High-Resolution Chirplet Transform

arXiv:2108.00572v1 [eess.SP] 2 Aug 2021 inlaayi,coeysae ntnaeu frequenci instantaneous closely-spaced multi-r analysis, representation, signal time-frequency high-resolution edblt fteMC.Nmrcleprmnso simulate effectiveness. on its experiments verify signals Numerical com- real MrCT. the and the improv develo deriving of to further By approach readability we CT. post-processing TF equation, the high-concentration frequency of instantaneous representation limitati the bined single overcoming by any different resolution geometrically of TF with combined improved CTs an are obtain multiple These combining combinations. develo chi parameter by multi-resolution for a (MrCT) guidance propose transform then theoretical We paramete CT. given certain high-resolution Gaussi The rotating provides transform. and a CT Fourier employs analysis very of fractional that analysis a by extension shape window the window play the with the give it ana and first in compare (TF) we CT paper, time-frequency present this high-resolution of In parameters in an- performance role The for the important tool signals. determine valuable chirp window a linear provides alyzing window Gaussian modulated APPLICATION ITS AND TIME-FREQUENCY ia ioogUiest,X’n 109 hn emi:l (e-mail: [email protected] China [email protected]; 710049, Xi’an, University, Jiaotong Xi’an nvriy ia,704,Cia(-al [email protected] (e-mail: Engineeri China Information 710049, and Xi’an, university, Electronic of School ploration, r oyehia nvriy ia 102 hn (e-mail China 710072, Xi’an University, [email protected]) Polytechnical ern ne rn o U20B2075. no. Grant under otwl nw Fmto stelna Ftransform, TF linear the is Th method topic. non- TF research for hot known past representations a well been the TF most has reliable In analysis of signal content. stationary study frequency the time-varying decades, with non-stat signals multi-component characterizing for tool it fective predict and system data the behavior. application understand structure future to various complex help the for into signal, insight significance the provide great can that of accurat because is as signals possible non-stationary as charact multi-component the of capture To tics [4-6,9]. (IFs)) closely-spaced frequencies compl or taneous fast-varying express (like often component content and several oscillation characteristics contain time-varying signals with Such mechan and [7,8]. [5,6], biomedicine engineering [3,4], sonar and radar [1,2], M ihrslto hrlttasom rmparameters from transform: chirplet High-resolution .L,Z hn n .L r ihSho fMteaisadSta and Mathematics of School with are Li W. and Zhang Z. Li, B. .Goi ihteNtoa niern aoaoyfrOffsh for Laboratory Engineering National the with is Gao J. .Zui ihSho fMteaisadSaitc,Northwe Statistics, and Mathematics of School with is Zhu X. hswr a upre yNtoa aua cec Foundat Science Natural National by supported was work This ne Terms Index Abstract iefeuny(F nlssmtos[0 rvd nef- an provide [10] methods analysis (TF) Time-frequency evdi ayra-ol plctos uha seismic as such applications, ob- real-world many widely in are served signals non-stationary Ulti-component Tesadr hrlttasom(T ihachirp- a with (CT) transform chirplet standard —The Tm-rqec nlss hrlttransform, chirplet analysis, —Time-frequency nlsst aaeescombination parameters to analysis .I I. inxagZu e i hohn hn,WnigL,Jinghu Li, Wenting Zhang, Zhuosheng Li, Bei Zhu, Xiangxiang NTRODUCTION n). u.edu.cn). [email protected]; g ia Jiaotong Xi’an ng, zhuxiangxi- : o fChina of ion es. r i Ex- Oil ore esolution instan- ionary tistics, the e lysis. rplet eris- ping ical a p ons ely an ex of st- rs to d e s s s n[3,a dpieSF sn asinwno function window proposed. Gaussian been using have STFT adaptive methods an analysis [33], In TF wavelet. adaptive or window many global with a associated by th representation, given limitation resolution TF TF linear intrinsic fixed a the on How- from operate [6,7,23,27,28,31,32]. suffer they post- fields methods many TF these in ever, The increasingl adapted been [31,32]. and have used SST above introduced time-reassigned transf methods processing the chirplet and synchroextracting [30], the squee [24,29], multiple the transform [26,28], SST high-order the demodulate [25,27], with the ma SST involving dealing drawback, presented when this been resolution have address TF improvements To SST low [25,26]. with a signals associated from fast-varying drawback suffers one it that that SS known is is chirplet-based the It the [22], etc. [24], transform [23], curvelet S-transform synchrosqueezed synchrosqueezing STFT-bas the the [21], including SST frameworks, ha transform ext SST been different for the has to allows It decade, attentions. also last research the but considerable received Over concentration [20]. TF whi reconstruction direction, the signal frequency improves fr the only instantaneous in not the only trajectory into transform (IF) coefficients mid-1990s synchrosqueezing quency TF the in the the Maes squeezes RM, and [19], Daubechies of dis- by yielde forward energy is case put signal’s representation (SST), special TF the a sharpened of As a gravity that of such center origin tribution the the from to coefficients TF position the repr reallocates meth [17,18] reassignment TF The (RM) linear studied. widely of been have methods sentations post-processing the but distribution. readability, [10,16], TF the class blurs Affine again or smoothing class kernel inferences, Cohen’s the unwanted the kernel to the introducing leads reduce by which to developed WVD are artifact of smoothing [ versions (TF applications practical smoothed cross-terms some Various for from unusable modulated it suffers renders frequency which it WVD non-linear the analysis, or by signals obtained multi-component be in can but resolution a TF [14], Good (WVD) representati distribution presented. Wigner-Ville order TF the bilinear In by the simultaneously. Heisenberg represented resolution, frequency TF the and high time achieve is to both l method The in precisely is not of signal basis. can signal a one kind pre-assigned i.e., the [5,10], this a principle analysis, uncertainty of with TF limitation products linear main inner chirplet the In via the [11], [13]. studied and (STFT) [12], (CT) transform (CWT) transform Fourier transform wavelet short-time continuous the as such nodrt mrv h Frslto oeessentially, more resolution TF the improve to order In TF good obtain and concentration TF the increase To iGao ai ocalize ended ons, orm 15]. zes od ny s), ed ch e- re e- d. at al T y d a s 1 TIME-FREQUENCY AND ITS APPLICATION 2 with time-varying variance (i.e., window width) dependent [58], the general linear CT [56], the synchro-compensating CT on the chirp rate (CR) of the input signal is proposed. A [59], and the double-adaptiveCT [54]. The adaptability of such more powerful adaptive STFT is developed in [34,35], where methods is achieved by modifying the window width or/and the variance used in the window function is time-frequency- CR parameter depending on the instantaneous signal’s nature, varying because there exist multiple different CRs at the same consequently, they are also confronted with the challenges of time instant for multi-component signals. S. Pei et. al [36] also high computational complexity and parameter matching, as presented an improved version of adaptive STFT with a chirp- most adaptive TF analysis methods do. To better deal with modulated Gaussian window. In [37], L. Li et al. proposed the signals with non-linear IFs, the CT has been extended the adaptive SST and its second-order extension based on the to general parameterized TF transforms, which involve the adaptive STFT with a time-varying window to further enhance warblet transform [60,61], the polynomial CT [62], the spline- the TF concentration. In addition to adaptive STFT-based kernelled CT [63], and the scaling-basis CT [57]. Besides, the methods, the adaptive wavelet transform (AWT) [38] is devel- CT also can be seen as a three-dimensional analysis method as oped by adjusting the wavelet parameter. The bionic wavelet it represents a signal in the joint time-frequency-CR domain transform, proposed by Yao and Zhang [39], introduces an when considering the CR parameter as a variable [13,53]. In extra parameter to adaptively adjust the TF resolution not only this sense, the CT enables separating the intersected signals by the frequency but also by the instantaneous amplitude and with cross-over IFs by utilizing the CR information. The its first-order differential. The energy concentration of the S- theoretical analysis and high-concentration representation of transform has been addressed in [40,41] by optimizing the three-dimensional CT are recently introduced in [64-66]. width of the window function used. Moreover, the adaptive Differing from the extensions of CT above, here we intro- WVD has been developed by designing different types of duce a high-resolution analysis approach by combining multi- filters in the ambiguity domain. L. Stanković[42] proposed L- ple CTs with different window widths and CR values, which class WVD; Boashash and O’Shea [43] proposed the polyno- can help to achieve a high-resolution TF representation by mial WVD for analyzing polynomial phase signals; Katkovnik overcoming limitations of any single representation of the CT. [44] presented the local polynomial WVD; Jones and Baraniuk In this paper, we first theoretically analyze the effect of vari- [45] proposed the adaptive optimal kernel TF distribution by ance and CR parameters on the TF resolution of CT and prove introducing a signal-dependent radially Gaussian kernel that that a narrow window limits the matching capacity of chirp adapts over time. Since 2013, Khan et al. conducted a series basis. We compare the CT with its extension that employs of research in developing adaptive WVD, mainly involving the a rotating Gaussian window by fractional Fourier transform adaptive fractional spectrogram [46], the adaptive directional [67,68]. By comparison, we conclude that the parameters of TF distribution [47], the locally optimized adaptive directional rotation-window CT have a clearer geometric meaning than TF distribution [48]. Despite the high TF resolution, the that of the standard CT, but this rotation extension is actually adaptive TF representations, especially the adaptive non-linear the CT with a special parameter combination and thus can TF analysis, generally have high computational complexity, not localize a signal in both time and frequency precisely. and determining the matching parameters remains challenging To obtain a high-resolution TF representation, we propose a [48,49]. multi-resolution CT (MrCT) by computing the geometric mean Other high-resolution methods are based on the combination of multiple CTs with different parameter combinations, which of multiple spectrograms or wavelet transforms [5,50,51]. The localizes the signal in both time and frequency better than it combined methods compute the geometric mean of multiple is possible with any single CT. Finally, we develop a TF post- estimates with short and long windows or a set of wavelets, processing method to improve the readability of MrCT based and have been shown to result in good joint TF resolution on a combined IF equation. for signals consisting of mixtures of tones and pulses [5,50]. The remainder of the paper is organized as follows. In Such combinations of conventional spectrograms or wavelet Section II, the parameter analysis of the standard CT and transforms do not work well, however, for signals containing rotation-window CT is presented. In Section III, we describe mixtures of chirp-like signals; moreover, how to improve the the details of the proposed MrCT method. The post-processing TF readability of the combinated transforms is an important TF method, named multi-resolution synchroextracting chirplet problem to be solved. transform, is presented in Section IV. Experimental results and The chirplet transform (CT), proposed by Mann and Haykin comparative studies are introduced in Section V. Finally, the [13], Milhovilovic and Bracewell [52] almost at the same time, conclusions are drawn in Section VI. is a popular TF analysis method for characterizing chirp-like signals, and it has been widely applied in different areas such II. WINDOW SHAPE ANALYSIS OF CT AND ITS as radar [53,54], biomedicine [55], and mechanical engineer- ROTATION-WINDOW EXTENSION ing [56,57]. This method uses the extra CR parameter than The following sections provide parameters analysis of CT STFT (i.e., using a chirp-modulated window function, also and compare it with a rotation-window CT. termed chirp basis) to characterize linear frequency variation. When the parameter equals to the chirp rate of a signal and the input window length is matching, the CT will generate a A. Signal model highly concentrated TF representation. This fact leads to the In this paper we consider the amplitude-modulation and development of many adaptive CTs, such as the self-tuning CT frequency-modulation (AM-FM) model to describe the time- TIME-FREQUENCY AND ITS APPLICATION 3 varying features of multi-component non-stationary signals, the energy distribution of the CT in multi-component non- which is given by stationary signal analysis. A matching one with the TF feature

K K of an observed signal can help for obtaining a high-resolution jφk (t) TF representation [54,56,58,59]. f(t)= fk(t)= Ak(t)e , (1) To further analyze the effect of chirp-based window on CT, k=1 k=1 X X we consider a concrete window, i.e., the Gaussian window 1 t2 where K is a positive integer representing the number of 2 g(t) = (√2πσ)− e− 2σ2 (σ > 0 is the variance of the win- AM-FM components, √ 1 = j denotes the imaginary unit, dow), as this function leads to the optimal TF resolution and A (t) > 0 and φ (t) are− the instantaneous amplitude and k k based on it the analytic expression of some TF transformations instantaneous phase of the k-th component (or mode), respec- is easy to compute. Denote the chirp-based Gaussian window tively. The first and the second derivatives of the phase, i.e., as φk′ (t), φk′′(t), are referred to as the instantaneous frequency 1 β 2 t2 β 2 (IF) and the chirp rate (CR) of the k-th component. To model j t 2 j t hσ,β(t)= g(t)e 2 = (√2πσ)− e− 2σ2 e 2 , (6) multi-component non-stationary signals as (1) is important to extract information hidden in f(t), and this representation then the corresponding CT can be expressed as has been used in many applications including seismic wave h C σ,β (t,ω) analysis, medical data analysis, mechanical vibration diagnosis f + t−µ 2 and speech recognition, see for example [20,38]. 1 ∞ ( ) j β (µ t)2 jω(µ t) = (√2πσ)− 2 f(µ)e− 2σ2 e− 2 − e− − dµ. The Fourier transform of a given signal f(t) L2(R), i.e. Z−∞ its correlation with a sinusoidal wave ejωt, is defined∈ as (7) + The WVD of this window h t is calculated as ∞ jωt σ,β( ) fˆ(ω)= f(t)e− dt. + ∞ τ τ Z−∞ jωτ Whσ,β (t,ω)= hσ,β(t + )hσ,β∗ (t )e− dτ 2 − 2 (8) Z−∞ 2 B. Window shape analysis of CT t σ2(ω βt)2 = √2e− σ2 − − . The chirplet transform (CT) [13,52] generalizes the STFT by using an extra CR parameter and the specific definition is Obviously, the shape of Whσ,β (t,ω) is an oblique ellipse cen- given by tered at the origin (see Fig. 1), and affected by two parameters, i.e., β and σ. Maybe, the parameter β determines the direction + β 2 hβ ∞ j (µ t) jω(µ t) of the ellipse axis, and the parameter σ determines the size C (t,ω)= f(µ)g∗(t µ)e− 2 − e− − dµ, f − of the ellipse. To make this issue clear, let us consider a level Z−∞ (2) curve of Whσ,β (t,ω), which is given by where β is the CR parameter/variable, g(t) is a window t2 + σ2(ω βt)2 = C, (9) function in the Schwartz class, the superscript denotes the σ2 − j β t2 ∗ complex conjugate, and hβ(t) = g(t)e 2 called the chirp- where C is a positive constant (i.e., percentage of the maxi- modulated window. If β =0, then the CT reduces to the STFT. mum energy). Importantly, the squared modulus of CT can be interpreted as a smoothed WVD, resulting from the smoothing of the WVD 1 of the signal by the WVD of the chirp-modulated window, as 4 presented in Lemma 1. 0.8 Lemma 1. For a signal f(t), its squared modulus of the CT 2 L l can be equivalently expressed as 0.6 0 hβ 2 1 Cf (t,ω) = Wf Whβ (t,ω), (3) 0.4 | | 2π ∗∗ -2 where denotes the 2D convolution operator, and Wf (µ, η) ∗∗ 0.2 and Whβ (µ, η) are the WVDs of f(t) and hβ(t) respectively -4 defined as -4 -2 0 2 4 + t ∞ τ τ jητ Wf (µ, η)= f(µ + )f ∗(µ )e− dτ, (4) 2 − 2 Fig. 1: Plot of Whσ,β (t,ω) when σ = 1.5 and β = 1. Z−∞+ ∞ τ τ jητ Whβ (µ, η)= hβ(µ + )hβ∗ (µ )e− dτ. (5) The following theorem explains the influence of the param- 2 − 2 Z−∞ eters σ and β on the shape of ellipse (9). The proof of Lemma 1 is similar to the result in [16,18] for the spectrogram. Theorem 1. For ellipse (9), the length of the long axis is that From Lemma 1, we can know that the window shape, 2 L = C(Aσ2 + + A2σ4 +4β2), (10) as a TF filter or kernel, is critical because it determines l σ2 p TIME-FREQUENCY AND ITS APPLICATION 4

2 1 where A =1+ β σ4 , and the angle θ between t-axis and Furthermore, the angle between t-axis and the long axis of the long axis (see Fig.− 1) satisfies ellipse (9), denoted by θ, satisfies ω 2β lim tan θ = β, (11) tan θ = 1 = + β. (18) σ + 2 2 4 2 2 → ∞ t1 σ ( A σ +4β + Aσ ) 1 tan θ 1, for 0 <σ< 1. (12) Based on expression (18),p and A 1+ β2 when σ + , 4 → → ∞ ≥ rσ − we have Proof. We first calculate the vertices of ellipse (9). Since the lim tan θ = β. (19) σ + vertice is the point on the ellipse that is maximum (minimum) → ∞ distance from the center point, then it can be obtained by Given the σ such that σ < 1 (i.e, a small variance). Since solving the following optimization problem: 4β2 2β2 A2σ4 +4β2 = A σ2 1+ A σ2(1 + ), max t2 + ω2, | | A2σ4 ≤ | | A2σ4 t,ω r thenp t2 (13) s.t. σ2 ω βt 2 C. 2β 2 + ( ) = σ − tan θ 2 + β. (20) ≥ σ2( A σ2(1 + 2β )+ Aσ2) Due to that the points on the ellipse satisfy the following | | A2σ4 relation: When A 0, i.e., 0 < β 1 1, equation (20) can be ≤ ≤ σ4 − C t2 further expressed as q ω = + βt, t [ √Cσ, √Cσ], (14) σ2 − σ4 ∈ − A 1 r tan θ | | + β 1. (21) ≥ β ≥ σ4 − C t2 r here we only consider the positive part of σ2 σ4 and − 1 assume β 0 due to the symmetry of the ellipse, then problem When A 0, i.e., β 4 1, we have ≥ q ≥ ≥ σ − (13) can be further written as an unconstrained optimization q β 1 problem as follows: tan θ + β β 1. (22) ≥ 4 β2 ≥ ≥ σ4 − Aσ + A r C t2 C At2 βt , (15) Based on the situations analyzed above, we can obtain results max +2 2 4 + 2 t rσ − σ σ (10-12), which finishes the proof. 2 1 √ √ where A =1+ β σ4 , and t [ Cσ, Cσ]. From (15), Theorem 1 throws up some interesting results. On the one one can calculate its− stable points∈ as− below: hand, the length Ll increases with the value of β increasing, and also exhibiting an increase with the decrease in value of 1 Aσ2 t1 = σ√C + , (16) σ, which easily leads the CT to generate a bad TF result when s2 2 A2σ4 +4β2 dealing with strong modulation signals because a large β and 1 p Aσ2 small σ are required to match its fast-varying feature. On the t2 = σ√C . (17) other hand, from expression (18), we can know that there does 2 − A2σ4 β2 s 2 +4 not exist a correspondence relationship that tan θ = β between Substituting (16) and (17) into (14),p the vertices of the ellipse the rotation angle θ and parameter β. But a large value of (9) are thus obtained as follows: σ is more advisable for CT because based on it the major

axis of Whσ,β (t,ω) can rotate along any direction to approach Aσ2 that tan θ = β. Moreover, relation (12) shows that tan θ is √ 1 (t1,ω1) = (σ C + , bounded and it can not be taken as a small value when the s2 2 A2σ4 +4β2 input variance is small (e.g., σ 0.1), which limits the slope √C 1 Aσ2 p 1 Aσ2 range of W (t,ω), leading to≤ inaccurate match in dealing + βσ√C + ), hσ,β σ s2 − 2 A2σ4 +4β2 s2 2 A2σ4 +4β2 with short signal segments containing smaller CR values. Fig. 2 shows the WVDs of the chirp-based Gaussian win- p 1 Aσ2 p (t2,ω2) = (σ√C , dow using various values of β and σ. The ﬁrst row in Fig. 2 is s2 − 2 A2σ4 +4β2 the results corresponding to a large variance (i.e., σ =2) over different values of β (β = 0, β = 1, β = 5). It can be seen √C 1 Aσ2 p 1 Aσ2 + + βσ√C ). from the results that the length of the major axis increases with σ 2 4 2 2 4 2 s2 2 A σ +4β s2 − 2 A σ +4β the increasing of β, and the rotation angle θ approximately satisﬁes the relationship that tan θ = β. While using a small It is easy to verifyp that point (t1,ω1) locates onp the long axis σ (e.g., σ = 0.3), the length of the major axis also increases of ellipse (9), thus the length of the long axis, denoted by Ll, can be computed by inserting (16) into (15), which is given with the increasing of β, but the range of rotation angle θ is limited in a neighborhood of π (see Fig. 2(d-f)). by 2 To solve the existing problem of the standard CT, we 2 will discuss the performance of a rotation-window CT in the L = C(Aσ2 + + A2σ4 +4β2). l σ2 following setion. p TIME-FREQUENCY AND ITS APPLICATION 5

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Fig. 2: The WVDs of hσ,β(t) using various values of σ and β. (a) The WVD with σ = 2 and β = 0, (b) the WVD with σ = 2 and β = 1, (c) the WVD with σ = 2 and β = 5, (d) the WVD with σ = 0.3 and β = 0, (e) the WVD with σ = 0.3 and β = 5, (f) the WVD with σ = 0.3 and β = 20.

C. CT with rotation window To discuss parameters σ and β are how they affect the shape of W˜ (t,ω), we also consider a level curve of W˜ , which In order to obtain a more canonical TF ﬁlter, i.e., parameters hσ,β hσ,β is σ and β have a clear role in determining the shape of 2 t 2 2 Whσ,β (t,ω), we use the fractional Fourier transform (FrFT) +ˆσ (ω βtˆ ) = C, (27) [67,68] to rotate Gaussian window g(t) with parameter β, σˆ2 − which is given by where C is a positive constant. The follwing theorem shows the determination of the shape of ellipse (27) by parameters 1 2 2 2 − σ(1+β2) β(1−σ2) σ and β. 1+ σ β t2 j t2 ˜ √ − 2(1+σ2 β2) 2(1+σ2β2) hσ,β(t)= 2π 2 e e . sσ(1 + β )! Theorem 2. For ellipse (27), the length Ll of the long axis (23) is that 2C σ , if σ < 1, Using this rotation window, the corresponding CT, named Ll = (28) 2σC, if σ 1. rotation-window CT, is that ( ≥ + The angle θ between t-axis and the long axis satisﬁes ˜ hσ,β ∞ jω(µ t) C (t,ω)= f(µ)h˜σ,β(t µ)e− − dµ. (24) f − β, if σ < 1, Z−∞ tan θ =  0, if σ =1, (29) h˜σ,β Based on Lemma 1, the squared modulus of C (t,ω) can  1 , if σ > 1. f − β be expressed as Proof. From Theorem 1, the length of the long axis of ellipse  h˜ 1 (27) is that C σ,β (t,ω) 2 = W W (t,ω), (25) f f h˜σ,β | | 2π ∗∗ 2 2 2 4 2 Ll = C(Aˆσˆ + + Aˆ σˆ +4βˆ ), (30) ˜ σˆ2 where Wh˜ is the WVD of hσ,β(t) as follows: q 2 2 2 σ,β ˆ ˆ2 1 2 1+σ β ˆ β(1 σ ) where A =1+ β σˆ4 . Due to σˆ = σ(1+β2) , β = 1+σ−2β2 , 2 − t σˆ2(ω βtˆ )2 then we have W˜ (t,ω)= √2e− σˆ2 − − , (26) hσ,β 2 2 2 ˆ 2 (1 σ )(1 β σ ) 2 Aσˆ = − − . (31) 2 1+σ2β2 ˆ β(1 σ ) σ(1 + σ2β2) with σˆ = σ(1+β2) , β = 1+σ−2β2 . TIME-FREQUENCY AND ITS APPLICATION 6

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Fig. 3: The WVDs of h˜σ,β(t) using various values of σ and β. (a) The WVD with σ = 2 and β = 0, (b) the WVD with σ = 2 and β = 1, (c) the WVD with σ = 2 and β = 5, (d) the WVD with σ = 0.3 and β = 0, (e) the WVD with σ = 0.3 and β = 5, (f) the WVD with σ = 0.3 and β = 20.

When σ < 1, then and the parameter β determines its direction. This property makes the role of β and σ more clear and is helpful for 1 σ2 Aˆ2σˆ4 +4βˆ2 = − . (32) parameter setting. Further explanation is given from numerical σ q experiments, which are presented in Fig. 3. From the results, Substituting (31) and (32) into (30), we have we can see that the size of the WVDs is unchangeable when 2C ﬁxing the value of σ. Moreover, when using different values Ll = . ˜ σ of β, the rotation of the WVD of hσ,β(t) satisﬁes the relation that θ β if σ < , and θ 1 if σ > . When σ 1, then tan = 1 tan = β 1 ≥ − σ2 1 Aˆ2σˆ4 +4βˆ2 = − . (33) σ D. Comparison q Substituting (32) and (34) into (31), we have The section above mainly analyzes the effect of the chirp- based Gaussian window and the rotating Gaussian window Ll =2Cσ. on the CT from theory, which provides important theoretical guidance for CT to choose suitable parameters so as to obtain a Similar to equation (18), the angle θ between the t-axis and high-resolution TF representation. In the following, we test the the major axis of ellipse (27) also meets the relation that performance of CT and rotation-window CT by an example. 2βˆ The waveform of the test signal is illustrated in Fig. 4(a), θ β.ˆ (34) tan = + which contains two chirp signals (CR=7) and two transient 2 ˆ2 4 ˆ2 ˆ 2 σˆ ( A σˆ +4β + Aσˆ ) signals. We compute the TF representations of this test signal Combining (31), (32) andq (34), we can obtain tan θ = β when using CT and rotation-window CT with different parameters 1 (see Fig. 4(b-g)). From the results, we can see that the CT σ < 1. Likewise, we derive that tan θ = β when σ > 1 by − can fully separate the two transient signals when employing a inserting (31) and (33) into (34). Especially, since βˆ =0 when smaller value of σ, but a small one results in a poor frequency σ =1, then tan θ =0 for σ =1. To sum up, results (28) and resolution such that the TF mixing of chirp components occurs (29) are derived. (see Fig. 4(a)). A relative large σ following with good CR For the rotation window h˜σ,β(t), Theorem 2 proves that estimation can resolve the two chirps, and yet obscuring the the parameter σ determines the size of the WVD of h˜σ,β(t), TF information of the two pulses (see Fig. 4(d)). Moreover, TIME-FREQUENCY AND ITS APPLICATION 7

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0 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Time / s Time / s Time / s (e) (f) (g) Fig. 4: Evaluation of TF resolution on a known signal structure. (a) Waveform of the test signal, (b) CT with σ = 0.1 and β = 7, (c) CT with σ = 0.1 and β = 1000, (d) CT with σ = 0.3 and β = 7, (e) the rotation-window CT with σ = 0.1 and β = 7, (f) the rotation-window CT with σ = 0.1 and β = 1000, (g) the rotation-window CT with σ = 0.3 and β = 7. using a large β to match the fast-varying feature of transient especially when dealing with signals with closely-spaced IFs. signals degrades the TF performance of the CT, as presented More effective possibilities should be considered to overcome in Fig. 4(c). By contrast, it follows that the rotation-window the TF resolution problem. The following section utilizes CT provides a better time resolution when using a small σ the combination of multiple CTs with different parameter and a large β (see Fig. 4(f)), and can separate the two chirps combinations to improve the TF resolution of the CT, as it even with a small σ (see Fig. 4(e)). In fact, an advantage can localize the signal in both time and frequency better than of the rotation-window CT over the standard CT is that the it is possible with any single CT. Moreover, the combined one parameters in the window become relative, forming effective effectively reduces the computational complexity of adaptive combinations such that the window shape can rotate along any CTs in which the window function is adjusted at every TF direction in TF plane only by adjusting the parameter β. location. Nevertheless, the rotation-window CT is actually the standard CT with a special parameter combination (i.e., σˆ and βˆ), thus it is still limited by the Heisenberg uncertainty principle A. Multi-resolution chirplet transform and can not localize precisely a signal in both time and Let us consider a two-component non-stationary signal frequency. As presented in Fig. 4, the window of these two including a transient signal and a chirp signal, which is given methods matches the pulse (chirp) components well but smears by the chirp (pulse) components. This illustrates the fundamental tradeoff of the CT: it is difficult to get both good time and f(t)= f1(t)+ f2(t), (35) good frequency resolution using a single fixed window. j(at+ b t2) f (t)= A δ(t t ), f (t)= A (t)e 2 , 1 1 − 0 2 2 III. MULTI-RESOLUTIONCHIRPLETTRANSFORM where A1 > 0, A2(t) 0, a,b R, and δ( ) is the Dirac From the discussion in the previous section, it is known delta function, a generalized≥ function∈ formulated· as δ(t)=1 that parameter selection is a significant challenge for CT, when t =0, δ(t)=0 when t =0. 6 TIME-FREQUENCY AND ITS APPLICATION 8

Assuming that around µ = t the amplitude A2(t) can be 60 60 50 50 well approximated by A2(t+µ) A2(t), then the CT of signal ≈ 1 t2 40 40 − 2σ2 (35) under the Gaussian window g(t) = (√2πσ) e− is 30 30 Fre / Hz Fre / Hz computed as 20 20 + 10 10 β 2 hσ,β ∞ 0 0 j 2 (µ t) jω(µ t) Cf (t,ω)= f(µ)g∗(t µ)e− − e− − dµ 0 0.5 1 1.5 2 0 0.5 1 1.5 2 − Time / s Time / s Z−∞ + 2 (a) (b) 1 ∞ µ j β µ2 jωµ = (√2πσ)− f (t + µ)e− 2σ2 e− 2 e− dµ 1 Fig. 5: The results of the combined CT with σ = 0.1 and σ = 0.3 Z−∞ (here β = 7). (a) The product of the two CTs, (b) the geometric + 2 1 ∞ µ β 2 − 2 j 2 µ jωµ mean of the two CTs. + (√2πσ) f2(t + µ)e− 2σ e− e− dµ

Z−∞ − 2 1 (t0 t) β 2 − 2 j 2 (t0 t) jω(t0 t) = (√2πσ) (A1e− 2σ e− − e− − + increase of the number of product-times, i.e., a small TF + µ2 (β−b) energy becomes smaller and smaller, and a large one is getting j(at+ b t2) ∞ j µ2 j(ω a bt)µ e 2 A2(t + µ)e− 2σ2 e− 2 e− − − dµ) bigger and bigger. Therefore it is undesirable to multiply

Z−∞ − 2 too many CTs for the combined CT. An effective way to 1 (t0 t) β 2 − 2 j 2 (t0 t) jω(t0 t) (√2πσ) A1e− 2σ e− − e− − solve the magnitude issue is to compute its geometric mean ≈ + 2 1 ∞ µ (β−b) 2 [5,50,51]. However, the mean step inevitably reduces time and − 2 j 2 µ j(ω a bt)µ + (√2πσ) f2(t) e− 2σ e− e− − − dµ frequency resolution of the TF representation (see Fig. 5(b)).

Z−∞ − 2 Nevertheless, we can derive the following conclusion based on β=b 1 (t0 t) b 2 − 2 j 2 (t0 t) jω(t0 t) ====(√2πσ) A1e− 2σ e− − e− − (37) and (38), which shows the superiority of the combined 2 σ (ω a bt)2 CT over any single CT. + f2(t)e− 2 − − . (36) Theorem 3. For signal (35), given any σ1 > 0, σ2 > 0, If f1(t) and f2(t) are further separable, e.g., A2(t) 0 when hσ,β ≈ σ1 = σ2, there does not exist σ > 0 such that Cf (t,ω) t [t0 σ, t0 + σ], then the modulus of CT can be expressed 6 |h | ∈ − σ1,β as has the same time and frequency resolution as Cf (t,ω) hσ ,β 1 | × h C 2 (t,ω) 2 . C σ,b (t,ω) f | | f | 1 1 2 σ2 2 (t0 t) (ω a bt) The proof of Theorem 3 is evident based on (37) and − 2σ2 2 (√2πσ) A1e− − + A2(t)e− − − . (37) hσ,β ≈ (38). If we assume Cf (t,ω) has the same frequency hσ ,β | hσ |,β 1 From (37), we can know that high time-resolution can be 1 2 2 resolution as Cf (t,ω) Cf (t,ω) for f2(t), which achieved for f (t) with a small value of σ, but that causes | σ2 σ2× | 1 means that σ2 = 1 + 2 . In this case, the duration of a degradation of frequency resolution. Conversely, a large σ 2 2 2 hσ,β σ1 +σ2 increases frequency resolution for f2(t) but at the expense of Cf (t,ω) for f1(t) is 2 . While the time spread of | h | h 1 temporal resolution of the CT of f (t) (see Fig. 4(b,d) for σ1,β σ2,β √2σ1σ2 1 C (t,ω) C (t,ωq) 2 for f1(t) is satisfying f f σ2+σ2 numerical illustration). | × | √ 1 2 σ2+σ2 To overcome this problem, let us consider the product of two √2σ1σ2 < σ (here σ 1 2 ) for σ σ , which 2 2 = 2 1 = 2 √σ1 +σ2 6 CTs with different parameters σ1 and σ2, which is calculated 1 hσ1,β qhσ2,β as means that Cf (t,ω) Cf (t,ω) 2 has a better time | hσ,β × | 2 2 resolution than C t,ω for f t if they have the same σ1+σ2 2 f ( ) 1( ) (t0 t) hσ ,b hσ ,b 1 2 2σ2σ2 | | 1 2 − 1 2 − frequency resolution for f t . Similarly, we can prove that Cf (t,ω) Cf (t,ω) A1e 2( ) | × |≈ 2πσ σ hσ ,β hσ ,β 1 1 2 1 2 2 2 2 Cf (t,ω) Cf (t,ω) provides a better frequency res- σ1 +σ2 2 2 (ω a bt) | ×hσ,β | + A2(t) e− 2 − − . (38) olution than C (t,ω) for f (t) if they have the same time | f | 2 resolution for f1(t). Furthermore, when specifying σ2 =2σ1, Comparing (37) with (38), we ﬁnd that the product has better hσ ,β hσ ,β 1 the time-bandwidth product of C 1 (t,ω) C 2 (t,ω) 2 time resolution for f1(t) and increased frequency resolution | f × f | can be computed approximately as 2 , smaller than the value for f2(t). To make this point more clear, we give the combined 5 h CT of the test signal (see Fig. 4(a)) in Fig. 5(a). Obviously, of C σ,β (t,ω) that corresponds to 1 . This theorem in fact | f | 2 the combined result achieves a sharper TF representation than shows that combined multi-resolution CT resolves the joint TF that by the single CT with σ = 0.1 or σ = 0.3, as presented density better than it is possible with a single CT represen- in Fig. 5(a) and Fig. 4(b,d). tation in dealing with signal (35). The numerical experiment Unlike the adaptive CTs [56,59], the combined CT only also veriﬁes the theoretical results, one can see Fig. 4(b,d) uses two parameters σ1 and σ2 to yield a high-resolution and Fig. 5. We should also highlight the fact that the combined result, thus it reduces the computational complexity effectively. procedure may not break the uncertainty principle because the Meanwhile, it is worth pointing out that, when computing principle does not hold for all kinds of TF representations. the product of multiple CTs to increase the TF resolution, In practical application, most signals contain multiple com- one existing issue is that the magnitude of the combined ponents with time-varying information, so we should extend representation may become increasingly polarized with the the combined CT based on two CTs to multi-resolution case. TIME-FREQUENCY AND ITS APPLICATION 9

K Define Sσ,β as a set of hσ,β(t) with different σ and β: utilize the chirp-Fourier transform (CFT) [70] to detect the CR m information of the signal for the sake of simplicity, which is Sσ,β = hσ,β(t) (σ, β) = (σ1,β1), (σ2,β2), , (σm,βm) , defined as { | ··· (39)} + ∞ j β µ2 jωµ where m is a positive integer denoting the number of chirp- CFTf (ω,β)= f(µ)e− 2 e− dµ. (43) based Gaussian windows. The multi-resolution chirplet trans- Z−∞ form (MrCT) of a signal f t with regard to SK is then ( ) σ,β When the chirp rate and the harmonic frequency is matched defined as to the signal, the magnitude of CFT appears a peak [70]. 1 m m Therefore we can obtain m CR estimations, which basically hσi,βi MrCf (t,ω)= Cf (t,ω) , (40) correspond to the varying tendency of the frequency of the sig- i=1 ! nal, by detecting the first m peaks of CFTf (ω,β) , denoted Y | | MrCT is to compute the geometric mean of m CTs, which as (ω∗,β∗) (i =1, 2, ,m). Parameters βi (i =1, 2, ,m) i i ··· ··· corresponds to the optimal solution in the Kullback-Leibler of MrCT are then taken as divergence [69] sense. h βi = βi∗, i =1, 2, , m. (44) Theorem 4. Given m CTs: C σi,βi (t,ω), i = 1, 2, ,m, ··· f ··· the MrCf (t,ω) is the optimal solution of the following The CR estimations, in turn, can be utilized in selecting | | optimization problem: the appropriate σ. Cohen in [10] presented the relationship m between an optimal window width and the CR of a signal. For hσi,βi min P (t,ω) Cf (t,ω) GKL, (41) the Gaussian window case, a width is described as a standard P (t,ω) 0 k − | |k ≥ i=1 deviation and can be defined as follows: X P (t,ω) where P (t,ω) C(t,ω) GKL = P (t,ω) ln 1 k − k C(t,ω) − σ(t)= , (45) P (t,ω)+ C(t,ω)dtdω for P (t,ω) 0 and C(t,ω) 0. 2π φ (t) ≥ R R ≥ | ′′ | Proof. The derivative with respect to P (t,ω) of optimization where φ (t) = 0. Based onp this fact, parameters σ (i = problem (41) must equal zero at the solution point: ′′ i 1, 2, ,m) of6 MrCT are determined by m ··· P (t,ω) ln h =0. (42) Cσ σi,βi σi = , i =1, 2, , m, (46) i=1 Cf (t,ω) | | 2π βi∗ ··· X | | Solving this equation for P (t,ω) yields the optimal solution where Cσ > 0 isp used to adjust the σ in a more rea- 1 m m sonable interval. Hence, the values of parameters (σi,βi) h P (t,ω)= C σi,βi (t,ω) . (i =1, 2, ,m) in MrCT is determined by | f | i=1 ! ··· Y Cσ (σi,βi) = ( ,βi∗), i =1, 2, , m. (47) 2π βi∗ ··· It is remarkable here that Theorem 4 provides a more | | p profound interpretation for the MrCT (it can be seen as Alternatively, the parameters σi (i =1, 2, ,m) also can ··· the“optima” way to combine individual CTs). There are, of be chosen multiplicatively or additively, i.e., σi = iσ1 or σi = course, other measures, such as L2-norm and maximum cross- σ1 + i∆σ, as presented in [5], and further matched with the entropy [51], used to combine information from multiple CTs, detected βi (i =1, 2, ,m) depending on the relation (45), which are worthy of continued investigation. even though the equality··· does not hold at this time. Number m is generally determined based on prior infor- B. Parameters selection mation or estimation from a coarse TF representation. This number is tolerant and usually much smaller than the length Determine the values of parameters (σ ,β ) (i = 1, 2, , i i of the analyzed signal, which is one important reason to reduce m) is crucial for MrCT. We discuss how to select effective··· the computational complexity of adaptive CTs. values in this section. It is well known that the optimal values of parameters σ and β are signal-dependent such that the window can C. Computational complexity adapt to the signal being analyzed. Several estimation methods of signal features (e.g., CR) are available in the literature The MrCT is to compute the geometric mean of m CTs, [35,46,54,56,58], while which generally focus on its contin- so its computational complexity mainly comes from the CT uously varying characteristics. We propose that the MrCT calculations with different windows. To be specific, we assume is computed based on several discrete values (σi,βi) (i = that a signal of N samples is utilized, then the CTs with 1, 2,...,m), over a range of values that includes its maxi- different (σi,βi) (i = 1, 2, ,m) can be implemented by ···2 mum and minimum acceptable values. In order to get robust FFT and they require O(mN log2 N) operations. Therefore, 2 estimates that contain the main information of the signal, we the total computing complexity of MrCT is O(mN log2 N). TIME-FREQUENCY AND ITS APPLICATION 10

IV. MULTI-RESOLUTIONSYNCHROEXTRACTINGCHIRPLET respect to different parameters. The MrIF (t,ω) is derived TRANSFORM from multi-resolution chirplet representation, which provides In this section, we will develop a novel analysis approach a more accurate IF estimation due to the high-resolution to enhance the TF readability of the proposed MrCT. of original transform. More importantly, the transient signal For the convenience of the following explanation, we con- satisﬁes the combined IF equation, as stated in the following sider the case of a chirp signal f(t) = Aejφ(t), where theorem. φ(t) = at + b t2. The CT of chirp signal f(t) under the 2 Theorem 5. For transient signal f(t) = Aδ(t t0), its TF 1 t2 2 − Gaussian window g(t) = (√2πσ)− e− 2σ2 is given by energy center t = t0 satisﬁes

hσ,β Cf (t,ω) MrIF (t0,ω)=0. (54) + 1 (t−µ)2 β 2 ∞ j (µ t)2 jω(µ t) = (√2πσ)− f(µ)e− 2σ2 e− 2 − e− − dµ Proof. The CT of the transient signal f(t)= Aδ(t t0) with 1 t2 − Z−∞ 2 √ − 2σ2 2 2 the Gaussian window g(t) = ( 2πσ) e− is computed √2πσ σ (ω φ′(t)) = f(t) exp − , as s1+ jσ2(β b) −2(1 + jσ2(β b)) 1 2 − − (t −t) β 2 hσ,β 2 0 j (t t) jω(t t) (48) √ − 2σ2 2 0 0 Cf (t,ω)= A( 2πσ) e− e− − e− − . where φ′(t)= a + bt. From (48), we can obtain that (55) 2 ∂ hσ,β hσ,β bσ (ω φ′(t)) From (55), we can obtain that C (t,ω)= C (t,ω) jφ′(t)+ − . ∂t f f 1+ jσ2(β b) − ∂ h h 1 (49) C σ,β (t,ω)= C σ,β (t,ω) ( + jβ)(t t)+ jω . ∂t f f σ2 0 − Due to that (56) ∂ Chσ,β (t,ω) ∂t f Combining (56) and (50) yields

+ t−µ 2 ∂ 1 ∞ ( ) j β (µ t)2 jω(µ t) 2σ2 2 thσ,β hσ,β = f(µ)e− e− − e− − dµ C (t,ω)= C (t,ω)(t0 t). (57) ∂t √2πσ f f − Z−∞ ! 1 th h = ( +pjβ)C σ,β (t,ω)+ jωC σ,β (t,ω), (50) Therefore, for different combinations (σi,βi) (i = 1, 2, , σ2 f f m), we have ··· 2 th 1 + (µ−t) σ,β 2 2 where C (t,ω) = (√2πσ)− ∞ f(µ)(µ t)e− 2σ m thσ ,β 1 f i i m β ( C (t0,ω)) j (µ t)2 jω(µ t) −∞ − i=1 f e 2 e dµ. Combining (49) and (50), we can MrIF (t ,ω)= =0. − − − − R 0 MrC (t ,ω) obtain Q f 0

1 ( + jβ)Cthσ,β (t,ω) σ2 f 2 (51) hσ,β bσ (ω φ′(t)) =C (t,ω) j(φ′(t) ω)+ − . Theorem 5 shows a very interesting result that the combined f − 1+ jσ2(β b) − IF equation not only can cover the instantaneous characteristic For different combinations (σi,βi) (i =1, 2, ,m), we can of chirp-like signals but also covers the instantaneous infor- ··· derive the following expression from equation (51) as mation of transient signals, which uniﬁes the well-known IF

m thσ ,β 1 and group delay detectors proposed in the SST [20,21] and ( C i i (t,ω)) m i=1 f the time-reassigned SST [31,32]. Q MrCf (t,ω) Based on the combined IF equation (53), a novel TF 1 m 2 m representation called the multi-resolution synchroextracting 1 1 bσi (ω φ′(t)) = ( 2 + jβi)− j(φ′(t) ω)+ 2− , chirplet transform (MrSECT) is proposed, which is σ − 1+ jσ (βi b) i=1 i i ! Y − (52) MrSEC(t,ω)= MrCf (t,ω)δ(MrIF (t,ω)). (58) where MrCf (t,ω) > γ, the parameter γ > 0 is a hard | | Considering the calculation error, it is suggested that (58) be threshold on MrCf (t,ω) to overcome the shortcoming that | | expressed as MrCf (t,ω) 0. Obviously, the set of points ω = φ′(t) satisﬁes| | ≈ MrSEC(t,ω) thσ ,β 1 m i i thσ ,β 1 ( C (t,ω)) m m C i i t,ω m i=1 f (Qi=1 f ( )) ∆ω MrIF (t,ω) := =0. (53) MrCf (t,ω), < , (59) MrC (t,ω) = MrCf (t,ω) 2 Q f  0, otherwise,  We call equation (53) the combined IF equation because it can be seen as a combination of multiple IF equations with where∆ω is the discrete frequency interval. TIME-FREQUENCY AND ITS APPLICATION 11

V. NUMERICAL VALIDATION ﬁgure, the spectrogram with a short window provides the To illustrate the proposed methods, we employ several best representation for this signal, and MrCT is better than examples involving multi-component chirp signals, cosine other two. Above all, the combination provides a better overall modulated signals and impulsive-like signals which with fast- representation than any of the spectrograms considering both varying frequency characteristics. For comparison, the STFT time and frequency resolutions. with different window width, the adaptive CTs (i.e., GLCT [56] and double-adaptive CT (DACT) [54]), and the combined 120 120 100 100 multi-resolution spectrogram (MrSTFT for short) [50] have 80 80 also been considered. In addition, the state-of-the-art highly 60 60 Fre / Hz Fre / Hz concentrated schemes, namely the SST [21], RM [18], time- 40 40 reassigned SST (TSST for short) [31], second-order SST 20 20 0 0 (SSST for short) [26], and SECT [30], have also been used. 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Time / s Time / s The window function used in these methods is uniﬁed as (a) (b) a Gaussian window, in which the parameter σ is adjusted manually with the aim of providing the best achievable repre- 120 120 sentations according to the visual inspection of the resulting 100 100 80 80

TF plots. 60 60 Fre / Hz Fre / Hz

40 40 A. Example 1 20 20 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 We ﬁrst use the proposed methods to detect close chirp Time / s Time / s components, which is given by (c) (d) Fig. 6: TF results using STFT and MrCT methods. (a) The STFT f(t)= f1(t)+ f2(t)+ f3(t)+ f4(t)+ n(t), with σ = 0.1, (b) the STFT with σ = 0.16, (c) the STFT with sin(2π (20t + 40t2)), t [0, 1], σ = 0.25, (d) the MrCT (m = 3). f1(t)= × ∈ 0, t (1, 2], ( ∈ 2 4 4 π t t , t . , . , STFT ( =0.1) STFT ( =0.1) sin(2 (34 + 40 )) [0 1 0 78] 3.5 STFT ( =0.16) 3.5 STFT ( =0.16) f2(t)= × ∈ STFT ( =0.25) STFT ( =0.25) ( 0, t [0, 0.1) (0.78, 2], 3 MrCT 3 MrCT ∈ ∪ 2.5 2.5 2 f3(t)= FT (0.02 exp(2πj(430t 5t ))), t [0, 2], 2 2 − ∈ Amplitude 1.5 Amplitude 1.5 f (t)= FT (0.02 exp(2πj(445t 5t2))), t [0, 2], 4 − ∈ 1 1 (60) 0.5 0.5 0 0 30 40 50 60 70 80 90 1.2 1.3 1.4 1.5 1.6 1.7 where n(t) is the Gaussian noise with the SNR = 10 dB, and Fre / Hz Time / s FT denotes the Fourier transform. The sampling frequency is (a) (b) 256 Hz. Fig. 7: TF energy curves by selecting various frames. (a) Spectrum To illustrate the resolution of MrCT, we ﬁrst compare of the time frame t = 0.5 s, (b) time evolution of the frequency bin this method to the STFT with different window width (i.e., Fre = 46 Hz. σ = 0.1, 0.15, 0.25) to analyze signals f1(t) and f4(t). The

TF results are displayed in Fig. 6. From the results, we can 120 120 observe that the STFT using a smaller value of σ (i.e., σ =0.1) 100 100 can achieve a highly concentrated characterization for f4(t), 80 80

60 60 Fre / Hz but spreads the energy distribution of f1(t). In contrast, a Fre / Hz relative large σ brings the opposite results for STFT used to 40 40 address these two signals, as presented in Fig. 6(c). Finding 20 20 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 the optimal value of σ is important for STFT, particularly Time / s Time / s when dealing with close signals. Unfortunately, it is not easy (a) (b) to determine the best one in practice. In this regard, the multi- 120 120 resolution representations reduce the difﬁculty of tuning the 100 100 parameters and can provide a sharped TF energy distribution 80 80

60 60

(see Fig. 6(d)). To further show the energy concentration more Fre / Hz Fre / Hz clearly, we list the TF energy curves by selecting two time 40 40 and frequency frames (see Fig. 7). The results show that: the 20 20 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 frequency peak is better deﬁned by the MrCT than that by Time / s Time / s the spectrograms computed with various analysis windows, (c) (d) which implies the effectiveness of MrCT to introduce the Fig. 8: TF results of signal (60) using various analysis methods. (a) extra CR parameter, besides the window width parameter; GLCT, (b) DACT, (c) MrSTFT, (d) MrCT. on the other hand, as for signal f4(t), seen in the second TIME-FREQUENCY AND ITS APPLICATION 12

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0 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Time / s Time / s Time / s (d) (e) (f) Fig. 9: TF results by various post-processing methods. (a) TF plot by SST, (b) TF plot by RM, (c) TF plot by TSST, (d) TF plot by SSST, (e) TF plot by SECT, (f) TF plot by MrSECT.

Fig. 8 presents the TF representations by two adaptive CTs where n(t) is the Gaussian noise with the SNR= 8 dB. The (i.e., GLCT and DACT) and two combined methods (i.e., sampling frequency is 512 Hz, and time duration is [0 1s]. MrSTFT and MrCT, here these two combinations use the same Fig. 10 provides the TF plots of signal (61) obtained by values of σ). From the results, we can see that the MrCT yields the GLCT, DACT, MrSTFT, and MrCT (m = 6). It is seen a high-concentration TF result, and the close components are that serious cross-terms appear between the close components clearly separated. While the other three methods lead to the TF in the TF representations of GLCT and DACT, and these two plots being difficult to interpret. In addition, the computational methods are unable to fully segregate time and frequency. By times required for the GLCT, DACT, MrSTFT, and MrCT in contrast, MrSTFT and MrCT provide a faithful local repre- addressing signal (60) are 0.312 s, 0.031 s, 0.027 s, and 0.041 sentation, with high resolution in both time and frequency. s, respectively (the tested computer configuration: Intel Core Compared to MrSTFT, the MrCT has more concentrated TF i5-6600 3.30 GHz, 16.0 GB of RAM, and MATLAB version energy at the modulation part (see TF plot in the time interval R2020b). The last three methods need much less calculation [0.1, 0.3], [0.7, 0.9]), showing good performance in handling comparing with the GLCT in which the window function is cosine modulation signals. adjusted at every TF location. Fig. 9 displays the TF representations obtained by six TF 250 250 post-processing methods, involving the SST, RM, TSST, SSST, SECT, and MrSECT. It can be seen from the results that, 200 200 there are heavy cross-terms between the first two modes in the 150 150 TF plane for the first five methods. The MrSECT is clearly Fre / Hz 100 Fre / Hz 100 superior to others, yielding high concentration of both close 50 50

0 0 components. Compared to the TF result of MrCT (see Fig. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Time / s Time / s 8(d)), MrSECT effectively improves its TF readability. (a) (b) B. Example 2 250 250 200 Next, let us consider a four-component AM-FM signal 200 with two cosine modulated modes and two impulsive-like 150 150 Fre / Hz components as 100 Fre / Hz 100 50 50 f(t)= f1(t)+ f2(t)+ f3(t)+ f4(t)+ n(t), 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 f1(t)=(1+0.05 cos(20πt)) cos(2π(9 sin(2πt)+80t)), Time / s Time / s (c) (d) f2(t)=(1+0.1 cos(20πt)) cos(2π(9 sin(2πt)+115t)), 2 Fig. 10: TF results of signal (61) using various analysis methods. f3(t) = 5exp( 10000π(t 0.46) ) cos(340πt), (a) GLCT, (b) DACT, (c) MrSTFT, (d) MrCT. − − 2 f4(t) = 5exp( 10000π(t 0.52) ) cos(340πt), − − (61) TIME-FREQUENCY AND ITS APPLICATION 13

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Fre / Hz 100 Fre / Hz 100 Fre / Hz 100

50 50 50

0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Time / s Time / s Time / s (d) (e) (f) Fig. 11: TF results by various post-processing methods. (a) TF plot by SST, (b) TF plot by RM, (c) TF plot by TSST, (d) TF plot by SSST, (e) TF plot by SECT, (f) TF plot by MrSECT.

Fig. 11 gives the TF representations generated by the six Fig. 13 shows the analysis results of ECG signal estimated post-processing methods. It clearly shows that the TF result by four different methods: (a) RM, (b) TSST, (c) SSST, (d) of SST is blurred. The SSST and SECT provide the TF plots MrSECT. The SSST presents a blurry TF representation, based with high energy concentration for f1(t) and f2(t), but fail on which, it is difﬁcult to make further analysis and judgment. to characterize impulsive-like signals effectively. The TSST Fig. 13(a,b,d) indicate that RM, TSST, and MrSECT achieve provides a better TF localization for the transient signals, while good energy concentration for the spikes of the ECG signal. degrading the TF sharpness of the cosine modulation signals These methods (especially the MrSECT) can resolve the spikes at the stationary part (see TF plot in the time interval [0.4, and describe the transient characteristics clearly. We also give 0.6]). The RM is to reassign the spectrogram from the TF the time-domain marginal distribution of each TF result as direction such that the corresponding time resolution is more well as locally-zoomed results for the parts marked with the concentrated than SST-based methods, but it is not as clear green boxes. As illustrated in Fig. 13, the result obtained by as the MrSECT. By comparison, the MrSECT has the best our MrSECT approach achieves superior effectiveness in TF TF readability in handling signal (61). It should be pointed dynamic estimation because it reﬂects the dynamic change of out that the proposed MrSECT is subject to the boundary the ECG signal more accurately. effects, missing a small amount of boundary information when pursuing a high-concentration TF representation. VI. CONCLUSION

C. Application In this paper, we focused on improving the TF resolution of the standard chirplet transform (CT) for multi-component A typical biomedical signal, ECG signal from the BIDMC non-stationary signal analysis. We ﬁrst theoretically analyzed the Dataset [71], is employed to validate our proposed method. the effect of the chirp-based Gaussian window on the CT, and The sample rate of the ECG signal is 125 Hz. About 8 s ECG compared the CT with the rotation-window CT. The given segment and its STFT are displayed in Fig. 12. analysis provides theoretical instruction for CT to choose suitable parameters so as to obtain high-resolution TF rep- 1.5 15 resentation. To overcome the limitation of CT in dealing

1 with closely-spaced signals, we proposed the multi-resolution 10 chirplet transform (MrCT) by computing the geometric mean 0.5 Fre / Hz Amplitude of multiple CTs, which takes advantage of multiple estimates 5 0 at a range of temporal resolutions and frequency bandwidths

-0.5 0 and localizes the signal in both time and frequency better 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 Time / s Time / s than it is possible with any single CT. Moreover, based on (a) (b) the combined IF equation, we developed a TF post-processing Fig. 12: The time series of the ECG signal and its spectrogram. approach to improve the readability of the MrCT. Finally, nu- TIME-FREQUENCY AND ITS APPLICATION 14

15 15

10 10 Fre / Hz Fre / Hz

5 5

0 0 1.5 1.5 1 1 0.5 0.5

15 15

10 10 Fre / Hz Fre / Hz

5 5

0 0 1.5 1.5 1 1 0.5 0.5

Amplitude 0 Amplitude 0 -0.5 -0.5 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 Time / s Time / s 0.4 1.5 0.4 1.5 0.2 1 0.2 1 0 0.5 0 0.5 0 0 Amplitude -0.2 Amplitude -0.2 3.3 3.4 3.5 3.6 3.9 4 4.1 4.2 3.3 3.4 3.5 3.6 3.9 4 4.1 4.2 (c) (d) Fig. 13: The ECG’s analysis results calculated by different methods: (a) RM, (b) TSST, (c) SSST, (d) MrSECT. In the below of each TF result, there are time distribution of each TF representation (red -), the original signal (blak -), and locally-zoomed results, as marked with the green boxes.

merical examples and real signal application of the theoretical ACKNOWLEDGEMENTS derivations were presented. We thank K. Abratkiewicz for sharing the implementation Although the proposed methods have a good performance of his work. in analyzing closely-spaced signals, some works should be considered. REFERENCES

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