# how to calculate the fourier transform of this input?

I have been trying to find a relation between dominant frequency and the time period 'T'. The input is a digital ramp with an increase of 1mV per step. And each step has a time period 'T'. Will I get the dominant frequency as a function of time period 'T'? if yes how will the equation be? DC can be neglected.

• What dominant frequency? All I see is a sampled ramp. Apr 20, 2021 at 7:05
• @aconcernedcitizen The signal has a step every time period T so it must have a periodic frequency. To put it another way, that is a ramp signal sampled with a train of unit impulses with period of T so also quite periodic. And the zero order hold of such a sampled signal has a sinc frequency envelope. Apr 20, 2021 at 7:33
• Can you please tell me more about it ? In that case there will be a dominant frequency right? Can you tell me how you got the sinc envelope? calculation for that, did you assume it as sum of rectangular wave of different amplitude? @Justme Apr 20, 2021 at 10:17
• Does the ramp continue on forever ? Is the signal 0 for t<0 ? Subtract a continuous time ramp from this signal. you will be left with a (periodic?) triangular wave. It's Fourier transform will be found in text books. Add that Fourier transform to the Fourier transform of the continuous time ramp. Since Fourier transform has linearity property.
– AJN
Apr 20, 2021 at 12:20
• @Justme You are talking about the period of the sampling clock, T, not the period of the signal, itself, which is what I was talking about. If you say this is a ramp, then this is not clear, not from OP's question, not from OP's picture, so it must be a guess from your part. It could also be mine, or anyone else's, but for clarification, OP must say it in clear. And if it is a ramp, then it can't have a period, and therefore, no FT (i.e. what period will the lowest harmonic have?). Apr 20, 2021 at 12:49

It is not clear what the signal is outside the axis range shown in the question. Does the ramp continue on forever ? Is the signal 0 for t<0 ?

# Using linearity property of Fourier Transform

Subtract a continuous time ramp from this signal. you will be left with a (periodic?) triangular wave. It's Fourier transform will be found in text books. It will have a fundamental frequency and harmonics.

To find Fourier transform of the ramp signal, use the Fourier transform of the step signal and property of integration.

Add that Fourier transform to the Fourier transform of the continuous time ramp to get Fourier transform of the total signal (Linearity property).

• Comments about energy of the signal by other posters need to be also considered. Fourier transform may not even exist.
– AJN
Apr 20, 2021 at 12:26

If you write the function as the infinite sum of step functions shifted by a multiple if T, you'll see that there is no fundamental frequency.

You also can't ignore DC because what you described is a signal with infinite energy, so it is non-physical. If the signal you described repeats, then you'd have a fundamentalist frequency of the repeat.

• Oh thanks again, I am stuck with this idea for more than a week. Ideally its infinite length but incase I take it for like 100T(in other words upto 100mV). Will the answer vary? now its not infinite energy. Apr 20, 2021 at 7:20
• Fourier transform assumes periodicity, take a finitely spanned part of this non-periodic infinite signal and assume its periodic and compute fourier @Hari Apr 20, 2021 at 8:00
• A Fourier series assumes periodicity, but Fourier transform does not. So if you do just go up from 0 to 100mV then back down to zero (and do not repeat), you can think of it as the sum of square impulses, each shifted with a delay. That won't have infinite energy, but also does not have a fundamental frequency. You need it to repeat to have a fundamental frequency. Apr 20, 2021 at 17:23

If a signal x(t) has finite energy E, than it can be Fourier transformed.

E is defined as the integral from -∞ to +∞ of the square of the absolute value of the signal x(t):

    +∞
E = ∫ |x(t)|^2 * dt
-∞


If your signal x(t) goes to zero at a certain point T in time, than it can be Fourier transformed.

If the energy E of your signal x(t) is not a finite number, than you might consider taking the Laplace transform.

You can write the signal as a sum of time shifted and stretched rect() signals. Then find the transform from a Fourier transform table.