12.10 Initialization of the wavelet transform

The filterbanks developed in the module on the filterbanks interpretation of the DWT start with the signal representation $\{c_0(n)\colon n\in \mathbb{Z}\}$ and break the representation down into wavelet coefficients and scaling coefficients at lower resolutions( i.e. , higher levels $k$ ). The question remains: how do we get the initial coefficients $\{c_0(n)\}$ ?

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

Practically speaking, however, it is very difficult to build an analog filter with impulse response $\phi (-t)$ for typical choices of scaling function.

The most often-used approximation is to set $c_0(n)=x(n)$ . The sampling theorem implies that this would be exact if $\phi (t)=\frac{\sin (\pi t)}{\pi t}$ , though clearly this is not correct for general $\phi (t)$ . Still, this technique is somewhat justified if we adopt the view that the principle advantage of the wavelettransform comes from the multi-resolution capabilities implied by an iterated perfect-reconstruction filterbank (with good filters).