Before we leave the case of amplitude-complementary, two-channel, critically sampled, perfect reconstruction filter banks, let's see what happens when is the simplest possible lowpass filter having unity dc gain, i.e.,
This case is obtained above by setting , , and . The polyphase components of are clearly
Choosing , and choosing and for aliasing cancellation, the four filters become
Thus, both the analysis and reconstruction filter banks are scalings of the familiar Haar filters (``sum and difference'' filters ). The frequency responses are
which are plotted in Fig.11.16.
Polyphase Decomposition of Haar Example
Amplitude-Complementary 2-Channel Filter Bank