DESIGN AND ANALYSIS OF ANALOG FILTERS A Signal ...

40 DESIGN AND ANALYSIS OF ANALOG FILTERS:
Summary of the Paley-Wiener Theorem
Some of the more important concepts of this section are as follows:
A given magnitude frequency response that is square-integrable
and satisfies (2.11) has, with an appropriate phase response,
a corresponding causal impulse response.
The inequality of (2.11), for square-integrable magnitude
frequency responses, is a necessary and sufficient condition in
order for the impulse response to be causal.
If the magnitude frequency response is not square-integrable,
then the inequality of (2.11) is neither necessary nor sufficient
for the impulse response to be causal.
A consequence of the Paley-Wiener Theorem is that no finiteenergy
signal can be both time-limited and frequency-limited.
Therefore all causal impulse response filters cannot be frequencylimited,
i.e., there can be no bands where the frequency response
is zero.
2.3 TIME-BANDWIDTH PRODUCTS
For the ideal filters presented in Section 2.1, the location of the center of the
impulse response is obviously at t = 0 for the zero phase response case, or at
if an ideal linear phase response of is assumed, since h(t) has
even symmetry about But if h(t) does not have even symmetry, the location of
the center of the impulse response could be defined in various ways, such as the time
at which the response is maximum. A more useful definition for the center is the time
where the “center of gravity” of occurs along the time axis:
Chapter 2 Analog Filter Design and Analysis Concepts