Time Scaling as a Vibration Removal 5
5.1 Time Frequency Analysis
5.1.3 Continuous Wavelet Transform
Another important family of time–frequency transforms which overcomes many of the WFT pitfalls is thewavelet transform. The time–frequency atoms of the wavelet transform are called waveletsand defined by
whereψis called themother wavelet,ais the dilation (scale) parameter andbis the translation parameter, cf. [53].
In contrast to the WFT atoms, wavelets have a varying time–frequency resolution which de-pends on the wavelet dilation parametera. In the time–frequency plane, the Heisenberg box of a wavelet atomψb,ais a rectangle centered at(b, µf/a), with time and frequency spreads, re-spectively, proportional toaand1/a. As the wavelet dilation parameteravaries, the time and
5.1. Time Frequency Analysis 67 frequency widths of the wavelet Heisenberg box changes, but its area remains constant achiev-ing the best compromise between the time and frequency resolutions. Figure 5.6 illustrates the energy spread of two wavelets centered at(b, µf/a)and(b0, µf/a0).
Fig. 5.6:Heisenberg boxes representing the energy spread of two wavelets
The wavelet transform exists in both continuous and discrete forms. In this thesis, only the continuous wavelet transform is of interest. The CWT is the correlation of a time signal with a dictionary of dilated and translated versions of the analyzing mother waveletψ. It decomposes a signal into a time–scale representation that elucidates the transient characteristics of that signal. For a finite energy signalx(t)∈L2(R), the continuous wavelet transform is defined by
Wψx(b,a) =hx, ψb,ai=
where(·)∗denotes the complex conjugate operator andWψx(b,a)is the wavelet transform of x(t). Since the continuous wavelet transform is complete and maintains an energy conserva-tion, an inverse continuous wavelet transform exists and is given by
x(t) = 1
as long asψsatisfies theadmissibility condition
Cψ =
As an example on CWT, Figure 5.7 shows the CWT scalogram of the three frequencies sine wave signal of Figure 5.2.
0 1 2 3 4
(a) Time signal (b) Continuous wavelet transform scalogram
Fig. 5.7:Continuous wavelet transform example
As has been discussed in the Heisenberg boxes of the wavelets, the scalogram in Figure 5.7.b il-lustrates the effectiveness of the CWT in optimally adjusting the time and frequency resolution according to the tested frequencies.
Another feature that makes the wavelet transform attractive for practical use is the possibility to select the wavelet functionψsuch that it fits the application at hand. Indeed, in the literature a number of well–developed wavelet functions can be found which cover a wide range of applications. The most commonly used wavelets are:
1. The Haar wavelet:is the simplest existing wavelet and the first function that relates to the wavelet transform. The Haar wavelet is given by
ψ(t) =
cf. [53]. The Haar wavelet provides an excellent time localization, but a poor frequency local-ization [52].
2. The Mexican hat wavelet: it offers a good time and frequency localization since by con-struction it is obtained from the second time derivative of the time–frequency optimal Gaus-sian atom as
5.1. Time Frequency Analysis 69
whereFbis the wavelet bandwidth parameter.
3. The Morlet wavelet: is the complex version of the Mexican hat wavelet. It is built based on a complex modulation of the second time derivative of the time–frequency optimal Gaus-sian atom as
ψ(t) = 1
√πFb ei2πFct e
−t2
Fb , (5.16)
where Fc is the wavelet center frequency andFb is the wavelet bandwidth parameter. The Morlet wavelet as defined above dose not satisfy the admissibility condition. To overcome this, it is customary to add a correction term to the original definition as, cf. [53]
ψ(t) = 1
√πFb e−π2FbFc2 e
−t2 Fb h
ei2πFcteπ2FbFc2 −1i
. (5.17)
The Morlet wavelet is by construction an analytic wavelet and thus is best suited for applica-tions where the phase and amplitude information of a signal are to be separated.
Besides the above three classical wavelets, the wavelet literature offers a variety of wavelets customized for application since by definition any function can be a wavelet as long as it sat-isfies the very mild admissibility condition of Equation (5.12).
To identify the critical oscillation regions, several wavelet candidates were tested, e.g. Mexican hat wavelet, Morlet wavelet and impulse response wavelet. All tested wavelets suffer from two main pitfalls: first, an additional optimization process is always required to optimize the wavelet shape parameters in order to achieve satisfactory results. Second, most of the available wavelets arerelativelysymmetric and two-sided wavelets. Thus, whenever such wavelets are used to capture some anti-symmetric single-sided system characteristics such as the system oscillations, additional spurious side effects will show up. Therefore, a new wavelet will be designed to overcome the existing pitfalls and to be more tailored for identifying the oscillation regions.
Since the oscillatory behavior of an elastic system is characterized by its impulse response, the new wavelet functionψ will be constructed based on the system impulse response. This specific wavelet will be called hereafter as thebalanced impulse response wavelet. The impulse re-sponse wavelet itself is not new, different forms of such a wavelet are available in the literature.
The starting point for building such a wavelet is the impulse response of an underdamped sec-ond order system
h(t) = ωn
p1−ζ2 e−ζωntsin (ωdt), (5.18) where ωd = ωnp
1−ζ2 is the damped natural frequency of the system. Since the system impulse response usually does not satisfy the admissibility condition, modifications have to be applied. Junsheng [54], for example, modified the impulse response via direct mirroring to
achieve the admissible impulse response wavelet
whereωcis the wavelet center frequency andβis a damping or control parameter. The Fourier transform of the impulse response wavelet is given by
Ψ (f) = βωc The resulting impulse response wavelet in time and frequency domains is shown in Figure 5.8.
−5 −4 −3 −2 −1 0 1 2 3 4 5
Fig. 5.8:Impulse response wavelet in time domain (left) and frequency domain (right) withωc = 1 Hz, β = 0.2
By construction, the impulse response wavelet is an odd function and thus has zero mean which is the essential part of the admissibility condition. On the other hand, the reflection clearly results in a two-sided wavelet that will not reproduce the original system response which, for example, had no symmetry in the beginning.
Our wavelet construction will complete a function with a damped oscillation behavior inR+
by adding a function with controllable support inR−such that the resulting function satisfies not only the admissibility condition but also provides a certain amount of smoothness, so that ψ andΨboth decay sufficiently fast. Consequently, with a generalized form of the impulse responseh(t)of an underdamped second order system where the damping ratioζis replaced with a general control parameterβas
g(t) =e−
√βωct 1−β2
sin (ωct), (5.21)
5.1. Time Frequency Analysis 71 the new wavelet function is given by
ψ(t) =
whereT < 0is a freely chosen parameter that defines the support extension to the negative axis and controls the time localization properties of the resulting wavelet. Moreover, q is a function from a finite dimensional space that has to satisfy the balancing condition
0
One way to buildqis to use polynomial completions. The2n+ 2conditions in Equations (5.24) and (5.25) form the well-known Hermite conditions which always have a unique solution in Π2n+1. Thus, the complete problem defined by Equations (5.23) to (5.25) can be solved in Π2n+1if the solutionpof Equations (5.24) and (5.25) happens to satisfy the balancing condition of Equation (5.23). Otherwise, the solution belongs toΠ2n+2and is given by
q=p− r
For example, for the case n = 1, q can be only constructed using Equation (5.26) and the coefficientsa0, . . . , a4of the polynomial completion are the solutions of the system
The Fourier transform of the resulting wavelet is
For a wavelet function constructed according to the above method and with an arbitrary poly-nomial order of2n+ 2, the Fourier transform can be explicitly computed as
Ψ (f) = χ[T,0]
where(·)∧denotes the Fourier transform operator. The first part of the above expression is sin-gular atf = 0and therefore hard to sample in the neighborhood of the origin. The singularity is only removable due to the choice of the coefficientsakwhich guarantees thatΨis uniformly continuous. This dependency of the coefficients which requires that the numerator isprecisely zero in order to apply the l’Hôpital rule cannot be maintained in floating point computations, hence this formula is numerically very unstable in the neighborhood of the origin. Fortunately, there is a series expansion of the truncated polynomial which can be used close to the origin.
Lemma. For the truncated polynomial function
q=χ[T ,0]
2n+2
X
m=0
am(·)m,
A convergent series representation is defined as
Q(f) =−
5.1. Time Frequency Analysis 73 Substituting this into the Taylor series
Q(f) =
which exists since q is compactly supported, henceQ ∈ C∞(R), gives Equation (5.29). The
sum 2n+2
X
m=0
|am| |T|m+1 m + k + 1,
is bounded independently ofkand the remainder of the series is the series expansion ofeif T, hence the series converges absolutely.
For small values of|T f|, the series in Equation (5.29) converges very fast and so Equation (5.29) is suitable and a very stable way for samplingΨclose to the origin, while for large values of
|T f|, Equation (5.28) is the more appropriate expression to evaluate. This observation suggests the use of small values of|T|which is in accordance with our application of completing a one–
sided wavelet without adding too much support on the negative side.
The resulting balanced impulse response wavelet for the casen= 1in time and frequency do-main is shown in Figure 5.9. By construction, the wavelet is real single-sided one that satisfies the admissibility condition.
Fig. 5.9:Balanced impulse response wavelet in time domain (left) and frequency domain (right) with ωc= 1 Hz,β= 0.2,T =−0.5 s
To compare this wavelet to the behavior of the conventional impulse response wavelet given
by Equation (5.19), a test signal is used which contains two impulse responses of a second order underdamped system with damped natural frequencyFd = 25 Hz and damping ratio ζ = 0.1, defined as
whereH(·) is the Heaviside function. For this signal, two wavelet transforms were consid-ered with an analyzing frequency of 25 Hzand identical center frequency and damping pa-rameter, ωc = 25 Hz andβ = 0.1. A normalized version of the test signal and the modulus of the wavelet coefficients is shown in Figure 5.10. As the results demonstrate, the wavelet constructed according to the above method outperforms the conventional one in catching the impulse amplitude envelopes and their time locations. Thus, it provides a much better al-ternative for applications where accurate detection of impulses’ amplitude and time location are needed such as the identification of the oscillation regions. Our paper in [55] reviews the construction of the balanced impulse response wavelet and its application for extracting the vibrational error in control systems.
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Fig. 5.10:Time domain comparison between conventional impulse response wavelet (red) and the balanced impulse response wavelet (green) withωc= 25 Hz,β= 0.1,T =−0.5 s
To compare the behavior of both wavelets in the frequency domain, Figure 5.11 shows a normalized version of the frequency response of a second order underdamped system, the conventional impulse response wavelet and the balanced impulse response wavelet. Both wavelets satisfy the admissibility condition Ψ (0) = 0, however, the the effects of mirroring in the conventional impulse response wavelet can be clearly seen by the smaller bandwidth of the wavelet around its center frequency in comparison to the system response. On the other hand, the balanced impulse response wavelet fits the system response perfectly.