Finding frequency of data given in time domain

Question

Think there is an analog signal source and we are recording it with a data logger. Sampling rate isn't important in here. We know the analog signal is 0 mV when time 0 and goes to 1000mV in 1000 milliseconds. It increases linearly like this. 1 mV for each millisecond. What we can say about its frequency?

To take up the questions from the comments:

(1) What exactly do you mean by "what can we say about its frequency?"

I'm not sure if this signal have measurable frequency. But the question can be answered in any way.

You will probably receive more helpful answers if you provide some more detail about your application or what you are trying to achieve by measuring the "frequency" of a "ramp".

I'm trying to find my analog signal source's frequency.

(2) what do you mean by "frequency is free from sampling rate?"

Its clear, I don't want to get limited by sampling rate. If you say that I can't get true frequency of an analog signal with slow sampling rates, forget it. Think there is no any limitation.

(3) What happens with the signal after the 1000ms. Does it drop to 0 Volt? Does it grow linearly until infinity?

Infinity or not. Its never important. I stop data acquiring after 1 second.

(4) add a sketch of the graph, where the time ranges over a wider time than 0ms-1000ms.

You mean 0ms-1000ms range is not enough to estimate?

(5) specify if you're looking for the spectrum of the analog or the discrete signal. They are not the same.

I said there is an analog signal source and we are recording it with a data logger. It's in time domain. I didn't get what you said. It's analog signal by itself.

Answer 1

Regarding the frequency content of this ramp, I have a two solutions in mind on how we can describe this, and yes what we assume the signal is beyond the time window shown will have an impact, but read on as I believe this will make sense:

Assumption 1: The signal is 0 outside of the interval shown (given what you describe, I believe this is the best assumption).

For this case, the solution is simply the Fourier Transform as follows:

$x(t) = t$, for $ 0<=t<=1$, 0 elsewhere

$$X(\omega) = \int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt = \int_0^1 t e^{-j\omega t}dt$$

Resulting in

$$X(\omega) = \frac{-1+e^{-j\omega}(1+j\omega)}{w^2} $$

To help illustrate, I have included a plot of the magnitude of the spectrum in log scale:

Note that the spectrum is continuous, which is expected when the time domain waveform does not repeat*.

Assumption 2: The signal repeats (is periodic) as shown beyond the interval shown (I don't think this is what you described, but including this explanation in case you were searching for a description that included discrete frequencies). If a time domain signal is periodic, then the frequency domain WILL be discrete, with frequencies that can only exist (if they do) at DC, the repetition rate, and multiples of the repetition rate. The solution above will be the envelope on which these discrete frequencies will exist (the magnitude and phase at each of the frequencies from the solution above). This is the Fourier Series Expansion.

*Note: I talk about the relationship between repetition in one domain and continuity or discreteness in the other domain in more detail in this post: Intuition for sidelobes in FFT

Also for completion and comparison, below is the general form or the Fourier Transform of a ramp; but that would both necessitate that the signal continues to grow beyond the interval shown, AND that we are able to observe it for an infinite amount of time, so I would not agree this is the best answer in comparision to what I provided above.

$$\mathcal{F}\{\text{ramp}(x)\}=(j\pi)\frac{1}{(2\pi)^2}\delta'(f)-\frac{1}{4\pi^2f^2}=\frac{j}{4\pi}\delta'(f)-\frac{1}{4\pi^2f^2}$$

For more details on that see:

https://math.stackexchange.com/questions/1920602/how-does-one-derive-the-fourier-transform-of-the-ramp-function?newreg=a2bfe3515df543b1ade526bfe4db7441

or

http://mathworld.wolfram.com/RampFunction.html