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
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 [
84].
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:
DTFT |
(8.1) |
Applying this relation to

gives
 |
(8.2) |
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,
![$ \omega T \in[-\pi,\pi]$](http://www.dsprelated.com/josimages_new/filters/img334.png)
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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