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 <t...@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. _______________________________________________ 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.