# [time-nuts] Tbolt issues

Bob kb8tq kb8tq at n1k.org
Fri Sep 2 08:59:11 EDT 2016

```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 at 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.
>>
>>
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