Frequency Response

**Definition. **The *frequency response* of an LTI filter is 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 is given by the
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

A basic property of the *z* transform is that, over the unit circle
,
we find the *spectrum* [83]. To show this, we set
in the definition of the *z* transform, 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

or

By Eq. (7.1), 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

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 a mathematical process called
*analytic continuation* (see §B.2). Since analytic
continuation is unique (for all filters encountered in practice), we
get the same results going either direction.

Because 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:

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