TABLE OF CONTENTS The wavelet transforms above form orthonormal bases, meaning that they are both orthogonal and normalized. We saw above that normalization means that the output of a CQF is a delayed version of the input, instead of a scaled version. We can show this as the inner product of the lowpass filter with itself (or with the highpass filter with itself), as in < lpf, lpf >. Given that the lowpass filter coefficients are [ a, b, c, d], we calculate the inner product as the multiplication of the first parameter with the transpose of the complex conjugate of the second one. Here, both parameters are the same and real-valued, and we get a 2 + b 2 + c 2 + d 2, which must be 1 for the transform to be normalized.
Orthogonality, in two dimensions, simply means that the components are at right angles. It goes without saying in the Cartesian coordinate system that a point ( x, y) can be plotted by moving x units to the right (or left if x is negative), and then moving y units up or down (up/down being 90 degrees away from right/left) to find the point's location. The bases for the Cartesian point ( x, y) are (1, 0) and (0, 1). If we find the inner product, < [1 0], [0 1] >, we see that it results in...
