Is Frequency a Fundamental Property of Signals or a Mathematical Construct?

[fisico30]

We all know what temporal frequency is. It is measured in Hz. it is the inverse of the period.
It tells the number of cycles per second etc...
In signal analysis there is the concept of instantaneous frequency as the time derivative of the phase angle (see angle modulations of signals).

When a finite time signal (like all real, physical signal) is Fourier analyzed, it shows a band of frequencies (the shorter the signal the larger the band). There is some sort of uncertainty.

Take the case of a cosine modulated by a Gaussian: [exp(-t^2)]*cos(t).
The time distance between peak is constant, and the instantaneous frequency is constant: equal to 1.
The maximum amplitude is changing in time however (increasing and decreasing to zero).

The finite time signal does not have a pure constant frequency, in the sense that it is not a pure sinusoid.
Why do we have to measure a signal for an infinite amount of time to say that it has a specific, single frequency? Is the idea of frequency strictly tied to the idea of infinite sinusoidal functions?
But those sinusoids are just unphysical, math constructs.
Where is the good in saying that a function representing (the best it can) a finite time signal does not have a single frequency, when compared to an ideal sinusoid?
An emitter starts and ends emitting radiation. Done. Why is it so useful to compare it to ideal, unreal sinusoids? It works mathematically, but maybe I am missing something philosophical about uncertainties...

thanks

 

[Xezlec]

We all know what temporal frequency is. It is measured in Hz. it is the inverse of the period.
It tells the number of cycles per second etc...
In signal analysis there is the concept of instantaneous frequency as the time derivative of the phase angle (see angle modulations of signals).

When a finite time signal (like all real, physical signal) is Fourier analyzed, it shows a band of frequencies (the shorter the signal the larger the band). There is some sort of uncertainty.

Take the case of a cosine modulated by a Gaussian: [exp(-t^2)]*cos(t).
The time distance between peak is constant

Really? I find that hard to believe. Can you prove that?

and the instantaneous frequency is constant: equal to 1.

Again, can you give a proof?

 

[Andy Resnick]

<snip>
Why do we have to measure a signal for an infinite amount of time to say that it has a specific, single frequency? Is the idea of frequency strictly tied to the idea of infinite sinusoidal functions?
But those sinusoids are just unphysical, math constructs.
Where is the good in saying that a function representing (the best it can) a finite time signal does not have a single frequency, when compared to an ideal sinusoid?
An emitter starts and ends emitting radiation. Done. Why is it so useful to compare it to ideal, unreal sinusoids? It works mathematically, but maybe I am missing something philosophical about uncertainties...

thanks

Conceptually, you are correct. Real emitters have a non-zero bandwidth.

However, the utility of Fourier Analysis, of plane waves and spherical waves, sines, cosine and exp(ikz) is too great to simply throw out as a poor approximation. In fact, as long as the time/spatial region of interest is sufficiently large (alternatively, the bandwidth is much less than the central frequency), then it's possible to have a good mathematical approximation using sinusoids, and in so doing gain all the mathematical tools available for signal analysis.

Even so, sometimes it's best to work in so-called 'reciprocal space' (frequency space), because then the detailed temporal or spatial profile of a pulse is less relevant.

 

[fisico30]

Thank you Dr. Resnick.
So I guess I need to get rid of my idea

A real signal oscillates in time the way it wants, and for how long it wants.
The mathematical language of Fourier theory is just a tool, useful to use other processing tools.
Single Frequency, per se, then only indicates a constant frequency (forever so) that only belongs to pure sinusoidal functions.
I like the instantaneous frequency concept, representing how fast a signal changes its current state (its instantaneous phase) with time.

I guess, when talking about lifetime, correlation and so on in radiation processes,
saying that an emitter has a short lifetime means that it gets "disturbed" in its radiating action. As a mathematical consequence, its frequency "linewidth" is large.
The lifetime idea is then very physical to me, while the linewidth idea is more mathematical, and does not directly reflect the ondulatory behavior of the emitted signal.

Instantaneous frequency and the frequency band of the signal are not correlated.
thanks once again!

 

[fisico30]

Hi Xezlec,
this is what i meant

take the signal cos(5*t). The phase is 5t. Take its time derivative and you get 5, a constant. taht is the instantaneous freq.

take the signal cos(5*t^2). Do the same: take the time derivative and get 5t. This means that the inst freq is a function of time.

Bye

 

[fisico30]

actually... 10t sorry

 

[fisico30]

Hello Dr. Resnick,

check this out, in regards to bandwidth. Dr. Paschotta shows that the spectral bandwidth of a signal is physical.


http://www.rp-photonics.com/spotlight_2007_10_11.html"


but I am not sure I get the gendanken experiment described before the beginning of the "Tow pulses section".

.Do you?