For example, if I have two signals, $\cos(2\pi ft+\frac\pi4)+\cos(2\pi ft+\frac\pi3)$, what would be different in both transforms (Fourier and cosine) how would the spectrum changes?

And What would happened for a signal like $\cos(2\pi ft)+\cos(2\pi ft+0.99\pi)$.


These integral transforms are linear, which means the transform of a sum is identical to the sum of the individual transforms, i.e.

$\mathcal F\left\{\cos(a)+\cos(b)\right\}= \mathcal F\left\{\cos(a)\right\}+\mathcal F\left\{\cos(b)\right\}$,

and also, that when multiplied with a constant, that constant can be "inside" or "outside" the transform:

$\mathcal F\left\{\alpha g(t) \right\}=\alpha\mathcal F\left\{g(t) \right\}\, \forall \alpha \text{ const.}$

No magic happening here; since (Euler's formula! It's important!):

$$\begin{align} \mathcal F\left\{\cos(a)\right\}&=\mathcal F\left\{\frac 12\left(e^{i a}+e^{-ia}\right)\right\}\\ &=\frac12\mathcal F\left\{e^{i a}+e^{-ia}\right\}\quad\quad\quad\quad\quad||\,a=2\pi f t + \varphi\\ &=\frac12\mathcal F\left\{e^{i(2\pi ft + \varphi)}+e^{-i(2\pi ft + \varphi)}\right\}\\ &=\frac12\mathcal F\left\{e^{i 2\pi ft}\cdot e^{i\varphi}+e^{-i 2\pi ft }\cdot e^{-i\varphi}\right\}\\ &=\frac12\mathcal F\left\{e^{i 2\pi ft}\cdot e^{i\varphi}\right\}+\frac12\mathcal F\left\{e^{-i 2\pi ft }\cdot e^{-i\varphi}\right\}\\ &=\frac{e^{i\varphi}}2\mathcal F\left\{e^{i 2\pi ft}\right\}+\frac{e^{-i\varphi}}2\mathcal F\left\{e^{-i 2\pi ft }\right\}\tag{*}\\ \end{align}$$

we see that for any fixed phase $\varphi$ of a cosine, for example $\frac\pi 3$ or $\frac\pi 4$, the Fourier Transform "consists" of two dirac impulses $\delta$ with positions that only depend on the cosine's frequency $f$ and with amplitudes that depend only on the phase offset.

Now, just insert your different $\varphi$ in $(*)$ and add up those sums of Fourier Transforms to see the very basic point:

What you're modelling here, mathematically, is the interference of two waves at the same frequency. Tada! Physical significance added.

Now: good luck with the rest of your homework.

  • $\begingroup$ Dear @Marcus , I am graduated and this is not homework, please note that my question is related to compare Fourier with cosine regarding the phase shift? is the fourier transform invarient to phase shift? $\endgroup$ Jul 19 '16 at 17:43
  • $\begingroup$ I answered the question you're asking in my answer. You can clearly see the phase shift $\varphi$ in the result. Hence, things obviously can't be phase-shift invariant. $\endgroup$ Jul 19 '16 at 20:39
  • $\begingroup$ But what would happen if we are interested in the absolute value of the result, it will be the same amplitude in frequency domain no matter how much is the phase, right? $\endgroup$ Jul 22 '16 at 19:55
  • $\begingroup$ I gave you a formula. Applying $|\cdot|$ to it is kind of in your hands, isn't it. $\endgroup$ Jul 22 '16 at 21:04
  • $\begingroup$ and no, they will not be the same. I don't know what you graduated in, but basic signal theory should tell you immediately that adding two waves in phase will yield a drastically different result than adding two waves 180° out of phase. $\endgroup$ Jul 22 '16 at 21:09

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