Audio Signal Processing and Coding

2.7 THE TRANSFER AND THE FREQUENCY RESPONSE FUNCTIONS

The z-transform of the impulse response of a filter is called the transfer function and is given by

Considering the difference equation, we can also obtain the transfer function in terms of filter parameters, i.e.,

The ratio of output over input in the z domain gives the transfer function in terms of the filter coefficients

For an FIR filter, the transfer function is given by

The frequency response function is a special case of the transfer function of the filter. That is for z = e^jΩ, then

By considering the difference equation associated with the LTI digital filter, the frequency response can be written as the ratio of two polynomials, i.e.,

Note that for an FIR filter the frequency response becomes

Example 2.3

Frequency responses ...