Michael Strothjohann wrote: > > Toni, > > First: Zero-padding isnt realy increasing the resolution of your > spectra. I'm very,very sorry about that, but > think twice: If it would possible to increase the resolution > by simply adding some zero-samples, you would be able to > get super-super-high resolution by adding a very large number of > zeros. Now comes the bad news: this dosnt work. But it does. The DFT samples a function in the frequency domain the same way the original data samples a function in the time domain. Both contain values between samples. They are just redundant and not needed to fully reconstruct the function from the samples. Sinc interpolation gives those values exactly if the function is band limited by the Nyquist criterion. A zero padded DFT gives the same values as if the higher resolution point had been calculated by sinc interpolation. A way to think of the DFT that makes this clearer is to look at the original definition. Each "complex" point is just the result of taking the inner product of a sampled time sequence with a sampled sinusoid (real part) and its quadrature cosinusoid (imaginary part) at each frequency that is an integer multiple of that frequency which has a wavelength equal to the length of the time sequence and limited by the Nyquist criterion. Each of those inner products is fully accurate regardless of how many zeros have been appended to the time sequence. The DFT is just a sampling method. The FFT does the same operation faster by factoring out redundant computation. As the length of the time series increases so does the number of sinusoids that will fit in the length of the time series. That increased resolution is quite real. Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein ////////////////////////////////////////\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ To contribute your unused processor cycles to the fight against cancer: http://www.intel.com/cure \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\///////////////////////////////////////
zeroing the FFT
Started by ●October 23, 2001
Reply by ●October 24, 20012001-10-24
Reply by ●October 25, 20012001-10-25
Jeff, It should be a signal, which has been mixed, i.e C = A + B, where you would want to resolve A and B. The point is to resolve two mixed signals and *not* take FFT of two signals separately and demonstrate the shift on a plot. You can't resolve A and B components by padding any number of zeros to the FFT of C. > The frequency response of an FIR filter can be calculated many ways; its use as > an FFT example is irrelevant. The fundamental use for FFTs is on real world > input data that needs to be analyzed. Zero padding -- and every other key FFT > technique -- applies to that, not FIR filters. Why *not*? -Akshay ----- Original Message ----- From: Jeff Brower <jbrower@jbro...> To: Akshay Joshi <ajoshi@ajos...> Cc: <audiodsp@audi...>; Tony Zampini <tony@tony...>; Suciu Radu <radusuciu@radu...>; Michael Strothjohann <strothjohann@stro...> Sent: Thursday, October 25, 2001 7:14 AM Subject: Re: [audiodsp] zeroing the FFT > Akshay- > > On Wed, 24 Oct 2001, "Akshay Joshi" <ajoshi@ajos...> wrote: > >Jeff, > > Tony has explained this. Adding zeros does not increase the resolution. > >It merely reduces the spacing between two frequency bins. If you have two > >sinusoids > >A = sin(2*pi*i/16) and B = sin(2*pi*i*255/4096). You can't resolve them with > >64 point data. You won't resolve them by taking a 4096 pt FFT by padding > >zeros over 64 point data. > > Absolutely, completely, 100% wrong. > > Click here to see plots: > > http://www.signalogic.com/sine.htm > > Sinea is your A above, sineb is your B. Clearly they cannot be resolved with a > 64 pt. FFT, and just as clearly they can with 4096 pt. FFT. Input data is the > same in each case -- except the 64 pt. FFT has none of those pesky zeros, and > the 4096 pt. FFT has 4032 of them. > > I have no idea why you think making the FFT longer and adding zeros will not > increase resolution and help resolve frequency domain detail. Any discrete, > finite Fourier transform is an approximation of the continuous transform; if you > make it longer the approximation improves. > > > Padding zeros is not useless. For taking FFT of radix 2 and so on, it is > >most convenient for computation speed.. you know all about it, but there > >should be no confusion. On a 900 pt data, you don't get more resolution out > >of 4096 pt FFT than a 1024 point FFT. > > > > Padding zeros is effective method to get a frequency response of say an > >FIR filter. It anyway has zeros at the tail.. so, this is the best place, > >where you can actually say that padding zeros "improves" resolution. > > The frequency response of an FIR filter can be calculated many ways; its use as > an FFT example is irrelevant. The fundamental use for FFTs is on real world > input data that needs to be analyzed. Zero padding -- and every other key FFT > technique -- applies to that, not FIR filters. > > Jeff Brower > Signalogic >
Reply by ●October 25, 20012001-10-25
FFT is just a representation of the original signal in time domain When you represent a point in 3D-space, you just need 3 vectors.. You can represent it , with 32 others vectors if you want . but using 32 vectors do not increase your resolution... FFT is the same.. In this case, you need 900 point to describe perfectly your signal.. If you do zero padding, you can use a 1024 point-FFT which algorithm must be faster (radix-2), You dont have the same representation (the same vectors in geometry space), but you dont have more resolution... Anyone agree with me ?? > Michael- > > >Your plots don't address the issue here. Generate a signal with two sine > >waves close together in frequency. Take a 64 sample block, a 64 sample > >block zero-padded to 4096 and a 4096 sample block, and plot these signals > >to see the difference. > > They exactly address the issue. Given the two 64 pt. blocks of data as > described by Akshay, if the FFT was not lengthened and zeros added, the two > sinusoids could not be resolved. > > Stretching the two time samples to 4096 pts. and using 4096 pt. FFTs allows the > sinusoids to be resolved without further processing; I'm not sure what that adds > to the discussion except possibly to demonstrate in another way that 4k FFT > length is enough to resolve this particular example
Reply by ●October 25, 20012001-10-25
--- Jeff Brower <jbrower@jbro...> wrote:
> Michael-
>
> >Your plots don't address the issue here. Generate
> a signal with two sine
> >waves close together in frequency. Take a 64
> sample block, a 64 sample
> >block zero-padded to 4096 and a 4096 sample block,
> and plot these signals
> >to see the difference.
>
> They exactly address the issue. Given the two 64
> pt. blocks of data as
> described by Akshay, if the FFT was not lengthened
> and zeros added, the two
> sinusoids could not be resolved.
>
> Stretching the two time samples to 4096 pts. and
> using 4096 pt. FFTs allows the
> sinusoids to be resolved without further processing;
> I'm not sure what that adds
> to the discussion except possibly to demonstrate in
> another way that 4k FFT
> length is enough to resolve this particular example.
>
> Jeff Brower
> DSP sw/hw engineer
> Signalogic
>
How are you defining "resolve"?
If the waves are added together, they cannot be
"resolved" by zero padding.
Mark
markrages@mark...
Reply by ●October 25, 20012001-10-25
Akshay- On Wed, 24 Oct 2001, "Akshay Joshi" <ajoshi@ajos...> wrote: >Jeff, > Tony has explained this. Adding zeros does not increase the resolution. >It merely reduces the spacing between two frequency bins. If you have two >sinusoids >A = sin(2*pi*i/16) and B = sin(2*pi*i*255/4096). You can't resolve them with >64 point data. You won't resolve them by taking a 4096 pt FFT by padding >zeros over 64 point data. Absolutely, completely, 100% wrong. Click here to see plots: http://www.signalogic.com/sine.htm Sinea is your A above, sineb is your B. Clearly they cannot be resolved with a 64 pt. FFT, and just as clearly they can with 4096 pt. FFT. Input data is the same in each case -- except the 64 pt. FFT has none of those pesky zeros, and the 4096 pt. FFT has 4032 of them. I have no idea why you think making the FFT longer and adding zeros will not increase resolution and help resolve frequency domain detail. Any discrete, finite Fourier transform is an approximation of the continuous transform; if you make it longer the approximation improves. > Padding zeros is not useless. For taking FFT of radix 2 and so on, it is >most convenient for computation speed.. you know all about it, but there >should be no confusion. On a 900 pt data, you don't get more resolution out >of 4096 pt FFT than a 1024 point FFT. > > Padding zeros is effective method to get a frequency response of say an >FIR filter. It anyway has zeros at the tail.. so, this is the best place, >where you can actually say that padding zeros "improves" resolution. The frequency response of an FIR filter can be calculated many ways; its use as an FFT example is irrelevant. The fundamental use for FFTs is on real world input data that needs to be analyzed. Zero padding -- and every other key FFT technique -- applies to that, not FIR filters. Jeff Brower Signalogic >----- Original Message ----- >From: Jeff Brower <jbrower@jbro...> >To: Akshay Joshi <ajoshi@ajos...> >Cc: <audiodsp@audi...>; Tony Zampini <tony@tony...>; Suciu >Radu <radusuciu@radu...>; Michael Strothjohann ><strothjohann@stro...> >Sent: Wednesday, October 24, 2001 8:58 PM >Subject: Re: [audiodsp] zeroing the FFT > > >> Akshay- >> >> > I think there is some confusion. Basically Toni and Michael are >right. >> >You don't resolve any further. Given a window of data, you dont resolve >any >> >further by padding zeros. >> >> Adding zeros = increasing the FFT length, which increases resolution -- >you can >> see more detail. Why else would you do it? >> >> Jeff Brower >> DSP sw/hw engineer >> Signalogic >> >> >> >----- Original Message ----- >> >From: Jeff Brower <jbrower@jbro...> >> >To: Tony Zampini <tony@tony...> >> >Cc: <audiodsp@audi...>; Suciu Radu <radusuciu@radu...>; >Michael >> >Strothjohann <strothjohann@stro...> >> >Sent: Wednesday, October 24, 2001 7:43 PM >> >Subject: Re: [audiodsp] zeroing the FFT >> > >> > >> >> Tony- >> >> >> >> >On the first issue, I was trying to say that by zero padding, >> >> >you end up with a finer resolution of your bin spacing. >> >> >In other words, the resulting frequency spectrum is >> >> >sampled at more closely spaced points. To put it yet another way, >> >> >the frequency bins are more closely spaced in frequency. >> >> >Although my words seem to indicate, I did NOT MEAN to imply >> >> >that, for example, zero-padding would allow you to better >> >> >resolve two closely spaced sinusoids. >> >> >> >> Why not? If the time data was sampled often enough then what else can >you >> >do? >> >> The sinusoids either have the same frequency or they do not. A >continuous >> >> Fourier transform would allow you to distinguish, so a finite FFT >> >transform of >> >> sufficient length would also allow you to distinguish. >> >> >> >> (Please note that I'm avoiding a situation where you have to >distinguish a >> >phase >> >> difference between same or similar sinusoids -- a different problem and >> >one FFTs >> >> do not help with much.) >> >> >> >> Jeff Brower >> >> DSP sw/hw engineer >> >> Signalogic >> >> >> >> >> >> >On the second matter, I'm still not sure of the original >> >> >question. However, clearly, a DFT on exactly the 900 original >> >> >samples (with no zero padding), whether implemented with a >> >> >brute force DFT algorithm, or some form of FFT algorithm >> >> >(which, you are correct, is doable on a non-power of 2 length), >> >> >should yield exactly the same results. It is simply an issue >> >> >of the method of computation. >> >> > >> >> >Now, if you wish to compare the FFT done on only the 900 points >> >> >to the FFT done on the 1024 points (900+124 zeros), you will get >> >> >two different samplings of the same underlying spectra. This >> >> >is analagous to sampling a 1000Hz sinewave at a sampling rate of >> >> >8000Hz, and comparing these samples to the samples of a 1000Hz >> >> >sinewave sampled, say, at 9000Hz. The samples will not look the same, >> >> >although, they represent the same underlying 1000Hz sinewave. >> >> > >> >> >By the way, Michael, I am a mathematician. >> >> > >> >> >Regards, >> >> >Tony >> >> > >> >> >----- Original Message ----- >> >> >From: Michael Strothjohann <strothjohann@stro...> >> >> >To: Tony Zampini <tony@tony...> >> >> >Cc: <Audiodsp@Audi...>; Suciu Radu <radusuciu@radu...> >> >> >Sent: Wednesday, October 24, 2001 3:23 AM >> >> >Subject: Re: [audiodsp] zeroing the FFT >> >> > >> >> > >> >> >> Toni, >> >> >> >> >> >> First: Zero-padding isnt realy increasing the resolution of your >> >> >> spectra. I'm very,very sorry about that, but >> >> >> think twice: If it would possible to increase the resolution >> >> >> by simply adding some zero-samples, you would be able to >> >> >> get super-super-high resolution by adding a very large number of >> >> >> zeros. Now comes the bad news: this dosnt work. >> >> >> >> >> >> Second: FFT with (say) 900 samples is possible. >> >> >> The number of samples is NOT required to be a power of 2. >> >> >> In most textbooks FFT is presented to the students >> >> >> in the form of the very well known butterfly-diagra. >> >> >> Reading the orginal papers in your library, you will >> >> >> find out that sometimes the prime-factorisation will helb you >> >> >> to do a FFT of a number of samples NOT a power of two. >> >> >> Have a nice time in reading these papers presented to >> >> >> the community some 30 years ago. Stop - you may also >> >> >> ask an old mathematican if you know one. >> >> >> ( sorry: the papers are 30 years old, so you >> >> >> would need to find a very, very old mathematican ) >> >> >> >> >> >> michael strothjohann >> >> >> >> >> >> >> >> >> >> >> >> Tony Zampini schrieb: >> >> >> > >> >> >> > Radu, >> >> >> > >> >> >> > Adding zeros to the end of your data is called zero-padding and >> >> >> > is commonly done to increase the resolution in the resulting >> >> >> > FFT frequency data. I'm not sure what you mean by your statement: >> >> >> > "The results are different from the FFT done on the 900 samples". >> >> >> > How did you do an FFT on the 900 samples? Did you do a DFT (which >> >> >> > is not limited to powers of 2)? Give us some more information, and >> >> >> > we can help you further. >> >> >> > >> >> >> > Best Regards, >> >> >> > Tony >> >> >> > ______________________________ >> >> >> > Tony Zampini (tony@tony...) >> >> >> > >> >> >> > ----- Original Message ----- >> >> >> > From: Suciu Radu <radusuciu@radu...> >> >> >> > >> >> >> > > I have an FFT algorithm that works for example only for 1024, >but I >> >> >have >> >> >> > > only , say 900 discrete values to calculate the FFT on. >> >> >> > > I put the other 124 values to zero (from position 901 to 1024). >> >> >> > > The results are different from the FFT done on the 900 samples >> >(1024 >> >> >> > > frequency samples). >> >> >> > > Does anybody know how to get it done? >> >> >> > > >> >> >> > > Thanks, >> >> >> > > Radu
Reply by ●October 25, 20012001-10-25
Michael-
>Your plots don't address the issue here.
Generate a signal with two sine
>waves close together in frequency. Take a 64 sample block, a 64 sample
>block zero-padded to 4096 and a 4096 sample block, and plot these signals
>to see the difference.
They exactly address the issue. Given the two 64 pt. blocks of data as
described by Akshay, if the FFT was not lengthened and zeros added, the two
sinusoids could not be resolved.
Stretching the two time samples to 4096 pts. and using 4096 pt. FFTs allows the
sinusoids to be resolved without further processing; I'm not sure what that
adds
to the discussion except possibly to demonstrate in another way that 4k FFT
length is enough to resolve this particular example.
Jeff Brower
DSP sw/hw engineer
Signalogic
Reply by ●October 26, 20012001-10-26
Jeff, Michael, Tony: On 24 Oct 2001 audiodsp@audi... wrote: > Date: Wed, 24 Oct 2001 09:40:51 -0400 > From: "Tony Zampini" <tony@tony...> > > On the first issue, I was trying to say that by zero padding, > you end up with a finer resolution of your bin spacing. > In other words, the resulting frequency spectrum is > sampled at more closely spaced points. To put it yet another way, > the frequency bins are more closely spaced in frequency. Zero-padding in time domain results in harmonic interpolation in frequency domain. Frequency bins are closer to each other, and > Although my words seem to indicate, I did NOT MEAN to imply > that, for example, zero-padding would allow you to better > resolve two closely spaced sinusoids. yes, since adding a bunch of zeroes does not add additional information to the sampled signal, thus zero padding would not increase frequency resolution. This is what any interpolation process would do, more closely spaced table, the same information about original signal (spectrum). > Regards, > Tony > Date: Wed, 24 Oct 2001 14:13:49 GMT > From: Jeff Brower <jbrower@jbro...> > > >Although my words seem to indicate, I did NOT MEAN to imply > >that, for example, zero-padding would allow you to better > >resolve two closely spaced sinusoids. > > Why not? If the time data was sampled often enough then what else can > you do? The sinusoids either have the same frequency or they do not. > A continuous Fourier transform would allow you to distinguish, so a > finite FFT transform of sufficient length would also allow you to > distinguish. Correct, but you have to sample the signal at enough rate to resolve close frequencies. Zero padding does not resolve them. > Jeff Brower > DSP sw/hw engineer > Signalogic > Regards, Andrew
Reply by ●October 26, 20012001-10-26
"Andrew V. Nesterov" wrote: > > > yes, since adding a bunch of zeroes does not add additional > information to the sampled signal, thus zero padding would > not increase frequency resolution. What do you mean then by resolution? If it covers the same space with more samples of it is this not increased resolution? The sampling theorem shows how to determine any value between samples exactly by sinc interpolation when the sampled function is band limited by the Nyquist criterion. That the new information which comes from zero padding is redundant when it comes to reconstructing the function does not negate the validity of the new information. While it is usually stated in the time domain the sampling theorem applies equally to the frequency domain. Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein ////////////////////////////////////////\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ To contribute your unused processor cycles to the fight against cancer: http://www.intel.com/cure \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\///////////////////////////////////////
Reply by ●October 28, 20012001-10-28
Akshay- > It should be a signal, which has been mixed, i.e C = A + B, where you >would want to resolve A and B. The point is to resolve two mixed signals and >*not* take FFT of two signals separately and demonstrate the shift on a >plot. > You can't resolve A and B components by padding any number of zeros to >the FFT of C. In the example you gave, if you add the sinusoids together as you indicate (later but not in the original example), no length of FFT will allow the sinusoids to be resolved. There is not enough time history data. Either more time data has to be provided, or a non-FFT technique has to be used, for example a phase shift detection, interpolated FFT, maximum entropy, etc. Do you know of a DSP-friendly (i.e. real-time) technique that can be used to resolve your mixed your example? Jeff Brower DSP sw/hw engineer Signalogic >----- Original Message ----- >From: Jeff Brower <jbrower@jbro...> >To: Akshay Joshi <ajoshi@ajos...> >Cc: <audiodsp@audi...>; Tony Zampini <tony@tony...>; Suciu >Radu <radusuciu@radu...>; Michael Strothjohann ><strothjohann@stro...> >Sent: Thursday, October 25, 2001 7:14 AM >Subject: Re: [audiodsp] zeroing the FFT > > >> Akshay- >> >> On Wed, 24 Oct 2001, "Akshay Joshi" <ajoshi@ajos...> wrote: >> >Jeff, >> > Tony has explained this. Adding zeros does not increase the >resolution. >> >It merely reduces the spacing between two frequency bins. If you have two >> >sinusoids >> >A = sin(2*pi*i/16) and B = sin(2*pi*i*255/4096). You can't resolve them >with >> >64 point data. You won't resolve them by taking a 4096 pt FFT by padding >> >zeros over 64 point data. >> >> Absolutely, completely, 100% wrong. >> >> Click here to see plots: >> >> http://www.signalogic.com/sine.htm >> >> Sinea is your A above, sineb is your B. Clearly they cannot be resolved >with a >> 64 pt. FFT, and just as clearly they can with 4096 pt. FFT. Input data is >the >> same in each case -- except the 64 pt. FFT has none of those pesky zeros, >and >> the 4096 pt. FFT has 4032 of them. >> >> I have no idea why you think making the FFT longer and adding zeros will >not >> increase resolution and help resolve frequency domain detail. Any >discrete, >> finite Fourier transform is an approximation of the continuous transform; >if you >> make it longer the approximation improves. >> >> > Padding zeros is not useless. For taking FFT of radix 2 and so on, it >is >> >most convenient for computation speed.. you know all about it, but there >> >should be no confusion. On a 900 pt data, you don't get more resolution >out >> >of 4096 pt FFT than a 1024 point FFT. >> > >> > Padding zeros is effective method to get a frequency response of say >an >> >FIR filter. It anyway has zeros at the tail.. so, this is the best place, >> >where you can actually say that padding zeros "improves" resolution. >> >> The frequency response of an FIR filter can be calculated many ways; its >use as >> an FFT example is irrelevant. The fundamental use for FFTs is on real >world >> input data that needs to be analyzed. Zero padding -- and every other key >FFT >> technique -- applies to that, not FIR filters. >> >> Jeff Brower >> Signalogic
Reply by ●October 29, 20012001-10-29
Jeff, > Do you know of a DSP-friendly (i.e. real-time) technique that can be used to > resolve your mixed your example? It is impossible, to resolve them. .. without making assumptions on the nature of signal. -Akshay ----- Original Message ----- From: Jeff Brower <jbrower@jbro...> To: Akshay Joshi <ajoshi@ajos...> Cc: <audiodsp@audi...>; Tony Zampini <tony@tony...>; Suciu Radu <radusuciu@radu...>; Michael Strothjohann <strothjohann@stro...> Sent: Sunday, October 28, 2001 10:31 PM Subject: Re: [audiodsp] zeroing the FFT > Akshay- > > > It should be a signal, which has been mixed, i.e C = A + B, where you > >would want to resolve A and B. The point is to resolve two mixed signals and > >*not* take FFT of two signals separately and demonstrate the shift on a > >plot. > > You can't resolve A and B components by padding any number of zeros to > >the FFT of C. > > In the example you gave, if you add the sinusoids together as you indicate > (later but not in the original example), no length of FFT will allow the > sinusoids to be resolved. There is not enough time history data. Either more > time data has to be provided, or a non-FFT technique has to be used, for example > a phase shift detection, interpolated FFT, maximum entropy, etc. > > Do you know of a DSP-friendly (i.e. real-time) technique that can be used to > resolve your mixed your example? > > Jeff Brower > DSP sw/hw engineer > Signalogic > > > >----- Original Message ----- > >From: Jeff Brower <jbrower@jbro...> > >To: Akshay Joshi <ajoshi@ajos...> > >Cc: <audiodsp@audi...>; Tony Zampini <tony@tony...>; Suciu > >Radu <radusuciu@radu...>; Michael Strothjohann > ><strothjohann@stro...> > >Sent: Thursday, October 25, 2001 7:14 AM > >Subject: Re: [audiodsp] zeroing the FFT > > > > > >> Akshay- > >> > >> On Wed, 24 Oct 2001, "Akshay Joshi" <ajoshi@ajos...> wrote: > >> >Jeff, > >> > Tony has explained this. Adding zeros does not increase the > >resolution. > >> >It merely reduces the spacing between two frequency bins. If you have two > >> >sinusoids > >> >A = sin(2*pi*i/16) and B = sin(2*pi*i*255/4096). You can't resolve them > >with > >> >64 point data. You won't resolve them by taking a 4096 pt FFT by padding > >> >zeros over 64 point data. > >> > >> Absolutely, completely, 100% wrong. > >> > >> Click here to see plots: > >> > >> http://www.signalogic.com/sine.htm > >> > >> Sinea is your A above, sineb is your B. Clearly they cannot be resolved > >with a > >> 64 pt. FFT, and just as clearly they can with 4096 pt. FFT. Input data is > >the > >> same in each case -- except the 64 pt. FFT has none of those pesky zeros, > >and > >> the 4096 pt. FFT has 4032 of them. > >> > >> I have no idea why you think making the FFT longer and adding zeros will > >not > >> increase resolution and help resolve frequency domain detail. Any > >discrete, > >> finite Fourier transform is an approximation of the continuous transform; > >if you > >> make it longer the approximation improves. > >> > >> > Padding zeros is not useless. For taking FFT of radix 2 and so on, it > >is > >> >most convenient for computation speed.. you know all about it, but there > >> >should be no confusion. On a 900 pt data, you don't get more resolution > >out > >> >of 4096 pt FFT than a 1024 point FFT. > >> > > >> > Padding zeros is effective method to get a frequency response of say > >an > >> >FIR filter. It anyway has zeros at the tail.. so, this is the best place, > >> >where you can actually say that padding zeros "improves" resolution. > >> > >> The frequency response of an FIR filter can be calculated many ways; its > >use as > >> an FFT example is irrelevant. The fundamental use for FFTs is on real > >world > >> input data that needs to be analyzed. Zero padding -- and every other key > >FFT > >> technique -- applies to that, not FIR filters. > >> > >> Jeff Brower > >> Signalogic