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I'm trying to understand what frequency domain is. I found general explanations on the Internet, for example:

  • frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies
  • Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time.
  • Fourier series transform a signal from time domain to frequency domain.

But I could not find an example which shows how I can obtain the frequency domain graph using the formula of a signal.

The graphs I found are sometimes discrete like this one: enter image description here

and sometimes continuous like this one:

enter image description here

which seems confusing.

Suppose I have a signal $x:\Bbb R\to \Bbb R$ defined by the formula $x(t) = \cos(6\pi t)e^{-\pi t^2}$.

It's clear how to plot its time domain graph, But how can I find its frequency domain function or graph?

Should I apply fourier transform on it?:

$$\widehat{x}(\tau)=\int_{-\infty}^\infty \cos(6\pi t)e^{-\pi t^2}e^{-2\pi t \tau i}dt$$

Does plotting this new function gives the frequency domain graph? (but it is complex valued, how can it be plotted?)

Or should I find the Fourier series of $x(t)$ and plot the series coefficients discretely?

So my only question is: How can I mathematically obtain the the frequency domain function (and then plot it to get the frequency domain graph) using the formula of $x(t)$?


marked as duplicate by Stanley Pawlukiewicz, MBaz, lennon310, hotpaw2, A_A Jul 30 '18 at 12:36

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  • $\begingroup$ I would say that given the fact your signal has an analytical form, you want to calculate the continuous Fourier Transform. Simply calculate that integral and you can plot the frequency domain graph by taking its absolute value. $\endgroup$ – jojek Jul 26 '18 at 20:14
  • $\begingroup$ Hi: I think you're thinking that the frequency function itself results in something different than the time domain representation of the function. it's doesn't. you still get the same function but it's REPRESENTED by a formula in the frequency domain. Specifcally, In continuous time. $x_t = \int_{-\pi}^{\pi} exp^{i t \lambda} dz (\lambda)$. There is a proof of this seemingly odd relationship in Hannan's "Time Series Analysis" that I usually read one sentence of each month. ( it's not easy ). For an easier explanation , I would read "An Introduction To Fourier Analysis" by R.D Stuart. $\endgroup$ – mark leeds Jul 26 '18 at 20:17
  • $\begingroup$ @markleeds I know that frequency domain representation is equivalent to time domain representation. But how can I find the frequency domain function? The plots seem to represent the frequency domain as another function $\Bbb R \to \Bbb R$. $\endgroup$ – nano Jul 27 '18 at 0:49
  • $\begingroup$ Sorry for my wrong assumption.What you describe is in the book I referenced by Stuart. I'm sure there are others that might be good also but I remember liking that one. Hopefully others can give possibly better recommendations. $\endgroup$ – mark leeds Jul 27 '18 at 7:39

enter image description here

In simple terms, Image shown here speaks for itself.

Before speaking about Fourier Transform black magic lets understand idea behind it. Work of the Mathematician Joseph Fourier demonstrated any arbitrary periodic (this is important) signal can be decomposed into bunch of sine-waves at with their corresponding amplitude and relative phases.

Essentially, he meant basic component to build any signal is a sine-wave. In other words you can build any arbitrary signal exactly if you have information about how many different frequencies are present, amplitude of the each sine wave, and there relative phases. This job can be done by inverse Fourier transform. And as should be clear by now, Fourier Transform does the reverse give it composed time domain signal and it will tell you sine-waves amplitude for corresponding frequencies and there relative phases. Hence, the output is a complex. Which makes sense in polar co-ordinates, which gives you amplitude and phase.

E.G. When you strum a guitar strings, it vibrates at multiple frequencies with different amplitude which makes up a beautiful sound, otherwise you know how a single frequency sounds(watch this :https://www.youtube.com/watch?v=qNf9nzvnd1k). one with the highest amplitude is known as tone.

So, as you can see in the figure attached here, Fourier transform tells you exactly the same information.

Suppose I have a signal $x:\Bbb R\to \Bbb R$ defined by the formula $x(t) = \cos(6\pi x)e^{-\pi x^2}$.

For to obtain Fourier transform of the function you specified it should be function of time (t). So, I will just re-write your function above slightly differently replace x with t, since x(t).

$x(t) = \cos(6\pi t)e^{-\pi t^2}$.

Also, Take a look at this, http://www.thefouriertransform.com/pairs/rightSidedSinusoids.php

Hope this helps.

  • $\begingroup$ that's a pretty cool figure you posted. $\endgroup$ – robert bristow-johnson Jul 26 '18 at 23:25
  • $\begingroup$ I know the general idea that the frequency domain is to show that amplitudes of the sine waves that add up to $x(t)$. But I don't understand what a frequency domain is exactly. Is it the coefficient of the sines and cosines in Fourier Series? Is it the complex valued function obtained by fourier transform? I need a more accurate definition. For example how do you obtain and plot the frequency domain of $x(t)=\cos(6\pi t) e^{-\pi t^2}$? $\endgroup$ – nano Jul 27 '18 at 1:07
  • $\begingroup$ Yes in magnitude plot is the co-efficient of sine/ cosine waves in series. FT is just an analog of Fourier series applied on function which periodic with infinite time (unknown time). Also, check the link I have provided (check equation (4) its a transformered equation to plot frequency domain plug in values for f and A and you get the spectrum shown). It is set of frequencies and A is the amplitude plug in values and you have the spectrum. $\endgroup$ – rahul_b Jul 27 '18 at 22:41

How can it be plotted?

Commonly, only the magnitude of the complex FT result is plotted (or the magnitude and the phase in two separate plots).

  • $\begingroup$ So what are the discrete plots? (magnitude of) Fourier Transform gives a continuous function. And what about the equivalence between the time-domain and frequency-domain plots: if only the magnitude is used for frequency-domain plot there is a huge loss of information and the resulted frequency domain has less information than the time domain and cannot be reversed. $\endgroup$ – nano Jul 27 '18 at 18:29
  • $\begingroup$ Yes. Magnitude-only plots are information lossy (unless the time domain function is purely symmetric). In general, there is no time-frequency equivalence unless you also plot phase. $\endgroup$ – hotpaw2 Jul 28 '18 at 3:26

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