Frequency Response

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

A basic property of the *z* transform is that, over the unit circle
,
we find the *spectrum* [84].^{8.1}To 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

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

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:

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University