The problem is that "frequency" has more than one meaning. The main dictionary definitions have to do with the frequency of occurrence of some items in a category with respect to a larger set, or the frequency of occurrence of some repeating event per unit of time. But we also use mathematical representations of waveforms containing a "frequency" or "angular frequency" parameter, and we can also define waveforms where the frequency parameter is itself a function over time. In these cases there obviously is an instantaneous frequency which for example represents the value of f at a particular value of t in sin(2 pi f t), where f = somefunction(t). So you have discrete events (a rising edge, or the positive zero crossing of a sinusoidal waveform) which define a "frequency" property which only has meaning when we compare the time values of at least two of these events, but we also have an equation defining a sinewave, where the instantaneous angular frequency describes the derivative of the phase change vs time. You have to consider continuous as well as discrete systems. In modern modulation theory the concept of vector modulation is used. This involves a carrier wave frequency and amplitude, then I/Q or vector modulation which instantaneously varies the amplitude (vector length) and phase (vector angle) of the signal. For a constant amplitude signal, the derivative of the vector modulation phase (arctangent of the I/Q ratio) corresponds to the instantaneous frequency. At work I deal with equipment which generates RF signal using a 50 GS/s maximum sampling rate D/A converter, which provides one sample every 20 ps. I can create a linear frequency up-chirp using this instrument with a frequency modulation slope of 2 MHz per us (microsecond) at a center frequency of 1 GHz. So there are 50,000 D/A samples each us, and although the average frequency over that us is 1 GHz (50 D/A samples/cycle), the start of the chirp is at 999 MHz (about 50.05 D/A samples/cycle) while the end of the chirp 1 us later is at 1001 MHz (about 49.95 D/A samples/cycle). In this case, the value of somefunction(T0 - 1 us) = 999 MHz and somefunction(T0 + 1 us) = 1001 MHz, where T0 is the time at the middle of the chirp. There are obviously not an integral number of D/A samples per sinewave cycle, but that is no problem. The D/A has 10 bits of resolution and is not perfect, and the combination of jitter and other errors produces wideband noise and spurs smeared over the frequency range of DC to the Nyquist rate, but these errors are very small (many 10's of dB down from the desired signal). The signal I just described creates the 2 MHz chirp in a 1 us time interval using 50,000 D/A samples. The 10-bit resolution voltage values of each of those samples (spaced by 20 ps) select the closest D/A values which represent the sine function with an "instantaneous frequency" given by somefunction (which in this case is a linear ramp). So you can think of this as a discrete system which is changing the instantaneous frequency every 20 ps (with instrument errors due to the limited 10-bit voltage resolution, amplitude errors, jitter errors, and errors from other sources). On the measurement side, I have an instrument with a 16-bit 400 MS/s A/D which can sample a superheterodyne downconverted signal at an IF frequency over a 165 MHz span. Those samples are run through a DDC (digital downconverter using a Hilbert filter) to create two 200 MS/s streams (I and Q waveforms). For the example above, the 1 us 2 MHz wide linear chirp is sampled with 200 I/Q points, and calculating the derivative (slope) of the phase - which is arctangent(I/Q) - results in a frequency vs time trace. So the instantaneous frequency can be measured with 5 ns resolution (1/200 MS/s I/Q rate) in time across that 1 us wide frequency chirp. So yes, the concept of "instantaneous frequency" is valid and is used everyday in many practical measurements on phase locked loop frequency synthesizers, radars, testing Bluetooth FSK transmitters, and for many other applications. -- Bill Byrom N5BB On Thu, Sep 1, 2016, at 10:39 PM, jimlux wrote: > On 9/1/16 5:51 PM, Charles Steinmetz wrote: >> Nick wrote: >> >>> On a theoretical basis, can one speak of the limit of the frequency >>> observed as tau approaches zero? >>> Might that in some way be the "instantaneous frequency" which people >>> often think of? >> >> That is (or is "something like") what it **would** be, but a little >> thought experiment will show that (and why) the linguistic >> construction >> is meaningless. >> >> The period of a 10MHz sine wave is 100nS. Think about observing >> it over >> shorter and shorter (but still finite) time intervals. >> >> When the time interval is 100nS, we see one complete cycle (360 >> degrees, >> 2 pi radians) of the wave. At this point we still have **some** >> shot at >> deducing its frequency, because no matter at what phase we >> start, we are >> guaranteed to observe two peaks (one high, one low) and at least one >> midpoint (e.g., zero-cross). Our deduction (inference) will be less >> accurate as the noise and distortion (harmonic content) >> increases, and >> it won't be all that good under the best of circumstances. >> >> Now shorten the observation time to 20nS. We see 1/5 of a complete >> cycle (72 degrees, 0.4 pi radians) of the wave. No matter which >> particular 72 degrees we see, we simply don't have enough >> information to >> reliably deduce the frequency. > > in fact, there's a whole literature on how accurate (or more > precisely, > what's the uncertainty) of the frequency estimate is. > > We often measure frequencies with less than a cycle - but making some > assumptions - measuring orbital parameters is done using a lot > less than > a complete orbit's data, but we also make the assumption of the > physics > involved. > > > --- > > Instantaneous frequency does have a theoretical meaning, even if not > measureable.. > > If I'm processing a linear frequency chirp, I can say that the > frequency at time t is some (f0 + t*slope). the frequency at time > t+epsilon is different, as is the frequency at time t-epsilon. > > > _________________________________________________ > time-nuts mailing list -- time-nuts@febo.com > To unsubscribe, go to > https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts > and follow the instructions there.

Hi The gotcha in your approach is that you are using more than one sample out of the system to get frequency. Thus you are measuring over a time period. To get instantaneous frequency you need to base it on a single sample. There are some other restrictions (infinite bandwidth being the big one). Bob > On Sep 2, 2016, at 2:25 AM, Bill Byrom <time@radio.sent.com> wrote: > > The problem is that "frequency" has more than one meaning. The main > dictionary definitions have to do with the frequency of occurrence of > some items in a category with respect to a larger set, or the frequency > of occurrence of some repeating event per unit of time. But we also use > mathematical representations of waveforms containing a "frequency" or > "angular frequency" parameter, and we can also define waveforms where > the frequency parameter is itself a function over time. In these cases > there obviously is an instantaneous frequency which for example > represents the value of f at a particular value of t in sin(2 pi f t), > where f = somefunction(t). > > So you have discrete events (a rising edge, or the positive zero > crossing of a sinusoidal waveform) which define a "frequency" property > which only has meaning when we compare the time values of at least two > of these events, but we also have an equation defining a sinewave, where > the instantaneous angular frequency describes the derivative of the > phase change vs time. You have to consider continuous as well as > discrete systems. > > In modern modulation theory the concept of vector modulation is used. > This involves a carrier wave frequency and amplitude, then I/Q or vector > modulation which instantaneously varies the amplitude (vector length) > and phase (vector angle) of the signal. For a constant amplitude signal, > the derivative of the vector modulation phase (arctangent of the I/Q > ratio) corresponds to the instantaneous frequency. > > At work I deal with equipment which generates RF signal using a 50 GS/s > maximum sampling rate D/A converter, which provides one sample every 20 > ps. I can create a linear frequency up-chirp using this instrument with > a frequency modulation slope of 2 MHz per us (microsecond) at a center > frequency of 1 GHz. So there are 50,000 D/A samples each us, and > although the average frequency over that us is 1 GHz (50 D/A > samples/cycle), the start of the chirp is at 999 MHz (about 50.05 D/A > samples/cycle) while the end of the chirp 1 us later is at 1001 MHz > (about 49.95 D/A samples/cycle). In this case, the value of > somefunction(T0 - 1 us) = 999 MHz and somefunction(T0 + 1 us) = 1001 > MHz, where T0 is the time at the middle of the chirp. There are > obviously not an integral number of D/A samples per sinewave cycle, but > that is no problem. The D/A has 10 bits of resolution and is not > perfect, and the combination of jitter and other errors produces > wideband noise and spurs smeared over the frequency range of DC to the > Nyquist rate, but these errors are very small (many 10's of dB down from > the desired signal). > > The signal I just described creates the 2 MHz chirp in a 1 us time > interval using 50,000 D/A samples. The 10-bit resolution voltage values > of each of those samples (spaced by 20 ps) select the closest D/A values > which represent the sine function with an "instantaneous frequency" > given by somefunction (which in this case is a linear ramp). So you can > think of this as a discrete system which is changing the instantaneous > frequency every 20 ps (with instrument errors due to the limited 10-bit > voltage resolution, amplitude errors, jitter errors, and errors from > other sources). > > On the measurement side, I have an instrument with a 16-bit 400 MS/s A/D > which can sample a superheterodyne downconverted signal at an IF > frequency over a 165 MHz span. Those samples are run through a DDC > (digital downconverter using a Hilbert filter) to create two 200 MS/s > streams (I and Q waveforms). For the example above, the 1 us 2 MHz wide > linear chirp is sampled with 200 I/Q points, and calculating the > derivative (slope) of the phase - which is arctangent(I/Q) - results in > a frequency vs time trace. So the instantaneous frequency can be > measured with 5 ns resolution (1/200 MS/s I/Q rate) in time across that > 1 us wide frequency chirp. > > So yes, the concept of "instantaneous frequency" is valid and is used > everyday in many practical measurements on phase locked loop frequency > synthesizers, radars, testing Bluetooth FSK transmitters, and for many > other applications. > > -- > Bill Byrom N5BB > > > >> On Thu, Sep 1, 2016, at 10:39 PM, jimlux wrote: >>> On 9/1/16 5:51 PM, Charles Steinmetz wrote: >>> Nick wrote: >>> >>>> On a theoretical basis, can one speak of the limit of the frequency >>>> observed as tau approaches zero? >>>> Might that in some way be the "instantaneous frequency" which people >>>> often think of? >>> >>> That is (or is "something like") what it **would** be, but a little >>> thought experiment will show that (and why) the linguistic >>> construction >>> is meaningless. >>> >>> The period of a 10MHz sine wave is 100nS. Think about observing >>> it over >>> shorter and shorter (but still finite) time intervals. >>> >>> When the time interval is 100nS, we see one complete cycle (360 >>> degrees, >>> 2 pi radians) of the wave. At this point we still have **some** >>> shot at >>> deducing its frequency, because no matter at what phase we >>> start, we are >>> guaranteed to observe two peaks (one high, one low) and at least one >>> midpoint (e.g., zero-cross). Our deduction (inference) will be less >>> accurate as the noise and distortion (harmonic content) >>> increases, and >>> it won't be all that good under the best of circumstances. >>> >>> Now shorten the observation time to 20nS. We see 1/5 of a complete >>> cycle (72 degrees, 0.4 pi radians) of the wave. No matter which >>> particular 72 degrees we see, we simply don't have enough >>> information to >>> reliably deduce the frequency. >> >> in fact, there's a whole literature on how accurate (or more >> precisely, >> what's the uncertainty) of the frequency estimate is. >> >> We often measure frequencies with less than a cycle - but making some >> assumptions - measuring orbital parameters is done using a lot >> less than >> a complete orbit's data, but we also make the assumption of the >> physics >> involved. >> >> >> --- >> >> Instantaneous frequency does have a theoretical meaning, even if not >> measureable.. >> >> If I'm processing a linear frequency chirp, I can say that the >> frequency at time t is some (f0 + t*slope). the frequency at time >> t+epsilon is different, as is the frequency at time t-epsilon. >> >> >> _________________________________________________ >> time-nuts mailing list -- time-nuts@febo.com >> To unsubscribe, go to >> https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts >> and follow the instructions there. > _______________________________________________ > time-nuts mailing list -- time-nuts@febo.com > To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts > and follow the instructions there.

Bill wrote: [lots and lots of snippage - the original message was posted on 9/2 for those who want to review it] > we can also define waveforms where the > frequency parameter is itself a function over time. Agreed. > In these cases there obviously is an instantaneous frequency "Obviously"? Hardly. > the instantaneous angular frequency describes the derivative of the > phase change vs time. Right. As I said, the so-called "instantaneous frequency" is a mathematical fiction that describes the result of a differentiation with respect to a waveform that is constantly changing in frequency. Like many mathematical concepts, it is abstract -- it has no real existence in the world. > At work I deal with equipment which generates RF signal using a 50 GS/s > maximum sampling rate D/A converter, which provides one sample every 20 > ps. I can create a linear frequency up-chirp using this instrument with > a frequency modulation slope of 2 MHz per us > The 10-bit resolution voltage values > of each of those samples (spaced by 20 ps) select the closest D/A values > which represent the sine function with an "instantaneous frequency" > given by somefunction (which in this case is a linear ramp). So you can > think of this as a discrete system which is changing the instantaneous > frequency every 20 ps Yes, you "can think of it as..." and "it represents" "instantaneous frequency." But again, that is just a mathematical fiction, *not* a real feature of the signal in the world. There is no instantaneous frequency, but (like many mathematical constructs) it can be a useful fiction. > On the measurement side, I have an instrument with a 16-bit 400 MS/s A/D > which can sample a superheterodyne downconverted signal at an IF > frequency over a 165 MHz span. Those samples are run through a DDC > (digital downconverter using a Hilbert filter) Note that both of these hardware examples operate in the finite time domain, not the instantaneous (infinitessimal time) domain. They are no different from my example at 10MHz, except that the decimal point has been moved several decades. But no matter how short a finite interval you use, it is still an infinity away from "instantaneous" (a single point in time of zero duration). In both cases, the so-called "instantaneous frequency" is derived by differentiating a finite-time measurement. In neither case is the frequency measured instantaneously. It *cannot* be, either in practice or, more importantly, in theory, for a very good reason -- it is not a real entity, it is only a mathematical fiction. Useful to we engineers, but not real. Best regards, Charles