DESIGN AND ANALYSIS OF ANALOG FILTERS A Signal ...

A Signal Processing Perspective 73
and
From (2.64) it is clear that and are both real for and the
poles are complex conjugates for Also note that for Q = 1/2,
For Q > 1/2, and
By differentiating the argument of the second logarithm on the right side of
(2.62), it is clear that monotonically deceases as a function of
if If Q then there will be a peak in the magnitude response;
the frequency of that peak and the corresponding magnitude value is as follows:
and
The magnitude response of (2.62) for K = 1, and several values of Q
is shown in Figure 2.33. The phase response of (2.63) for K = 1, and
several values of Q is shown in Figure 2.34.
The unit impulse response of (2.56), with G(s) = 1, and with (2.60) and
(2.61) applied, is as follows for Q > 1/2: 12
where
and
The corresponding unit step response is as follows:
12 If Q < 1/ 2 , then the roots are real and the impulse response is the sum of two simple exponential
functions: If Q = 1/2, then there is a repeated root, and
Section 2.9 Second-Order Transfer Functions