Frequently Asked Questions (F.A.Q.)..... 1. (A Frequently Added Quotation, F.A.Q., is appended below.) Assuming that we were able to generate a Diracian, and then produce a comb of them by delays and by superposition, there wouldn't be a factor of "T" in such superposition, so where does yours come from? Where did the factor of "T" come from in the "sampling function" in your opening lines? Consider a 16-bit ADC capable of 100 M Samples per sec. In the first instance we'll use it to sample a geophysical signal of bandwidth limited to 300 HZ and sample at 1 kHz, with suitable analogue instrumentation to match the input signal to the full range of the ADC. If we keep the circuit the same, but now sample at 65.536 MHZ, your claimed factor of "T" will result in the 16 bit range being compressed down to one bit, because the new sampling frequency is 2^(16) faster than the old. We know this doesn't happen - there will be more samples, but they'll still be of the same magnitude and over the same full-scale 16-bit range. So, where does the factor of "T" come from? 2. Where have your Diracian impulses come from? The Diracian has some interesting properties - zero width, area of unity, infinite sum of all possible cosines, a height which is not discussed but which appears to be greater in magnitude than any voltage appearing in your circuits. In the systems that you deal with, what experimental evidence do you have that there are pulses with the attributes of Diracians upon which to base your theory? The Diracian, or Unit Impulse is a very good mathematical tool to analyse the response of systems once a mathematical model of those systems had been produced. It is, however, a poor mathematical claim to make that such impulses are found to be part of a system when neither the area nor the magnitude of such impulses are found anywhere in such systems. To those who ask, "Who cares? I get good results." , I suggest that their approach is unscientific and compares to the religious loonies who sacrifice goats and virgins to stop the Sun falling out of the sky and justify the continuing practice by the Sun remaining in the sky. So.....is the world of DSP a world of scientific men, or is it a world of snake-oil charlatans and of religious loonies? Where do these Diracian impulses come from?> robert bristow-johnson wrote: > > x(t)*q(t) = T*SUM{x[k]*d(t-k*T)} .------. > x(t)--->(*)------------------------------------->| H(f) |---> x(t) > ^ '------' > | > | +inf > '------- q(t) = T * SUM{ d(t - k*T) } > k=-inf > > > where: d(t) = 'dirac' impulse function > and T = 1/Fs = sampling period > Fs = sampling frequency > > > +inf > q(t) = T * SUM{ d(t - k*T) } is the "sampling function", is periodic > k=-inf with period T, and can be expressed as a > Fourier series. It turns out that ALL of > the Fourier coefficients are equal to 1.
comp.dsp - a FAQ
Started by ●October 13, 2005
Reply by ●October 13, 20052005-10-13
What is the link for what you appended? Thanks, Dirk Polymath wrote:> Frequently Asked Questions (F.A.Q.)..... > > 1. (A Frequently Added Quotation, F.A.Q., is appended below.) > Assuming that we were able to generate a Diracian, and then produce > a comb of them by delays and by superposition, there wouldn't be a > factor > of "T" in such superposition, so where does yours come from? > > Where did the factor of "T" come from in the "sampling function" in > your opening lines? > > Consider a 16-bit ADC capable of 100 M Samples per sec. In the first > instance we'll use it to sample a geophysical signal of bandwidth > limited to 300 HZ and sample at 1 kHz, with suitable analogue > instrumentation to match the input signal to the full range of > the ADC. > > If we keep the circuit the same, but now sample > at 65.536 MHZ, your claimed factor of "T" will result in the > 16 bit range being compressed down to one bit, because the new > sampling frequency is 2^(16) faster than the old. We know this > doesn't happen - there will be more samples, but they'll still > be of the same magnitude and over the same full-scale 16-bit range. > > So, where does the factor of "T" come from? > > 2. Where have your Diracian impulses come from? > > The Diracian has some interesting properties - zero width, > area of unity, infinite sum of all possible cosines, a height > which is not discussed but which appears to be greater in magnitude > than any voltage appearing in your circuits. In the > systems that you deal with, what experimental evidence do you > have that there are pulses with the attributes of Diracians upon > which to base your theory? > > The Diracian, or Unit Impulse is a very good mathematical tool > to analyse the response of systems once a mathematical model of > those systems had been produced. > > It is, however, a poor mathematical claim to make that such > impulses are found to be part of a system when neither the > area nor the magnitude of such impulses are found anywhere in such > systems. > > To those who ask, "Who cares? I get good results." , I suggest that > their approach is unscientific and compares to the religious > loonies who sacrifice goats and virgins to stop the Sun falling > out of the sky and justify the continuing practice by the Sun > remaining in the sky. > > So.....is the world of DSP a world of scientific men, or is > it a world of snake-oil charlatans and of religious loonies? > > Where do these Diracian impulses come from? > > > robert bristow-johnson wrote: > > > > x(t)*q(t) = T*SUM{x[k]*d(t-k*T)} .------. > > x(t)--->(*)------------------------------------->| H(f) |---> x(t) > > ^ '------' > > | > > | +inf > > '------- q(t) = T * SUM{ d(t - k*T) } > > k=-inf > > > > > > where: d(t) = 'dirac' impulse function > > and T = 1/Fs = sampling period > > Fs = sampling frequency > > > > > > +inf > > q(t) = T * SUM{ d(t - k*T) } is the "sampling function", is periodic > > k=-inf with period T, and can be expressed as a > > Fourier series. It turns out that ALL of > > the Fourier coefficients are equal to 1.
Reply by ●October 13, 20052005-10-13
Don't know - it's something that Bristow-Johnston reposts periodically with bits added on here and there - I got it by searching google for bits of its content - it'll be just as quick for you to find it again as it would be for me. dbell wrote:> What is the link for what you appended? > > Thanks, > > Dirk > > > Polymath wrote: > > Frequently Asked Questions (F.A.Q.)..... > > > > 1. (A Frequently Added Quotation, F.A.Q., is appended below.) > > Assuming that we were able to generate a Diracian, and then produce > > a comb of them by delays and by superposition, there wouldn't be a > > factor > > of "T" in such superposition, so where does yours come from? > > > > Where did the factor of "T" come from in the "sampling function" in > > your opening lines? > > > > Consider a 16-bit ADC capable of 100 M Samples per sec. In the first > > instance we'll use it to sample a geophysical signal of bandwidth > > limited to 300 HZ and sample at 1 kHz, with suitable analogue > > instrumentation to match the input signal to the full range of > > the ADC. > > > > If we keep the circuit the same, but now sample > > at 65.536 MHZ, your claimed factor of "T" will result in the > > 16 bit range being compressed down to one bit, because the new > > sampling frequency is 2^(16) faster than the old. We know this > > doesn't happen - there will be more samples, but they'll still > > be of the same magnitude and over the same full-scale 16-bit range. > > > > So, where does the factor of "T" come from? > > > > 2. Where have your Diracian impulses come from? > > > > The Diracian has some interesting properties - zero width, > > area of unity, infinite sum of all possible cosines, a height > > which is not discussed but which appears to be greater in magnitude > > than any voltage appearing in your circuits. In the > > systems that you deal with, what experimental evidence do you > > have that there are pulses with the attributes of Diracians upon > > which to base your theory? > > > > The Diracian, or Unit Impulse is a very good mathematical tool > > to analyse the response of systems once a mathematical model of > > those systems had been produced. > > > > It is, however, a poor mathematical claim to make that such > > impulses are found to be part of a system when neither the > > area nor the magnitude of such impulses are found anywhere in such > > systems. > > > > To those who ask, "Who cares? I get good results." , I suggest that > > their approach is unscientific and compares to the religious > > loonies who sacrifice goats and virgins to stop the Sun falling > > out of the sky and justify the continuing practice by the Sun > > remaining in the sky. > > > > So.....is the world of DSP a world of scientific men, or is > > it a world of snake-oil charlatans and of religious loonies? > > > > Where do these Diracian impulses come from? > > > > > robert bristow-johnson wrote: > > > > > > x(t)*q(t) = T*SUM{x[k]*d(t-k*T)} .------. > > > x(t)--->(*)------------------------------------->| H(f) |---> x(t) > > > ^ '------' > > > | > > > | +inf > > > '------- q(t) = T * SUM{ d(t - k*T) } > > > k=-inf > > > > > > > > > where: d(t) = 'dirac' impulse function > > > and T = 1/Fs = sampling period > > > Fs = sampling frequency > > > > > > > > > +inf > > > q(t) = T * SUM{ d(t - k*T) } is the "sampling function", is periodic > > > k=-inf with period T, and can be expressed as a > > > Fourier series. It turns out that ALL of > > > the Fourier coefficients are equal to 1.
Reply by ●October 13, 20052005-10-13
in article 1129241475.055814.273160@o13g2000cwo.googlegroups.com, dbell at dbell@niitek.com wrote on 10/13/2005 18:11:> What is the link for what you appended?it's something of mine. the last version i posted was back last June: http://groups.google.com/group/comp.dsp/msg/e9b6488aef1e2580?hl=en&fwc=1 but i have posted this many times. particularly when the topic is about resampling. i have Beanie killfiled so i only see responses (like yours) to his perennial troll. (looks like you got caught by the trawling, so what am i doing jumping into the net?) dunno if you wanna touch it not, but the issue (for me) is that the scale factor of T (the sampling period) should be tossed into the sampling function rather than in the reconstruction LPF as it is done in nearly every textbook that explicitly proves the Nyquist/Shannon Sampling and Reconstruction Theorem. i do this mostly for dimensional reasons (what kind of filter has a dimensionful gain of "T" in the passband?) but it also has use when you try to explain the effect of the zero-order hold on frequency response. when people leave it out, sometimes they mess up and their answer is off by a scaling factor (and has the wrong dimension of physical quantity).> Thanks,FWIW. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge." Polymath wrote:> Frequently Asked Questions (F.A.Q.)..... > > 1. (A Frequently Added Quotation, F.A.Q., is appended below.) > Assuming that we were able to generate a Diracian, and then produce > a comb of them by delays and by superposition, there wouldn't be a > factor > of "T" in such superposition, so where does yours come from? > > Where did the factor of "T" come from in the "sampling function" in > your opening lines? > > Consider a 16-bit ADC capable of 100 M Samples per sec. In the first > instance we'll use it to sample a geophysical signal of bandwidth > limited to 300 HZ and sample at 1 kHz, with suitable analogue > instrumentation to match the input signal to the full range of > the ADC. > > If we keep the circuit the same, but now sample > at 65.536 MHZ, your claimed factor of "T" will result in the > 16 bit range being compressed down to one bit, because the new > sampling frequency is 2^(16) faster than the old. We know this > doesn't happen - there will be more samples, but they'll still > be of the same magnitude and over the same full-scale 16-bit range. > > So, where does the factor of "T" come from? > > 2. Where have your Diracian impulses come from? > > The Diracian has some interesting properties - zero width, > area of unity, infinite sum of all possible cosines, a height > which is not discussed but which appears to be greater in magnitude > than any voltage appearing in your circuits. In the > systems that you deal with, what experimental evidence do you > have that there are pulses with the attributes of Diracians upon > which to base your theory? > > The Diracian, or Unit Impulse is a very good mathematical tool > to analyse the response of systems once a mathematical model of > those systems had been produced. > > It is, however, a poor mathematical claim to make that such > impulses are found to be part of a system when neither the > area nor the magnitude of such impulses are found anywhere in such > systems. > > To those who ask, "Who cares? I get good results." , I suggest that > their approach is unscientific and compares to the religious > loonies who sacrifice goats and virgins to stop the Sun falling > out of the sky and justify the continuing practice by the Sun > remaining in the sky. > > So.....is the world of DSP a world of scientific men, or is > it a world of snake-oil charlatans and of religious loonies? > > Where do these Diracian impulses come from? > > > robert bristow-johnson wrote: > > > > x(t)*q(t) = T*SUM{x[k]*d(t-k*T)} .------. > > x(t)--->(*)------------------------------------->| H(f) |---> x(t) > > ^ '------' > > | > > | +inf > > '------- q(t) = T * SUM{ d(t - k*T) } > > k=-inf > > > > > > where: d(t) = 'dirac' impulse function > > and T = 1/Fs = sampling period > > Fs = sampling frequency > > > > > > +inf > > q(t) = T * SUM{ d(t - k*T) } is the "sampling function", is periodic > > k=-inf with period T, and can be expressed as a > > Fourier series. It turns out that ALL of > > the Fourier coefficients are equal to 1.
Reply by ●October 13, 20052005-10-13
r-b-j, I was not caught in his net. I read his post. It seemed on topic. There is no problem with your mathematics, but his comments about 'T' are not without merit in that moving the 'T' factor into the frequency domain would make the sampling model more 'representative' of real world sampling. To answer his question, the 'T' factor is there because it is not in the frequency domain. You can pretty much put it where you want. Mathematicians, physicists, and engineers don't all use the same definitions for Fourier transforms, but if they are consistent it all works out. As to the second topic. If he rereads his question, I think he will find that he has answered it himself. Dirk
Reply by ●October 14, 20052005-10-14
dbell wrote:> r-b-j, > > I was not caught in his net. I read his post. It seemed on topic. > > There is no problem with your mathematics, but his comments about 'T' > are not without merit in that moving the 'T' factor into the frequency > domain would make the sampling model more 'representative' of real > world sampling. To answer his question, the 'T' factor is there > because it is not in the frequency domain. You can pretty much put it > where you want. Mathematicians, physicists, and engineers don't all > use the same definitions for Fourier transforms, but if they are > consistent it all works out.You can't put it anywhere you want and be dimensionally consistent. An impulse isn't dimensionless. It has whatever units are required by what it represents. For example, a the impulse of hammer striking a pendulum is assumed (like all other impulses) to have zero duration, but it's magnitude has the units of force times time. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●October 14, 20052005-10-14
Jerry, The published scaling on Fourier transforms runs from a) all scaling on the forward transform, to b) the sqrt of the scaling applied to both the forward and inverse transforms, to c) all applied to the inverse transform. Seems to be a function of your field (mathematics, physics, different engineering disciplines). The scaling can be shifted between forward and inverse transforms in the definitions. Knowing the definition being used, and consistency, makes them all work. As far as the sampling goes, if we think in terms of loosely defining x(n)=Xanalog(nT) then the impulses conceptually corresponding to the sampling are not individually functions of 'T', only their spacing is. That corresponds to Polymath's view, and is more consistent with real-world ADCs. But as jbr pointed out, it would require a reconstruction filter whose gain depends on T, which understandably jbr doesn't like, so he finds it convenient to define x(n)=T*Xanalog(nT), effectively resulting from scaling the impulses up by factor 'T', as in his example. The two x(n) definitions are scaled differently, and would have to be filtered differently to get the same reconstructed result Xanalog(t). To have the same dimensions for the x(n) definitions, 'T' would have to be dimensionless when applied as a scale factor. So to sample and reconstruct with no gain change, for different values of 'T', gain goes on the reconstruction filter input signal or the filter. Or it could be split between. It can be put where it is convenient. Clearly you would have to take how the gain was applied into account if you need to calibrate any intermediate calculations. Dirk
Reply by ●October 14, 20052005-10-14
dbell wrote:> Jerry, > > The published scaling on Fourier transforms runs from a) all scaling on > the forward transform, to b) the sqrt of the scaling applied to both > the forward and inverse transforms, to c) all applied to the inverse > transform. Seems to be a function of your field (mathematics, physics, > different engineering disciplines). The scaling can be shifted between > forward and inverse transforms in the definitions. Knowing the > definition being used, and consistency, makes them all work.Scaling is multiplication by a dimensionless constant. One can do that wherever one likes. If we want the equation f = ma to come out in pounds when a is feet/sec^2, then m must have the dimension lb-sec^2/ft to maintain dimensional consistency. Do you remember your physics teacher reminding you that if the dimensions don't work out, there must be a mistake? Have your work or study exposed you to dimensional analysis? (Reynolds, Froude, Mach numbers and more are dimensionless constants and hence independent of scale.) For convenience, we cite mass in either slugs or poundals, depending on which system of units we adopt, but we need to be consistent. Likewise, we cite inductance in Henrys, rather than in more fundamental units, volt-seconds/ampere. (Interestingly, volt-second is a measure both of magnetic flux and the strength of a voltage impulse. Foot-pound can be torque or energy.)> As far as the sampling goes, if we think in terms of loosely defining > x(n)=Xanalog(nT) then the impulses conceptually corresponding to the > sampling are not individually functions of 'T', only their spacing is.The area of a voltage impulse has units of volt-seconds, same as a Weber. When you don't account for dimensions, contradictions appear. When it comes to a mathematical description of sampling, those contradictions have been plastered over rather than resolved. ... Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●October 14, 20052005-10-14
in article 1129271883.643327.198370@o13g2000cwo.googlegroups.com, dbell at dbell@niitek.com wrote on 10/14/2005 02:38:> As far as the sampling goes, if we think in terms of loosely defining > x(n)=Xanalog(nT) then the impulses conceptually corresponding to the > sampling are not individually functions of 'T', only their spacing is. > That corresponds to Polymath's view, and is more consistent with > real-world ADCs. But as jbr pointed out,who is "jbr"?> it would require a reconstruction filter whose gain depends on T, which > understandably jbr doesn't like,if by "jbr" you mean me, i don't like the scaling by T because of dimensional reasons. i would like to conceptualize my reconstruction LPF as such that the species of animal coming out is the same species going in. ya know: voltage in -> voltage out. i don't want voltage in -> volt-seconds out.> so he finds it convenient to define x(n)=T*Xanalog(nT), > effectively resulting from scaling the impulses up by factor 'T',or "down", but since T is of dimension "seconds", it is meaningless to try to differentiate "scaling up" vs. "scaling down".> as in his example. The two x(n) definitions are scaled differently, and would > have to be filtered differently to get the same reconstructed result > Xanalog(t). To have the same dimensions for the x(n) definitions, 'T' > would have to be dimensionless when applied as a scale factor.but it isn't. "T" is not dimensionless and never has been. we measure time in units of seconds or nanoseconds or years or Planck Times. T is not dimensionless and that is the main point you might have missed here, Dirk.> So to sample and reconstruct with no gain change, for different values > of 'T', gain goes on the reconstruction filter input signal or the > filter. Or it could be split between. It can be put where it is > convenient. Clearly you would have to take how the gain was applied > into account if you need to calibrate any intermediate calculations.i am not sure what you mean by "Xanalog(nT)". i am using the convention that x(t) is a continuous-time function and x[k] is a discrete-time function (or, more precisely, a "sequence" of numbers). even though the arguments of x(t) and x[k] have different dimension, the values of x(t) and x[k] have the same dimension. probably the dimensionless number inside the DSP or computer (assuming no mu-law) is: Quantize( x[k]/Xref ) = x[k]/Xref + e[k] where e[k] is the quantization error. with "my" dimensionally corrected sampling function: � � � � � � � �+inf � � q(t) = T * SUM{ d(t - k*T) } � � � � � � � k=-inf where d(t) is the "Dirac delta function" or "impulse function" defined in the traditional engineering sense. the sampled function: � � � � � � � � � � � � � +inf � � x(t)*q(t) = x(t) * T * SUM{ d(t - k*T) } � � � � � � � � � � � � �k=-inf � � � � � � � � � �+inf � � � � � � � = T * SUM{ x[k] * d(t - k*T) } � � � � � � � � � k=-inf where x[k] =def x(k*T) when the sampled function x(t)*q(t) is passed through a brickwall LPF with bandwidth 1/(2T) and passband gain of 1, then the output is the reconstructed x(t). on the other hand, if the T is left out of q(t) (and consequenctly x(t)*q(t)), then the reconstruction LPF must have the dimensionful gain of T. that means a different animal comes out of the filter than goes in. now here is where it makes a difference. what if the output is the piece-wise constant function that we see coming out of conventional D/A converters: � � � � � � � � � �+inf � � � ��y(t)� = SUM{ x[k] * p((t - k*T)/T) } � � � � � � � � � k=-inf where { 1 for |u| < 1/2 p(u) =def { (i'm not anal about |u| = 1/2) { 0 for |u| > 1/2 note that there is not ambiguous scaling here: y(k*T) = x(k*T) for all integers, k. now, Dirk, ask yourself what hypothetical filter has to go in between the ideally sampled input x(t)*q(t) and the piece-wise constant D/A output y(t)? and then ask the same question if you left the "T" out of the definition of the sampling impulse train, q(t) (and subsequently the sampled input x(t)*q(t) ) ? if you have access to the Journal of the Audio Engineering Society, in Nov 1988, i make use of this to come up with a hypothetical filter that tells us the effect on frequency response of an RC-limited "deglitched" D/A output. this was preceded by Barry Blesser and guess what? he was off by the factor T. that is why putting this scaling factor in the correct place has salience. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●October 14, 20052005-10-14
