Impulse Response Approximation

Hi Chris, Thanks for pointing this filter design method out to us. It's certainly interesting. By the way, I found a downloadable copy (is "downloadable" a word?) of reference [1] at: http://www.ux.uis.no/norsig/norsig2002/Proceedings/papers/cr1018.pdf While quickly reviewing reference [1] I noticed: [I] One thing that initially puzzled me was that the authors wrote: "This paper presents a simple FIR lowpass filter design procedure which requires only one multiplication operation." The filter design procedure requires hundreds, or maybe thousands, of multiplies. What they should have written was: "This paper presents a simple FIR lowpass filter design procedure that yields a FIR filter requiring only one multiplication per output sample." [II] Also, I noticed that the variables 'N' and 'S' were defined in Section 2 of reference [1]. Then, sadly, their design example contradicted their own definitions of variables 'N' and 'S'. [II] The authors used the phrase "stepped triangular" to describe the designed filter's impulse response. I realized that the word triangular seems inappropriate. The designed filter's impulse response will only be triangular if the filter's transition region is fairly wide relative to the filter's sample rate. For filters with a narrow (sharp) transition region width the impulse response will not be triangular. OK, having said (griped about) all of this, I think the authors' filter design method is quite interesting and deserves our attention. I hope to experiment with their design idea as time allows. Thanks again, Chris, for alerting us to this FIR filter design technique. [-Rick-]

Thanks for the comments. The referenced paper isn't the best quality, which is surprising since the co-author, Mitra, is the same Mitra that has published a DSP book? I orignally ran across the stepped-triangula in [2]. Then I found the reference above which I used for this brief blog post. I don't know if this is a decent method for approximating LPF and I see your point on the "stepped-triangular" name. My conclusion thus far has been no, it is not really a good method for approximating a LPF even with the computation gains. The part that I thought was interesting was the cascading of the, what I call, the sparse running sum. H(z) = [1 - z^{sN}] / [1 - z^{-s}] Where this is similar to the RRS but the integrator stage has an extra delay. What this ends up doing is adding poles and canceling out some of the zeros. This can be observed in the blogs frequency response plots, where the Hrrs response each lobe is more attenuated and the Hrss has a "bump" in the higher frequencies. Because of this I haven't found a good use for Hrss. Now in [2] the author shows a better response using an additional term (1 + z^-R) / 2, this might be for a later post :). This is an example of the pole-zero plot with s=3 and N=5, and illustrates the canceling of the zeros in the higher frequencies. (The poles, red x, are a little difficult to see on this plot) Also, after reading your comments I realize I had made a mistake in the blog post, how I intended to represent the variables. In general N is the number of steps and s is the number of samples per step, when trying to define the "stepped-triangular". And sN = D. I have made the corrections to the equations in this short blog post. I thought this could possibly be exploited in conjunction with the use of CIC filters. But I don't think it is as useful as my original thoughts. [2] Jovanovic-Dolecek, Gordana, "Modified CIC Filter for Rational Sample Rate Conversion", ECTI Transactions on Computer and Informatioin Technology Vol.4, No.1 May 2010 Chris

From the first glance, it looks like a first order Taylor approximation, which is like you said, an approximation with constant function.