[time-nuts] Tbolt issues

[Bill Byrom]

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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