, it is easy to deduce that the squared magnitude of the Chirp(let) transform iswhere is the Wigner–Ville distribution [2] of (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]where 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 , i.e.,where and .
The last chirp-based transform considered here is the warping operator [13], [14], [15], defined aswhere is a continuous differentiable time mapping and its derivative. An equivalent formulation to this warped-time Fourier transform iswhere .1 Eq. (7) represents the inner product of signal 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 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.
Section snippets
The FChT
The analysis formula of the FChT of signal is defined aswhere t is time, f is frequency,2 and is the second-order polynomial controlled by the so-called chirp rate The FChT involves the inner product between and the complex signalswhich are chirps whose IF, defined as the time derivative of the exponent, varies linearly over time
Properties
In this section we derive the most relevant property of the FChT regarding the marginalization of the WD; Parseval's theorem along with other basic properties are also derived; the TF resolution over harmonically related chirplets covers the final part of the section.
Prior related contributions
The FChT compares seamlessly against the Chirplet [10], the fractional Fourier [12], and the Fourier transforms, all yielding the marginalization of the TF plane along different straight line geometries (see Fig. 1 again for an illustrative comparison). Additionally, it is important to address here previous works [19], [20], [21], [22] related to the FChT to a larger or lesser extent.
The work [19] proposes the so-called Harmonic fractional Fourier transform (HFT)as
Chirp rate estimation
The most important aspect of the FChT in practical scenarios regards the adequacy of the law to the actual TF characteristics of the signal. Signals with fan geometry are found in practice only in short segments, such as in case of speech [17] or the song of some mammals [18]. In that sense, two options are at hand: either to use a warping function with more degrees of freedom than for matching the possible non-linear geometry, or to parse the signal into short segments and
Discrete-time formulation
In analogy to the discrete-time Fourier transform, the formulation of the discrete-time FChT could be thought as the continuous-time transform of signalwhere is the discrete-time signal and is the sampling interval. This way of proceeding results inwhere n is discrete time, is frequency, and the analysis chirp rate is the discrete counterpart of the chirp rate , that is,Likewise, frequency is related to
Results
The first experiment provides a comparison among the CT, FrFT and FChT on a toy synthetic example. The synthetic signal corresponds to a train of pulses non-equidistantly spaced, in such way that the fundamental frequency changes in a linear fashion; for the sake of clarity, the spectral envelope delineated by the harmonics is not flat. The mentioned transforms were applied over that signal, in such a way that the resolution achieved around the fourth harmonic were the highest. The results
Conclusions
The fan-chirp transform (FChT) is an effective method for representing signals with fan time–frequency (TF) structure. This type of signals, denoted here as chirp-periodic signals, are common in nature, such as segments of the song of mammals and human speech in natural intonation. The FChT possesses the property of marginalizing the WD along straight line paths according to a fan geometry. This geometry is entirely described by the user-defined chirp rate , or by its inverse, the focal point
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