Beginning with Eq. (2.2.1), we have
where is the -transform of the filter input, is the -transform of the output signal, and is the filter transfer function.
Since every complex number can be represented as a magnitude and angle, the frequency response may be decomposed into two real-valued functions, the amplitude response and the phase response. Formally, we may define them as follows:
Thus is the magnitude (or complex modulus) of , and is the phase (or complex angle) of .
Definition. The real valued function in Eq. (49) is called the filter amplitude response and it specifies the amplitude gain that the filter provides at each frequency.
If the filter input and output signals are and respectively, then