A Friendly Guide to Wavelets

In this chapter we review some elements of linear algebra, function spaces, Fourier series, and Fourier transforms that are essential to a proper understanding of wavelet analysis. In the process, we introduce some notation that will be used throughout the book. All readers are advised to read this chapter carefully before proceeding further.

Fourier series are ideal for analyzing periodic signals, since the harmonic modes used in the expansions are themselves periodic. By contrast, the Fourier integral transform is a far less natural tool because it uses periodic functions to expand nonperiodic signals. Two possible substitutes are the windowed Fourier transform (WFT) and the wavelet transform. In this chapter we motivate and define the WFT and show how it can be used to give information about signals simultaneously in the time domain and the frequency domain. We then derive the counterpart of the inverse Fourier transform, which allows us to reconstruct a signal from its WFT. Finally, we find a necessary and sufficient condition that an otherwise arbitrary function of time and frequency must satisfy in order to be the WFT of a time signal with respect to a given window and introduce a method of processing signals simultaneously in time and frequency.

The WFT localizes a signal simultaneously in time and frequency by "looking" at it through a window that is translated in time, then translated in frequency (i.e., modulated in time). These two operations give rise to the "notes" gω,t(u). The signal is then reconstructed as a superposition of such notes, with the WFT ƒ(tω,t) as the coefficient function. Consequently, any features of the signal involving time intervals much shorter than the width T of the window are underlocalized in time and must be obtained as a result of constructive and destructive interference between the notes, which means that "many notes" must be used and ƒ(ω, t) must be spread out in frequency. Similarly, any features of the signal involving time intervals much longer than T are overlocalized in time, and their construction must again use "many notes," with ƒ(ω,t) spread out in time. This can make the WFT an inefficient tool for analyzing regular time behavior that is either very rapid or very slow relative to T. The wavelet transform solves both of these problems by replacing modulation with scaling to achieve frequency localization.

The reconstruction formula for windowed Fourier transforms is highly redundant since it uses all the notes gω,t to recover the signal and these notes are linearly dependent. In this chapter we prove a reconstruction formula using only a discrete subset of notes. Although still redundant, this reconstruction is much more efficient and can be approximated numerically by ignoring notes with very large time or frequency parameters. The present reconstruction is a generalization of the well-known Shannon sampling theorem, which underlies digital recording technology. We discuss its advantages over the latter, including the possibility of cutting the frequency spectrum of a signal into a number of "subbands" and processing these subbands in parallel.

In Chapter 3, we expressed a signal / as a continuous superposition of a family of wavelets Ψs,t, with the CWT ƒ (s,t) as the coefficient function. In this chapter we discuss discrete constructions of this type. In each case the discrete wavelet family is a subframe of the continuous frame { Ψs,t}, and the discrete coefficient function is a sampled version of ƒ (s,t). The salient feature of discrete wavelet analysis is that the sampling rate is automatically adjusted to the scale. That means that a given signal is sampled by first dividing its frequency spectrum into "bands," quite analogous to musical scales in that corresponding frequencies on adjacent bands are related by a constant ratio δ > 1 (rather than a constant difference v > 0, as is the case for the discrete WFT). Then the signal in each band is sampled at a rate proportional to the frequency scale of that band, so that high-frequency bands get sampled at a higher rate than that of low-frequency bands. Under favorable conditions the signal can be reconstructed from such samples of its CWT as a discrete superposition of reciprocal wavelets.

In all of the frames studied so far, the analysis (computation of ƒ (Ψ, t) or ƒ ( s , t) or their discrete samples) must be made directly by computing the relevant integrals for all the necessary values of the time-frequency or timescale parameters. Around 1986, a radically new method for performing discrete wavelet analysis and synthesis was born, known as multiresolution analysis. This method is completely recursive and therefore ideal for computations.

In this chapter we construct finite filter sequences leading to a family of scaling functions øN and wavelets ΨN where N = 1,2 … N = 1 gives the Haar system, and N = 2,3… give multiresolution analyses with orthonormal bases of continuous, compactly supported wavelets of increasing support width and increasing regularity. Each of these systems, first obtained by Daubechies, is optimal in a certain sense. We then examine some ways in which the dilation equations with the above filter sequences determine the scaling functions, both in the frequency domain and in the time domain. In the last section we propose a new algorithm for computing the scaling function in the time domain, based on the statistical concept of cumulants. We show that this method also provides an alternative scheme for the construction of filter sequences.

In this chapter we propose an application of electromagnetic wavelets to radar signal analysis and electromagnetic scattering. The goal in radar, as well as in sonar and other remote sensing, is to obtain information about objects (e.g., the location and velocity of an airplane) by analyzing waves (electromagnetic or acoustic) reflected from these objects, much as visual information is obtained by analyzing reflected electromagnetic waves in the visible spectrum. The location of an object can be obtained by measuring the time delay ז between an outgoing signal and its echo. Furthermore, the motion of the object produces a Doppler effect in the echo amounting to a time scaling, where the scale factor s is in one-to-one correspondence with the object's velocity. The wideband ambiguity function is the correlation between the echo and an arbitrarily time-delayed and scaled version of the outgoing signal. It is a maximum when the time delay and scaling factor best match those of the echo. This allows a determination of s and ז, which then give the approximate location and velocity of the object. The wideband ambiguity function is, in fact, nothing but the continuous wavelet transform of the echo, with the outgoing signal as a mother wavelet! When the outgoing signal occupies a narrow frequency band around a high carrier frequency, the Doppler effect can be approximated by a uniform frequency shift. The wideband ambiguity function then reduces to the narrow band ambiguity function, which depends on the time delay and the frequency shift. This is essentially the windowed Fourier transform of the echo, with the outgoing signal as a basic window. The ideas of Chapters 2 and 3 therefore apply naturally to the analysis of scalar-valued radar and sonar signals in one (time) dimension. But radar and sonar signals are actually waves in space as well as time. Therefore they are subject to the physical laws of propagation and scattering, unlike the unconstrained time signals analyzed in Chapters 2 and 3. Since the analytic signals of such waves are their wavelet transforms, it is natural to interpret the analytic signals as generalized, multidimensional wideband ambiguity functions. This idea forms the basis of Section 10.2.

In this chapter we construct wavelet representations for acoustics along similar lines as was done for electromagnetics in Chapter 9. Acoustic waves are solutions of the wave equation in space-time. For this reason, the same conformal group C that transforms electromagnetic waves into one another also transforms acoustic waves into one another. Hence the construction of acoustic wavelets and the analysis of their scattering can be done along similar lines as was done for electromagnetic wavelets. Two important differences are: (a) Acoustic waves are scalar-valued rather than vector-valued. This makes them simpler to handle, since we do not need to deal with polarization and matrix-valued wavelets, (b) Unlike the electromagnetic wavelets, acoustic wavelets are necessarily associated with nonunitary representations of C. This means, for example, that Lorentz transformations change the norm energy of a solution, and relatively moving reference frames are not equivalent. There is a unique reference frame in which the energy of the wavelets is lowest, and that defines a unique rest frame. The nonunitarity of the representations can thus be given a physical interpretation: unlike electromagnetic waves, acoustic waves must be carried by a medium, and the medium determines a rest frame. We construct a one-parameter family of nonunitary wavelet representations, each with its own resolution of unity. Again, the tool implementing the construction is the analytic-signal transform.