Jerry Avins <jya@ieee.org> writes:> Randy Yates wrote: > > ... > > > What's a BRM? > > A "binary rate multiplier", a circuit of counter length N that accepts a > master clock F and a digital input number n, and outputs a frequency > F*n/N, Every transition of the output occurs on an edge of the input > clock at such time that the jitter is the theoretical minimum. The > details are simple, but out of place in this group. If there's interest, > I'll start a separate thread.No, but thanks.> > Jerry, I would like to bring this piece of the thread to > > resolution. The frequency counter circuit can produce an accurate > > estimate in a very short amount of time, much less than 1/accuracy. > > Therefore, you have produced one and only one method to estimate > > > frequency that requires such a long time, i.e., the DFT. Don't you > > think that your original statement, > > I didn't think I had produced any.Now you're being forgetful. Remember this? By "straight-forward", I mean the time needed to determine the frequency with a frequency counter or an FFT. Other more indirect methods exist.> The simplest way to understand (and > therefore, in my view, the most straightforward) is simply counting the > number of cycles in 100 seconds and then display the results as Hz, with > a decimal point before the rightmost two digits.I agree this technique gives you your required 1/A time interval. Who said this was representative of reality? Why limit your count of cycles to integer values? There are techniques (use a PLL synth to multiply the input frequency up a factor of N?) one could easily use to reduce the required time period for a given accuracy by an order of magnitude.> > The straight-forward way to measure frequency to .01 Hz requires > > samples for 1/.01 = 100 seconds. > > was a little off? > > > I agree that what is "straightforward" is a matter of interpretation.Yes, this is the issue.> Many commercial frequency counters work exactly like the one that I > called above "the most straightforward" way; other types are considered > "sophisticated". The context in which I made my original assertion, the > analysis of samples taken at times unrelated to the signal being > measured, is already more complex. Getting a result faster than a DFT > can provide it doesn't seem straightforward to me. That said, I don't > expect my view to be universally shared.What is so difficult about digitizing the signal being measured, running the resulting real digital signal through a complex filter with frequency response H(f) = 1, f >=0, 0 otherwise, to obtain a complex signal r[n]e^(i*theta[n]) then reporting the frequency as (theta[n] - theta[n-1]/Ts), Ts = 1/Fs, Fs the sample rate? This can be done in Ts seconds to infinte accuracy in zero-noise conditions. -- Randy Yates Sony Ericsson Mobile Communications Research Triangle Park, NC, USA randy.yates@sonyericsson.com, 919-472-1124
Frequency offset estimation.
Started by ●May 12, 2004
Reply by ●May 17, 20042004-05-17
Reply by ●May 18, 20042004-05-18
Randy Yates wrote:> Jerry Avins <jya@ieee.org> writes: > > >>Randy Yates wrote: >>...>>>Jerry, I would like to bring this piece of the thread to >>>resolution. The frequency counter circuit can produce an accurate >>>estimate in a very short amount of time, much less than 1/accuracy.A true counter, the original and most obvious way, needs a time of 1/accuracy. The frequency indicators that measure period and calculate its reciprocal are not, strictly speaking, counters, but they're called that because counting came first (and is, I think, more straightforward, especially when it comes to explaining in detail how they work).>>>Therefore, you have produced one and only one method to estimate >>>frequency that requires such a long time, i.e., the DFT. Don't you >>>think that your original statement, >> >>I didn't think I had produced any. > > > Now you're being forgetful. Remember this? > > By "straight-forward", I mean the time needed to determine the > frequency with a frequency counter or an FFT. Other more indirect > methods exist.I had indeed forgotten. But notice: that's two ways, not one.>>The simplest way to understand (and >>therefore, in my view, the most straightforward) is simply counting the >>number of cycles in 100 seconds and then display the results as Hz, with >>a decimal point before the rightmost two digits. > > > I agree this technique gives you your required 1/A time interval. Who said > this was representative of reality?It is representative of many commercial frequency counters, or at least was when I dealt with them. It's the way the inexpensive one I bought a few years ago works.> Why limit your count of cycles to > integer values? There are techniques (use a PLL synth to multiply the > input frequency up a factor of N?) one could easily use to reduce the > required time period for a given accuracy by an order of magnitude.PLLs either suffer from FM or need a long time to lock up. I don't see one as part of a good way to measure frequency rapidly and accurately. My bias comes from hardware, so maybe you can enlighten me with a software counterexample.>>> The straight-forward way to measure frequency to .01 Hz requires >>> samples for 1/.01 = 100 seconds. >>> was a little off? >> >> >> I agree that what is "straightforward" is a matter of interpretation. > > > Yes, this is the issue.So is the contention about whether to say tomayto or tomahto? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●May 18, 20042004-05-18
Jerry Avins <jya@ieee.org> writes:> Randy Yates wrote: > >> Jerry Avins <jya@ieee.org> writes: >> >>>Randy Yates wrote: >>> > > ... > >>>>Jerry, I would like to bring this piece of the thread to >>>>resolution. The frequency counter circuit can produce an accurate >>>>estimate in a very short amount of time, much less than 1/accuracy. > > A true counter, the original and most obvious way, needs a time of > 1/accuracy. The frequency indicators that measure period and calculate > its reciprocal are not, strictly speaking, counters, but they're called > that because counting came first (and is, I think, more straightforward, > especially when it comes to explaining in detail how they work). > >>>>Therefore, you have produced one and only one method to estimate >>>>frequency that requires such a long time, i.e., the DFT. Don't you >>>>think that your original statement, >>> >>> I didn't think I had produced any. >> Now you're being forgetful. Remember this? >> By "straight-forward", I mean the time needed to determine the >> frequency with a frequency counter or an FFT. Other more indirect >> methods exist. > > I had indeed forgotten. But notice: that's two ways, not one.Yes, I concede that now. At the time I mae the statement I was thinking the frequency counter example could be easily made to operate in much less than 1/A time.>>>The simplest way to understand (and >>>therefore, in my view, the most straightforward) is simply counting the >>>number of cycles in 100 seconds and then display the results as Hz, with >>>a decimal point before the rightmost two digits. >> I agree this technique gives you your required 1/A time >> interval. Who said >> this was representative of reality? > > It is representative of many commercial frequency counters, or at least > was when I dealt with them. It's the way the inexpensive one I bought a > few years ago works.That's reasonable. I see your point.>> Why limit your count of cycles to >> integer values? There are techniques (use a PLL synth to multiply the >> input frequency up a factor of N?) one could easily use to reduce the >> required time period for a given accuracy by an order of magnitude. > > PLLs either suffer from FM or need a long time to lock up. I don't see > one as part of a good way to measure frequency rapidly and accurately.Good point. You trade off one lengthy operation (averaging time) for another (lock-up time).> My bias comes from hardware, so maybe you can enlighten me with a > software counterexample.I already did, which you snipped and ignored. Digitize the signal being measured, run the resulting real digital signal through a complex filter with frequency response H(f) = 1, f>=0, 0 otherwise, to obtain a complex signal r[n]*e^(i*theta[n]) then estimate the frequency as (theta[n] - theta[n-1]/Ts), Ts = 1/Fs, Fs the sample rate. This can be done in Ts seconds to infinte accuracy in zero-noise conditions.>>>> The straight-forward way to measure frequency to .01 Hz requires >>>> samples for 1/.01 = 100 seconds. >>>> was a little off? >>> >>> >>> I agree that what is "straightforward" is a matter of interpretation. >> Yes, this is the issue. > > So is the contention about whether to say tomayto or tomahto?Not really. I'd rather not phrase this as a personal contention between you and I, Jerry, because at its root, it really isn't. It's about me trying to understand what the hell's going on in that ever-instructive sphere called reality. You are helping (I know it's painful, but I appreciate your bearing with me) because I'm starting to see that, from the perspective of a hardware/analog implementation POV, it really isn't that easy to estimate the frequency quickly. I put forth a digital method that seems to be able to obtain perfect estimates in Ts seconds. Something seems amiss in that the analog and digital worlds would be this disparate in their ability to measure frequency. It is precisely that disparity that has my antenna/red flags up and I'd like to resolve. Perhaps the transient response of the complex filter I proposed would make it impossible to measure the signal immediately and again we would end up having to wait on the order of 1/A to get a measurement of accuracy A? -- % Randy Yates % "Watching all the days go by... %% Fuquay-Varina, NC % Who are you and who am I?" %%% 919-577-9882 % 'Mission (A World Record)', %%%% <yates@ieee.org> % *A New World Record*, ELO http://home.earthlink.net/~yatescr
Reply by ●May 18, 20042004-05-18
Randy Yates <yates@ieee.org> writes:> [...] > Perhaps the transient response of the complex filter I proposed would > make it impossible to measure the signal immediately and again we > would end up having to wait on the order of 1/A to get a measurement > of accuracy A?To clarify, consider two signal cases. In case 1, the theoretical, infinite-extent sinusoid, x1(t) = sin(wt), is input into this measurement system. Then everything is at steady-state and we can measure frequency perfectly in Ts seconds. In case 2, a "gated" sinusoid, x2(t) = u[t - t0] * sin(wt), is input into this system. The conjecture is that the filter, having an impulse response with length > 1, will take some amount of time to stabilize, and during this time, the frequency of the sinusoid will be uncertain. This goes back to that time-frequency uncertainty stuff that I never really understood very well and that I thought didn't apply here. Leon Cohen, the author of a book on time-frequency uncertainty, and I are in communication on this very topic, but no conclusions have been made as yet. -- % 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 ●May 18, 20042004-05-18
Randy Yates wrote:> Jerry Avins <jya@ieee.org> writes: > > >>Randy Yates wrote:...>>>Why limit your count of cycles to >>>integer values? There are techniques (use a PLL synth to multiply the >>>input frequency up a factor of N?) one could easily use to reduce the >>>required time period for a given accuracy by an order of magnitude. >> >>PLLs either suffer from FM or need a long time to lock up. I don't see >>one as part of a good way to measure frequency rapidly and accurately. > > > Good point. You trade off one lengthy operation (averaging time) for another > (lock-up time). > > >>My bias comes from hardware, so maybe you can enlighten me with a >>software counterexample. > > > I already did, which you snipped and ignored.I was unclear. I meant a software example using a PLL. The example I snipped,> Digitize the signal being measured, run the resulting real digital > signal through a complex filter with frequency response H(f) = 1, f>=0, > 0 otherwise, to obtain a complex signal r[n]*e^(i*theta[n]) then > estimate the frequency as (theta[n] - theta[n-1]/Ts), Ts = 1/Fs, Fs > the sample rate. This can be done in Ts seconds to infinte accuracy in > zero-noise conditions.needs a few comments: * Zero-noise conditions are rare at best :-). * Complex quantities in nature can't be measured in a single instant on a single wire. Your method needs two complex samples of a signal on a single wire -- the input to the frequency "counter". I think that needs more time than Ts, maybe much more. (How many taps in your HT?) * The effect of noise on accuracy is not trivially obvious, nor is the amount of noise riding on the signal to be measured necessarily known. * The method seems to me to be clever, to require sophisticated analysis for determining its worth with a given signal, and unlike the straightforward counter, understanding it uses mathematics well beyond cardinal numbers. It's neat, but in my book, a bit roundabout. ...>>So is the contention about whether to say tomayto or tomahto? > > > Not really. I'd rather not phrase this as a personal contention between > you and I, Jerry, because at its root, it really isn't.Contention doesn't even hint at animosity with me. An attempt to resolve a disagreement is a contention, That's often fun (as now) sometimes enlightening (and I love the suspense over who will be enlightened), and only rarely does it engender ill will. [ Two astronomical digressions.] On June 8, there will be a transit of Venus, the first since 1882. The next one will be relatively soon: June 9, 2012. On Cosmic Expansion There was a stargazer named Hubble Who said "We expand like a bubble." But fixing the rate Was a source of debate, Dissention, contention, and trouble. [End digression]> It's about me > trying to understand what the hell's going on in that ever-instructive > sphere called reality. You are helping (I know it's painful, but I > appreciate your bearing with me) because I'm starting to > see that, from the perspective of a hardware/analog implementation > POV, it really isn't that easy to estimate the frequency quickly.And I'm fascinated by the prospect of software doing what physical limitation prevents hardware from doing. It seems like a paradox, but I expect to see the paradox resolved by either 1), the invention of hardware methods that match the software capabilities, or 2), the discovery that for reasons unforeseen, the software methods won't work.> I put forth a digital method that seems to be able to obtain perfect > estimates in Ts seconds. Something seems amiss in that the analog and > digital worlds would be this disparate in their ability to measure > frequency. It is precisely that disparity that has my antenna/red flags > up and I'd like to resolve.I see that scheme as a "paradox" with a type-2 resolution, for some of the reasons noted in my comments about it.> Perhaps the transient response of the complex filter I proposed would > make it impossible to measure the signal immediately and again we > would end up having to wait on the order of 1/A to get a measurement > of accuracy A?Probably some of that on top of what I already mentioned. More astronomy, but not a digression: The ancient Egyptians had the length of a year pretty much down pat: 365 31/128 days. They reckoned time by dividing the period from dawn to dusk into twelve? "hours", so the length of an hour varied from day to day. They had no instruments capable of distinguishing the difference between the lengths of different days. So how did they measure the length of a year so precisely? (365 31/128 = 365.2421875. The accepted actual number is 365.242199.) The apparent accuracy may be partly an artifact of their method of calculating with fractions, but mostly, by observation. They were able to determine the date of the winter solstice give or take a day. Their records going back about 1,500 years left them with an uncertainty of one day in that length of time. Heroic, but as I interpret the word, straightforward. (Their number is about sixty times as accurate as they deserved.) Jerry -- The proof of the meter is in the measurement. �����������������������������������������������������������������������
Reply by ●May 18, 20042004-05-18
Randy Yates wrote:> Randy Yates <yates@ieee.org> writes: > >>[...] >>Perhaps the transient response of the complex filter I proposed would >>make it impossible to measure the signal immediately and again we >>would end up having to wait on the order of 1/A to get a measurement >>of accuracy A? > > > To clarify, consider two signal cases. In case 1, the theoretical, > infinite-extent sinusoid, x1(t) = sin(wt), is input into this > measurement system. Then everything is at steady-state and we can > measure frequency perfectly in Ts seconds. > > In case 2, a "gated" sinusoid, x2(t) = u[t - t0] * sin(wt), is input > into this system. The conjecture is that the filter, having an impulse > response with length > 1, will take some amount of time to stabilize, > and during this time, the frequency of the sinusoid will be uncertain. > > This goes back to that time-frequency uncertainty stuff that I never > really understood very well and that I thought didn't apply here. Leon > Cohen, the author of a book on time-frequency uncertainty, and I are > in communication on this very topic, but no conclusions have been made > as yet.Go out and try to build one (whatever it may be). Either it will work as planned or you'll be able to learn why not. In the first case you'll be triumphant; in the second, wiser. Reality is a patient and thorough teacher. This harks back to our earlier thrashing out of complex samples. They have nifty uses that you taught me, and I hope that I was able to teach you that you can't send them on a single wire, or acquire them in a single sample with a single physical A-to-D. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●May 18, 20042004-05-18
Jerry Avins wrote:> ... a few comments: > > * Zero-noise conditions are rare at best :-). > * Complex quantities in nature can't be measured in a single instant on > a single wire. Your method needs two complex samples of a signal on a > single wire -- the input to the frequency "counter". I think that > needs more time than Ts, maybe much more. (How many taps in your HT?) > * The effect of noise on accuracy is not trivially obvious, nor is the > amount of noise riding on the signal to be measured necessarily known. > * The method seems to me to be clever, to require sophisticated analysis > for determining its worth with a given signal, and unlike the > straightforward counter, understanding it uses mathematics well beyond > cardinal numbers. It's neat, but in my book, a bit roundabout.more: * Digitizing usually requires a signal to have been passed through an anti-alias filter, hardly a one-Ts operation. One of the lessons here is that how fast we can calculate on samples isn't the whole story. In fact, the time to acquire them may dominate. Measuring the frequency of days (the basic unit being the year) to the presently accepted accuracy by counting days would take about 60,000 years. Obviously, there are faster ways. But then, the ancients couldn't know or care about leap seconds. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●May 18, 20042004-05-18
Jerry Avins <jya@ieee.org> writes:> Randy Yates wrote: > > > Jerry Avins <jya@ieee.org> writes: > > > > >>Randy Yates wrote: > > ... > > >>>Why limit your count of cycles to > >>>integer values? There are techniques (use a PLL synth to multiply the > >>>input frequency up a factor of N?) one could easily use to reduce the > >>>required time period for a given accuracy by an order of magnitude. > >> > >>PLLs either suffer from FM or need a long time to lock up. I don't see > >>one as part of a good way to measure frequency rapidly and accurately. > > Good point. You trade off one lengthy operation (averaging time) for > > another > > > (lock-up time). > > > > >>My bias comes from hardware, so maybe you can enlighten me with a > >>software counterexample. > > I already did, which you snipped and ignored. > > > I was unclear. I meant a software example using a PLL.Gotcha.> The example I snipped, > > > > Digitize the signal being measured, run the resulting real digital > > signal through a complex filter with frequency response H(f) = 1, > > f>=0, 0 otherwise, to obtain a complex signal r[n]*e^(i*theta[n]) > > then > > > estimate the frequency as (theta[n] - theta[n-1]/Ts), Ts = 1/Fs, Fs > > the sample rate. This can be done in Ts seconds to infinte accuracy in > > zero-noise conditions. > > needs a few comments: > > * Zero-noise conditions are rare at best :-).I'm just comparing apples to apples. If you inject a noiseless signal into your frequency counter, it still takes 100 seconds to measure to 0.01 Hz.> * Complex quantities in nature can't be measured in a single instant on > a single wire. Your method needs two complex samples of a signal on a > single wire -- the input to the frequency "counter". I think that > needs more time than Ts, maybe much more. (How many taps in your HT?)I believe you're just restating my idea, i.e., the transient response of the filter might come into play.> * The effect of noise on accuracy is not trivially obvious, nor is the > amount of noise riding on the signal to be measured necessarily known.Just say No to noise. :) (See above.)> * The method seems to me to be clever, to require sophisticated analysis > for determining its worth with a given signal, and unlike the > straightforward counter, understanding it uses mathematics well beyond > cardinal numbers. It's neat, but in my book, a bit roundabout.It's a filter, an arctangent, and a division. In the realm of DSP, not too sophisticated, in my opinion. All real sinusoids have a symmetric frequency response. Hacking off one side (positive or negative) of the frequency-domain leaves a single complex sinusoid, from which you can get delta theta/delta t. -- Randy Yates Sony Ericsson Mobile Communications Research Triangle Park, NC, USA randy.yates@sonyericsson.com, 919-472-1124
Reply by ●May 18, 20042004-05-18
Jerry, Randy and others, You may find the following document on timekeeping very interesting. It applies to the frequency estimation in the sense of how well can we measure time. I.e., measure the period of a wave and then find its frequency. http://literature.agilent.com/litweb/pdf/5965-7984E.pdf -- Clay S. Turner, V.P. Wireless Systems Engineering, Inc. Satellite Beach, Florida 32937 (321) 777-7889 www.wse.biz csturner@wse.biz "Jerry Avins" <jya@ieee.org> wrote in message news:40aa545f$0$3035$61fed72c@news.rcn.com...> Jerry Avins wrote: > > > ... a few comments: > > > > * Zero-noise conditions are rare at best :-). > > * Complex quantities in nature can't be measured in a single instant on > > a single wire. Your method needs two complex samples of a signal on a > > single wire -- the input to the frequency "counter". I think that > > needs more time than Ts, maybe much more. (How many taps in your HT?) > > * The effect of noise on accuracy is not trivially obvious, nor is the > > amount of noise riding on the signal to be measured necessarily known. > > * The method seems to me to be clever, to require sophisticated analysis > > for determining its worth with a given signal, and unlike the > > straightforward counter, understanding it uses mathematics well beyond > > cardinal numbers. It's neat, but in my book, a bit roundabout. > more: > * Digitizing usually requires a signal to have been passed through an > anti-alias filter, hardly a one-Ts operation. > > One of the lessons here is that how fast we can calculate on samples > isn't the whole story. In fact, the time to acquire them may dominate. > > Measuring the frequency of days (the basic unit being the year) to the > presently accepted accuracy by counting days would take about 60,000 > years. Obviously, there are faster ways. But then, the ancients couldn't > know or care about leap seconds. > > Jerry > -- > Engineering is the art of making what you want from things you can get. > ����������������������������������������������������������������������� >
Reply by ●May 18, 20042004-05-18
Randy Yates wrote:> Jerry Avins <jya@ieee.org> writes: > > >>Randy Yates wrote: >> >> >>>Jerry Avins <jya@ieee.org> writes: >>> >> >>>>Randy Yates wrote: >> >> ... >> >> >>>>>Why limit your count of cycles to >>>>>integer values? There are techniques (use a PLL synth to multiply the >>>>>input frequency up a factor of N?) one could easily use to reduce the >>>>>required time period for a given accuracy by an order of magnitude. >>>> >>>>PLLs either suffer from FM or need a long time to lock up. I don't see >>>>one as part of a good way to measure frequency rapidly and accurately. >>> >>>Good point. You trade off one lengthy operation (averaging time) for >>>another >> >>>(lock-up time). >>> >> >>>>My bias comes from hardware, so maybe you can enlighten me with a >>>>software counterexample. >>> >>>I already did, which you snipped and ignored. >> >> >>I was unclear. I meant a software example using a PLL. > > > Gotcha. > > >>The example I snipped, >> >> >> >>>Digitize the signal being measured, run the resulting real digital >>>signal through a complex filter with frequency response H(f) = 1, >>>f>=0, 0 otherwise, to obtain a complex signal r[n]*e^(i*theta[n]) >>>then >> >>>estimate the frequency as (theta[n] - theta[n-1]/Ts), Ts = 1/Fs, Fs >>>the sample rate. This can be done in Ts seconds to infinte accuracy in >>>zero-noise conditions. >> >>needs a few comments: >> >>* Zero-noise conditions are rare at best :-). > > > I'm just comparing apples to apples. If you inject a noiseless signal > into your frequency counter, it still takes 100 seconds to measure to > 0.01 Hz.With a counter, either the noise creates extra zero crossings, or it doesn't matter. Some counters select a low-performance low-pass filter with the range (gate time) switch to suppress high-frequency noise that might create a spurious zero crossing very near the real one. A signal has to look dirty on a scope before a counter becomes inaccurate; say, 10 dB SNR. How much noise can the phi_2 - phi_1 method tolerate to get .01% accuracy? A counter doesn't really indicate cycles per second. In reality, it indicates zero crossings per second, which may not be exactly the same thing in the presence of large amounts of noise. Still a front end that squares up the signal (like the limiter in an FM receiver) adds much robustness, and a simple RC rolloff before the zero-crossing detector adds more. The noise performance of the period measurer (with or without conversion to frequency) is less robust and harder to assess. Noise can more easily move a zero crossing than it can create one. Measuring more periods gives more precision -- the number of them is in the denominator -- and divides the noise-induced jitter among them. Again. the longer the measurement, the more accurate the result. (There ain't no free lunch.)>>* Complex quantities in nature can't be measured in a single instant on >> a single wire. Your method needs two complex samples of a signal on a >> single wire -- the input to the frequency "counter". I think that >> needs more time than Ts, maybe much more. (How many taps in your HT?) > > > I believe you're just restating my idea, i.e., the transient response > of the filter might come into play.Yes. You mentioned that further down as a generality. I was specific here.>>* The effect of noise on accuracy is not trivially obvious, nor is the >> amount of noise riding on the signal to be measured necessarily known. > > > Just say No to noise. :) (See above.)How?>>* The method seems to me to be clever, to require sophisticated analysis >> for determining its worth with a given signal, and unlike the >> straightforward counter, understanding it uses mathematics well beyond >> cardinal numbers. It's neat, but in my book, a bit roundabout. > > > It's a filter, an arctangent, and a division. In the realm of DSP, > not too sophisticated, in my opinion.De Gustibus ... Which scheme would you more likely be able to explain to a group of Cub Scouts?> All real sinusoids have a symmetric frequency response. Hacking off > one side (positive or negative) of the frequency-domain leaves a > single complex sinusoid, from which you can get delta theta/delta t.Gold! How, exactly? What procedure will create analytic samples from real ones without the latency of a Hilbert transformer? The glimmering I have involves a complex-coefficient filter that, as you put it, hacks off the negative frequencies. But what about that filter's latency? Is it much shorter than the Hilbert transformer? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
