The wavelet expansion of a signal f(t) as first formulated in Section 2.2 is repeated here by

(10.1)

where the form a basis or tight frame for the signal space of interest (e.g., L₂). At first glance, this infinite series expansion seems to have the same practical problems in calculation that an infinite Fourier series or the Shannon sampling formula has. In a practical situation, this wavelet expansion, where the coefficients are called the discrete wavelet transform (DWT), is often more easily calculated. Both the time summation over the index k and the scale summation over the index j can be made finite with little or no error.

The Shannon sampling expansion 8, 6 of a signal with infinite support in terms of sinc expansion functions

(10.2)

requires an infinite sum to evaluate f(t) at one point because the sinc basis functions have infinite support. This is not necessarily true for a wavelet expansion where it is possible for the wavelet basis functions to have finite support and, therefore, only require a finite summation over k in Equation 10.1 to evaluate f(t) at any point.

The lower limit on scale j in Equation 10.1 can be made finite by adding the scaling function to the basis set as was done in Equation 3.28. By using the scaling function, the expansion in Equation 10.1 becomes

(10.3)

where j=J₀ is the coarsest scale that is separately represented. The level of resolution or coarseness to start the expansion with is arbitrary, as was shown in  Chapter: A multiresolution formulation of Wavelet Systems in Equation 3.19, Equation 3.20, and Equation 3.21. The space spanned by the scaling function contains all the spaces spanned by the lower resolution wavelets from j=–∞ up to the arbitrary starting point j=J₀. This means V_J0=W_–∞⊕⋯⊕W_J0–1. In a practical case, this would be the scale where separating detail becomes important. For a signal with finite support (or one with very concentrated energy), the scaling function might be chosen so that the support of the scaling function and the size of the features of interest in the signal being analyzed were approximately the same.

This choice is similar to the choice of period for the basis sinusoids in a Fourier series expansion. If the period of the basis functions is chosen much larger than the signal, much of the transform is used to describe the zero extensions of the signal or the edge effects.

The choice of a finite upper limit for the scale j in Equation 10.1 is more complicated and usually involves some approximation. Indeed, for samples of f(t) to be an accurate description of the signal, the signal should be essentially bandlimited and the samples taken at least at the Nyquist rate (two times the highest frequency in the signal's Fourier transform).

The question of how one can calculate the Fourier series coefficients of a continuous signal from the discrete Fourier transform of samples of the signal is similar to asking how one calculates the discrete wavelet transform from samples of the signal. And the answer is similar. The samples must be “dense" enough. For the Fourier series, if a frequency can be found above which there is very little energy in the signal (above which the Fourier coefficients are very small), that determines the Nyquist frequency and the necessary sampling rate. For the wavelet expansion, a scale must be found above which there is negligible detail or energy. If this scale is j=J₁, the signal can be written

(10.4)

or, in terms of wavelets, Equation 10.3 becomes

(10.5)

This assumes that approximately f∈V_J1 or equivalently, ∥f–P_J1f∥≈0, where P_J1 denotes the orthogonal projection of f onto V_J1.

Given f(t)∈V_J1 so that the expansion in Equation 10.5 is exact, one computes the DWT coefficients in two steps.

For J₁ large enough, φ_J1,k(t) can be approximated by a Dirac impulse at its center of mass since . For large j this gives

(10.6)

where is the first moment of φ(t). Therefore the scaling function coefficients at the j=J₁ scale are

(10.7)

which are approximately

(10.8)

For all 2-regular wavelets (i.e., wavelets with two vanishing moments, regular wavelets other than the Haar wavelets—even in the M-band case where one replaces 2 by M in the above equations, m₀=0), one can show that the samples of the functions themselves form a third-order approximation to the scaling function coefficients of the signal 2. That is, if f(t) is a quadratic polynomial, then

(10.9)

Thus, in practice, the finest scale J₁ is determined by the sampling rate. By rescaling the function and amplifying it appropriately, one can assume the samples of f(t) are equal to the scaling function coefficients. These approximations are made better by setting some of the scaling function moments to zero as in the coiflets. These are discussed in Section: Approximation of Scaling Coefficients by Samples of the Signal .

Finally there is one other aspect to consider. If the signal has finite support and L samples are given, then we have L nonzero coefficients 〈f,φ_J1,k〉. However, the DWT will typically have more than L coefficients since the wavelet and scaling functions are obtained by convolution and downsampling. In other words, the DWT of a L-point signal will have more than L points. Considered as a finite discrete transform of one vector into another, this situation is undesirable. The reason this “expansion" in dimension occurs is that one is using a basis for L² to represent a signal that is of finite duration, say in L²[0,P].

When calculating the DWT of a long signal, J₀ is usually chosen to give the wavelet description of the slowly changing or longer duration features of the signal. When the signal has finite support or is periodic, J₀ is generally chosen so there is a single scaling coefficient for the entire signal or for one period of the signal. To reconcile the difference in length of the samples of a finite support signal and the number of DWT coefficients, zeros can be appended to the samples of f(t) or the signal can be made periodic as is done for the DFT.

If f(t) has finite support, create a periodic version of it by

(10.10)

where the period P is an integer. In this case, 〈f,φ_j,k〉 and 〈f,ψ_j,k〉 are periodic sequences in k with period (if j≥0 and 1 if j<0) and

(10.11)

(10.12)

where (k modulo ) and . An obvious choice for J₀ is 1. Notice that in this case given L=2^J₁ samples of the signal, 〈f,φ_J1,k〉, the wavelet transform has exactly 1+1+2+2²+⋯+2^J₁–1=2^J₁=L terms. Indeed, this gives a linear, invertible discrete transform which can be considered apart from any underlying continuous process similar the discrete Fourier transform existing apart from the Fourier transform or series.

There are at least three ways to calculate this cyclic DWT and they are based on the equations Equation 10.25, Equation 10.26, and Equation 10.27 later in this chapter. The first method simply convolves the scaling coefficients at one scale with the time-reversed coefficients h(–n) to give an L+N–1 length sequence. This is aliased or wrapped as indicated in Equation 10.27 and programmed in dwt5.m in Appendix 3. The second method creates a periodic by concatenating an appropriate number of c_j(k) sections together then convolving h(n) with it. That is illustrated in Equation 10.25 and in dwt.m in Appendix 3. The third approach constructs a periodic and convolves it with c_j(k) to implement Equation 10.26. The Matlab programs should be studied to understand how these ideas are actually implemented.

Because the DWT is not shift-invariant, different implementations of the DWT may appear to give different results because of shifts of the signal and/or basis functions. It is interesting to take a test signal and compare the DWT of it with different circular shifts of the signal.

Making f(t) periodic can introduce discontinuities at 0 and P. To avoid this, there are several alternative constructions of orthonormal bases for L²[0,P]1, 4, 5, 3. All of these constructions use (directly or indirectly) the concept of time-varying filter banks. The basic idea in all these constructions is to retain basis functions with support in [0,P], remove ones with support outside [0,P] and replace the basis functions that overlap across the endpoints with special entry/exit functions that ensure completeness. These boundary functions are chosen so that the constructed basis is orthonormal. This is discussed in Section: Time-Varying Filter Bank Trees . Another way to deal with edges or boundaries uses “lifting" as mentioned in Section: Lattices and Lifting.

Given that the wavelet analysis of a signal has been posed in terms of the finite expansion of Equation 10.5, the discrete wavelet transform (expansion coefficients) can be calculated using Mallat's algorithm implemented by a filter bank as described in Chapter: Filter Banks and the Discrete Wavelet Transform and expanded upon in   Chapter: Filter Banks and Transmultiplexers . Using the direct calculations described by the one-sided tree structure of filters and down-samplers in Figure: Three-Stage Two-Band Analysis Tree allows a simple determination of the computational complexity.

If we assume the length of the sequence of the signal is L and the length of the sequence of scaling filter coefficients h(n) is N, then the number of multiplications necessary to calculate each scaling function and wavelet expansion coefficient at the next scale, and , from the samples of the signal, , is LN. Because of the downsampling, only half are needed to calculate the coefficients at the next lower scale, and , and repeats until there is only one coefficient at a scale of j=J₀. The total number of multiplications is, therefore,

(10.13)

which is linear in L and in N. The nu