Spectral transfer function

Filters are often used to change the spectra of given data. With input X(Z), filters B(Z), and output Y(Z), we have Y(Z) = B(Z)X(Z) and the Fourier conjugate $\overline{Y}(1/Z) = \overline{B}(1/Z) \overline{X}(1/Z)$. Multiplying these two relations together, we get

$$\overline {Y} Y \eq (\overline{B} B)(\overline{X} X) \nonumber$$

which says that the spectrum of the input times the spectrum of the filter equals the spectrum of the output. Filters are often characterized by the shape of their spectra; this shape is the same as the spectral ratio of the output over the input:

$$\overline{B} B \eq {\overline {Y} Y \over \overline{X} X }$$ (49)

EXERCISES:

1. Suppose a wavelet is made up of complex numbers. Is the autocorrelation relation s_k = s_-k true? Is s_k real or complex? Is $S(\omega)$ real or complex?