Digital Signal Processing Reference
In-Depth Information
1
ideal
0.9
0.8
F
c
= 0.75
0.7
0.6
0.5
0.4
0.3
0.2
0.1
F
c
= 0.4
0
0
0.2
0.4
0.6
0.8
1
Normalised Frequency,
F
Normalized Frequency,
F
Figure 6.13
Second-order differentiating filters derived from unity gain low-pass filters with cut-off
as shown.
their first-order counterparts.
Figure 6
.13 shows two filters derived from unity
gain low-pass filters with cut-off at 0.4 and 0.75. We note immediately that they
overpredict the ideal second-order frequency response in comparison to that found
for first-order differentiators. However, the acceptable bandwidth now extends to
about 0.4 for the filters derived from (6.26).
Whereas second-order differentiating filters designed using Fourier
coefficients tend to show poor attenuation coefficients, the second order
differentiators designed using this simple technique tend to imitate the attenuation
characteristics of the seed filter, and furthermore show good dc performance. As
such, this gives us a choice of second-order differentiators: (a) converted unity
gain filters with excellent dc and attenuation characteristics, but relatively large ac
error, and (b) window derived differentiators with excellent ac characteristics but
poor attenuation. Both differentiator types have been archived in this work. A
comparison of general filter properties is given in Table 6.1. A Matlab
function that performs the said conversion has been provided in Figure 6.14.
6.7 CHARTS AND TABLES
The layout of the differentiators presented in the rest of the chapter is perhaps best
depicted as shown in Figure 6.15. It is worthwhile to note that (1) only low-pass
second-order differentiators are given, and (2) second-order differentiators are
divided into Types a and b. Type a differentiators are those designed by
converting unity gain filters as in Section
6.6,
whereas Type b filters were
