Article Journal of Vibration and Control 1–20 ! The Author(s) 2016 A new wavelet family based Reprints and permissions: sagepub.co.uk/journalsPermissions.nav on second-order LTI-systems DOI: 10.1177/1077546316674089 jvc.sagepub.com
Tariq Abuhamdia1, Saied Taheri1 and John Burns2
Abstract In this paper, a new family of wavelets derived from the underdamped response of second-order Linear-Time-Invariant (LTI) systems is introduced. The most important criteria for a function or signal to be a wavelet is the ability to recover the original signal back from its continuous wavelet transform. We show that it is possible to recover back the original signal once the Second-Order Underdamped LTI (SOULTI) wavelet is applied to decompose the signal. It is found that the SOULTI wavelet transform of a signal satisfies a linear differential equation called the reconstructing differential equation, which is closely related to the differential equation that produces the wavelet. Moreover, a time-frequency resolution is defined based on two different approaches. The new transform has useful properties; a direct relation between the scale and the frequency, unique transform formulas that can be easily obtained for most elementary signals such as unit step, sinusoids, polynomials, and decaying harmonic signals, and linear relations between the wavelet transform of signals and the wavelet transform of their derivatives and integrals. The results obtained are presented with analytical and numerical examples. Signals with constant harmonics and signals with time-varying frequencies are analyzed, and their evolutionary spectrum is obtained. Contour mapping of the transform in the time-scale and the time- frequency domains clearly detects the change of the frequency content of the analyzed signals with respect to time. The results are compared with other wavelets results and with the short-time fourier analysis spectrograms. At the end, we propose the method of reverse wavelet transform to mitigate the edge effect.
Keywords SOULTI-wavelets, second-order systems wavelets, LTI-wavelets, second-order linear-time-invariant wavelets, time-fre- quency analysis, spectrogram, scalogram, chirp analysis, frequency-identification, edge-effect, forward-wavelet-transform, forward wavelet transform, reverse-wavelet-transform, reverse wavelet transform
1. Introduction features from a signal, then the analyzing wavelet family should also have these features. This is similar Wavelets provide a powerful tool to analyze signals to the way we measure the periodicity of a signal by and extract information from them. They are capable making inner product with the harmonic functions of extracting frequency, time, and nonharmonic infor- because of their periodicity. mation. These potentials lured many scholars to use This idea also implies that if we want to use time- them in the analysis of dynamic systems. Scholars frequency analysis on a dynamic system by analyzing have used wavelets for system identification, system its response, a better understanding can be developed if modeling, system response solution, and even control design. For broad and extensive survey on the use of 1Center for Tire Research (CenTire), Virginia Polytechnic Institute and wavelets in systems and control, the reader is referred State University, USA to Abuhamdia and Taheri (2015). 2Interdisciplinary Center for Applied Mathematics (ICAM), Virginia Mathematically, the wavelet transform is an inner Polytechnic Institute and State University, USA product between a function and a set of basis functions Received: 18 May 2016; accepted: 4 September 2016 which are all derived from a single function called the Corresponding author: mother wavelet. It measures how much parallelism Tariq Abuhamdia, Department of Mechanical Engineering, Virginia Tech, exists between the analyzed function and the set of 100T Randolph Hall, 460 Old Turner St, Blacksburg, VA, 24060, USA. basis functions. Therefore, if we seek to extract some Email: [email protected]
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 2 Journal of Vibration and Control the analyzing wavelet is close in characteristics to sys- as a pseudo wavelet and defined it by tems responses and behaviors. 8 < !2 This observation led to investigating the character- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 ! 0 2 2 2 2 istics of the underdamped second-order response of ð! ! Þ þð2 0!0!Þ ð!, !0, 0Þ¼: 0 ð2Þ Linear Time-Invariant (LTI) systems to find if it can 0 ! 5 0 serve as a mother wavelet. The underdamped second- order impulse response of LTI systems is oscillatory and used the continuous wavelet transform but in the and decaying exponentially, and it dies out to effective frequency domain to identify the parameters of zero well within the defined period. Furthermore, its dynamic systems by mapping the match between the frequency domain representation is a second-order system frequency response and the pseudo wavelet. filter that can effectively extract certain frequency In the following section, we show that it is possible bands from the signals. to construct a family of wavelets from the response of It was intuitive to try to construct families of wave- Second-Order Underdamped LTI (SOULTI) systems lets from the building blocks of systems responses, espe- that we call, for brevity, the SOULTI wavelets. We cially LTI systems. Such families of wavelets could be show that their inverse continuous wavelet transform useful in systems characterization and provide new per- exists and define the basic properties an analyst needs spective for understanding systems and how their to perform time-scale or time-frequency analysis. responses evolve. It was a remarkable coincidence In Section 2, we define the SOULTI wavelet families. that Robinson (1962) called the response of a second- Section 3 constructs and proves the existence of the order LTI systems a wavelet. However, the closest inverse wavelet transform for the SOULTI wavelets. point in this track was using the response of second- Section 4 explores the basic properties of the order LTI-systems as pseudo wavelets (Freudinger SOULTI wavelet and the associated transform and et al., 1998; Hou and Hera, 2001). They were con- lists the SOULTI wavelet transform for elementary sig- sidered pseudo wavelets because they failed to satisfy nals. Section 5 defines the time and the frequency prop- the reconstruction conditions, namely the inverse erties of the wavelet and derives different definitions wavelet transform was not possible. In addition to for the time-frequency resolution of the wavelet trans- those efforts, Newland (1993) proposed the harmonic form. Section 6 presents an application with numerical wavelets which possess the important advantages examples for analyzing signals with different frequency of being orthogonal and having excellent frequency characteristics, and Section 7 addresses the edge effect localization. Moreover, they can be viewed as per- and proposes a solution to reduce its influence on the fect band-pass filters. Jezequel and Argoul (1986) analysis. used a transfer function in the frequency domain (ratio of zeros and poles) as a kernel for an integral 2. Second-order underdamped LTI transform that transforms signals from the frequency wavelets domain to another two-dimensional domain whose axes represent some parameters of the model repre- Second-order LTI systems are very common in most sented by kernel. dynamic fields of science. The mechanical mass-spring- The response of second-order systems had been damper system, shown in Figure 1(a), and the used before to analyze signals for different purposes RLC-electrical circuit, shown in Figure 1(b), are typical and under different names but as a pseudo wavelet or examples of such systems. The response of the SOULTI 2 dictionary of wavelets. Freudinger et al. (1998) defined system in Figure 1, for the impulse input !n ðtÞ is the Laplace wavelet, by given by ! ( pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi2 #ðt Þ !n 2 j2 #ðt Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi !nt 2 Ae 1 e t 2 ½ , þ T hðtÞ¼ e sinð!n 1 tÞuðtÞð3Þ ð f, , , tÞ¼ s 1 2 0 else ð1Þ where <1 is the damping ratio, !n is the natural frequency and u(t) is the Heaviside step function. 2 where represents the damping ratio, # is the fre- The impulse input is scaled by !n to simplify the deriv- quency, and Ts is the effective duration of the wavelet ation of the frequency properties in Sections 4 and 5. that defines the effective compact support. They formed The damped frequency !d of the underdamped system a dictionary of wavelets but not a basis or frame. Hou is given by and Hera (2001) used the the magnitude of second- pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 order LTI systems response in the frequency domain !d ¼ !n 1 ð4Þ
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 Abuhamdia et al. 3
Figure 1. (a) Mass-spring-damper system; (b) RLC electrical circuit. so equation (3) can be rewritten in terms of !d as Suppose that J ða, 1Þ R, and let f ðtÞ : J ! R 1 and f ðtÞ2L and is exponentially bounded, see section !ffiffiffiffiffiffid !d p t (2.1). The SOULTI wavelet transform of f(t) can be now hðtÞ¼ e 1 2 sinð! tÞuðtÞð5Þ 1 2 d defined by the generic continuous wavelet transform definition Let the reciprocal of the damped frequency, defined in Z 1 equation (4), be the scaling parameter as ~ t W ff ðtÞg ¼ f ð , sÞ¼ f ðtÞ dt, 2 ð 1, 1Þ 1 s 1 ð Þ s ¼ ð6Þ 10 !d The SOULTI wavelet transform in equation (10) offers Substitute equation (6) into equation (5) to get a measurement of similarity between any signal and the response of second-order LTI systems for characteriza- t s 1 pffiffiffiffiffiffis t tion and identification purposes. In addition, The hðtÞ¼ e 1 2 sin uðtÞð7Þ 1 2 s SOULTI wavelet gives a direct and simple relationship between scale and frequency as shown in equation (6), which is the impulse response in terms of the scaling where the frequency is the reciprocal of the scale. parameter s where s 2ð0, 1Þ. Now, the SOULTI wave- let can be defined as 2.1. Region of convergence p p ffiffiffiffiffiffi t s, t s ðÞs t The region of convergence of the SOULTI transform ¼ ¼ e 1 2 sin uðt Þ s 1 2 s defines the region S T, where s 2 S ¼ ð8Þ ð0, 1Þ, 2 T ¼ ð 1, 1Þ, in which the transform in equation (10) converges to a finite value. Before explor- which represents the real part of the Laplace wavelet ing such region, notice that f(t) has to be exponentially defined by Freudinger et al. (1998). The parameter p is bounded in order for the transform in equation (10) to used to give the wavelet a preservation property. For converge. example, when the wavelet is scaled, its energy content Exponential boundedness is defined in the following; is also scaled, so to preserve the energy of the L2 norm define J ða, 1Þ R, and let f ðtÞ : J ! R,if9 , k 2 R under scaling we use p ¼ 1/2. However, to preserve the such that j f ðtÞj jke tj8t 2 J, then f(t) is exponentially L1 norm of the wavelet, namely bounded. f t Z Z If ( ) is exponentially bounded, then the SOULTI 1 t 1 transform is convergent in the scale region defined by dt ¼ j ðtÞjdt ð9Þ 1 s 1 0 5 s 5 pffiffiffiffiffiffiffiffiffiffiffiffiffi ð11Þ we use p ¼ 1. Figure 2 graphs the SOULTI wavelet 1 2 versus time showing its time properties. The wavelet function defined in equation (8) represents more than When the time domain is considered for convergence, one family of wavelets, where each family is linked to a i.e. considering the values of the time shift that ren- single value of , where 0 < <1. It retains the LTI ders the transform convergent, we have to be careful second-order response characteristics completely. about the uniqueness of the transform because
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 4 Journal of Vibration and Control
Figure 2. The SOULTI mother wavelet time function at ¼ 0.3, s ¼ 1, and p ¼ 1. the transform on finite time interval could be identical Using Plancherels theorem, equation (10) becomes for two different functions on a set of measure greater Z 1 than zero. Therefore, the time region has to be expli- ~ s j! f ð , sÞ¼ e ðs!ÞFð!Þd! ð14Þ citly indicated on the transform, and uniqueness is not 2 1 achieved in this case. In the next section, we will define j! the inverse continuous wavelet transform with respect where the conjugate of se ðs!Þ is substituted in the to the SOULTI wavelet. inner product. Note that the integral in the right side of equation (14) represents the inverse Fourier transform 3. SOULTI wavelet inverse transform of ðs!ÞFð!Þ. Applying the Fourier transform to equa- tion (14) yields Notice that equation (10) represents an inner product in Z 1 the time domain, which is equivalent to the inner prod- j! ~ s e f ð , sÞd ¼ ðs!ÞFð!Þð15Þ uct in the frequency domain according to Plancherel’s 1 2 theorem (Yoshida, 1965). Applying Plancherel’s the- orem to equation (10) yields In general, we cannot divide both sides by s ðs!Þ because it could vanish at some values of ! or s. ð Þ 1 However, in our case s s! is given by f~ ð , sÞ¼hf ðtÞ, ,si¼ hFð!Þ, ,sð!Þi ð12Þ 2 s1 p ,s ,s 1 2 where ð!Þ is the Fourier transform of , and using s ðs!Þ¼ ð16Þ 2 !s the Fourier transform shift and scale properties it can 2 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi ! s þ 1 2 j be expressed in terms of the Fourier transform of the 1 2 mother wavelet, ð!Þ,as
ÈÉ which does not vanish for any value of ! or s 2 (0,1). ,s j! F ¼ se ðs!Þð13Þ Figure 3 shows the wavelet spectrum magnitude, which is equivalent to its conjugate spectrum
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 Abuhamdia et al. 5
Figure 3. SOULTI wavelet in the frequency domain with the mean frequency !CG , the standard deviation-based frequency window SD, and the (half-power)-based frequency window BW. ¼ 0.5, s ¼ 1, and p ¼ 1. magnitude. The curve never crosses the zero axis and the SOULTI wavelet transform of f(t) be given by equa- decays asymptotically to zero. Consequently dividing tion (10), then the inverse wavelet transform satisfies the by s ðs!Þ is a legitimate operation. Therefore, we identity can solve for F(!) in equation (15) to get 2 2 ~ Z d f ðt, sÞ 1 1 f ðtÞ¼W 1ff~ ðt, sÞg ¼ s p 14ð1 2Þs2 j! ~ 2 Fð!Þ¼ e f ð , sÞd ð17Þ dt s ðs!Þ 1 3 ð19Þ ~ pffiffiffiffiffiffiffiffiffiffiffiffiffi d f ðt, sÞ 2 ~ 5 2 1 s þ f ðt, sÞ To retrieve f(t), take the inverse Fourier transform of dt equation (17), so the inverse wavelet transform with respect to the SOULTI wavelet becomes Proof. Substituting s ðs!Þ from equation (16) into Z Z equation (18) gives 1 1 j!t 1 ~ e j! ~ W ff ð , sÞg ¼ f ðtÞ¼ e f ð , sÞd d! 1 1 s ðs!Þ Z ÀÁ ! 1 ej!t 1 2 1 2 !s ð18Þ ð Þ¼ 2 2 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi f t 1 p ! s 2 j 1 s 1 1 2 Z ð20Þ 1 j!t ~ Equation (18) forms the inverse wavelet transform with e f ðt, sÞdtd! respect to the SOULTI wavelet or the reconstruction 1 formula of the original wavelet definition shown in Note that is substituted by t inside the second inte- equation (10). If f(t) is differentiable, we can use a sim- gral. Using the operator notation for the Fourier trans- pler and probably more practical inverse formula to form, equation (20) becomes retrieve f(t) back from its wavelet transform. noÀÁ pffiffiffiffiffiffiffiffiffiffiffiffiffi no p 1 1 2 2 2 2 ~ f ðtÞ¼s F ! s 1 þ 1 j 1 2 !s F f ðt, sÞ Theorem 1. Let J ða, 1Þ R, and let f(t): J ! R be differentiable and exponentially bounded, and let ð21Þ
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 6 Journal of Vibration and Control
Applying the linear operators properties (Naylor and SOULTIÀÁ wavelet transform with respect to the wavelet t Sell, 2000) to equation (21) gives family s of f(t) at scale s is a solution of the fol- lowing nonhomogeneous differential equation noÀÁno p 1 1 2 2 2 ~ f ðtÞ¼s F ! s 1 F f ðt, sÞ nono d2ð yðtÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi d ð yðtÞÞ f ðtÞ¼sp 1 ð1 2Þs2 2 1 2 s þ yðtÞ p 1 1 ~ 2 þ s F F f ðt, sÞ ð22Þ dt dt nopffiffiffiffiffiffiffiffiffiffiffiffiffi no ð23Þ p 1 1 2 ~ s F j 1 2 !sF f ðt, sÞ Proof. The proof follows by direct substitution. Fix s, and after applying the Fourier transform differentiation so it can be treated as a constant. Now, suppose that a property to equation (22) we arrive at equation (19). # solution of equation (23) is given by This provides a simple and direct method in the time domain to calculate the inverse wavelet transform with ~ respect to the SOULTI wavelet. However, in order for ypðtÞ¼f ðt, sÞð24Þ ~ the formula in equation (19) to apply, f ð , sÞ has to be at least twice differentiable with respect to time. When Substitute yp(t) back into the right side of equation (23) ~ considering the transform that defines f ð , sÞ in equa- to get tion (10), we find that f~ ð , sÞ is twice differentiable with respect to time if f(t) is differentiable. So if f(t) is expo- d2 Gðt, sÞ¼sp 1 ð1 2Þs2 f~ ðt, sÞ nentially bounded and f ðtÞ2C1, then its SOULTI dt2 ð25Þ wavelet transform is unique and f(t) can be retrieved pffiffiffiffiffiffiffiffiffiffiffiffiffi d 2 ~ ~ using equation (19). 2 1 s f ðt, sÞ þ f ðt, sÞ dt Equation (19) also provides information about the uniqueness of the SOULTI wavelet transform. The inverse wavelet transform given by equation (19) is a but we just proved by Theorem 1 that G(t,s) ¼ f(t). # linear second-order differential equation, which we will call the Reconstructing Differential Equation. The ori- 4. SOULTI transform of elementary ginal function f(t) is the input function and its wavelet signals and its properties transform at scale s is a solution or part of the response. However, the other conditions must be satisfied in Let us examine the validity of equation (19) with an order for equation (19) to server as inverse formula example. Let f ðtÞ¼e t, then its SOULTI wavelet for the SOULTI wavelet transform. transform is given by Z R R p 1 p ffiffiffiffiffiffi t Corollary 2. Let J ða, 1Þ , and let f ðtÞ : J ! s t ðÞs t ~ ð Þ¼ 1 2 ð Þ f , s 2 e e sin dt 26 be differentiable and exponentially bounded, then the 1 s
Figure 4. SOULTI Wavelet transform surface of the decaying exponential function at ¼ 0.7 and ¼ 2.
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 Abuhamdia et al. 7 which can be evaluated using the integration by parts Table 1. SOULTI wavelet transform for basic signals. technique to give # f(t) f~ ð , sÞ
1 p 1 p 1 u(t) s ~ s f ð , sÞ¼e pffiffiffiffiffiffiffiffiffiffiffiffiffi ð27Þ 2 2 2 1 p 2ffiffiffiffiffiffiffiffi s 1 ð1 Þ s þ 2 1 s þ 1 2 tu(t) s p þ 2 1 2 1 À which represents an analytical formula in terms of the 3 t2uðtÞ s1 p 2ð2 2 1Þð1 2Þs2 scale s, the time shift , and the wavelet damping ratio pffiffiffiffiffiffiffiffiffiffiffiffiffi þ4 1 2s þð1 2Þ 2 , in addition to the decay rate . t Note that the transform of e in equation (27) con- 4 e t e s1pp ffiffiffiffiffiffiffiffi sists of a multiplication of two functions, a function of ð1 2Þ s2þ2 1 2 sþ1 time and a function of scale. Note also, that the trans- s1 p form is very similar to the Laplace Transform of a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinð! þ Þ 5 sin(!t) A2 þ B2 delayed and scaled function. Figure 4 shows the wavelet A transform of the exponential function as described in ¼ tan 1 B equation (27). Aðs, !Þ¼1 ð1 2Þs2!2 Let us now evaluate the SOULTI inverse transform pffiffiffiffiffiffiffiffiffiffiffiffiffi by using the formula in equation (19). Differentiating Bðs, !Þ¼2 1 2s! equation (27) with respect to time twice and substitut- s1 p ing the result into the right hand side of equation (27) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosð! þ Þ 6 cos(!t) 2 2 and substituting by t yields A þ B B ¼ tan 1 hpffiffiffiffiffiffiffiffiffiffiffiffiffi A p 1 2 2 2 t t t s ð1 Þs e þ e þ 2 1 2 se Aðs, !Þ¼1 ð1 2Þs2!2 !# pffiffiffiffiffiffiffiffiffiffiffiffiffi s1 p Bðs, !Þ¼2 1 2s! pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ e t=# ð1 2Þ s2 þ 2 1 2 s þ 1 1 p s t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e cosð! þ Þ 7 e cosð!tÞ 2 2 ð28Þ A þ B B ¼ tan 1 A We can use the result in equation (27) to find the Aðs, !Þ¼1 þð1 2Þð 2 !2Þs2 SOULTI wavelet transform for the sinusoidal func- pffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 1 2 s tions. Table 1 lists the SOULTI wavelet transform for pffiffiffiffiffiffiffiffiffiffiffiffiffi some elementary signals. Figure 5 shows the wavelet Bðs, !Þ¼2 ð1 2Þ !s2 þ 2 1 2!s transform scalogram of the cos(!t) function. 1 p s t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e sinð! þ R R 8 e sinð!tÞ 2 2 Lemma 3. Let J ða, 1Þ , and let xðtÞ : J ! be A þ B differentiable and exponentially bounded as defined in B ¼ tan 1 Theorem 1, and the SOULTI wavelet transform of A x(t) be given by equation (10), then the SOULTI wave- Aðs, !Þ¼1 þð1 2Þð 2 !2Þs2 let transform of x_ðtÞ is given by pffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 1 2 s pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 p ffiffiffiffiffiffi Bðs, !Þ¼2 ð1 Þ !s þ 1 !s e 1 2 x~_ ð , sÞ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi x~ ð þ s , sÞ 2 s 1 ! pffiffiffiffiffiffiffiffiffiffiffiffiffi ð29Þ which can be evaluated by the integration by parts tech- 1 2 ¼ tan 1 nique to obtain p p ffiffiffiffiffiffi t 1 s 2ðÞs t x~_ ð , sÞ¼ xðtÞe 1 sin 2 s Proof. Substitute x_(t) in equation (10) to have 1 ! Z p 1 p ffiffiffiffiffiffi t s xðtÞ ðÞs t pffiffiffiffiffiffiffiffiffiffiffiffiffie 1 2 sin dt Z 2 1 t 1 s 1 2 s x~_ ð , sÞ¼ x_ðtÞ dt, 2 ð 1, 1Þ ð30Þ s ð31Þ
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 8 Journal of Vibration and Control
Figure 5. SOULTI wavelet transform of f(t) ¼ cos(!t), at ¼ 0.7.
where is given by is given by pffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffi ¼ ð Þ 2 1 p tan 32 ~ ð , sÞ¼s 1 2e 1 x~ ð s , sÞ s XðaÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 2 ð35Þ ¼ tan 1 Since x(t) is exponentially bounded, the first term in equation (31) vanishes, so equation (31) becomes
p ffiffiffiffiffiffi dXðtÞ 2 Proof. Since xðtÞ¼ , substitute x(t) in place of x_(t), e 1 dt x~_ ð , sÞ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi s 1 2 and X(t) in place of x(t) in equation (33), and the result Z p 1 p ffiffiffiffiffiffi t ð þs Þ s ðÞs t ð þ s Þ ð Þ 1 2 can be written as 2 x t e sin dt 1 s ð33Þ
Z p ffiffiffiffiffiffi Z 1 1 but the part inside the parenthesis is equal to t e 1 2 t ð þ s Þ xðtÞ dt ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi XðtÞ dt x~ ð þ s , sÞ, hence equation (33) is equivalent to equa- s s 1 2 s tion (29). # ð36Þ
Lemma 4. Let J (a,1) R, and let x(t): J ! R be exponentially bounded as defined in Theorem 1, and make the substitutions (t) þ X(a) ¼ X(t) and ¼ þ s the SOULTI wavelet transform of x(t) be given by into equation (36), then equation (35) follows. equation (10), then the SOULTI wavelet transform of (t), defined by 5. The time-frequency resolution and Z properties t ðtÞ¼XðtÞ XðaÞ¼ xðrÞdr ð34Þ The time-frequency resolution is an important property a of the wavelet transform. The time-frequency
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 Abuhamdia et al. 9 resolution is defined by Table 2. Time-frequency resolution based on the standard deviation definition at p ¼ 1. ¼ T ð37Þ z tCGz oCGz TSD SD where T is the time resolution or the time window and 0.05 20.075 0.48531 39.85 0.24577 9.794 is the frequency resolution or frequency window. 0.1 10.149 0.47298 19.704 0.33949 6.689 The time window represents the time interval that a 0.15 6.8878 0.46292 12.896 0.40724 5.252 frequency can be identified within, while represents 0.2 5.2909 0.45506 9.4305 0.46191 4.356 the range of frequencies within a time interval. There 0.25 4.3571 0.44939 7.3101 0.5088 3.719 are different ways to define the time and frequency reso- 0.3 3.7522 0.44595 5.8715 0.55084 3.234 lutions. One possible way is to use the standard devi- 0.35 3.3322 0.44484 4.8325 0.58997 2.851 ation definition, in which the resolutions are defined by 0.4 3.0245 0.44625 4.0534 0.62763 2.544 R 2 1 2 2 0.45 2.7882 0.45046 3.4579 0.66502 3 TSD ðt tCG Þ j ðtÞj dt ¼ 1 R ð38Þ 0.5 2.5981 0.4579 3 0.7033 2.11 2 1 j ðtÞj2dt 1 0.55 2.4372 0.46922 2.6493 0.74366 1.970 R 1 0.6 2.2933 0.48537 2.3828 0.78752 1.877 2 ð! ! Þ2j ð!Þj2d! SD ¼ 1 R CG ð39Þ 0.65 2.157 0.50783 2.1798 0.83668 1.824 2 1 2 1 j ð!Þj d! 0.7 2.02 0.53899 2.0203 0.89372 1.806 0.75 1.8741 0.58291 1.8838 0.96259 1.813 where tCG and !CG represent the center of mass of the 0.8 1.71 0.64715 1.7483 1.05 1.836 signal in time and frequency respectively, and they are 0.85 1.5153 0.74778 1.5888 1.1692 1.858 given by 0.9 1.2689 0.927 1.3714 1.3512 1.853 R 0.95 0.92196 1.3553 1.0296 1.7057 1.756 1 tj ðtÞj2dt R0 tCG ¼ 1 2 ð40Þ j ðtÞj dt 0 In the frequency domain, the frequency correspond- R 1 2 ing to attenuating the input power by a half is con- 0 !j ð!Þj d! !CG ¼ R ð41Þ sidered the frequency bandwidth of the system or the 1 j ð Þj2 0 ! d! cut-off frequency, so we can use the bandwidth to define the frequency window. Table 2 lists the results of calculating for some values The 2% settling time, tst is reached when the of 0 < <1atp ¼ 1. The values of do not depend on enveloping function enters within 2% of the final the scale value and they satisfy the Heisenberg principle value. Therefor, for a scaled wavelet, it is given by (Kaiser, 1994). Using the standard deviation, the reso- pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi lution satisfies the inequality >1/4 (Kaiser, 1994). s 1 2 4s 1 2 tst ¼ logð0:02Þ’ ð42Þ !CG is proportional to the scale s, while tCG is inver- sely proportional to s. However, the standard deviation does not offer meaningful time and frequency windows On the other hand, the wavelet bandwidth is the fre- of resolution. The SOULTI wavelet is not symmetrical quency at which the frequency spectrum of a scaled neither in time nor in frequency. Moreover, it has no wavelet satisfies compact support neither in time nor in frequency. So we would question the significance of the standard s2 2p 2 1 2 1 deviation window about the signal center in time js ðs!Þj ¼ ¼ ð43Þ 2 2 2 and the significance of the frequencies included in the 2 2 1 p2 !ffiffiffiffiffiffiffiffis s ! þ 1 2 þ standard deviation window and weather that is really 1 2 what is accentuated in the time-scale or time-frequency analysis. Solving for ! gives the bandwidth by We can attain an alternative definition for the SOULTI wavelet time-frequency resolution based on !jjs ðs!Þj2¼1 ¼ BW systems dynamics and control theory. The system 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi response is considered settled when it enters the 2% 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 þ 2s2 2p 4 2 þ 4 4 margin of the final value and never leaves it again. s 1 2 Therefore, we can use the 2% settling-time value to ð44Þ define the time window, namely T2% ¼ 2%tst.
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 10 Journal of Vibration and Control
d Substitute equation (42) and equation (44) into equa- Since d 5 0 8 2ð0, 1Þ, then tion (37) gives the resolution as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 2:65 5 ð Þ 5 1ð49Þ 1 2 2 þ 2s2 2p 4 2 þ 4 4 4 ¼ 4 ð45Þ
which means that the 2%tst BW definition satisfies Equation (45) gives us a way to determine an appro- the Heisenberg principle when p ¼ 1. Another advan- priate value for p based on the time-frequency reso- tage of having p ¼ 1, is preserving the wavelet frequency lution shape. In order for to be independent of s, function peak constant. This is sometimes useful since it we must have p ¼ 1, which yields guarantees that all the frequency bands are amplified at the same level, see Figure 6. This functions as a normal- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi izing factor though it does not preserve the wavelet 1 2 2 þ 2 4 2 þ 4 4 energy. Figure 2 shows the standard deviation-based ð Þ¼4 ð46Þ and the 2%tst time windows, while Figure 3 illustrates the standard deviation-based and the half-power band- width-based frequency windows for the SOULTI which indicates that the 2%tst BW resolution defin- wavelet. ition depends only on thus on the wavelet family , The 2%tst BW gives a better meaning for the time- so we may write the resolution as ( ). frequency resolution of the SOULTI wavelet, but when Now, let us investigate the values of in range is small, <0.4, the definition suffers from two prob- 0 < <1. When ! 0, we have lems. First, the variation in the frequency response magnitude varies significantly within the bandwidth, lim ð Þ¼1 ð47Þ which requires better focus on the resonance range. !0 t Secondly, as s decreases, the bandwidth of ðsÞ contains all the bandwidths corresponding to larger s, while when ! 1 we get i.e. BWð , s2Þ BWð , s1Þ when s1 < s2. For <0.4, another definition of the time-frequency lim ð Þ¼2:65 ð48Þ resolution, that better reflects the data on the time-scale !1 or the time-frequency analysis domain can be provided
Figure 6. Wavelet amplitude in frequency domain for different values of the scale s at ¼ 0.2.
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 Abuhamdia et al. 11 based on the quality factor half-power bandwidth From equation (52) we find definition. The quality factor is the peak value of the fre- quency response. For small , the quality factor for LTI 1 2 !2 ¼ ð53Þ second-order system can be approximated by 1 s2ð1 2Þ (Meirovitch, 1997) 1 þ 2 !2 ¼ ð54Þ 1 2 s2ð1 2Þ Q ¼ ð50Þ 2 ffiffiffiffiffiffiffiffi2 The half power points, q1 and q2 are the points when Using the approximation !1 þ !2 ’ p , which is s 1 2 j ðs!Þj ¼ pQffiffi, see Figure 7. The bandwidth of the fre- 2 quency response is valid for small values of , it is easy to show that
Q ¼ !2 !1 ð51Þ 2 Q ¼ !2 !1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ð55Þ s 1 2 where !1 is the corresponding frequency to q1, and !2 is the corresponding frequency to q2, as shown in Substituting equations (42) and (55) into equation (37), Figure 7. To find !1 and !2, we have to solve the wave- the new time-frequency resolution definition becomes let power in equation (52) for ! where p ¼ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 4s 1 2 2 ¼ T ’ pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 8 ð56Þ 1 1 s 1 2 j ðs!Þj2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 8 2 ð s2!2ð1 2ÞÞ2 þ 2 ! 1 2s Equation (56) shows a very interesting result where the ð52Þ time-frequency resolution is constant and does not
Figure 7. j (s!)j for different values of showing the quality factor and the half-power bandwidth.
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 12 Journal of Vibration and Control depend on . Note that this approximation is valid for it has an inverse values of <0.4, for larger values the quality factor is Z smaller, hence the time-frequency resolution defined by 1 ð!Þ 2 0 5 d! ¼ C 5 1ð64Þ equations (45) and (46) would be more meaningful and ! suitable to adopt. Of course one may require a wider 0 bandwidth than the half-power quality factor band- width, which will make the time-frequency resolution 6. Application: Frequency identification coarser. For example, if instead of the pQffiffi bandwidth 2 and spectrogram limit we use Q/x, where x < Q, then the bandwidth and the time-frequency resolution become To validate the capability of the SOULTI wavelet in pffiffiffiffiffiffiffiffiffiffiffiffiffi detecting features of signals, we produce the frequency 2 evolution or the scale evolution with respect to time of 2 pxffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q ¼’ ð57Þ some signals using the SOULTI wavelet. We present s 1 2 some examples of SOULTI wavelet analysis of finite pffiffiffiffiffiffiffiffiffiffiffiffiffi time signals with white noise added to them at different ¼ 2 ð Þ 8 x 1 58 Signal to Noise Ratio (SNR) levels. It is important to emphasize that the continuous ver- The frequency at Q represents the frequency at which sion of the wavelet transform is performed in these the wavelet filter is centered at. Moreover, it is easy to examples, where the transform integral is performed predict where the wavelet frequency is centered because numerically. In all the examples, ¼ 0.1 is used because the scale is directly linked to frequency as stated by it gives the wavelet a large quality factor value as shown equation (6). The peak occurs at (Meirovitch, 1997) in Figure 6. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 !Q ¼ !n 1 2 ð59Þ 6.1. Identifying constant frequencies in time-invariant frequency signals For small we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi First, two noisy signals with the same frequency are tested. 2 2 ¼ ¼ !Q ¼ !n 1 2 ’ !n 1 ¼ !d ð60Þ The first has SNR 15 dB, and the second has SNR 7.5 dB. Figure 8 shows the two signals. Figure 9 shows which with equation (6) shows that we can easily the contour map of the two signals wavelet transform. approximate to a good accuracy the wavelet peak fre- We notice that in both cases the ridges and the 1 quency by the relation peaks are distinctly recognized at s ¼ 2, which corres- ponds to ! ¼ 2 rad/s by the scale-frequency relation in 1 equation (6). The ridge of the wavelet transform is ! ’ ð61Þ Q s the set of points in the time-scale domain , where the wavelet integral has stationary points (t,s) 2 Figure 6 shows clearly the accuracy of equation (61) for such that ts(t,s) ¼ s, where ts is a stationary point, i.e. ~ t df ð , sÞ the 0:2ðsÞ SOULTI family with different scaling values. ¼ 0 (Tchamitchan and Torresani, 1992). For larger values of , i.e. >¼ 0.4, the approximation ds ts Notice also that at the end of the signal the transform in equation (61) is not valid and we have to use the is distorted and the peaks diminish due to the edge exact relation effect. Also notice that since the SOULTI wavelet is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 1 causal the edge effect appears at the end of the time !Q ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ð62Þ scale of the signal only and the noisy signal with 2 s 1 SNR ¼ 7.5 dB has slightly worse edge effect. In the second test, a signal carrying two different fre- The SOULTI wavelet has (0) ¼ 1 when p ¼ 1. For quencies is analyzed. The signal has SNR ¼ 15 dB and is other values of p, the wavelet magnitude depends on graphed in Figure 10. Figure 11 shows two mappings. The s1 p. Moreover, we have first maps the contours on the time-scale domain and it Z showsclearlytworidgesthatstretchalongtwolinesof 1 2 j ð!Þj constant scale s ¼ 0.125 and s ¼ 0.5, parallel to the time d! ¼1 ð63Þ axis. The second plots the contours on the time-frequency 0 ! domain. The scale-frequency conversion is performed which implies that the SOULTI wavelet does not satisfy using equation (6). The ridges stretch along the constant the admissibility condition stated in equation (64) but frequency values ! ¼ 2 rad/s and ! ¼ 8rad/s.
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 Abuhamdia et al. 13
Figure 8. Top: single harmonic with white noise of SNR ¼ 15 dB. Bottom: single harmonic with white noise of SNR ¼ 7.5 dB.
Figure 9. Contour mapping of the SOULTI wavelet Transform of a harmonic signal of frequency ¼ 2 rad/s; (a) SNR ¼ 15 dB, (b) SNR ¼ 7.5 dB.
6.2. Identifying the instantaneous frequency useful in analyzing time-varying and nonlinear oscilla- in time-varying frequency signal tions. In this example, we analyze a signal consisting of a combination of constant harmonics with linear chirp The advantage of the time-scale or time-frequency ana- as a time-varying frequency component. White noise is lysis over classic frequency analysis is that it is more added to the signal with SNR ¼ 20 dB. The signal is
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 14 Journal of Vibration and Control
Figure 10. Sum of two harmonics with white noise. !1 ¼ 2 rad/s, !2 ¼ 8 rad/s, and the SNR ¼ 15 dB.
Figure 11. SOULTI wavelet transform for the two harmonics signal in Figure 10. (a) Time-scale contour mapping. (b) Time- frequency contour mapping. given by equation (65) (FFT) and the Welch averaging of the frequency spec- trum. While the FFT identifies the constant harmonics xðtÞ¼10 sinð0:4t2Þþ5 cosð2tÞþ8 sin t þ þ DðtÞ with peaks at ! ¼ 1 and ! ¼ 2, it is not possible to dis- 7 tinguish the instantaneous frequency change from the ð65Þ FFT. The Welch averaging does not identify the con- stant harmonics because of the interference from the where D(t) represents the white noise or the disturbance frequency-changing component. term. Figure 12 plots the signal in the time domain. Figure 14 shows the SOULTI wavelet transform of Performing Fourier analysis to the signal does not the signal. The transform distinctly traces the instant- reveal the instantaneous frequency change in the aneous frequency with respect to time, where the signal. Figure 13 shows the Fast Fourier Transform dashed lines represents ridgelines that trace this
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 Abuhamdia et al. 15
Figure 12. Two constant harmonics with a time varying frequency component signal.
Figure 13. Frequency spectrum of the signal described in Figure 12. (Dashed line) FFT. (Solid line) Welsh spectrum averaging. frequency along time. When examining the time vary- along ridgelines of almost constant scales at s ¼ 0.5 and ing component in equation (65) we find that the instant- s ¼ 1 in Figure 14(a) and along ridgelines of almost aneous frequency is given by !(t) ¼ 0.8 t, which is the constant frequency at ! ¼ 1 and ! ¼ 2 in Figure 14(b). equation of the dashed line on the time-frequency On 14(a), the parabolic dashed line, which traces the wavelet mapping shown in Figure 14(b). The dashed instantaneous change of the chirp frequency, intersects curve in Figure 14(a) is the inverse of the line the s ¼ 1 and the s ¼ 0.5 lines at times t ¼ 0.26 s and ! ¼ 0.8 t, namely s(t) ¼ 1/0.8 t, which conforms to the t ¼ 1.6 s respectively. scale-frequency relation in equation (6). Notice also, As a comparison between the SOULTI wavelet and that the other two constant frequencies are identified other wavelets in resolving frequencies with respect to
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 16 Journal of Vibration and Control
Figure 14. SOULTI wavelet transform for the chirp signal described in Figure 12 and shown in Figure 12. (a) Scalogram (Time-scale) contour mapping (b) Spectrogram (Time-frequency) contour mapping.
Figure 15. Scalograms of the signal described in Figure 12 using different wavelets. (a) By Morlet wavelet; (b) by Complex Shannon wavelet (fb ¼ 1, fc ¼ 1); (c) by Mexican hat wavelet; (d) by Frequency B-Spline wavelet (order ¼ 2, fb ¼ 1, fc ¼ 1). fb: bandwidth fre- quency. fc: wavelet center frequency.
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 Abuhamdia et al. 17
Figure 16. Spectrograms of the linear chirp signal in Figure 12 at different Window widths (samples). (a) W ¼ 8 (b) W ¼ 16 (c) W ¼ 32 (d) W ¼ 64 (e) W ¼ 128 (f) W ¼ 256. time, the same chirp signal is analyzed with four differ- qualitative information about the shape of the instant- ent wavelets, Morlet, complex Shannon, Mexican hat aneous frequency change with respect to time. For each and the frequency B-Spline. The four wavelets scalo- wavelet, the relation between the scale and the fre- gram graphs are shown in Figure 15. Notice that the quency along the ridgeline is different, but one can four are able to resolve the chirp into curves parabolic argue that it is the reciprocal of some function of the in shape with different ridge widths. However, the instantaneous frequency. curves are not the reciprocal of the instantaneous fre- From the previous discussion, we conclude that it is quency. In addition, the constant scale ridges are not at difficult to construct a spectrogram for each scalogram scales that can be easily matched to frequencies. shown in Figure 15. However the SOULTI wavelet The Mexican hat wavelet gives the best match to the scalogram can be directly transformed into spectro- parabolic curve, but when resolving the constant fre- gram by applying the scale-frequency change in equa- quencies in the signal it shows large shifts. It is difficult tion (6). to infer the accurate frequencies in the signal from To evaluate the SOULTI wavelet spectrogram, we these wavelets scalogram maps, though one can infer compare it to the Short Time Fourier Transform
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 18 Journal of Vibration and Control
(STFT) spectrograms. Six spectrograms based on the measured signals are finite in duration and we do not STFT were computed for different window sizes (W), have information about the signal after or before the where the size is measured by the number of samples. times of recording. Second, many wavelets do not have To make the comparison compatible with the continu- compact support rather they have an effective window. ous wavelet transform, the window overlap is set as Third, at the beginning of the analysis, ( ¼ 0), the (W 1) to perform window sweep over the time wavelet window is defined for negative and positive vector of the signal. Figure 16 plots the spectrograms range of time, t 5 0 and t 4 0, but the analyzed with the three straight lines that represent the instant- signal is defined only for t 4 0, so the inner product is aneous frequencies imposed on it. Note that the narrow computed between the signal and part of the wavelet. windows (W 64) are better in resolving the linear Similarly, at the end of the analysis, the wavelet effect- chirp than resolving the constant frequencies, while ive window will move out of the signal range and only the wider windows (W 128) are better in resolving part of it will take part in the inner product with the the constant frequencies. signal. This partial inner product gives inaccurate However, notice that when applying the STFT the results at both edges. wider the window the less time resolution is obtained, The SOULTI wavelet is a right sided wavelet or the more end effect occurs and the more time trunca- signal, i.e. the mother wavelet is zero for t < 0. tion from both sides of the signal is taken. For example, Therefore, when performing the wavelet transform, the Figure 16(f) only shows frequency information for the effective wavelet window sets fully inside the range of the time period 12.8 t 17.2 s of the signal. Frequency signal at the beginning of the analysis when ¼ 0. information for periods 0 t < 12.8 s and 17.2 < t 30 s However, at the end of the analysis, the effective is not available, while the SOULTI wavelet spec- window moves out of the signal range and the inner trogram provides information for the duration of the product is performed between the signal and part of signal as shown in Figure 14(b). Moreover, the the effective window. Therefore, though the SOULTI SOULTI spectrogram resolves both the linear chirp wavelet solves naturally the edge effect at the beginning and the constant harmonics, and its direct link between it does not solve the problem at the end, which makes frequency (spectrograms) and scale (scalograms) allows the analysis at the end inaccurate and distorted. checking the results for small or close frequencies. As a solution for the end edge effect, we propose in this section performing a Reverse Wavelet Transform 7. Edge effect mitigation (RWT) analysis starting from the end of the signal. So the mother wavelet is reflected about t ¼ 0, then it is Edge effect in harmonic and wavelet analysis of finite shifted to the end of the signal, and the wavelet analysis duration signals is caused by many factors. First, the is performed end-to-start. Then, we reflect the results
Figure 17. Reverse wavelet transform of the signal in Figure 10. (a) Scalogram, time-scale contour. (b) Spectrogram, time-frequency contour mapping.
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 Abuhamdia et al. 19
Figure 18. Reverse wavelet transform of the signal in Figure 12. (a) Scalogram, time-scale contour mapping. (b) Spectrogram, time- frequency contour mapping.
Figure 19. Average of FWT and RWT of the linear chirp in Figure 12. (a) Scalogram, time-scale contour mapping. (b) Spectrogram, time-frequency contour mapping. back with respect to time. The same result can be Figure 12. Note that at the end of the analysis there obtained by just reflecting the signal, performing the are clear ridges in both the scalograms and the spectro- wavelet analysis as usual and then reflecting the grams whereas the beginning shows distortions. This results back. result gives an indication that the distortion of Figure 17 shows the RWT of the signal in Figure 10 the ridges at the end of the studied signals is due to and Figure 18 shows the RWT of the signal in the edge effect.
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016 20 Journal of Vibration and Control
Instead of producing two scalograms or spectro- representation of basic signals. It also preserves all grams for each signal, the average of the Forward the important characteristics and parameters that Wavelet Transform (FWT) and the RWT can be com- exist in the time domain to the time-scale or time-fre- puted and mapped to give refined results at both ends quency domain. of the signal. Figure 19 shows the average of the FWT and RWT. The edges are better resolved and the dis- Declaration of Conflicting Interests tortion at the edges almost disappeared. However, a slight reduction in the ridges amplitude is notices. The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. 8. Remarks and conclusions It is shown that the impulse response of SOULTI sys- Funding tems can be used as a wavelet to obtain time-scale and The author(s) received no financial support for the research, time-frequency analysis directly. We also proved that authorship, and/or publication of this article. the transform can be reversed to obtain the original signal, hence an inverse wavelet transform for the References SOULTI wavelet exists. A region of convergence can Abuhamdia T and Taheri S (2015) Wavelets as a tool for be defined for the transform on the scale domain. This systems analysis and control. Journal of Vibration and region defines in which range of scales the SOULTI Control. Published online before print 16 December wavelet transform converges. 2015. DOI: 10.1177/1077546315620923. Moreover, it is shown that the original signal can be Freudinger LC, Lind R and Brenner MJ (1998) Correlation retrieved back by substituting the transform into the filtering of modal dynamics using the Laplace wavelet. In: conjugate differential equation. The SOULTI wavelet International modal analysis conference, volume 2, can be evaluated for most elementary functions and California (CA), Bethel, Connecticut (CT), USA, 2–5 basic signals. In addition, there is a direct relation between February, pp.868–877. Santa Barbara, CA: SEM. the SOULTI wavelet transform of a signal and the trans- Hou Z and Hera A (2001) A system identification technique form of its derivative or integral. For a wavelet scaling using pseudo-wavelets. Journal of Intelligent Material power p ¼ 1, we found that the time-frequency resolution Systems and Structures 12(10): 681–687. is preserved constant using the three definitions for com- Jezequel L and Argoul P (1986) New integral transform for linear systems identification. Journal of Sound and puting the time-frequency resolution, the standard devia- Vibration 111(2): 261–278. tion based, the -3dB bandwidth based, and the Q-factor Kaiser G (1994) A Friendly Guide to Wavelets. Boston, MA: bandwidth based. Birkhauser. The important result that the reconstruction differ- Meirovitch L (1997) Principles and Techniques of Vibrations. ential equation shows is extending the notion that the Vol. 1, Upper Saddle River, New Jersey: Prentice Hall. wavelet transform is the output of a filter bank from Naylor AW and Sell GR (2000) Linear Operator Theory in digital wavelets to crude noncompactly supported Engineering and Science. New York: Springer Science & wavelets. The reconstruction differential equation in Business Media. equation (19) shows that the SOULTI wavelet trans- Newland DE (1993) Harmonic wavelet analysis. Proceedings form at scale s is part of the output (particular solution) of the Royal Society of London. Series A: Mathematical of the second-order system modeled by the differential and Physical Sciences 443(1917): 203–225. equation itself. Robinson EA (1962) Random Wavelets and Cybernetic Systems. Vol. 9, New York: Hafner Publishing Company. The RWT can reveal whether the distortion at the end Tchamitchan P and Torresani B (1992) Ridge and skeleton of the time range on scalograms and spectrograms is due extraction from the wavelet transform. In: Ruskai MB, to the edge effect or not. Moreover, taking the average Gregory B and Ronald C (eds) Wavelets and Their between the FWT and RWT is a practical and simple Applications. Boston, MA: Jones and Bartlett Publishers, method to eliminate the edge effect on both edges. pp. 123–153. The SOULTI wavelet transform formula provides Yoshida K (1965) Functional Analysis. Vol. 123, 1st ed. an analytical tool for time-frequency or time-scale Heidelberg: Springer–Verlag.
Downloaded from jvc.sagepub.com at University Libraries | Virginia Tech on October 19, 2016