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The filterbanks developed in the module on the filterbanks interpretation of the DWT start with the signal representation c 0 n n 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:

c 0 n φ t n x t t φ t n x t t n φ t x t
which suggests that the proper initialization of wavelet transform is accomplished by passing the continuous-time input x t through an analog filter with impulse response φ t and sampling its output at integer times ( ).

Practically speaking, however, it is very difficult to build an analog filter with impulse response φ 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 φ t t t , though clearly this is not correct for general φ 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).

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Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
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