DESIGN AND ANALYSIS OF ANALOG FILTERS A Signal ...

38 DESIGN AND ANALYSIS OF ANALOG FILTERS:
Comment 1
Note that highpass and bandstop filters are directly excluded from the Paley-
Wiener Theorem since those filters do not have magnitude frequency responses that
are square-integrable. In both cases, however, the order of the numerator and of the
denominator of a rational transfer function will be the same, and polynomial
division will result in where will be square-integrable, and the
Paley-Wiener Theorem may be applied to it. Or, the theorem may be applied to the
lowpass prototype, rather than directly to the highpass or bandstop magnitude
frequency response, in which case the theorem is applicable indirectly.
Comment 2
The Paley-Wiener Theorem does not guarantee that an that satisfies
(2.11) will have a causal inverse (Papoulis, 1962). Rather, given a that
satisfies (2.11), the theorem states that there exists an associated phase response such
that has a causal inverse.
Comment 3
If is not square-integrable, then (2.11) is neither necessary nor
sufficient (Papoulis, 1962).
Comment 4
The Paley-Wiener Theorem must be satisfied in order for a given magnitude
frequency response to be realizable, since a causal h(t) is required. However, as
outlined in Sections 2.6 and 2.7, there are more constraints to be imposed than
causality before realizability is assured. For example, h(t) must also be constrained
to be real, a constraint that was not imposed in the above statement of the Paley-
Wiener Theorem. If is constrained to be an even function of then, due
to fundamental properties of the Fourier transform, h(t) will be real. To be
realizable, however, other constraints also need to be imposed, as discussed below.
Comment 5
While there may be points along the axis where there
cannot be a band of frequencies where the magnitude frequency response is zero,
otherwise (2.11) will not be satisfied, i.e., there is no corresponding causal impulse
response. In more detail, if
then
5
Rational transfer functions are considered, in general, later in this chapter. Highpass and bandstop
transfer functions are not specifically considered until Chapter 9.
Chapter 2 Analog Filter Design and Analysis Concepts