5.7 TRANSFER FUNCTIONS AND FREQUENCY RESPONSE

The z domain notation introduced in the previous section may be extended further. Rather than simply generating waveforms with known frequency, amplitude and phase in response to an impulse input, we wish to have a system which can “shape” the frequency content of any given signal. Ultimately, this will lead to digital filters, which have numerous applications in audio, communications, noise removal, and elsewhere. Digital filters are covered in Chapters 8 and 9.

Suppose a system has a transfer function:

(5.41)

The output is simply found by inference:

(5.42)

That is, a zero-frequency gain of 0.5 and a delay of four samples. Suppose the input is a sampled complex exponential of frequency ω radians,