Olin Lathrop > 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. >=20 > 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. >=20 > In the single pole example with FF =3D 1/16, a new sample has a weighting > of 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 after the first, and 1/16 after the last. >=20 > 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. >=20 > 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. The gain of multiple filters is the product of each of their > individual gains, which is the product of all FFs. >=20 > 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 using multiple lighter filters versus a single heavy one. OK, so you're saying that if two filters have the same response at Fs/2, they have the same "random noise" attenuation? At least I now know what you mean, but I don't think that's actually a useful criterion in real- world applications. Just considering the single-sample impulse response of the two filters, your 4-pole filter has a response much higher (2.5x peak) and broader than that of the 1-pole "heavy" filter -- even though the response in the first iteration is indeed identical. -- Dave Tweed --=20 http://www.piclist.com PIC/SX FAQ & list archive View/change your membership options at http://mailman.mit.edu/mailman/listinfo/piclist .