By Shields A. L.
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Extra resources for Cyclic vectors in spaces of analytic functions
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2 The Continuous Wavelet Transform (CWT) 31 phase factors, which are collected in the vector trans. In the definition of this vector oms denotes the sampling circular frequency. Fast CWT-reconstruction The reconstruction algorithm is implemented in the function file myicwt. A prototype call to this function reads [t,reco,matrix]=myicwt(matrix,T,start,step,stop); matrix is the result from a preceding call of mycwt, T again denotes the sampling distance. As with the mycwt-function from the respective input parameters the scale vector a=(start:step:stop)*T is constructed.
The top row symbolizes the input sequence {fk } (cf. 3), the bottom row the output sequence {fk1 }. Sequence f 1 is obtained as follows: The elements of the input sequence pointing toward a certain element of the output sequence with an arrow are multiplied with the corresponding numbers and added at the node, where the arrows meet. g. f01 = f0 12 + f1 12 . The diagram illustrates the 50 3 The Discrete Wavelet Transform 0 2 4 6 8 2 r r r r r ✲t ❄ a Fig. 3. One-step-version of the DWT: Subset of the t-a-plane.
Apply the IDFT to the product sequence √ 1 {Fk Ak }N k=0 to −1 obtain {Lψ f (a, kTS )}N k=0 . Schematically: cψ |a| 1 −1 IDF T N −1 {Fk Ak }N k=0 −→ {Lψ f (a, kTS )}k=0 . 24) We conclude this section with some remarks: 1. The algorithm can be performed for any scale factor a = 0. Practically relevant are positive scale factors; for the sampling considered here, one usually takes a = TS , 2TS , . . , (N − 1)TS . 2. Again, usually the sampling distance TS is given a priori. Thus for small scale factors one must be aware that eq.