Discrete Wavelet Transformations: An Elementary Approach with Applications

CHAPTER 8

ORTHOGONALITY AND FOURIER SERIES

In Chapter 7 we constructed wavelet transforms by building orthogonal matrices from a lowpass/highpass filter pair h, g, where the Fourier series H() satisfied orthogonality conditions and derivative conditions at = π (see (7.81)) and the highpass (wavelet) filter g was constructed from h using the rule g_k = (−1)^kh_{L − k}, k = 0, … , L.

While the derivative conditions are in terms of the Fourier series H(), the orthogonality conditions were obtained simply by insisting that the rows of W_N (7.62) are orthonormal. This led to the quadratic part of the system (7.81).

It is possible to develop the quadratic part of system (7.81) using only conditions on the Fourier series H(). In Section 8.1 we learn about these conditions and see how to write the orthogonality conditions entirely in terms of H(). In Chapter 7 we learned how to build the highpass filter g from the lowpass filter h. As we see in Section 8.2, once we know the Fourier series H() for the ...