The formulas presented in Equation 1
and Equation 2 from the module titled "Filter Sizing"
imply that αα and hence the required filter
order NN are independent of the cutoff frequency fcfc. The supporting
analysis showed that this is only true in the limit of high order filters,
i.e. when NN is large. The dependence for shorter filters is shown in
Figure 2 from the module titled "Filter Sizing".
Why should this occur? Consider the filter
design problem shown in Figure 1. Again the goal is a simple
lowpass filter with cutoff frequency fcfc. The frequency sampling points
at frequency multiples of fsNfsN are also shown as solid dots.
Instead of fixing the gains we presume that the filter gains h^nh^n,
or, equivalently, the graphic equalizer levers, are optimized, by whatever
means, to yield the best stopband ripple performance.
Figure 1(a) shows the combination of gains h^nh^n needed
to constrain the peak stopband ripple to a given level, say δ2¯δ2¯.
The frequency at which this equal ripple band starts is of course
fstfst and the difference between fstfst and fcfc is ΔfΔf.
Now suppose that fcfc is increased slightly, as shown in
Figure 1(b). Now a different set of the h^nh^n are
needed to make the peak ripple equal δ2¯δ2¯ and these result
in different values of fstfst and ΔfΔf. Pursuing this graphical
analysis we find that:
- Cutoff frequencies near multiples of fsNfsN result in
smaller transition bands, and hence smaller values of αα, than
those near the center of two bins. This occurs, to first order, since
two or more stopband basis filters are needed to
cancel the first sidelobe of the last basis passband filter when the passband
stops between two bins, while one is needed if the passband stops
near a bin.
- Because these “hard" and “easy" frequency ranges occur
for every bin, the number of the ranges, counting both positive and
negative frequencies, is about the same as the filter
orderNN.
- The variation in the transition band ΔfΔf is more pronounced
as NN decreases since there are fewer basis filters to use in
optimizing the response.
As an aside one might observe from Figure 1 from the module titled Performance Comparison with other FIR Design Methods that all
three methods perform about equally for high levels of stopband ripple.
Intuitively the reason for this should now be clear. Window-based methods
need not use much shaping if high levels of ripple are tolerable.
Similarly, frequency sampling need not use many adjustable coefficients.
Since this is true the equal-ripple techniques will not perform much better
since their only advantage is that of adjusting all of the filter gains.
The underlying point is that, for high-ripple designs, all of the methods
produce designs closely resembling the sum of simple, shifted
sinNqsinqsinNqsinq functions and produce a transition band
ΔfΔf of about the order of fsNfsN, hence an αα
of about unity. Only as the stopband ripple specification grows tighter
does the method and accuracy of adjusting the coefficients and the number
of them available for adjustment begin to affect the transition band
performance.
