Taking a time derivative after an FFT would be easy on SPICE if you had some curve that would FFT into a ramp: Just multiply the function by the ramp. The inverse transform of a ramp on Excel is complex, the real part being a small negative offset with a large positive zero frequency. The imaginary part may work but it looks like it would be hard to fashion with a circuit even if the curve was known. Just builting up 2**n voltage sources where frequency = amplitude = n is tedious. The FFT is, of course, just 2**n peaks that increase linearly in height on the linear - linear graph. To get a nice ramp would require an infinite number of voltage sources & frequencies and amplitudes. Is there anyway to get a nice smooth envelope -- "envelope" may have another technical meaning -- over the time domain curve to get a nice ramp in the FFT? Bret Cahill
Frequency Domain Ramp
Started by ●February 24, 2011
Reply by ●February 24, 20112011-02-24
Bret Cahill wrote:> Taking a time derivative after an FFT would be easy on SPICE if you > had some curve that would FFT into a ramp: Just multiply the function > by the ramp. > > The inverse transform of a ramp on Excel is complex, the real part > being a small negative offset with a large positive zero frequency. > The imaginary part may work but it looks like it would be hard to > fashion with a circuit even if the curve was known. > > Just builting up 2**n voltage sources where frequency = amplitude = n > is tedious. The FFT is, of course, just 2**n peaks that increase > linearly in height on the linear - linear graph. To get a nice ramp > would require an infinite number of voltage sources& frequencies and > amplitudes. > > Is there anyway to get a nice smooth envelope -- "envelope" may have > another technical meaning -- over the time domain curve to get a nice > ramp in the FFT? > > > Bret CahillI suggest getting a copy of "The Fourier Transform and Its Applications" by Bracewell. It's a good read and has all of this sort of stuff in it. Cheers Phil Hobbs -- Dr Philip C D Hobbs Principal ElectroOptical Innovations 55 Orchard Rd Briarcliff Manor NY 10510 845-480-2058 email: hobbs (atsign) electrooptical (period) net http://electrooptical.net
Reply by ●February 24, 20112011-02-24
> > Taking a time derivative after an FFT would be easy on SPICE if you > > had some curve that would FFT into a ramp: �Just multiply the function > > by the ramp. > > > The inverse transform of a ramp on Excel is complex, the real part > > being a small negative offset with a large positive zero frequency. > > The imaginary part may work but it looks like it would be hard to > > fashion with a circuit even if the curve was known. > > > Just builting up 2**n voltage sources where frequency = amplitude = n > > is tedious. �The FFT is, of course, just 2**n peaks that increase > > linearly in height on the linear - linear graph. �To get a nice ramp > > would require an infinite number of voltage sources& �frequencies and > > amplitudes. > > > Is there anyway to get a nice smooth envelope -- "envelope" may have > > another technical meaning -- over the time domain curve to get a nice > > ramp in the FFT? > > > Bret Cahill > > I suggest getting a copy of "The Fourier Transform and Its Applications" > by Bracewell. �It's a good read and has all of this sort of stuff in it.At $500 / copy it better be a good read. That's what university libraries pay for Russian books on tech esoterica. Anyway some nuke medicine page claimed the INV FT of a ramp was just 1/ t. It's somewhat curious that the INV FFT of a function is equal to the reciprocal of the function but nevertheless 1/t seemed to approach a ramp on Excel's FFT, at least at higher frequencies. I couldn't get anything linear on the SPICE FFT but then, I couldn't get anything to output a 1/t waveform except a crude 10 point plot.> Cheers > > Phil Hobbs > > -- > Dr Philip C D Hobbs > Principal > ElectroOptical Innovations > 55 Orchard Rd > Briarcliff Manor NY 10510 > 845-480-2058 > > email: hobbs (atsign) electrooptical (period) nethttp://electrooptical.net- Hide quoted text - > > - Show quoted text -
Reply by ●February 24, 20112011-02-24
Bret Cahill wrote:>>> Taking a time derivative after an FFT would be easy on SPICE if you >>> had some curve that would FFT into a ramp: Just multiply the function >>> by the ramp. >> >>> The inverse transform of a ramp on Excel is complex, the real part >>> being a small negative offset with a large positive zero frequency. >>> The imaginary part may work but it looks like it would be hard to >>> fashion with a circuit even if the curve was known. >> >>> Just builting up 2**n voltage sources where frequency = amplitude = n >>> is tedious. The FFT is, of course, just 2**n peaks that increase >>> linearly in height on the linear - linear graph. To get a nice ramp >>> would require an infinite number of voltage sources& frequencies and >>> amplitudes. >> >>> Is there anyway to get a nice smooth envelope -- "envelope" may have >>> another technical meaning -- over the time domain curve to get a nice >>> ramp in the FFT? >> >>> Bret Cahill >> >> I suggest getting a copy of "The Fourier Transform and Its Applications" >> by Bracewell. It's a good read and has all of this sort of stuff in it. > > At $500 / copy it better be a good read. That's what university > libraries pay for Russian books on tech esoterica. > > Anyway some nuke medicine page claimed the INV FT of a ramp was just 1/ > t. > > It's somewhat curious that the INV FFT of a function is equal to the > reciprocal of the function but nevertheless 1/t seemed to approach a > ramp on Excel's FFT, at least at higher frequencies. > > I couldn't get anything linear on the SPICE FFT but then, I couldn't > get anything to output a 1/t waveform except a crude 10 point plot. > >Five hundred bucks? Where do you find that? Try http://www.abebooks.com and you'll find a whole pile of the second edition (which is better than good enough) for about $22 plus shipping. Cheers Phil Hobbs -- Dr Philip C D Hobbs Principal ElectroOptical Innovations 55 Orchard Rd Briarcliff Manor NY 10510 845-480-2058 email: hobbs (atsign) electrooptical (period) net http://electrooptical.net
Reply by ●February 24, 20112011-02-24
On Feb 24, 9:10�am, Bret Cahill <Bret_E_Cah...@yahoo.com> wrote:> Is there anyway to get a nice smooth envelope -- "envelope" may have > another technical meaning -- over the time domain curve to get a nice > ramp in the FFT?No. The usual FFT for finite sampled sequences is done with circular boundary conditions, and you must deal with the enormous discontinuity at that boundary that any frequency-dependent ramp creates. It isn't a suitable 'test function' in the sense of nicely behaved functions for the transformation, so it is going to look very nasty in the time domain (after the inverse transformation of the 'ramp').
Reply by ●February 24, 20112011-02-24
> > Is there anyway to get a nice smooth envelope -- "envelope" may have > > another technical meaning -- over the time domain curve to get a nice > > ramp in the FFT? > > No. �The usual FFT for finite sampled sequences is done with circular > boundary conditions, and you must deal with the enormous discontinuity > at that boundary that any frequency-dependent ramp creates.Would this be true if every point used by the FFT -- SPICE has a 256 minimum -- was plotted and used by the FFT?> It isn't a suitable 'test function' in the sense of nicely behaved > functions for the transformation, so it is going to look very nasty in > the time domain (after the inverse transformation of the 'ramp').Plotting 1/t with about 8 points, (0.125, 8) (.25, 4) . . . (8, 0.125) on SPICE then taking the FFT and then taking the reciprocal -- not sure why this step works or is necessary -- isn't a bad ramp, at least at the lower frequencies. It would be really convenient if a time derivative could be taken mathematically without a derivative circuit in either domain. You cannot take a FFT of a wave form after you've taken its time derivative in the time domain -- it won't appear in the box -- and in the frequency domain you only get a derivative w/ respect to frequency, whatever that is. There doesn't seem to be an easy way to create a time signal on SPICE that equals 1/t. Bret Cahill
Reply by ●February 25, 20112011-02-25
In article <y8L9p.18$aC6.13@newsfe03.iad>, Martin Brown <|||newspam|||@nezumi.demon.co.uk> wrote:>Bad things happen when this basic assumption of periodic boundary >conditions is for whatever reason invalid. A saw tooth has very obvious >boundary discontinuity problems.Indeed. In that, it is exactly the same as polynomial fitting of curves, or any such infinite approximation. Go outside the domain of validity and things can go badly wrong. I am often amazed at the things people use Fourier approximations for - not because they do, but because they seem to work more often than a naive analysis would expect. But, as you say, you don't get that by just rushing in, blindly. Regards, Nick Maclaren.
Reply by ●February 25, 20112011-02-25
On 24/02/2011 22:18, Bret Cahill wrote:>>> Is there anyway to get a nice smooth envelope -- "envelope" may have >>> another technical meaning -- over the time domain curve to get a nice >>> ramp in the FFT? >> >> No. The usual FFT for finite sampled sequences is done with circular >> boundary conditions, and you must deal with the enormous discontinuity >> at that boundary that any frequency-dependent ramp creates. > > Would this be true if every point used by the FFT -- SPICE has a 256 > minimum -- was plotted and used by the FFT?It is always true that an FFT has an implicit assumption of periodic boundary conditions at the length of the transform which are most commonly a tiled circular wrap around at the edges, but in some implementations may be Dirichlet or mirror boundary conditions leading to a DCT variant. In addition there is also two plausible choices of origin exactly in the centre of each cell or at the edge. Bad things happen when this basic assumption of periodic boundary conditions is for whatever reason invalid. A saw tooth has very obvious boundary discontinuity problems. Real applications of FFTs for imaging tend to spend a lot of time and effort ameliorating this potential aliasing effect at the boundaries.> >> It isn't a suitable 'test function' in the sense of nicely behaved >> functions for the transformation, so it is going to look very nasty in >> the time domain (after the inverse transformation of the 'ramp'). > > Plotting 1/t with about 8 points, (0.125, 8) (.25, 4) . . . (8, > 0.125) on SPICE then taking the FFT and then taking the reciprocal -- > not sure why this step works or is necessary -- isn't a bad ramp, at > least at the lower frequencies. > > It would be really convenient if a time derivative could be taken > mathematically without a derivative circuit in either domain. > > You cannot take a FFT of a wave form after you've taken its time > derivative in the time domain -- it won't appear in the box -- and in > the frequency domain you only get a derivative w/ respect to > frequency, whatever that is. > > There doesn't seem to be an easy way to create a time signal on SPICE > that equals 1/t.What are you trying to do? Regards, Martin Brown
Reply by ●February 25, 20112011-02-25
Bob Masta wrote:> On Thu, 24 Feb 2011 09:10:50 -0800 (PST), Bret Cahill > <Bret_E_Cahill@yahoo.com> wrote: > >> Taking a time derivative after an FFT would be easy on SPICE if you >> had some curve that would FFT into a ramp: Just multiply the function >> by the ramp. > > Not sure what you are ultimately trying to do, but note that > you can obtain the FFT of the time derivative by taking the > FFT of the raw waveform and applying a +6 dB/octave > "envelope" to that... essentially, you just tilt the > spectrum up at a 6 dB/octave slope. > > This turns out to be very handy for measuring frequency > response of a system. Classically, one can apply an impulse > to the system and take the FFT to get the frequency > response. But an impulse is pretty narrow (one sample, in a > digital system), so it doesn't have much energy. A step > response, on the other hand, has a whole lot more. Since > the derivatve of a step is an impulse, you can get the > frequency response by applying a step, taking the FFT, and > tilting it. This is so handy that I built this feature into > my Daqarta software. See "Frequency Response Measurement - > Step Response" at<http://www.daqarta.com/dw_0a0s.htm>. >Are you windowing the data before taking the DFT? Could get ugly otherwise. Cheers Phil Hobbs -- Dr Philip C D Hobbs Principal ElectroOptical Innovations 55 Orchard Rd Briarcliff Manor NY 10510 845-480-2058 email: hobbs (atsign) electrooptical (period) net http://electrooptical.net
Reply by ●February 25, 20112011-02-25
On Thu, 24 Feb 2011 09:10:50 -0800 (PST), Bret Cahill <Bret_E_Cahill@yahoo.com> wrote:>Taking a time derivative after an FFT would be easy on SPICE if you >had some curve that would FFT into a ramp: Just multiply the function >by the ramp.Not sure what you are ultimately trying to do, but note that you can obtain the FFT of the time derivative by taking the FFT of the raw waveform and applying a +6 dB/octave "envelope" to that... essentially, you just tilt the spectrum up at a 6 dB/octave slope. This turns out to be very handy for measuring frequency response of a system. Classically, one can apply an impulse to the system and take the FFT to get the frequency response. But an impulse is pretty narrow (one sample, in a digital system), so it doesn't have much energy. A step response, on the other hand, has a whole lot more. Since the derivatve of a step is an impulse, you can get the frequency response by applying a step, taking the FFT, and tilting it. This is so handy that I built this feature into my Daqarta software. See "Frequency Response Measurement - Step Response" at <http://www.daqarta.com/dw_0a0s.htm>. Best regards, Bob Masta DAQARTA v6.00 Data AcQuisition And Real-Time Analysis www.daqarta.com Scope, Spectrum, Spectrogram, Sound Level Meter Frequency Counter, FREE Signal Generator Pitch Track, Pitch-to-MIDI Science with your sound card!
