Wavelet Transform (Mertins - Signal Analysis (Revised Edition))

Signal Analysis: Wavelets,Filter Banks, Time-Frequency TransformsandApplications. Alfred MertinsCopyright 0 1999 John Wiley & Sons LtdPrint ISBN 0-471-98626-7 Electronic ISBN 0-470-84183-4Chapter 8Wavelet TransformThe wavelettransform was introduced at the beginning of the 1980s byMorlet et al., who used it to evaluate seismic data [l05 ],[106]. Since then,various types of wavelet transforms have been developed, and many otherapplications ha vebeen found.

The continuous-time wavelet transform, alsocalled the integral wavelet transform (IWT), finds most of its applications indata analysis, where it yields an affine invariant time-frequency representation.The most famous version, however, is the discrete wavelet transform(DWT).This transform has excellent signal compaction properties for many classesof real-world signals while being computationally very efficient. Therefore, ithas been applied to almost all technical fields including image compression,denoising, numerical integration, and pattern recognition.8.1The Continuous-TimeWaveltTransformThe wavelet transform W,@,a) of a continuous-time signal x ( t ) is defined asThus, the wavelet transform is computed as the inner product of x ( t ) andtranslated and scaled versions of a single function $ ( t ) ,the so-called wavelet.If we consider t)(t) to be a bandpass impulse response, then the waveletanalysis can be understood as a bandpass analysis.

By varying the scaling210Continuous-TimeTransform8.1.WaveletThe211parameter a the center frequency and the bandwidth of the bandpass areinfluenced. The variation of b simply means a translation in time, so that fora fixed a the transform (8.1) can be seen as a convolution of z ( t ) with thetime-reversed and scaled wavelet:The prefactor lal-1/2 is introduced in order to ensure that all scaled functionsl ~ l - ~ / ~ $ * ( t with/ a ) a E IR have the same energy.Since the analysis function $(t)is scaled and not modulated like the kernelof the STFT,a wavelet analysis is often called a time-scale analysisrather thana time-frequency analysis. However, both are naturally related to each otherby the bandpass interpretation.

Figure 8.1 shows examples of the kernels ofthe STFT and the wavelet transform. As we can see, a variation of the timedelay b and/or of the scaling parameter a has no effect on the form of thetransform kernel of the wavelet transform. However, the time and frequencyresolution of the wavelet transform depends ona. For high analysis frequencies(small a) we have good time localization but poor frequency resolution. Onthe other hand,for low analysis frequencies, we have good frequencybut poortime resolution.

While the STFTis a constant bandwidth analysis, the waveletanalysis can be understood as a constant-& or octave analysis.When using a transform in order to get better insight into the propertiesof a signal, it should be ensuredthat the signal can be perfectly reconstructedfrom its representation. Otherwise the representation may be completely orpartly meaningless. For the wavelet transform the condition that must be metin order to ensure perfect reconstruction isC, =dw< 00,where Q ( w ) denotes the Fourier transform of the wavelet.

This condition isknown as the admissibility condition for the wavelet $(t). The proof of (8.2)will be given in Section 8.3.Obviously, in order to satisfy (8.2) the wavelet must satisfyMoreover, lQ(w)I must decrease rapidly for IwI+ 0 and for IwI + 00. That is,$(t)must be a bandpass impulseresponse. Since a bandpass impulse responselooks like a small wave, the transform is named wavelet transform.212Chapter 8. WaveletTransformFigure 8.1. Comparison of the analysis kernels of the short-time Fourier transform(top, the real part is shown) and the wavelet transform (bottom, real wavelet) forhigh and low analysis frequencies.Calculation of the WaveletTransformfromUsing the abbreviationthe Spectrum X ( w ) .the integralwavelet transform introducedby equation (8.1) can also be writtenasa) = ( X ’?h,,>(8.5)With the correspondences X ( w ) t)z ( t ) and Q ( w ) t)$(t),and the timeand frequency shift properties of the Fourier transform, we obtainBy making use of Parseval’s relation we finally getContinuous-TimeTransform8.1.WaveletThe213Equation (8.7) statesthatthewavelet transformcan also be calculatedby means of an inverse Fourier transformfromthewindowed spectrumX ( w )Q*(aw).Time-Frequency Resolution.

Inorder to describe the time-frequencyresolution of the wavelet transform we consider the time-frequency windowassociated with the wavelet. The center ( t o , W O ) and the radii A, and A, ofthe window are calculated according to (7.8) and (7.11). This givesand(8.10)(8.11)For the center andthe radii of the scaled function @($)lalQ(aw) wehave {ado, + W O } and { a .

A t , +A,}, respectively. This means that the wavelettransform W,@,a ) provides information on a signal ~ ( tand) its spectrumX ( w ) in the time-frequency window[ b + a . t o - a . A t ,b + a . t o + a . A t ]XWOA,[---,aaWO-+-l,aA,a(8.12)The area 4 A t A , is independent of the parameters a and b; it is determinedonly by the used wavelet $(t). The time window narrows when a becomessmall, and it widens when a becomes large. On the other hand, thefrequencywindow becomes wide when a becomes small, and it becomes narrow when abecomes large. As mentioned earlier, a short analysis window leads to goodtime resolution on the one hand, buton the otherto poor frequencyresolution.Accordingly, a long analysis window yields good frequencyresolution but poortime resolution. Figure 8.2 illustrates thedifferent resolutions of the short-timeFourier transform and the wavelet transform.Affine Invariance.

Equation (8.1) shows that if the signal is scaled ( z ( t )+W,(b,a) is scaled as well; except this,W,(b, U ) undergoes no other modification. For this reason we also speak of anz ( t / c ) ) , the wavelet representation214Chapter 8. WaveletTransformW00 2ZlzZ2Figure 8.2. Resolution of the short-time Fourier transform (left) and the wavelettransform (right).afine invariant transform. Furthermore, the wavelet transform is translationinvariant, i.e. a shift of the signal ( x ( t ) + x ( t - t o ) ) leads to a shift ofthe wavelet representation W z ( b , a ) by t o , but W z ( b ,U ) undergoes no othermodification.8.2Wavelets for Time-ScaleAnalysisIn time-scale signal analysis one aims at inferring certain signal propertiesfrom the wavelet transform in a convenient way.

Analytic wavelets are especially suitable for this purpose. Like an analytic signal, they contain onlypositive frequencies. In other words, for the Fourier transform of an analyticwavelet $ ~ b , ~ ( tthe) following holds:%,a(W)=0forw0.(8.13)Analytic wavelets have a certain property, which will be discussed brieflybelow. For this consider the real signal z ( t ) = cos(w0t). The spectrum isX ( w ) = 7r [S(w - WO)+ S(w +WO)]x ( t ) = cos(w0t).t)(8.14)Substituting X ( w ) according to (8.14) into (8.7) yieldsW&U) = 12.

Iu~;/~(S@- w0)+ S(W + w 0 ) ) Q*(aw) ejwb dw-cc=+(8.15)l a [ ; [ q * ( a w o ) ejuob+ ~ * ( - a w o )e-juobI.Hence, for an analytic wavelet:12w z ( b , a ) = - la[; ~ * ( a w oej'ob.)(8.16)8.2. Wavelets215 Analysisfor Time-ScaleSince only the argument of the complex exponential in (8.16) depends on b,the frequency of z ( t )can be inferred from the phase of W,(b, a ) . For this, anyhorizontal line in the time-frequency plane can beconsidered. The magnitudeof W,(b,a) is independent of b, so that the amplitude of z ( t ) can be seenindependent of time.

This means that the magnitude of W , (b, a ) directlyshows the time-frequency distribution of signal energy.The Scalogram. A scalogram is thesquaredmagnitudetransform:of the waveletScalograms, like spectrograms, can be representedas images in which intensityis expressed by different shades of gray. Figure 8.3 depicts scalograms for~ ( t=)d ( t ) . We see that here analytic wavelets should be chosen in order tovisualize the distribution of the signal energy in relation to time and frequency(and scaling, respectively).The Morlet Wavelet.

The complex wavelet most frequently used in signalanalysis is the Morlet wavelet, a modulated Gaussian function:(8.18)Note that the Morlet wavelet satisfies the admissibility condition (8.2) onlyapproximately. However, by choosing proper parameters WO and /3 in (8.18)one can make the wavelet at least “practically” admissible. In order to showthis, let us consider the Fourier transform of the wavelet, which, for W = 0,does not vanish exactly:(8.19)By choosingWO2 2.rrP(8.20)we get Q ( w ) 5 2.7 X 10-9 for W 5 0, which is sufficient for most applications[132].

Often W O 2 5/3 is taken to be sufficient [65], which leads to Q ( w ) 510-5,5 0.Example. The exampleconsidered belowis supposed to give a visualimpression of a wavelet analysis and illustrates thedifference from a short-timeFourier analysis. The chosen test signal is a discrete-time signal; it contains216TransformChapter 8. WaveletImaginary component(b)t -Figure 8.3. Scalogram of a delta impulse ( W s ( b , a ) = l$(b/a)I2);(a) real wavelet;(b) analytic wavelet.two periodic parts and two impulses.' An almost analytic, sampled Morletwavelet is used.

The signal is depicted in Figure 8.4(a). Figures 8.4(b) and8.4(c) show two corresponding spectrograms (short-time Fourier transforms)with Gaussian analysis windows. We see that for a very short analysis windowthe discrimination of the two periodic components is impossible whereas theimpulses are quite visible. A long window facilitates good discrimination ofthe periodic component, but the localization of the impulses is poor.

This isnot the case in the wavelet analysis represented in Figure 8.4(d). Both theperiodic components and the impulses are clearly visible. Another propertyof the wavelet analysis, which is well illustrated in Figure 8.4(d), is that itclearly indicates non-stationarities of the signal.'In Section 8.8 the question of how the wavelet transform of a discrete-time signal canbe calculated will be examined in more detail.2178.3. Integral and Semi-DiscreteReconstructionlogcL(4t-Figure 8.4. Examples of short-time Fourier and wavelet analyses; (a) test signal;(b) spectrogram (short window); (c) spectrogram (long window); (d) scalogram.8.3Integral and Semi-Discrete ReconstructionIn this section, two variants of continuous wavelet transforms will be considered; they onlydifferin the way reconstruction is handled.