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: which suggests that the proper initialization of wavelet transform is accomplished by passing the continuous-time input xt x t through an analog filter with impulse response φ-t φ t and sampling its output at integer times (Figure 1).

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=xn c 0 n x n . The sampling theorem implies that this would be exact if φt=sinπtπt φ t t t , though clearly this is not correct for general φt 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).