Hi, for my diploma thesis I am studying different resampling techniques. Just now I wrote a Farrow resampler in Matlab. The structure of the resampler is quite easy to implement, but I struggle with the polynomial coefficients. I tried to derive them from FIR filters, but although the filter approximations look good in time domain, the attenuation is quite low (20dB-40dB for an original filter designed with nearly 80dB attenuation). I currently use 10 polynoms of 4th order with a sufficiently (or what I considered sufficiently) shaped filter. I tried different filter shapes (equiripple (doesn't work so well), different windows, least- squares). Does anyone have recommendations for the practical filter coefficients? Can you guess what I did wrong?

# Farrow Filter implementation in Matlab

Started by ●November 16, 2007

Reply by ●November 19, 20072007-11-19

> Can you guess what I did wrong?For example, trying to use a filter designed using a least-squares method (such as Parks McClellan)? Check the impulse response of your prototype, whether the outermost taps fall on a "smooth" impulse response with the others. Some methods give filters where the outermost coefficients are "discontinuous", and that can't be approximated with a polynomial. That's a guess, as you asked, so it may well be something completely different. -mn

Reply by ●November 28, 20072007-11-28

Hi, Thanks for the answer. I think I figured it out now... I also tried out Kaiser and other windows to design a filter -- and then I decided to use the example coefficients from a paper which give me 50dB attenuation. And I use the filter in a place in my architecture, where this is enough... a filter guy from our department said, that 50dB ar probably quite good already, because the coefficients are only an approximation and small errors have big effects in high attenuation. -- Imke

Reply by ●November 28, 20072007-11-28

>>and small errors have big >effects in high attenuation.yes, small errors have big effects on small numbers :) You could compare the performance of your Farrows implementation with that of the filter prototype. Or even easier: if the prototype alone doesn't meet the specs you know where you have to look for. Otherwise, it's the polynomials.

Reply by ●November 28, 20072007-11-28

> Otherwise, it's the polynomials.I found a suitable place in my processing architecture for the farrow -- if I put it in front of a shaper FIR, that one can do the rest of the attenuation.

Reply by ●November 28, 20072007-11-28

Reply by ●November 28, 20072007-11-28

On Nov 16, 8:29 am, Imke Huismann <pikku.kor...@gmail.com> wrote:> Hi, > > for my diploma thesis I am studying different resampling techniques. > Just now I wrote a Farrow resampler in Matlab. The structure of the > resampler is quite easy to implement, but I struggle with the > polynomial coefficients. I tried to derive them from FIR filters, but > although the filter approximations look good in time domain, the > attenuation is quite low (20dB-40dB for an original filter designed > with nearly 80dB attenuation). > > I currently use 10 polynoms of 4th order with a sufficiently (or what > I considered sufficiently) shaped filter. I tried different filter > shapes (equiripple (doesn't work so well), different windows, least- > squares). > > Does anyone have recommendations for the practical filter > coefficients? Can you guess what I did wrong?You may want to take a look at Harris' book "Multirate Signal Processing for Communication Systems" pgs 190-191 which discuss the Farrow Filter. The following is essentially paraphrasing the book with the numbers changed for your application: Assuming out of band sidelobe levels are bounded by 5dB /bit. If you need 80 dB attenuation then this is 16 bits. Then the precision needed is 2/(2^16) ~= 3.05e-5. Thus the error in your polynomial approximations need to be less than this. He also warns about the Remez filters which have constant sidelobe levels. You get an 6dB /octave falloff in the sidelobe's for every continuous n'th derivative (this is an asymptotic value). Obviously the usual Parks McCellan(Remez) filters don't have this, so there is an impulse at the ends of the time domain filter, which subsequently precludes using low order polynomials near this area. Hope that helps. Cheers, David

Reply by ●November 29, 20072007-11-29

On Nov 28, 6:02 pm, Dave <dspg...@netscape.net> wrote:> On Nov 16, 8:29 am, Imke Huismann <pikku.kor...@gmail.com> wrote: > > > > > Hi, > > > for my diploma thesis I am studying different resampling techniques. > > Just now I wrote aFarrowresampler in Matlab. The structure of the > > resampler is quite easy to implement, but I struggle with the > > polynomial coefficients. I tried to derive them from FIR filters, but > > although thefilterapproximations look good in time domain, the > > attenuation is quite low (20dB-40dB for an originalfilterdesigned > > with nearly 80dB attenuation). > > > I currently use 10 polynoms of 4th order with a sufficiently (or what > > I considered sufficiently) shapedfilter. I tried differentfilter > > shapes (equiripple (doesn't work so well), different windows, least- > > squares). > > > Does anyone have recommendations for the practicalfilter > > coefficients? Can you guess what I did wrong? > > You may want to take a look at Harris' book "Multirate Signal > Processing for Communication Systems" pgs 190-191 which discuss theFarrowFilter. The following is essentially paraphrasing the book with > the numbers changed for your application: > > Assuming out of band sidelobe levels are bounded by 5dB /bit. If you > need 80 dB attenuation then this is 16 bits. Then the precision needed > is 2/(2^16) ~= 3.05e-5. Thus the error in your polynomial > approximations need to be less than this. > > He also warns about the Remez filters which have constant sidelobe > levels. You get an 6dB /octave falloff in the sidelobe's for every > continuous n'th derivative (this is an asymptotic value). Obviously > the usual Parks McCellan(Remez) filters don't have this, so there is > an impulse at the ends of the time domainfilter, which subsequently > precludes using low order polynomials near this area. > > Hope that helps. > Cheers, > DavidHi, I read this book (or at least the resampling chapter) and also some of Harris' papers. I tried several window types and also equiripple (remez) filters as prototype filters -- but I don't believe that it is possible to get much more than 50dB attenuation. In the paper posted by mnentwig (http://yehar.com/dsp/deip.pdf) there are several polynomial filtering methods reviewed and non of them got more than about 50 dB. Although the pictures in the paper "Performance and design considerations of the Farrow filter when used for arbitrary resampling of sampled time series" from Harris, F. look more promising (60 dB). I used the coefficients and got 50dB. I think it is not the right approach to try to get 80dB with a Farrow resampler -- if it is possible it is probably a very cost intensive implementation. Please correct me when I'm wrong. Cheers Imke

Reply by ●December 4, 20072007-12-04

On Nov 29, 5:28 am, Imke Huismann <pikku.kor...@gmail.com> wrote:> On Nov 28, 6:02 pm, Dave <dspg...@netscape.net> wrote: > > > > > On Nov 16, 8:29 am, Imke Huismann <pikku.kor...@gmail.com> wrote: > > > > Hi, > > > > for my diploma thesis I am studying different resampling techniques. > > > Just now I wrote aFarrowresampler in Matlab. The structure of the > > > resampler is quite easy to implement, but I struggle with the > > > polynomial coefficients. I tried to derive them from FIR filters, but > > > although thefilterapproximations look good in time domain, the > > > attenuation is quite low (20dB-40dB for an originalfilterdesigned > > > with nearly 80dB attenuation). > > > > I currently use 10 polynoms of 4th order with a sufficiently (or what > > > I considered sufficiently) shapedfilter. I tried differentfilter > > > shapes (equiripple (doesn't work so well), different windows, least- > > > squares). > > > > Does anyone have recommendations for the practicalfilter > > > coefficients? Can you guess what I did wrong? > > > You may want to take a look at Harris' book "Multirate Signal > > Processing for Communication Systems" pgs 190-191 which discuss theFarrowFilter. The following is essentially paraphrasing the book with > > the numbers changed for your application: > > > Assuming out of band sidelobe levels are bounded by 5dB /bit. If you > > need 80 dB attenuation then this is 16 bits. Then the precision needed > > is 2/(2^16) ~= 3.05e-5. Thus the error in your polynomial > > approximations need to be less than this. > > > He also warns about the Remez filters which have constant sidelobe > > levels. You get an 6dB /octave falloff in the sidelobe's for every > > continuous n'th derivative (this is an asymptotic value). Obviously > > the usual Parks McCellan(Remez) filters don't have this, so there is > > an impulse at the ends of the time domainfilter, which subsequently > > precludes using low order polynomials near this area. > > > Hope that helps. > > Cheers, > > David > > Hi, > > I read this book (or at least the resampling chapter) and also some of > Harris' papers. I tried several window types and also equiripple > (remez) filters as prototype filters -- but I don't believe that it is > possible to get much more than 50dB attenuation. > > In the paper posted by mnentwig (http://yehar.com/dsp/deip.pdf) there > are several polynomial filtering methods reviewed and non of them got > more than about 50 dB. Although the pictures in the paper > "Performance and design considerations of the Farrow filter when used > for arbitrary resampling of sampled time series" from Harris, F. look > more promising (60 dB). I used the coefficients and got 50dB. > > I think it is not the right approach to try to get 80dB with a Farrow > resampler -- if it is possible it is probably a very cost intensive > implementation. Please correct me when I'm wrong. > > Cheers > ImkeFrom Harris' description he uses a least squares polynomial fit, which will be dependent on the prototype filter coefficients used. You also have some control over the decomposition of the filter used in the Farrow. The polynomials are calculated off line and fixed - they only need to evaluated at different values. The computational cost is only upfront and remains fixed during runtime. This is fundamentally different than using fixed polynomials which try to fit the input of the filter. The fit and resulting error will be dependent on the actual input data - this is different than in the Farrow, where the fit is predetermined by the filter. I don't think the results in the paper posted by mnentwig are applicable to the results of the Farrow. I believe there is a paper by some scientists in FMV (Swedish Defence Research) which essentially says that it is best to upsample and filter first before applying polynomial approximations for data fits. Even the paper posted hints at that. Consider if the -60 dB stop band already sits on top of sidelobes 80 dB down - you still end up with sidelobes that are atleast 80 dB down - so the polynomial filter isn't what is driving the performance. That said, I'll qualify - this isn't my exact area of expertise so what I'm saying could be wrong. Also, your result could be correct - it may be difficult to achieve the 80 dB attenuation via a Farrow implementation, the goal of my post was simply to provide the original poster with some quantitative measure of what was needed in order to achieve the desired attenuation levels - whether that is actually achievable is another question. Cheers, David

Reply by ●December 4, 20072007-12-04

I just wrote some matlab code to do a polynomial interpolation of the filter coefficients. To make it work properly I had to handle the end of each subfilter properly - essentially this means including the next tap in the filter in the polynomial fit. In my case the original filter had 72 dB attenuation. In the evaluations the resulting interpolated filter does meet these requirements. See the line b2=[b2;b2(1,2:end),0]; . Note: I had to use a 5th order polynomial to do the fits. Here's my matlab code: % b = coefficients of a predesigned filter % In my test case b has 300 taps % Designed using 72 dB attenuation Kaiser window filter Ntaps=length(b); D=50; Npoly=Ntaps/D; b2= reshape(b,D,Npoly); %For a good polynomial fit we need to handle the end points in a smooth %fashion b2=[b2;b2(1,2:end),0]; x=(0:D)'; p_ord = 5; %Order of polynomials for fitting p=zeros(Npoly,p_ord+1); for k =1:Npoly k p(k,:) =polyfit(x,b2(:,k),p_ord); %This plots the error in the polynomial evaluation %plot(b2(:,k)-polyval(p(k,:),x)) %pause end %Lets evaluate the resulting filter at some other sample point. x2 = x(1:(length(x)-1))+0.5; h2=[]; for k =1:Npoly h2=[h2,polyval(p(k,:),x2)]; end plot(20*log10(abs(fft(h2(:),2048)))) Hope that helps. Cheers, David