The frequency response of an LTI filter may be defined as the spectrum of the output signal divided by the spectrum of the input signal. In this section, we show that the frequency response of any LTI filter is given by its transfer function evaluated on the unit circle, i.e., . We then show that this is the same result we got using sine-wave analysis in Chapter 1.
Beginning with Eq. (6.4), we have
where X(z) is the z transform of the filter input signal , is the z transform of the output signal , and is the filter transfer function.
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 z transform, Eq. (6.1), to obtain
which may be recognized as the definition of the bilateral discrete time Fourier transform (DTFT) when is normalized to 1 [59,84]. When is causal, this definition reduces to the usual (unilateral) DTFT definition:
This immediately implies the following:
We can express this mathematically by writing
By Eq. (7.2), the frequency response specifies the gain and phase shift applied by the filter at each frequency. Since , , and are constants, the frequency response is only a function of radian frequency . Since is real, the frequency response may be considered a complex-valued function of a real variable. The response at frequency Hz, for example, is , where is the sampling period in seconds. It might be more convenient to define new functions such as and write simply instead of having to write so often, but doing so would add a lot of new functions to an already notation-rich scenario. Furthermore, writing makes explicit the connection between the transfer function and the frequency response.
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
whenever is an integer. Since every discrete-time spectrum repeats in frequency with a ``period'' equal to the sampling rate, we may restrict to one traversal of the unit circle; a typical choice is [ ]. For convenience, is often allowed.
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: