Randy Yates wrote:> Then you can conclude that f = (fs/2pi) * asin(y2).Clearly, y2 = -y0 when y1 = 0. How is amplitude accounted for? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
minimum cycles for fft, and limits of filling short sample out with zeros
Started by ●September 4, 2005
Reply by ●September 8, 20052005-09-08
Reply by ●September 8, 20052005-09-08
Randy Yates wrote:> Firstly, did I say these notions had a direct connection to a > real-world > system? > > Secondly, did I say you could do this with a single ADC?Most of our discussions are about things that can be measured. When that's not the case, I expect that to be explicit. You gave a hint when you wrote "analytic"; I suppose that was enough. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●September 8, 20052005-09-08
"Jerry Avins" <jya@ieee.org> wrote in message news:Hpadnfw5Xtgcrb3eRVn-rw@rcn.net...> Clay S. Turner wrote: >> "Jerry Avins" <jya@ieee.org> wrote in message>> >> Hello Jerry and others, >> >> In the case of evenly temporally spaced real values, the 3 pt formula for >> frequency is: >> >> f = (fs/2pi) *acos( (y0+y2)/(2*y1) ) >> >> where fs is the sampling rate and y0, y1, y2 are three consecutive >> samples all assumed to be within >> 360 degrees of each other. > > And when y1 happens to be zero? >Hello Jerry, This is the pathological case where the three independent equations becomes only two independent equations. In this case the unknown amplitude and unknown frequency become entangled in a way that won't allow separation. So you get to do one of a couple of things. Either just punt because the argument to the acos() is 0/0 or if you were applying this 3 pt formula to a stream of data, just fill in the gap with the average of the values found on either side. Or if you know the amplitude and y1==0, f = (fs/2pi)*asin(y2/A) If one is implementing an FSK demodulator, one probably already has good knowledge of the amplitude. Also in the case of FSK, skip the acos() and work directly in the inverse trig function space to save computation time - doesn't help the /0 though. Clay
Reply by ●September 8, 20052005-09-08
glen herrmannsfeldt wrote:> Stan Pawlukiewicz wrote: > > (snip) > >> The problem with this, is that in quantum mechanics the location of >> the particle is governed by a probability density which is related to >> the observation. In signal processing, the transform broadens but >> the location of peak governs the frequency measurement. The frequency >> content of a truncated signal increases but its not a probability >> density, i.e. the true frequency isn't a random parameter governed by >> chance, like a quantum particle. > > > Say I have a signal that in frequency space has a broad peak, > maybe not so peak shaped at all. What do I call the frequency > of that signal? Somewhere within the peak, but where? > > -- glen >There's a difference between uncertainty and trying to stuff a square peg in a round hole. In discussing the uncertainty principle, in this forum and in the past, the context has been either measuring the frequency of a single sine wave or the ability to measure the frequencies of two closely spaced sine waves. Would you agree that there is a difference between uncertainty and ambiguity?
Reply by ●September 8, 20052005-09-08
Stan Pawlukiewicz <spam@spam.mitre.org> writes:> Now, I know for a fact Peter that you know about parametric estimation > and the CRLB, so what's up with the Heisenberg witchcraft.:-) Just stirring the pot a little to see what comes out. Ciao, Peter K.
Reply by ●September 8, 20052005-09-08
"Clay S. Turner" <Physics@Bellsouth.net> writes:> I think the Fourier uncertainty principle has an application in signal > processing, but I also think one has to be cognizant of the underlying > assumptions. When a theory says something can't be done, it only means that > it can't be done given the assumptions are true.Precisely. I'm just not sure what assumptions are being made where. This discussion has shed some light. Thanks, Peter K.
Reply by ●September 8, 20052005-09-08
Jerry Avins <jya@ieee.org> writes:> Randy Yates wrote: >> Firstly, did I say these notions had a direct connection to a >> real-world >> system? >> Secondly, did I say you could do this with a single ADC? > > Most of our discussions are about things that can be measured. When > that's not the case, I expect that to be explicit. You gave a hint > when you wrote "analytic"; I suppose that was enough.Jerry, I didn't mean to be cruel, but it sometimes seems that I must make my logical as sharp as possible in order to get through to you. I didn't want to get sucked into a long discussion regarding practical limitations because in my opinion that's not what this thread is about. But, though it is highly contrived, here is a real system that allows you to do something close to the sort of measurement I was alluding to. Have a DSP generate a complex exponential signal in software. Convert that into analog using two DACs, one for I and one for Q. Feed that into a pair of ADCs, again an I and a Q, which feed DSP #2. Come up with software on DSP #2 to estimate the frequency of the signal DSP #1 is producing as quickly as possible. If we go with the uncertainty voo-doo, folks would say that we can't measure the frequency accurately in fewer than xyz samples. I say we can measure it (the instantaneous frequency) accurately in two samples. -- % Randy Yates % "Ticket to the moon, flight leaves here today %% Fuquay-Varina, NC % from Satellite 2" %%% 919-577-9882 % 'Ticket To The Moon' %%%% <yates@ieee.org> % *Time*, Electric Light Orchestra http://home.earthlink.net/~yatescr
Reply by ●September 8, 20052005-09-08
Randy Yates <yates@ieee.org> writes:> Jerry Avins <jya@ieee.org> writes: > >> Randy Yates wrote: >>> Firstly, did I say these notions had a direct connection to a >>> real-world >>> system? >>> Secondly, did I say you could do this with a single ADC? >> >> Most of our discussions are about things that can be measured. When >> that's not the case, I expect that to be explicit. You gave a hint >> when you wrote "analytic"; I suppose that was enough. > > Jerry, > > I didn't mean to be cruel, but it sometimes seems that I must make my > logical as sharp as possible in order to get through to you. I didn't > want to get sucked into a long discussion regarding practical > limitations because in my opinion that's not what this thread is > about. > > But, though it is highly contrived, here is a real system that allows > you to do something close to the sort of measurement I was alluding > to. Have a DSP generate a complex exponential signal in > software. Convert that into analog using two DACs, one for I and one > for Q. Feed that into a pair of ADCs, again an I and a Q, which feed > DSP #2. Come up with software on DSP #2 to estimate the > frequency of the signal DSP #1 is producing as quickly as possible. > > If we go with the uncertainty voo-doo, folks would say that we can't > measure the frequency accurately in fewer than xyz samples. I say we > can measure it (the instantaneous frequency) accurately in two samples.In fact we can do better than this, in case you or anyone else were considering making arguments based on, e.g., the finite response of the DAC reconstruction filters and the ADC anti-aliasing filters. Transfer the digital samples from DSP #1 to DSP #2 via a digital interface (e.g., a UART). Voila! No analog anomalies to deal with. -- % Randy Yates % "How's life on earth? %% Fuquay-Varina, NC % ... What is it worth?" %%% 919-577-9882 % 'Mission (A World Record)', %%%% <yates@ieee.org> % *A New World Record*, ELO http://home.earthlink.net/~yatescr
Reply by ●September 9, 20052005-09-09
Randy Yates wrote:> Jerry Avins <jya@ieee.org> writes: > > >>Randy Yates wrote: >> >>>Firstly, did I say these notions had a direct connection to a >>>real-world >>>system? >>>Secondly, did I say you could do this with a single ADC? >> >>Most of our discussions are about things that can be measured. When >>that's not the case, I expect that to be explicit. You gave a hint >>when you wrote "analytic"; I suppose that was enough. > > > Jerry, > > I didn't mean to be cruel, but it sometimes seems that I must make my > logical as sharp as possible in order to get through to you. I didn't > want to get sucked into a long discussion regarding practical > limitations because in my opinion that's not what this thread is > about. > > But, though it is highly contrived, here is a real system that allows > you to do something close to the sort of measurement I was alluding > to. Have a DSP generate a complex exponential signal in > software. Convert that into analog using two DACs, one for I and one > for Q. Feed that into a pair of ADCs, again an I and a Q, which feed > DSP #2. Come up with software on DSP #2 to estimate the > frequency of the signal DSP #1 is producing as quickly as possible. > > If we go with the uncertainty voo-doo, folks would say that we can't > measure the frequency accurately in fewer than xyz samples. I say we > can measure it (the instantaneous frequency) accurately in two samples.Except for how we count samples, we seem to agree. (I count two analytic samples as four samples. Two DACs each sample twice.) Our strongest point of agreement lies in what you describe as "voodoo". The circumstance that in the general case, the longer a signal is observed, the more accurately it can be described is entirely unrelated to any notion of quantum mechanics, superficial similarities to some features of the Uncertainty Principle notwithstanding. (What analog of Planck's constant shall one use in DSP?) Nevertheless, practical considerations -- noise, jitter, distortion, and more -- make accurate measurements difficult with short sequences and impossible with the shortest theoretical ones. (When sampling a signal ideally limited to 4 KHz at 8 KHz with a perfect sampler, how many samples are needed to accurately reconstruct a component at 3.999 kHz?) Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●September 9, 20052005-09-09
Stan Pawlukiewicz wrote: (snip)> There's a difference between uncertainty and trying to stuff a square > peg in a round hole. In discussing the uncertainty principle, in this > forum and in the past, the context has been either measuring the > frequency of a single sine wave or the ability to measure the > frequencies of two closely spaced sine waves. Would you agree that > there is a difference between uncertainty and ambiguity?I will agree that they are different, but maybe not as different as some people seem to think. In real systems you never find a single sine wave, that is, from t= negative infinity to positive infinity. When I first learned all this there was the concept of wave packet, which tends to be a sine multiplied by a gaussian or some similar peak shaped function. The peak shaped function then has a width, and, amazingly enough, so does the fourier transform of the combined function. Atomic emission spectra tend to be closer to lorentzian than gaussian, but it works out about the same. If you try to measure the frequency, is the result uncertain or ambiguous? As for your question in this context, I would say you can have ambiguity without uncertainty, but not uncertainty without ambiguity. -- glen
