### Haar Example

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.,

 (12.29)

This case is obtained above by setting , , and . The polyphase components of are clearly

 (12.30)

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.

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Polyphase Decomposition of Haar Example
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Amplitude-Complementary 2-Channel Filter Bank