# [time-nuts] "Instantaneous frequency" [WAS: Tbolt issues]

Charles Steinmetz csteinmetz at yandex.com
Sat Sep 10 04:55:05 EDT 2016

```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

```