Dave Tweed wrote:
> Olin, I have to ask, in what sense do these filters have "the same
> random noise attenuation"? It's pretty clear that they have different
> bandwidths
> and different cutoff slopes. I don't see the significance of having
> the FFs multiply out to the same value in both cases.

Think of the weighting of any one sample.  That weighting is the product of
all FFs of the cascaded filters.  With a single pole that is simply the
single FF value.

In the single pole example with FF =3D 1/16, a new sample has a weighting o=
f
1/16, which then decreases from there.  In the 4 pole example, a new sample
has a weighting of 1/2 after the first pole, 1/4 after the second, 1/8 afte=
r
the first, and 1/16 after the last.

Another way of saying this is that the unit step response after one
iteration will be the same for any filter with the same total FF product,
and that value will be the combined FF.

Yet another way to look at this is to imagine alternating samples of -1 and
+1 fed into the filter.  The output amplitude will have the input amplitude
times FF.  The "gain" to this square wave input is FF per filter stage.  Th=
e
gain of multiple filters is the product of each of their individual gains,
which is the product of all FFs.

Of course the filters will differ in characteristics when something other
than this special input signal is fed in, but that's part of the point usin=
g
multiple lighter filters versus a single heavy one.

By the way, as you make the number of filter stages large and the individua=
l
FFs closer to 1 to compensate, you approach a transmission line.


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