The Altes Family of Log-Periodic Chirplets and the Hyperbolic Chirplet Transform

The Altes F amily of Log-Periodic Chirplets and the Hyperbolic Chirplet T ransform Donnacha Daly ∗ and Didier Sornette Nov ember 13, 2018 Abstract This work re visits a class of biomimetically inspired log-periodic wave- forms first introduced by R.A. Altes in the 1970s for generalized target de- scription. It was later observed that there is a close connection between such sonar techniques and wa velet decomposition for multiresolution analysis. Moti vated by this, we formalize the original Altes wa veforms as a family of hyperbolic chirplets suitable for the detection of accelerating time-series oscillations. The formalism results in a flexible set of wav elets with desir- able properties of admissibility , regularity , v anishing moments, and time- frequency localization. These “ Altes wa velets” also facilitate efficient im- ∗ Dr . Daly is corresponding author . E-mail: donnacha@ieee.org . He conducted this w ork as Senior Reearcher at ETH Zurich, Switzerland. Prof. Sornette holds the Chair of Entrepreneurial Risks at ETHZ. E-mail: dsornette@ethz.ch . 1 plementation of the scale in v ariant hyperbolic chirplet transform (HCT). A synthetic application is presented in this report for illustrati ve purposes. keywords : wa v elet transform, chirp, complex systems, discrete scale inv ariance, critical failure, phase transition, Doppler radar 2 1 Intr oduction Over the course of the 1970’ s, Richard A. Altes de v eloped the theory behind a new family of wa v eforms with optimal Doppler tolerance for sonar applica- tions. Inspired by mammalian acoustic echo-location calls such as those of bats and dolphins, the constructed class of time-frequency concentrated pulses con- sists of carefully parameterized hyperbolic chirps [1 – 5]. Some time later , Patrick Flandrin and his colleagues at the French national center for scientific research (CNRS) made a beautiful e xposition of the close mathematical parallels between generalized tar get description using these chirps, and wa velet decomposition. i.e., e v aluation of the wa velet transform of the tar get impulse response [6]. One of the contrib utions of the current work is to extend these analyses by observ- ing that such chirps are Lo g-P eriodic (LP), a property closely associated with the deep symmetry of discrete scale in variance in physical systems [7]. By making the bridge between the chirplet approach and the LP property , we extend the toolset for detecting criticality in complex systems, with the mature body of knowledge from wa velet theory . Before exploring this howe ver , it is proposed to formalize the original Altes wa v e- forms as wavelets . Building on the work of Flandrin [6], which considered ad- missibility conditions, to include other desirable wav elet properties such as the degree of regularity (smoothness), the number of vanishing moments and the de- gree of time-frequency ( TF ) localization. The practicalities of time-discretization for efficient implementation in a Hyperbolic Chirplet Transform (HCT) are also 3 explored. De veloping the HCT for detection of log-periodicty is cumbersome, when using Altes’ original formulation. As a preliminary , therefore, a re-parameterization more intuitiv e to the signal processing practitioner is proposed. W e specify the wa velets in terms of center- and cut-of f frequency (or bandwidth) and chirp-rate. In addition to familiarity , these parameters hav e the adv antage of simplifying TF localization requirements and the study of other wav elet properties. The result is a po werful and practical e xtension to the e xisting body of wa v elet tools for signal analysis. W e should stress that the HCT does not claim to be a general substitute to existing wa velet transforms. Thus, our purpose is not to perform a comparati ve analy- sis or horse-race between the HCT and the very mature set of existing wav elet transforms [8, 9]. Our goal is rather to present and study this specific Chirplet because it possesses distinct properties, namely log-periodicity resulting in some optimality characteristics summarised below . From a practical perspectiv e, log- periodic oscillations with an acceleration towards criticality can serve as indica- tors of an incipient bifurcation. Such signals abound in nature, often as precursors to phase transitions in the non-linear dynamics of complex systems [7]. For ex- ample, the authors’ interest lies in automatic detection of the well documented phenomenon of log-periodic price dynamics during financial b ubbles and preced- ing market crashes [10]. Ho we ver , the methodology presented here is more widely applicable in such di v erse domains as prediction of critical failures in mechanical 4 systems [11], and fault detection in electrical networks. Examples beyond failure diagnostics include animal species identification via call recordings, commercial & military radar , and there are probably many more. The next section presents a revie w of relev ant background material to the current work. A recap of the Altes wa v eform together with our re-parameterization is provided in Section 3. The conditions under which it can be used as a wav elet in a HCT are presented in Section 4, along with a discussion of its wa velet prop- erties. Section 5 examines parameter selection and wa v elet design criteria, while Section 6 is an empirical look at the performance of the Altes wa v elet in detecting noisy LP-oscillations, in comparison with other TF methods. 2 T ransf orms f or Log-Periodic Chir p Detection 2.1 Chirplet T ransf orms The idea of using chirps as wav elets, ie. chirplets, was introduced in [12, 13], leading to the Gaussian chirplet transform as well as the warblet transform. A hardware implementation applied to detection of bat-calls was provided in [14]. An adaptiv e version of the Gaussian chirplet was designed to linearize curves in the TF -plane in [15]. This has been used for detection of bat echo-location signals and compared with the Gaussian wa velet transform, the short time F ourier transform and the W igner V ille Distrib ution (WVD) [16], noting that 5 “...the Gaussian chirplet decomposition linearizes the curving chirps. Suc h ap- pr oximation is quite coarse. Hyperbolic chirps pr obably can better model signals, such as the sound of bat. ” W av elet based LP-detection methods were also compared in [17]. The Cauchy wa velet performed worst, the Mexican Hat better and the Morlet best. More comprehensi ve attempts to capture general non-linear TF modulations using the Polynomial Chirplet T ransform were dev eloped in [18] and applied to bat-sonar detection in [19]. None of these used a hyperbolic chirp in a wa velet transform for the detection of log-periodicity . “Hyperbolic wa velets” do appear else where e.g. [20, 21] but with a different meaning. T o a void confusion, we refer exclusi v ely to hyperbolic chirplets in the current context. 2.2 P ower Law Chir ps and the Mellin T ransf orm The hyperbolic chirplet was introduced as a special case of the power law chirp in [22]. Its use for the optimal detection of chirping gra vitational wav es was pre- sented in [23]. It is seen that, while the optimal detection of a linear chirp is via the WVD, for po wer law chirps the optimal detector uses the Bertrand distrib u- tion, which can have prohibiti ve computational requirements. As an alternati ve, the Mellin transform can be considered, which represents a signal as a projection onto a family of hyperbolic chirps, analogous to the Fourier transform projec- tion on a basis of comple x exponentials. This idea was e xtended to LP-detection in [24]. It was sho wn that the Mellin transform, coupled with a Fourier transform 6 can be used to replace a wa velet transform, and is suited to problems of scale in- v ariance. Similarly , in [25] the Mellin T ransform has been used to provide the LP decomposition of the most general solution of the renormalization group equation. 2.3 The Hyperbolic Chirplet T ransf orm Here, we choose to implement the wav elet transform directly using the Altes chirp as mother wav elet. W e call this a Hyperbolic Chirplet Transform (HCT), and label our parameterization of the Altes wa veform as the Altes chirplet . Some pre vious studies hav e come close to our no vel approach. In [26] a strong case, and a powerful methodology , were both presented for tai- loring wav elets to the signals being detected. While the example of LP-detection was examined, the study did not extend to using a dedicated hyperbolic chirplet for the purpose. In [7], a direct link was made between the functional form of the Altes wa velet (our eq. 6) and the renormalization group encountered in [25, 27]. The solution to the homogeneous Altes wav elet equation was thus identified as self-similar with discrete scale in v ariance. Self similar wav elets were introduced in [28] for fractal modulation. Ho we ver , these were neither identified as hyper- bolic, nor used for LP-detection. Generalized target detection was shown in [6] to share important features with wa v elet decomposition. This qualitati v e work proposed the autocorrelation function of the Altes chirp as the wav elet in the de- composition, b ut did not go so far as demonstrating the utility of these results. It can be considered as a useful launching point for the current work. 7 2.4 Motiv ation: Log P eriodicity & Discr ete Scale In variance Apart from radar/sonar applications in target detection, there are other strong mo- ti v ations for an interest in hyperbolic chirplets and their ability , via the wa velet transform, to detect log-periodic oscillations in noisy time-series. T o the best of our knowledge, LP-constructs first appeared in the 1960s for modeling shocks to layered systems [29] and the discrete hierarchy of vortices in hydrodynamic tur - bulence [30]. In the 1970s, log-periodicity was recognized in the self-similarity of propagating wa v es [31]. Around the same time, the renormalization group theory of critical phenomena introduced solutions for the statistical mechanics of critical phase with complex critical exponents, characterized by log-periodicity [32 – 34]. In the 1980s, phase transitions occurring on hierarchical lattices were shown to exhibit discrete scale in variance, with its signature of complex exponents and LP- oscillation [35 – 38]. Since then, literature on the topic has expanded rapidly . This potted historical perspecti ve is condensed from [7] which the reader is in- vited to pursue for further insight. A central result is that LP-signatures indicate that a system and/or its underlying physical mechanisms hav e a hierarchy of char - acteristic scales corresponding to discrete scale in variance. This is interesting as it provides important insights into the underlying physics, which may allo w us to make forecasts of rupture such as earthquakes [39, 40], mechanical failure [41] or the bursting of bubbles in financial markets [10, 42, 43]. In fact, any system with built in geometrical hierarchy will lead to log-periodicity , e.g. wa ve propagation in fractal systems [44], Ising and Potts models on hierarchical structures [45, 46] 8 and sandpile models on discrete fractal lattices [47] to pick a fe w . Gi ven our gro wing understanding of the ubiquity of LP-signatures in complex systems, it is useful to equip ourselves with reliable tools for their e xtraction and diagnosis. 3 Re-parameterizing the Altes W a vef orm Altes’ early w ork on chirps sho wed that a TF -localized pulse which is periodic in the logarithm of progressing time has optimal Doppler tolerance [2]. W e find it more instructi ve here to introduce the topic using his reasoning from [3]. 3.1 Matched-Filtering Detection of Echo Components Consider a w av eform described in the Fourier domain by U ( ω ) , which produces a set of echoes when reflected from a tar get in a sonar detection setting. Depending on the w a velength, its angle of incidence, the velocity and reflectivity of the target and other factors, reflections may be in- or out-of-phase with the incident wa ve- form. By superposition, an echo is thus hypothesized to be the weighted sum of time-integrated (when in-phase) or -dif ferentiated (when out-of-phase) versions of the original signal at dif ferent lags. The goal is to design this transmit signal U ( ω ) such that it can be reliably re- cov ered from these echo components V n ( ω ) under constraints on recei ver com- plexity . Since time-integrated and -differentiated versions of U ( ω ) are giv en in the frequenc y domain by ( j ω ) n U ( ω ) , n ∈ Z , detection can be achieved by using 9 a bank of filters V ∗ n ( ω ) , each matched to one of these ener gy normalized echo components, written as V n ( ω ) = ( j ω ) n U ( ω ) 1 2 π R ∞ −∞ | ( j ω ) n U ( ω ) | 2 . (1) 3.2 Constant- Q Filter -Banks Altes’ insight was that the complexity of the required filter bank could be con- strained if these filters hav e a constant time-bandwidth product for all v alues of n . This is because the component V ∗ n ( ω ) is repeatedly dif ferentiated as n in- creases, which, from eq. (1) and the Cauchy-Schwarz inequality , results in band- width expansion. If this is not compensated by a corresponding compression of the time-domain impulse response, then the required matched-filter complexity gro ws rapidly with n . On the other hand, for a bank of filters with constant time-bandwidth product, also kno wn as Constant- Q [48], filter complexity is con- stant. Consider matched filters V ∗ n ( ω ) which satisfy a scaling contraint defined by 1 k > 1 : V ∗ n ( ω ) ∝ V ∗ n − 1  ω k  . (2) If the root mean square filter delay spread of V ∗ n ( ω ) is τ n and its bandwidth is B n , then eq. (2) gi ves us B n = k B n − 1 and its in verse Fourier transform gi ves 1 Undesirably , this k introduced by Altes tunes both the bandwidth and chirp-rate of the final wa veform, one of the reasons we later re-parameterize. 10 τ n = τ n − 1 /k . Therefore, τ n B n = τ n − 1 B n − 1 = τ m B m ∀ m, n ∈ Z . (3) This is a sufficient constant- Q condition on V ∗ n ( ω ) and stipulates a fixed time- bandwidth product for all n , as well as fixing the same ratio of center-frequenc y to bandwidth for all filters. From eq. (2) V ∗ n ( ω ) ∝ V ∗ 0  ω k n  (4) ⇒ ω n U ( ω ) (1) ∝ U  ω k n  (5) ⇒ ω n U ( ω ) = C ( n ) U  ω k n  , (6) where C ( n ) is a proportionality constant dependent on n but independent of ω . From relativ ely simple arguments, we get to eq. (6), an explicitly solvable homo- geneous functional in U ( ω ) . 3.3 The F amily of Altes W a vef orms Performing analytic continuation from integer to real values of n , then taking the deri v ati ve with respect to n of eq. (6) at n = 0 and noting that C (0) = 1 , we get d U ( ω ) U ( ω ) = C 0 (0) − log ω ω log k d ω , ω 6 = 0 . (7) 11 Integration and some simplification leads to the pulse U ( ω ) designed specifically to hav e echo components that are easily detectable. This is the original Altes wa veform, U ( ω ) = Aω ν exp  − 1 2 log 2 ω log k  exp  j 2 π c log ω log k  , (8) which can be verified to satisfy eq. (6) by choosing C ( n ) = k nν + n 2 2 exp ( − j 2 π nc ) , (9) The set of real constants { A, ν , k , c } in eq. (8) arises through grouping of fixed terms, and can be considered as the parameterization of a family of wa v eforms. As observed by Altes, these wav eforms are optimally Doppler tolerant, suitable for radar applications. Howe ver , the abo ve parameterization is not straightforward to work with from a wa velet design perspectiv e, which is why , in the following, a more familiar set of filtering parameters is presented. 3.4 Center Fr equency , Cutoff Fr equency & Bandwidth Eq. (8) represents a bandpass w av eform and should therefore be specific to a cen- ter frequenc y ω 0 and a pair of upper and lower cutoff frequencies ω ± c , equi v alent 12 Figure 1: The reparamaeterized Altes chirplet: a) Log-magnitude frequency re- sponse, with center frequency ω 0 , bandwidth B . and upper/lo wer cutoff frequen- cies ω ± c . W e specify a unit passband response, | U ( ω 0 ) | = 1 . b) The log-magnitude time-domain en velope, concentrated at τ 0 with delay spread τ . The wa veform sup- port is bounded at τ ± c . to some bandwidth B . Let the log-magnitude response be defined as M ( ω ) ∆ = log | U ( ω ) | (10) (8) = log A + ν log ω − 1 2 log 2 ω log k (11) and, without loss of generality , be specified with a unit passband response (0 dB gain). This yields a maximum response M ( ω 0 ) = 0 at M 0 ( ω 0 ) = 0 from which we get ω 0 = k ν (12) as pre viously noted in [6], and thereby A = k − ν 2 / 2 . (13) 13 As illustrated in Fig. 1a, the magnitude response | U ( ω ) | drops a way from unity at ω 0 to some lo wer le vel K c > 0 at cutof f frequency ω c gi ving, from equations (11), (12) and (13) M ( ω c ) = ν log ω c − ν 2 log ω 0 − ν 2 log 2 ω c log ω 0 (14) ! = log( K c ) , 0 < K c < 1 . (15) By defining the positi ve constant κ c ∆ = − log K c log 2 ω c ω 0 > 0 . (16) we can easily reparameterize the Altes constants ν and k as ν = 2 κ c log ω 0 , (17) k = exp  1 2 κ c  . (18) Substituting eq. ’ s (17), (18) & (13) in eq. (11) giv es the much simplified magni- tude response expression M ( ω ) = − κ c log 2 ω ω 0 . (19) The definition of κ c allo ws us to write ω ± c (16) = ω 0 exp ± r − log K c κ c ! rad/s . (20) 14 W e can specify the upper and lo wer cutoff frequencies by respectiv ely choosing ω c greater or less than center frequency ω 0 . W e recommend to disambiguate in fa v or of the upper one, and speak of the cutoff frequency ω c ∆ = ω + c > ω 0 . The upper cutoff ex erts control ov er decay of U ( ω ) at higher frequencies, and hence wa veform regularity (Section 4.2). There is less need to worry about the lo wer cutof f because of the zero in U ( ω ) at ω = 0 and its extremely rapid decay at lo w frequencies. Accordingly , the bandwidth is defined simply B ∆ = ω + c − ω − c (21) (20) = ω 0  ω c ω 0 − ω 0 ω c  rad/s (22) with the lo wer cutof f frequency gi v en by ω − c = ω 2 0 /ω c . For the presented formulation to be useful in practice, a value must be placed on K c = | U ( ω c ) | used in the definition of cutoff frequency in eq. (15). For the twin objecti ves of wa v elet frequency localization and a voidance of discrete-time aliasing elaborated in Section 5, it makes sense to place tight restrictions on K c . W e use a lev el corresponding to − 40 dB throughout, which of course can be v aried depending on application requirements. T o be pedantic, this le vel represents 20 log 10 | U ( ω c ) | = − 40 dB (23) ⇒ K c = 10 − 40 20 = 0 . 01 . (24) 15 3.5 Chirp Rate λ and Re-parameterization of the Altes W av elet As foreseen in Section 3.2, the phase and magnitude response behaviours of the Altes’ wa veform in eq. (8) are both gov erned by single parameter k . In order to decouple them, and allo w for simpler chirplet design, we introduce log λ ∆ = log k c . (25) The phase response of U ( ω ) then corresponds to hyperbolic chirping in the time domain, with λ controlling the chirp rate. The time-domain pulse gi ven by in verse Fourier transform of eq. (8) u ( t ) = F − 1 { U ( ω ) } (26) is sho wn in [49] to hav e phase response φ ( t ) ∝ log t, (27) a logarithmic function of time. Instantaneous frequency is ω I ( t ) ∆ = d φ ( t ) d t ∝ 1 t (28) which describes a h yperbola in the time-frequenc y plane. Since the instantaneous period T I ( t ) = 2 π /ω I ( t ) ∝ t is linear in time, signals of type u ( t ) are inter- changeably (and correctly) referred to as ha ving linear period modulation (LPM), 16 hyperbolic frequency modulation (HFM) or logarithmic phase modulation [50], or as being log-periodic, which is preferred in the financial and physics litera- ture [51]. The proposed re-parameterization no w emerges from eq. (8) through equations (12), (15) and (25), as U ( ω ) = exp log K c log 2 w w 0 log 2 ω c ω 0 ! exp  j 2 π log ω log λ  , ∀ ω > 0 . (29) with ω 0 > 0 and λ > 0 , or more concisely from eq. (16) U ( ω ) = exp  − κ c log 2 w w 0  exp  j 2 π log ω log λ  , ∀ ω > 0 . (30) This parameterisation makes clear that λ is the scaling ratio of the discrete scale in variance symmetry . It is the ratio of the local periods of the successi ve oscil- lations in the chirp. ω 0 is the center frequency for which | U ( ω ) | is maximum (here normalised to 1 ). κ c controls the lower and upper cut-of f frequencies ω ± c in eq. (20) beyond which the signal amplitude falls off by more than a certain pre- specified le vel K c from eq. (24). For a giv en K c , the Altes chirplet can thus be parameterised by the triplet { ω 0 , κ c , λ } . Alternativ ely , κ c can be replaced by the bandwith B from eq. (22) (see figure 3) or cut-off frequency ω c from eq. (20) as in eq. (29) (see figures 4 and 5). As the chirp rate parameter λ → 1 in eq. (30), the oscillations become faster and faster until a singularity occurs at λ = 1 . T o av oid it, we must consider 0 < λ < 1 17 or λ > 1 . Note also, howe v er , that replacing λ by 1 λ simply changes the sign of the complex exponential in eq. (30), equiv alent to a frequency domain conjugation. This in turn represents time rev ersal and conjugation of the chirplet in the time domain, an example of which will be presented in Fig. 5. Making the dependence on chirp rate explicit by subscript, we can write u 1 λ ( t ) = u ∗ λ ( − t ) . (31) Later , in considering the effect of λ on time-frequency localization, we thus need only examine λ ∈ [0 , 1) . The effects for λ ∈ (1 , ∞ ) are identical, and found by reciprocation of λ . In contrast to the original Fourier domain specification of U ( ω ) [3], we propose to zero the non-positi ve frequencies and define U ( ω ) ∆ = 0 , ∀ ω ≤ 0 (32) While Altes prefered to impose Hermitian symmetry on U ( ω ) to ensure a real wa velet, preserv ation of the analytic form retains useful phase and en velope prop- erties — in particular , this allo ws natural definitions for instantaneous amplitude and instantaneous frequency of arbitrary signals being analyzed [22]. From (32) and (30), U ( ω ) is continuous at ω = 0 since lim ω → 0 + U ( ω ) = lim ω → 0 − U ( ω ) = U (0) = 0 . (33) 18 Figure 2: Example 1 is taken from the original Altes paper [3] using his parameter set { ν, k , c } = {− 0 . 55 , 1 . 8 , − 0 . 35 } . There is no frequenc y normalization, and it is not obvious how the parameters relate to the observed wa v eform. W e find { ω 0 , B , λ } = { 0 . 7328 , 7 . 344 , 0 . 1865 } . This Altes chirp U ( ω ) is bandpass and frequency localized, a minimum require- ment for formal admissibility as a wa velet. 3.6 Examples Four e xamples are no w presented to illustrate the behavior of the Altes wa veform as a function of its parameters. In each case, we sho w the analytic time-domain wa veform u ( t ) from eq. (26), as well as the magnitude and phase responses of U ( ω ) in the frequenc y domain from eq. (30). In the first example, Fig. 2 recreates a w av eform from Altes’ original paper with { ν, k , c } = {− 0 . 55 , 1 . 8 , − 0 . 35 } . A primary difference to the original exposi- 19 Figure 3: Example 2 demonstrates tunability of f amiliar wav elet parameters such as center frequency ω 0 , bandwidth B and chirp rate λ , as a result of our re- parameterization. In this case { ω 0 , B , λ } = { π 6 , π 5 , 3 4 } , and a unit sampling interv al is chosen. tion is that our chirp is complex, and both the imaginary component and com- plex env elope can be observed in addition to the real wa veform. As tri via, we can ev aluate from eq. (12) that this bandpass wav eform has center frequency ω 0 = 0 . 7328 rad/s, as well as providing from equations (25), (20) and (22), the pre viously unav ailable chirp rate λ = 0 . 1865 , cutoff frequency ω c = 7 . 4146 rad/s, and bandwidth B = 7 . 3440 rad/s, assuming as we hav e a -40dB cutoff, i.e. K c = 10 − 2 . In the second example, we choose to show in Fig. 3 how our parameterization al- lo ws practical specification of center -frequency , bandwidth and chirp rate, in this case { ω 0 , B , λ } = { π 6 , π 5 , 3 4 } . In the graphic, it can be seen that the − 40 dB band- width B is 0 . 2 π as specified. When implementing in discrete time, it is useful, 20 Figure 4: In example 3, the chirplet is tuned via its cutoff frequency rather than its bandwidth: { ω 0 , ω c , λ } = { π 10 , 9 π 10 , 3 4 } . This allows tight control of magnitude response decay at high frequencies, which influences regularity (wa v elet smooth- ness) and aliasing in a discrete time implementation. although not necessary , to work in units of normalized frequency , such that the sampling frequency ω s = 2 π corresponds to a unit sampling interval. This con- vention is adopted in the remainder which fixes the Nyquist rate at ω Nyq = π for the coming discussion on discrete-time implementation. The third example illustrates the tuning of wav eform cutoff frequency using pa- rameters { ω 0 , ω c , λ } = { π 10 , 9 π 10 , 3 4 } . In Fig. 4, it can be seen that the -40 dB cutoff is tightly tuned to ω c = 9 π 10 as specified. The wider bandwidth opens a greater range of frequencies over which to chirp. This is visible in the wav eform, when compared with Fig. 3. Our final example is chosen to show the ef fect of the chirp rate parameter λ . From eq. (30), it is clear that 0 < λ < 1 results in ne gati v e phase response, while λ > 1 21 Figure 5: Example 4 is chosen to show ho w a reciprocal chirp-rate λ affects the phase- of the chirplet in the time-domain. W e ha ve used { ω 0 , ω c , λ } = { π 10 , 9 π 10 , 4 3 } . Compared with example 3, This results in time-re versal and conjugation, as ex- plained in Section 3.5. yields positi ve phase response. W e sho wed that replacing λ by 1 λ amounts to con- jugation and time rev ersal of the Altes chirp. The effect is clear to see in Fig. 5 in which the Altes chirplet is parameterized identically to example 3, with the ex- ception of the chirp rate, which is in v erted: { ω 0 , ω c , λ } = { π 10 , 9 π 10 , 4 3 } . The effects of chirp-rate on wa v elet delay-spread will be examined further in Section 5.2 on time-frequency localization. 4 The Hyperbolic Altes Chirplet T ransf orm Up to this point, we ha ve arri ved at the description of a family of waveforms reparameterized in eq. (30) from Altes’ original frequency domain representation 22 in eq. (8). In this Section, we sho w that we are actually dealing with a family of wavelets in the formal sense, with desirable additional properties. 4.1 Admissibility The continuous wav elet transform (CWT) of a real or complex, square integrable signal s ( t ) at position b and scale a is C ψ ( a, b ) = Z R s ( t ) 1 √ a ψ ∗  t − b a  d t, a ∈ R + , b ∈ R (34) where ψ ( t ) is said to be an admissible wav elet if it satisfies the fairly loose con- ditions that it is square integrable and sufficiently band-limited [52]. Calder ´ on’ s reproducing identity tells us that in this case the original signal s ( t ) can be recov- ered exactly from its w av elet coef ficients C ψ ( a, b ) by the in verse transform s ( t ) = 1 C Ψ Z R Z R + C ψ ( a, b ) 1 √ a ψ  t − b a  d a d b, (35) where C Ψ is the admissibility constant. The formal admissibility conditions on ψ ( t ) are E ψ ∆ = Z R | ψ ( t ) | 2 d t < ∞ (36) C Ψ ∆ = Z R | Ψ( ω ) | 2 | ω | < ∞ (37) 23 corresponding to square-integrability and transform in v ertability , respecti vely . E ψ is the wa v elet ener gy and we ha ve used the Fourier T ransform F { ψ ( t ) } = Ψ( ω ) . The CWT in eq. (34) represents the correlation of s ( t ) with scaled and shifted versions of the w av elet ψ a,b ( t ) = 1 √ a ψ  t − b a  . (38) The factor 1 √ a ensures that k ψ a,b ( t ) k is independent of { a, b } , and often the mother wa velet ψ ( t ) is normalized so that E ψ = k ψ ( t ) k 2 = k ψ a,b ( t ) k 2 = 1 . (39) The admissibility conditions tell us that ψ ( t ) is a finite energy pulse with a fre- quency response that decays at high frequencies, and which must ha ve no DC component. In simpler words, it is a frequency localized bandpass wa v eform. Proposition 1 The Altes waveform u ( t ) = F − 1 { U ( w ) } is an admissible wavelet. Proof . A log-normal random variable X with log ( X ) ∼ N ( µ, σ 2 ) has, by defini- tion, probability density function and expected v alue given by p X ( x ) = 1 xσ √ 2 π exp − (log x − µ ) 2 2 σ 2 ! , x > 0 (40) E [ X ] = exp  µ + σ 2 2  . (41) 24 The mean is defined by E [ X ] ∆ = Z ∞ −∞ xp X ( x ) d x (42) so the equality (41) translates to 1 σ √ 2 π Z ∞ 0 exp − (log x − µ ) 2 2 σ 2 ! d x = exp  µ + σ 2 2  . (43) Replacement x → ω , µ → log ω 0 and σ 2 → 1 4 κ c gi ves r 2 κ c π Z ∞ 0 e − 2 κ c log 2 ω ω 0 d ω = e log ω 0 + 1 8 κ c (44) and so from eq. (30) 1 2 π Z ∞ −∞ | U ( ω ) | 2 d ω = ω 0 √ 8 π κ c exp  1 8 κ c  (45) From Parse val’ s Theorem Z ∞ −∞ | u ( t ) | 2 d t = 1 2 π Z ∞ −∞ | U ( ω ) | 2 d ω (45) < ∞ (46) proving square integrability by eq. (36). As an aside, equations (45) and (46) can be used for normalization in eq. (39). 25 By making the substitution x ← log ω ω 0 in eq. (37) we get C U (30) = Z ∞ 0 exp 2  − κ c log 2 ω ω 0  d ω ω (47) = Z ∞ −∞ exp  − 2 κ c x 2  d x (48) = r π 2 κ c (49) < ∞ (50) proving in vertability and, thereby , admissibility . Not only can the Altes wa veform no w be correctly called a wa v elet, but since it is a log-periodic chirp, we can also refer to it to as the Altes chirplet, and its application within a CWT as the Hyperbolic Chirplet T ransform, consistent with prior taxonomy . W av elets that only have the property of admissibility are kno wn as crude wav elets, because admissibility is a weak condition and does not guarantee usefulness for signal processing. This leads us to examine other wav elet properties which can enhance their applicability . 4.2 Regularity Regularity describes the smoothness of a wa v elet ψ ( t ) in the time domain. The order of regularity corresponds to the number of times ψ ( t ) is continuously dif- ferentiable. W e say that ψ ( t ) is bounded and has uniform Lipschitz regularity of order α > 0 ov er R if its frequency weighted magnitude response is Lebesgue 26 integrable according to Z R | Ψ( ω ) | (1 + | ω | α ) d ω < ∞ . (51) This captures the fact that smoothness in time is directly related to the rate of decay at high frequencies. Intuitiv ely , signals with higher frequency content vary more rapidly , and are therefore less regular . Proposition 2 The Altes chirplet u ( t ) has infinite r e gularity . Proof . Using U ( ω ) from eq. (30), we define I 0 (51) = Z ∞ 0 exp  − κ c log 2 ω ω 0  d ω + Z ∞ 0 ω α exp  − κ c log 2 ω ω 0  d ω (52) ∆ = I 1 + I 2 . (53) Replacing x ← log ω ω 0 gi ves I 2 = ω α +1 0 Z ∞ −∞ exp  −  κ c x 2 − (1 + α ) x  d x (54) = ω α +1 0 r π κ c exp  (1 + α ) 2 4 κ c  (55) The limit α → 0 gi v es I 1 allo wing us to compute the definite integral I 0 = ω 0 r π κ c e 1 4 κ c  1 + ω α 0 e α 2 +2 α 4 κ c  < ∞ (56) 27 for all finite α > 0 . W av elets are often used in compression applications, whereby a signal is represented by a truncated set of its CWT coef ficients. Regular wav elets hav e the adv antage that, when used in a CWT for such application, the undesirable artifacts arising from truncation (e.g. audio distortion) are less noticeable when compared to those produced using less smooth wa v elets, ev en if the compression error magnitude is similar . Infinitely re gular wa v elets are thus suitable for use in information-coding. The Altes chirplet joins an august family including the Mor- let, Me xican Hat, Me yer , Gauss and Shannon wa velets, all infinitely continuously dif ferentiable in time. 4.3 V anishing Moments If wa v elet ψ ( t ) has M vanishing moments, then it is orthogonal to all polynomials of order M − 1 . A richer set of signals can be represented with a sparser set of coef ficients when the mother wa velet of the analyzing CWT has higher M . The number of vanishing moments is the highest integer M such that in the time domain Z ∞ −∞ t m ψ ( t ) d t = 0 ∀ m ∈ { 0 , 1 , 2 , . . . , M } (57) or equi v alently in the frequency domain d m Ψ( ω ) d ω m     ω =0 = 0 ∀ m ∈ { 0 , 1 , 2 , . . . , M } . (58) 28 The latter version makes it clear that v anishing moments flatten the wa v elet re- sponse Ψ( ω ) around DC. Ho we ver , steeper response decay at low frequencies narro ws the wav elet bandwidth from belo w , and there must be a corresponding time dilation. Thus, a higher number of vanishing moments comes at the cost of increasing support in the time domain i.e. longer wa velets. This will be addressed in Section 5. Proposition 3 The Altes chirplet has an infinite number of vanishing moments. Proof . Since U ( ω ) = 0 , ω ≤ 0 it suf fices to sho w that lim ω → 0 + d n U ( ω ) d ω n = 0 , ∀ n ∈ Z + . (59) It is obvious that the e xponential form of the Altes wa velet U ( ω ) from eq. (30) renders it infinitely continuously dif ferentiable over the positi v e frequencies ω > 0 . While computing these deriv ati ves is cumbersome, by re writing U ( ω ) = exp( Q ( ω )) , ω > 0 (60) it can be readily found from chain and product rules that they tak e the form d n U ( ω ) d ω n = U ( ω ) n X p =1 p X q =1 a p Q b p p ( ω ) a q Q b q q ( ω ) (61) 29 where Q i ( ω ) = d i Q ( ω ) d ω i (62) (30)(60) = c i, 0 log ω + c i, 1 + j c i, 2 ω i . (63) There are a finite number of terms in the double sum of eq. (61). This number is independent of ω , as are the integer constants a i , b i and the real constants c i,k . These coefficients must be found by computation, but we do not need them to observe that, as ω shrinks to zero in eq. (61), the magnitude of U ( ω ) shrinks far more quickly than the products of Q b i i ( ω ) explode. The former has order of gro wth exp( − log 2 ω ) which dominates the latter , whose order of growth is only (log ω ) /ω n at the origin. This dominance holds true for any n ∈ Z + . 4.4 Scale In variance When the Altes Chirplet is used as the mother wa velet in the CWT , we call this the Hyperbolic Chirplet T ransform (HCT). The self-similarity of the Altes Chirplet, as it emer ges from the homogeneity of eq. (6), leads to the extraordinary property of transform scale in v ariance, ie. the HCT can be computed trivially at any scale, from kno wledge of one scale only . Proposition 4 The HCT is scale in variant. 30 Proof . The continuous wa v elet transform of eq. (34) can also be defined in the frequency domain as C ψ ( a, b ) = √ a 2 π Z R S ( ω )Ψ ∗ ( aω ) e j ω b d ω . (64) T aking n = 1 in eq. ’ s (6) and (9) and using eq. (25) giv es ω U ( ω ) = k ν + 1 2 exp  − j 2 π log k log λ  U  ω k  . (65) For k → 1 a in eq. (65) with U ( ω ) → Ψ( ω ) in eq. (64) and U ( ω ) gi v en by (30), we get C u ( a, b ) = g ( a ) 2 π Z R S ( ω ) ω U ∗ ( ω ) e j ω b d ω (66) with g ( a ) ∆ = a 1+2 κ c log ω 0 exp  j 2 π log a log λ  (67) i.e. independent of ω . W e ha ve used eq. (17) to replace ν in line with the new parameterization. From the abo ve, it can readily be shown that, for any real mul- tiplier m > 0 , we hav e C u ( ma, b ) (66)(67) = g ( m ) C u ( a, b ) (68) with g ( m ) independent of a and b , pro ving scale in v ariance. The integrand in eq. (66) is independent of scale a which means that the Hyperbolic Chirplet T rans- form C u ( a, b ) can be computed once at scale a , and the coef ficients C u ( ma, b ) can 31 be found at all other wa velet scales ma via eq. (68). This results from the self- similarity of the Altes wa v elet that performs in a sense a full multiscale analysis with just one scale of magnification a . Of course the chirplet transform values must be re-computed for dif ferent shift v alues b . Note that, at unit scale, a = 1 and g (1) = 1 . In this case eq. (66) has the time domain equi v alent C u (1 , b ) (34)(38) = s ( t ) ? u ( t − b ) , (69) where ? is cross-correlation. In words, we find the CWT coefficients at unit scale by measuring the correlation of signal s ( t ) with the delayed Altes chirplet, and use eq. (68) to find the coefficients at other scales. In practice, the HCT is imple- mented digitally , with the w a veform being discretized in both time and amplitude. As such a full, infinite-support, infinite-scale, multiresolution analysis will not be achie v able from eq. (68), but will depend on the specific implementation. Ne v- ertheless, the HCT can be implemented as an extremely efficient transform as a result of the self-similarity of the Altes chirplet. 5 Parameter Selection The Altes chirplet was sho wn in Section 4.3 to ha ve an infinite number of vanish- ing moments. This requires infinite support in the time domain, suggesting very long wa velets. In this section, it is seen that, through suitable parameterization, the chirplet’ s time-decay can be tailored for desired localization, in trade-of f with 32 its frequency domain bandwidth and chirping flexibility . In addition, the CWT , when implemented in software or hardware, must be approximated in discrete- time with a finite number of samples using appropriate time-frequency sampling. The influence of discrete-time implementation on parameter selection is there- fore also discussed. As there is no tri vial time domain representation of the Altes chirplet from eq. (26), this part of the study is numerical. 5.1 Delay Spr ead W e start by defining the delay-spread of the wa veform in a manner similar to ho w its bandwidth is specified in the frequency domain in Section 3.4. T ime localization implies that the time domain en velope | u λ ( t ) | rises gradually from zero to some peak at t = τ 0 where the wa v eform energy is concentrated, before falling away again to zero, as in Figures 2 – 5 for example, and illustrated clearly in Fig. 1b . W e define the delay spread empirically as the time τ between the earliest appearance of significant energy at t = τ − c , and the later time t = τ + c beyond which the energy appears to ha v e v anished: τ = τ + c − τ − c . (70) For this to be precisely specified, some threshold amplitude is needed, which de- fines the presence or absence of wa v e ener gy at cutof f points τ ± c . W e use the same -40 dB lev el (24) used in defining the wa veform bandwidth in Section. 3.4. Normalizing to the peak magnitude of the time domain wa veform, we define this 33 amplitude threshold as | u λ ( τ ± c ) | | u λ ( τ 0 ) | ! = K c = 0 . 01 . (71) While this is not the most con v entional definition of delay spread, it provides consistency with eq. (24) for time-frequenc y localization analysis, and is certainly v alid. The questions can now be asked: how does the Altes w av elet delay-spread τ de- pend on the parameterization { ω 0 , ω c , λ } from eq. (29), and ho w does it trade off against the chirplet bandwidth? Indeed, what are useful values of, or limits on, these parameters for signal processing applications? 5.2 The Efficient Fr ontier of Delay Spread vs. Bandwidth In Section 3.5, it was seen that in vestigation of λ ∈ (0 , 1) is suf ficient. All results deri ved apply then also for 1 /λ . Furthermore, as outlined in Section 3.6, we consider unit sampling and need only consider ω 0 ∈ (0 , π ) and ω c ∈ (0 , π ] > ω 0 . The results of a dense parameter sweep within these domains are presented in Fig. 6. This figure sho ws the measured delay spread and bandwidth for each grid point, i.e. for multiple realizations of the Altes chirplet at distinct parameter settings. It can be immediately seen that there is an ef ficient frontier to the lower left, along which there is an optimal trade-off between time-localization and frequency-localization. 34 Figure 6: Each dot in the figure represents one instance of the Altes chirplet u ( t ) at a gi ven v alue of parameter set { ω 0 , ω c , λ } for the representation eq. (29). The parameters are each swept in a dense grid ov er their domains. For each instance, the delay spread τ and bandwidth B of the chirplet are charted. It can be seen that certain parameterizations fall on an ef ficient frontier , representing realizations with maximized time-frequency localization. Altes chirplets away from this Frontier are inef ficient, as they could be tuned to hav e smaller bandwidth without increasing the delay-spread and vice-v ersa. W e examine the Altes parameters when moving do wn this curve from upper left to lo wer right. In all cases we find that λ < 1 / 2 , a loose parameter bound for the efficient frontier . This is because the Altes chirplet in eq. (30) is ill-behav ed near λ = 1 and is certainly not efficiently localized there. The effects of this ill-behavior be gin to disappear for λ < 1 / 2 . At the upper left part of the frontier , ω 0 > π / 2 and B < π / 4 , and the wa ve- form becomes frequenc y localized (narrowband). There is little chirping possible, and in this case, we are dealing with a more con ventional wa velet, rather than a chirplet. 35 Moving do wn the curve to the right, we see that the cutoff frequency quickly con verges to the Nyquist rate, ω c → π , while the center frequency slides down to ω o → π / 2 , as the Altes chirplet becomes less frequency localized. Finally , the ef ficient frontier flattens out at B = 3 π / 4 . This is the point of minimum delay spread, which occurs for chirplets with { ω 0 , ω c } = { π / 2 , π } . This is consistent with equation (22). T able 1 summarizes the efficient parameterizations discussed in this section, which maximize time-frequency localization. 5.3 Inefficiently Localized Chirplets At this point, we could call it a day , having identified parameterizations of the Altes Chirplet for ef ficient time-frequency localization. Howe ver , it appears that nature values chirping flexibility abov e localization-efficienc y when it comes to ubiquitous log-periodicities. For example, the parameterization in Fig. 2 that Altes used to model bat chirps has a high ratio of cutoff to center frequency ω c /ω 0 (ie. high bandwidth) and is not frequency localized. Howe ver , it is also not suf ficiently time-localized to lie on the ef ficient frontier of Fig. 6. More generally , we can say that a) significant chirping occurs away from the fre- quency localized regime, such that there is a sufficient range of frequencies ov er which to sweep; and similarly , b) the chirplet must be long enough li ved to to af ford numerous oscillations of the log periodicity . Excessiv e time localization impedes desirable chirping. 36 T able 1: Parameterizations of the Altes Chirplet for Ef ficient T ime-Frequency Localization ω 0 ω c or B λ Ef ficient Frequency ω 0 > π 2 ω c < π λ < 1 2 Localization ( B < π 4 ) ⇔ (minimal chirping) λ > 2 Ef ficient T ime-Freq. ω 0 > π 2 ω c = π λ < 1 2 Localization ( π 4 < B < 3 π 4 ) ⇔ (the frontier) λ > 2 Ef ficient T ime ω 0 = π 2 ω c = π λ < 1 2 Localization ( B = 3 π 4 ) ⇔ (min. delay-spread) λ > 2 Experience and research has taught us that the chirp wa v eforms encountered in practice, such as those used by animals for echolocation, as well as those encoun- tered in the signatures of discrete-scale in v ariance such as fracture dynamics, ex- hibit inefficient time-frequency localization. In simpler words, we seek chirplets with relati v ely wide bandwidths and long delay spreads, whose parameterizations are selected away from the efficient frontier of Fig. 6. This opens up interesting questions on possible impro vements that hav e remained unearthed until no w . W e leav e this problem for future analysis. 5.4 Discr ete-Time Implementation In specifying which parameterizations to use, we start by fixing cutof f frequency ω c = π . In Section 5.2, it was found that this is required for ef ficient localization. No w that inef ficiently localized chirps are also in consideration, there remains a 37 Figure 7: Altes chirplet delay spread τ as a function of the parameters { ω 0 , λ } for inef ficiently localized chirps. The cutoff frequency is fixed at ω c = π . At these settings, the delay spread increases more or less linearly with bandwidth, which increases complexity of implementation. Note that increasing ω 0 /ω c is used as a proxy for increasing bandwidth as per equation (22). The vertical dotted line sho ws by example that a 1024-point Fourier Transform would be the minimum size required to implement all chirplets with λ < 0 . 9 , ω 0 > π 10 . good reason to fix ω c = π . In order to minimize the complexity of the discrete- time implementation, the chirplet should be represented by a sampling that is as sparse as possible, such that the wa v elet dynamics are fully captured without aliasing. Assuming a unit sampling interv al (a sampling frequency of ω s = 2 π ), this translates directly to the requirement that the chirplet cutoff equals the Nyquist sampling rate ω c = ω Nyq = π . This, together with Section 5.2, requires that ω 0 < π 4 (i.e. B > 3 π 4 ) for frequency delocalization i.e. such that wideband chirping can occur . Before moving on to chirp-rate λ , we take a short aside on complexity . W ith critical-rate unit-sampling, the number of time-samples required to represent a 38 chirplet is equal to the delay spread τ since there is approximately no energy out- side this windo w , as described in Section 5.1. Now consider Fig. 7, in which it seen that, counterintuiti vely , the delay-spreads of wideband Altes chirplets in- cr ease more or less linearly with bandwidth (i.e. when parameterized for inef fi- cient localization). This puts a burden on comple xity , which must be bounded. T o generate the time-domain wa v eform using the In verse Fast Fourier Transform (IFFT) in eq. (26), we would select a transform size that is large enough to capture the full delay spread. Since the IFFT is implemented cost-effecti vely when its size is an integer power of two, the y -axis abscissa of Fig. 7 are labeled dyadically , directly allowing the choice of transform size for discrete-time implementation. Gi ven bounds on the Altes parameter set, we can directly select the appropriate IFFT transform size from the chart. Recall from Section 4.4 that the Hyperbolic Chirplet T ransform should be com- putable over a range of chirp-rate values for 0 < λ < 1 (or equiv alently λ > 1 for accelerating rather than decelerating chirps). In addition, as described above, in aiming for wideband, chirping ω c /ω 0 should be large. Examining Fig. 7, it ap- pears that, for a fixed delay-spread, these requirements compete for complexity , and a trade-off must be reached. For example, selecting ω c ω 0 = 10 and the chirp rate λ = 0 . 75 w ould require an FFT size of at least 512 for alias-free computation of the Altes time-domain chirplet u ( t ) in eq. (26). This comple xity analysis abo v e is crucial for choosing the F ourier transform size in discrete-time implementation of the Altes chirplet. 39 Figure 8: This chart sho ws the number of full time-domain oscillations of the Altes chirplet within the limits of its delay spread, as a function of its parameters { ω 0 , ω c , λ } . The settings are the same as those used in Fig. 7. For 0 < λ < 1 , the larger λ , the larger the number of oscillations, which scales approximately as − 1 / ln( λ ) . 5.5 Number of Chir plet Oscillations: Discrete Scale In v ariance W e can place bounds on chirping parameter λ by e xamining the number of os- cillations of the chirplet. Consider the chirplets in figures 2-3. The first example appears to hav e about two full oscillations of the wa veform, while the second has about six. W e can put more precision on this by counting the number of oscilla- tions of the Altes chirplet ov er a relev ant parameter sweep. Repeating the sweep of Fig. 7, the chart of Fig. 8 sho ws the measured number of full oscillations of the chirplet, rather than the measured delay spread. It can be seen that the number of oscillations becomes lar ge for v alues of chirp-rate close to unity . This is relev ant for two main reasons. Firstly , the well kno wn wav elets such as Meyer , Shannon, Gauss, Morelet, Me xi- 40 can Hat, etc. all ha ve a limited number of oscillations o v er their re gion of support (or within their delay spread, for infinite support wa velets). It has been found that a lo w number of oscillations, from two or three up to a dozen or so, has gi v en useful results in their application domains, such as data compression and signal analysis. This is because different frequencies and scales can be analyzed using a small number of oscillations, by appropriate dilations. It is purported that this should also hold for the Altes chirplet. A more compelling reason stems from the concept of discrete scale in v ariance introduced in Section 2.4. As explained there, signals which are log-periodic are discrete scale in variant, or self-similar . Such signals are in v ariant under a discrete subset of dilatations. Our Altes chirplet is not perfectly discrete scale in v ariant, because of its time domain en v elope, which forces time-localization, and hence usefulness as a w av elet. Howe ver , the chirplet can still be used to detect discrete scale in v ariance 2 by its pseudo log-periodicity . The ratio of intervals between the peaks of its oscillation is constant, as in genuine log-periodic signals. This has the subtle advantage that only a few oscillations of the chirplet are necessary for LP- detection, since further oscillations only detect extensions of the log-periodicity at higher or lower (discrete) scales. More simply , we do not need a lar ge number of chirplet oscillations to detect a log-periodicity . A fe w cycles will do, plus appropriate dilations and scaling, as in the hyperbolic chirplet transform. The implications of this become apparent in Fig. 8, which suggests that we need 2 The discr ete scale inv ariance of a log periodic signal discussed here is different to the full scale in v ariance of the HCT discussed in Section 4.4. 41 T able 2: P arameter Bounds on the Altes Chirplet for Discrete-Time Implementa- tion and use in the Hyperbolic Chirplet T ransform Parameter V alue(s) Comment Center Frequency ω 0 ω 0 < π 4 W ideband Chirping Cutof f Frequency ω c ω c = ω Nyq = π Critical sampling with unit sampling interv al Chirp-Rate λ 1 / 4 < λ < 3 / 4 Bounded Number ( ⇔ 4 / 3 < λ < 4 ) of oscillations Fourier T ransform Use Fig 7 N FFT =512 will work Size N FFT to select for most cases a tighter upper bound than λ < 1 . In order to remain below 20 oscillations of the chirp, we can retain all of the previous parameter bounds but need to tighten the upper chirp-rate bound to λ < 0 . 75 . In addition, we also propose a lower bound on the useful number of oscillations. Log-periodicity can occur spuriously in noisy data and it is desirable to a v oid falsely reporting such events as significant. A deep study on the statistics of random-walk data (inte grated noise) has sho wn that the most likely number of spurious oscillations which occur is 1.5, and that 2.5 can occur with a likelihood as high as 10% ov er many realizations [53]. W e therefore suggest that usefully parameterized Altes chirplets will hav e at least 2 oscillations, implying a practical lo wer bound λ > 0 . 25 from Fig. 8. 42 5.6 Summary of Altes Chirplet P arameter Selection In this Section, it has been shown ho w to parametrize the Altes chirplet for effi- cient time-frequency localization. Howe v er , it has also been noted that localization- ef ficiency should be sacrificed for chirping flexibility , and instead we seek param- eters that allo w wide-band chirping for a selection of chirp-rates with minimized sampling rate and F ourier transform size. A numerical examination of these prop- erties ov er the parameter space leads to the conclusions of T able 2 for parameter- izing the Altes chirplet. The snippet of M A T L A B T M belo w shows that the chirplet can be implemented in a straight-forward manner in a software en vironment, giv en the selected parameters. function [U,u] = altesChirplet(w0,wc,lam,Nfft) %% Altes Chirplet time domain u & freq domain U Kc = 0.01; Np = (Nfft/2)+1; wp = linspace(0,pi,Np); % positive frequencies kc = -log(Kc)/(log(wc/w0)ˆ2); Mwp = exp(-kc * (log(wp/w0).ˆ2)); Qwp = exp(2 * pi * j * log(wp)/log(lambda)); U = Mwp. * Qwp; Upad = [0 U(2:end) zeros(1,(Nfft/2)-1)]; u = ifft(Upad); 43 6 Empirical r esults using Synthetic Data In order to help demonstrate the applicability of the Altes wav elet to detection of log-periodicities via the Hyperbolic Chirplet T ransform, we hav e constructed a synthetic test signal in which three log-periodic chirps are b uried in white noise, along with a sine wa ve. The resulting signal for analysis is sho wn in Fig. 9a, having a signal to noise ratio (SNR) of 0 dB. For comparison, we have chosen to analyze this signal by three methods, a Short T ime Fourier T ransform (STFT), a Continuous W av elet T ransform (CWT) using the complex Morlet wav elet, and the Hyperbolic Chirplet T ransform using an Altes chirplet. The resulting spectrogram / scalogram outputs are sho wn in figures 9b-9d respecti vely . In Fig. 9b, the first thing to spot is that the STFT picks out a sinusoid, which has been injected at frequency π 3 . This is not detected by the wav elet or chirplet transforms, highlighting the imperativ e to pick a suitable transform for the prob- lem at hand. Assuming we are only interested in detecting log-periodicities, it is clear that the STFT is relatively poor , as is to be expected from its constant time- resolution at all frequencies. W e hav e implemented an overlap-add STFT using a 128-point Fourier transform with a Hamming windo w . Fig. 9c shows some improv ement by the CWT in detecting the LP-chirps. In Section 2.1, we noted that Morlet wav elets hav e been reported to exhibit superior performance o ver other wa v elets for chirp detection [17], so we use a Morlet here as the mother wav elet for the CWT . For fairness of comparison with the (complex) Altes chirplet, we use the complex Morlet, which mar ginally impro ves 44 localization in the scalogram. Furthermore, since the complex Morlet has much narro wer time-support than the Altes chirplet at unit scale, we use a higher set of scales in the CWT than the HCT , such that the time resolution of their scalograms are comparable. It is found that we need to choose a set of scales approximately 10 × those of the HCT , as can be seen from the y -axes of figures 9c-d. The CWT scalogram seems to suggest, correctly in fact, that there are three bursts of signal acti vity over the interval. Howe ver , while the result is partially clear around the timing and scaling of signal acti vity (the localisation of the three LP-chirps is quite poor) it would be a considerable stretch to conclude that this represents a set of log-periodicities. . Fig. 9d is more promising. There is a significant increase in sharpness identifying the timing and scaling of bursty signals b uried in the noise. But of greater interest are the three linear ridges, which sweep upwards from left to right. The HCT has captured the benefits of linearizing chirps in the time-scale plane, as identified in [15] [16], and has correctly isolated the placement of the chirps as synthesized in the artificial test-signal. This is despite the high le vels of noise and tonal- interference superimposed on the chirps to be detected. In parameterizing the Altes chirplet in this example, we ha ve kept within the bounds of T able 2, selecting { ω 0 , ω c , λ } = { π 5 , π , 1 2 } . In fact, many systems ex- hibiting discrete scale in v ariance ha ve a preferred scaling ratio λ = 2 ( ⇔ λ = 1 2 ). For example in [7], it has been observed that the mean field value of λ = 2 is obtained by taking an Ising or Potts model on a hierarchical lattice in the limit 45 of an infinite number of neighbors. Also, we hav e seen that chirplet parameter λ = 1 2 lies directly between the limits of applicability we found for the HCT 1 4 < λ < 3 4 . W e surmise that λ = 1 2 ⇔ λ = 2 is a natural choice for the chirp- rate. W e find that, when making this selection for our mother wav elet, the HCT will ne vertheless succeed in detecting chirps generated with dif ferent v alues of λ . This is because λ is a scale ratio for the distance between successiv e peaks in a log-periodic wav eform, while a is the scale ratio within the HCT that serves as a dilation factor , stretching the analyzing wav eform to find log-periodicities at other chirp rates. This is all to say that fixing λ = 1 2 in the HCT is not seen to limit its use for more general LP-detection. It is worth noting that the basic analyzing wa veforms for the STFT (a windowed sinusoid) and the CWT (the complex Morlet) are both symmetric in time about their center . Ho we ver , the Altes Chirplet is ske wed (Fig. 1b), which moves the center of ener gy when analyzing at dif ferent scales. For consistency with the CWT , the Altes chirplet is centered on its energy peak (ie. by setting τ 0 ! = 0 in Fig. 1b) for HCT implementation. T o conclude, we see that there is promise in the use of the Altes chirplet and the HCT for improving our ability to detect log-periodic signatures in noisy signals. Looking forward, our research is taking us do wn a more applied route than the theoretical framew ork of the current study , and there are certainly many real-world applications where the value of the methodology can be quantified more precisely . Such application is beyond the scope of this introductory paper . 46 7 Conclusion Building on the excellent sonar wa veform designs of R.A. Altes from the 1970s, this article has taken the step to make his work both more accessible to the sig- nal processing community , and more widely applicable in the context of wav elet transform analysis. T o achiev e the former , a reparameterization allows simple specification of a f amily of chirplets in terms of bandwidth, center frequenc y , and chirp-rate. It is demonstrated that these wa v elets are admissible, infinitely regular , hav e infinite vanishing moments, and furthermore, deliv er scale in v ariance when implemented in a continuous wa velet transform. For the latter , it has been shown ho w to design a useful parameterization of these chirplets for application in a discrete-time Hyperbiolic Chirplet Transform (HCT). 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Test Signal (unit sampling): 3 x Log − Periodic Chirps + Sinusoid + Noise Clean signal for illustration only (not used) Test signal with noise and tonal interference Normalized Freq. f (x π rad/s) b. Short Time Fourier Transform (STFT) 0 50 100 150 200 250 300 350 400 450 500 0 0.1 0.2 0.3 0.4 0.5 Wavelet Scale a c. Continuous Wavelet Transform (CWT) using Complex Morlet Wavelet 0 50 100 150 200 250 300 350 400 450 500 0 20 40 60 80 100 Chirplet Scale a d. Hyperbolic Chirplet Transform (HCT) using Altes Chirplet Time t (Samples) 0 50 100 150 200 250 300 350 400 450 500 0 2 4 6 8 10 Dashed verticals show time − centering of the chirps for detection Figure 9: Comparison of transform analyses for the detection of log-periodicity in a synthetic signal. Subplot a) shows the test signal: three log-periodic chirps and a sinusoid buried in noise. For illustration, the chirps are also sho wn without noise and tonal interference, and their centers of energy ( τ 0 from Fig. 1b) are marked with dashed verticals. The remaining subplots are log-magnitude contour charts of the transform outputs: b) Short T ime Fourier T ransform (STFT) spectrogram of the test signal using a 128-point Fourier Transform and Hamming window c) Con- tinuous W avelet Transform (CWT) scalogram using the complex Morlet wav elet and d) Hyperbolic Chirplet T ransform (HCT) scalogram of the test signal using an Altes chirplet. It can be seen that the HCT isolates the log periodic chirps for de- tection in the time-scale plane. The formation of linear ridges in the HCT output confirms the presence of log-periodicity . 56