Abstract
We saw in the previous chapter that the wavelet kernels G and H had some repeating structure in the rows and columns, similar to the circular Toeplitz structure in filters. Clearly the matrices are not filters, however. Nevertheless, we will now prove that wavelet kernels can be implemented easily in terms of filters. but that several filters are needed in the computation. Each of these filters will have an interpretation in terms of how the wavelet transform treats different frequencies. Much has been done in establishing efficient implementations of filters, and by expressing a wavelet transform in terms of filters we can take advantage of this.
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Notes
- 1.
The name windowing is a bit strange. Here this does not correspond to applying a window to the sound samples as we explained in Exercise 3.9. We will see that it rather corresponds to applying a filter coefficient to a sound sample.
- 2.
It seems strange to use the name matrixing, for something which obviously is matrix multiplication. The reason must be that the procedure has been established outside a linear algebra framework.
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Ryan, Ø. (2019). The Filter Representation of Wavelets. In: Linear Algebra, Signal Processing, and Wavelets - A Unified Approach. Springer Undergraduate Texts in Mathematics and Technology. Springer, Cham. https://doi.org/10.1007/978-3-030-01812-2_5
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