Digital Signal Processing Reference
In-Depth Information
(3.3)
τ
R
F
=
γ
c
where for Gaussian designs as described above,
= 0.9168. In general, there is
very little dependence of rise time on filter length, apart from filters of length 33.
Moreover, there is no more than a 5% difference in rise times for filters whose
lengths are between 55 and 255. The theoretical underpinnings of these
observations, that is (3.2) and (3.3), can be traced to the type of window used, the
transition width of the main lobe, and the fact that for higher cut-off frequencies
the faster the filter response (see e.g. [1], pp 189-190 and [8], pp 549-559).
γ
3.1.4
Pass-Band Ripple
R
and Attenuation
A
The pass-band ripple
R
is shown in Figure 3.2, and is defined as the maximum
excursion of the filter response from unity in the pass-band. This is usually written
as
R
=
δ
+
δ
(3.4)
max,
p
min,
p
where
min,p
correspond to the maximum and minimum values of the
response in the passband. Thus
R
is the largest absolute peak to peak swing in the
passband. For the filters given,
R
is usually between 10
−5
and 10
−6
in absolute
terms, or 20log
10
R
in terms of decibels. For example, Table 3.3 uses the latter for
coefficient bit size effects.
The attenuation
A
in the rejection band is given by
δ
max,p
and
δ
A
=
2
log
10
δ
(dB) (3.5)
max,
s
where in this case,
max,s
is the largest peak in the stopband. Because the filter
design treats the passband and attenuation ripples as free parameters, they do not
show any significant correlation with transition width, cut-off frequency, or filter
length. At the design stage, once the attenuation
A
has been chosen, the passband
ripple
R
will automatically follow. This happens because the symmetry of the
window means that the nature of the ripple seen in the passband is identical to that
in the rejection band (i.e.,
δ
δ
max,s
=
R
). Thus, the filters given here will exhibit
similar ripple and attenuation magnitudes.
A
is typically in the range of -95 to
-105 dB.
3.1.5
Overshoot
Q
The overshoot
Q
of a low-pass filter is the maximum excursion of the output from
a unit input step function (see Figure 3.6). An awareness of the overshoot helps us
to appreciate the influence of the Gibbs oscillatory phenomenon on small signals.
Figure 3.7(a) shows the relationship between
Q
and filter length.
Q
shows its
lowest values at
F
= 0.5 and 0.9 for filter lengths greater than or equal to 99.
