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Initialization of the Wavelet Transform

Module by: Phil Schniter. E-mail the author

The filterbanks developed in the module on the filterbanks interpretation of the DWT start with the signal representation c 0 ⁢n n∈Z c 0 n n and break the representation down into wavelet coefficients and scaling coefficients at lower resolutions (i.e., higher levels kk). The question remains: how do we get the initial coefficients c 0 ⁢n c 0 n ?

From their definition, we see that the scaling coefficients can be written using a convolution:

c 0 ⁢n=〈φ⁢t−n,x⁢t〉=∫−∞∞φ⁢t−n⁢x⁢tdt=φ⁢−t*x⁢t| t= n ′ c 0 n φ t n x t t φ t n x t t n ′ φ t x t
(1)
which suggests that the proper initialization of wavelet transform is accomplished by passing the continuous-time input x⁢t x t through an analog filter with impulse response φ⁢−t φ t and sampling its output at integer times (Figure 1).

Figure 1
Figure 1 (waveletsampling.png)

Practically speaking, however, it is very difficult to build an analog filter with impulse response φ⁢−t φ t for typical choices of scaling function.

The most often-used approximation is to set c 0 ⁢n=x⁢n c 0 n x n . The sampling theorem implies that this would be exact if φ⁢t=sinπ⁢tπ⁢t φ t t t , though clearly this is not correct for general φ⁢t φ t . Still, this technique is somewhat justified if we adopt the view that the principle advantage of the wavelet transform comes from the multi-resolution capabilities implied by an iterated perfect-reconstruction filterbank (with good filters).

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