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An Approach to Time-Frequency Analysis with Ridges of the Continuous Chirplet Transform Mikio Aoi, Kyle Lepage, Yoonsoeb Lim, Uri T

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An Approach to Time-Frequency Analysis with Ridges of the Continuous Chirplet Transform Mikio Aoi, Kyle Lepage, Yoonsoeb Lim, Uri T IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 63, NO. 3, FEBRUARY 1, 2015 699 An Approach to Time-Frequency Analysis With Ridges of the Continuous Chirplet Transform Mikio Aoi, Kyle Lepage, Yoonsoeb Lim, Uri T. Eden, and Timothy J. Gardner Abstract—We propose a time-frequency representation based nated across frequencies (fast transients). Such phase continuity on the ridges of the continuous chirplet transform to identify both is reflected in the phase of a complex TFR that is continuous in fast transients and components with well-defined instantaneous time and frequency. In spite of the ubiquity of phase-continuity frequency in noisy data. At each chirplet modulation rate, every in natural signals, the most common discrete-time analyses of ridge corresponds to a territory in the time-frequency plane such that the territories form a partition of the time-frequency frame-based TFRs treat each point in the time-frequency plane plane. For many signals containing sparse signal components, independently of every other point, ignoring aprioriinfor- ridge length and ridge stability are maximized when the analysis mation regarding continuity of phase in time and frequency. kernel is adapted to the signal content. These properties provide In fact, many sparsity-enforcing algorithms impose penalties opportunities for enhancing signal in noise and for elementary for the inclusion of correlated coefficients in a representation stream segregation by combining information across multiple [4]–[7]. analysis chirp rates. The purpose of this paper is to propose a strategy for lever- Index Terms—Chirp, time-frequency, synchrosqueezing, ridges. aging phase continuity of signal components to develop an adaptive TFR in which each component in a signal is repre- sented parsimoniously, relative to the chirplet basis family. I. INTRODUCTION The work builds on a contour representation of sound derived HE continuous chirplet transform (CT) [1] is a time-fre- from the ridges of the Gabor transform [8], [9]. In the present T quency representation (TFR) that generalizes the Gabor study, we generalize the contour representation of sound to a transform to represent signals with linear frequency modulation derivation based on the ridges of the CT, and consider new [2], [3]. Since the quality of a TFR, reflected by such properties signal representation issues that arise in this context. as energy concentration and the variance of derived parameter Using the CT-based approach, bias in instantaneous fre- estimates, is dependent on how well-adapted the TFR is to the quency (IF) estimation of an amplitude-modulated component, signal, the CT offers more flexibility for parsimonious represen- with approximately linear frequency modulation, is reduced or tation than traditional TFR’s with an unmodulated basis. eliminated when the analysis chirp rate matches the modulation However, because many signals are formed by multiple com- rate of the signal. For components with nonlinear modulation, ponents, or display nonlinear frequency modulation, methods the bias reduction comes at the price of requiring the evaluation that adapt a single CT to the whole signal may fail to represent of the transform at various modulation rates. Thus, ridges each component optimally. An improvement can be achieved by that represent the same component, but were extracted using taking many transforms, and adapting the representation across multiple transforms, form complimentary estimates that need these representations to each component separately by lever- to be combined to contribute all available information to a aging properties of the signal class of interest. One such prop- given estimate. This approach of combining estimates across erty is phase continuity. transforms also reduces the need to search for a single analysis Many physical signals display continuous phase advance- parameter (such as a “best” window width [8], [10]). ment that persists for several cycles (oscillations) or is coordi- The organization of this paper is as follows: In Section II we will review the basic notation and concepts used in the rest of Manuscript received January 29, 2014; revised July 22, 2014; accepted Oc- the paper, including development of the theory of chirplet ridges tober 01, 2014. Date of publication October 28, 2014; date of current version for both intrinsic-mode-type (IMT) [11] and click-like signals. January 08, 2015. The associate editor coordinating the review of this man- uscript and approving it for publication was Prof. Trac D. Tran. This work In Section III we conduct an analytic case study of the ridges ex- was supported in part by NSF grant NSF-DMS-1042134, the NSF Science of tracted from a deterministic, noise-free chirp and compare the Learning Center CELEST (SBE-0354378) and by a Career Award at the Sci- properties of the proposed method with the previous method of entific Interface to TJG from the Burroughs Wellcome Fund. (Corresponding [8], [9]. In Section IV we demonstrate properties of ridges that author: M. Aoi.) M. Aoi, K. Lepage, and U. T. Eden are with the Department of Mathematics can be used to identify oscillatory signal components. We show, and Statistics, Boston University, Boston, MA 02215 USA (e-mail: mcaoi@bu. consistent with the notion of physical signals displaying smooth edu; [email protected]; [email protected]). phase progression over extended regions of the time-frequency Y. Lim is with the Center for Bionics, Biomedical Engineering Research Institute, Korea Institute of Science and Technology, Seoul 136-791, Korea plane, that the length of a ridge is a useful metric for signal (e-mail: [email protected]). detection. We further show that the stability of ridge estimates T. J. Gardner is with the Department of Biology, Boston University, Boston, with respect to analysis parameters can be used to adaptively MA 02215 USA (e-mail: [email protected]). enhance the representation of signal components on a per-com- Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. ponent basis. We illustrate these principles using both simulated Digital Object Identifier 10.1109/TSP.2014.2365756 data and sample data from zebra finch vocalizations. Finally, 1053-587X © 2014 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/ redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 700 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 63, NO. 3, FEBRUARY 1, 2015 in Section V we discuss some properties of the chirplet ridge Commonly, the Gabor ridges are defined by equations, provide several interpretations and perspectives on them, and highlight the relationship with other ridge extraction (1) methods. or, equivalently [25], [26] (see Appendix A), II. NOTATION AND BACKGROUND (2) The contour representation of sound developed in [8], [9] where may represent either the Gabor or complex is a generalization of methods known more broadly as ridge wavelet transforms, and is the transform phase (for and skeleton methods. Ridge and skeleton methods [12]–[14] examples see [12], [13], [23], [27], [28]). We will show in are invertible TFRs which define ridges as curves in the time- Section III that ridges defined by (2) give biased estimates of frequency plane that represent the IF of a signal component. the IF of frequency- and amplitude-modulated components. The restriction of signal resynthesis to these ridges is called the Furthermore, the ridges of (2) are not useful for the identifica- “skeleton”. Ridge and skeleton methods have been used for es- tion of fast transients. timation of component IF, phase, and amplitude in fields such as In [8], the ridges (their “contours”) of , with direc- mechanical and civil engineering [15]–[17], analysis of chaotic tionality are defined by a linear combination of the attractors [18], and neuroscience [19], [20]. These methods are derivatives of . These linear combinations can be similar in spirit and application to high-resolution TFRs like described as directional derivatives of .Ifwede- time-frequency reassignment [21], [22], synchrosqueezing [23] and the Hilbert-Huang transform [24] in that they seek to iden- fine the rotated coordinates ,where tify, characterize, and decompose non-stationary, multicompo- is the standard rotation ma- nent signals by the IF of localized components. trix, then we can define the directional Gabor ridges in terms of Ridges, in the context of [8], [9], are used to explicitly link points that satisfy together associated points in the time-frequency plane resulting (3) in an “object-based” TFR. In this method, objects are defined by the phase (alternatively, modulus [25]) of the Gabor transform. 1 From this perspective, the fundamental objects of representation or, equivalently are not taken to be finer grain “atoms,” of the discrete trans- (4) form, but rather, the fundamental objects are the ridges them- selves (although in practice, the continuous transforms are esti- mated at discretely sampled times and frequencies). While the where ,and method implies no constraints on the signal beyond integra- . Note that and ,defining bility, parsimonious component representations are possible for the direction of the derivative as the direction with angle signals containing sparse, separable components, such as brief from the axis. Thus, (3) is a generalization of (1), where (1) transients, or IMT [11] components with well-defined IF. While is obtained from (3) by setting . “separability” of components traditionally refers to some notion The directed Gabor ridges can be used to improve detection of discriminabilty in the frequency dimension, in the present and estimation of both types of components [9]. context we generalize separability to refer to discriminability 1) Gabor Ridge Territories: Let be the set of points which in some direction in the time-frequency plane. satisfy (4), and be the subset of corresponding to In this section, we will define the ridges described in [8], [9] local maxima of with respect to the direction .
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