Initialization of the Wavelet Transform 2.2 2003/01/17 2005/09/30 20:20:58.604 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Charlet Reedstrom charlet@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu wavelet transform 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 wavelet transform comes from the multi-resolution capabilities implied by an iterated perfect-reconstruction filterbank (with good filters).