- applying an impulse to the system and measuring its response (see impulse response)
- sweeping a constant-amplitude pure tone through the bandwidth of interest and measuring the output level and phase shift relative to the input
- applying a signal with a wide frequency spectrum (for example digitally-generatedmaximum length sequence noise, or analog filtered white noise equivalent, like pink noise), and calculating the impulse response by deconvolution of this input signal and the output signal of the system.
Beginning with Eq.(6.4), we have
A basic property of the z transform is that, over the unit circle , we find the spectrum .8.1To show this, we set in the definition of the ztransform, Eq.(6.1), to obtain
Applying this relation to gives
Thus, the spectrum of the filter output is just the input spectrum times the spectrum of the impulse response . We have therefore shown the following:
This immediately implies the following:
We can express this mathematically by writing
Notice that defining the frequency response as a function of places the frequency ``axis'' on the unit circle in the complex plane, since . As a result, adding multiples of the sampling frequency to corresponds to traversing whole cycles around the unit circle, since
We have seen that the spectrum is a particular slice through the transfer function. It is also possible to go the other way and generalize the spectrum (defined only over the unit circle) to the entire plane by means of analytic continuation (§D.2). Since analytic continuation is unique (for all filters encountered in practice), we get the same result going either direction.
Because every complex number can be represented as a magnitude and angle , viz., , 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:
Frequency Response Ranges