Digital Signal Processing Reference
In-Depth Information
0.4
0
(
a
)
(
b
)
0.36
-20
0.32
-40
0.28
-60
0.24
-80
0.2
0.16
-100
0.12
-120
0.08
-140
0.04
0
-160
0
0.2
0.4
0.6
0.8
1
0.001
0.01
0.1
1
Normalised Frequency,
F
Normalised Frequency,
F
Normalized Frequency,
F
Normalized Frequency,
F
Figure 6.4
(a) Magnitude of a differentiating filter with cut off at 0.35. (b) Same filter as (a) but
showing extent of attenuation in rejection band.
6.3.3
Frequency Response
of First-Order Differentiating Filters
ˆ
The magnitude of the FFT of (6.2),
D
(
F
)
, will yield a linear response against
normalized frequency with slope
, modeled by (6.1), but with high-pass rejection
beyond
F
c
. This forms the frequency response of the differentiator. It has become
customary in using differentiating filters to plot the FFT magnitude divided by
against normalized frequency
F
. Figure 6.4 (a) shows a 99-tap first-order
differentiating filter with cut-off at 0.35 (
DIFF99F0.35
); while this gives us some
quick information about the transition width, there is little information about the
attenuation beyond 0.35. The attenuation information is recovered when
20log(magnitude/
) is plotted against log
F
(Figure 6.4 (b)); however, the
transition width is somewhat lost. As shown in Figure 6.4 (b), one must add
9.94dB (i.e., 20log
) to the maximum attenuation to compensate for normalizing
by
. This way of presenting the information allows for a clearer appreciation of
filter performance at low frequencies. In spite of this, we opted to present first-
order differentiators as shown in (a), recognizing that additional information
would need to be gathered from the accompanying table.
It is worth noting that the maximum attenuation
A
, given in the tables
associated with the differentiator, was calculated relative to the ideal differentiator
(from (6.1)) using the relation
ˆ
(6.4)
A
=
max{
20
log[
D
(
F
)
/
πF
]}
