# Calculating the Fourier transform of shifted scaled unit step function

I have $$x_1(t)$$ here. To get $$x_2(t)$$, I need to differentiate $$x_1(t)$$. Express $$x_2(t)$$ as $$2u(t+2)-4u(t)+2u(t-2)$$.
From Fourier transform definition integral, I got $$X_2(j\omega)=\frac{2e^{j\omega 2}}{j\omega}-\frac{4\pi \delta(\omega)}{j\omega}+\frac{2e^{-j\omega2}}{j\omega}$$.
Is this correct? It seems weird and would be complicated to calculate the magnitude and phase spectrum. Thanks!

• Simplifying it, I got $X_2(j\omega) = \frac{4\cos(2\omega)}{j\omega} - \frac{4\pi \delta(\omega)}{j\omega}$ – keanehui Apr 5 '20 at 8:42
• You are on the right path. Since $u(t)$ is not a stable function, it's fourier transform will have discontinuity. – jithin Apr 5 '20 at 9:49
• @jithin that's not right: the rect function (not stable) has the Fourier transform sinc, and that is as continuous as any function will ever get! – Marcus Müller Apr 5 '20 at 12:21
• @MarcusMüller Pssssssst, not all functions have plugged discontinuities. – Cedron Dawg Apr 5 '20 at 12:27
• @CedronDawg :) but really, the rect hint is all I'm willing to give here – I must have tried to derive the Fourier transform of the Heaviside function so many times that I forgot how to do it right, because because I don't get the (ugly) right result when I try to do it again. Keanehul, $u(t)$ is not the function you're looking for, honestly! – Marcus Müller Apr 5 '20 at 12:33

It looks like the idea of that exercise is to compute the Fourier transform of $$x_3(t)$$, and then from that derive the Fourier transform of the original function $$x_1(t)$$.
Note that $$x_3(t)$$ is just a sum of scaled and shifted Dirac impulses, so computing its Fourier transform is trivial. The Fourier transform of $$x_2(t)$$ is then easily derived by multiplying $$X_3(j\omega)$$ by $$\pi\delta(\omega)+1/j\omega$$, and noting that the Dirac delta is cancelled because $$X_3(0)=0$$. So you just end up dividing by $$j\omega$$. In the same way, $$X_1(j\omega)$$ is obtained from $$X_2(j\omega)$$. Also note that none of the three Fourier transforms $$X_1(j\omega)$$, $$X_2(j\omega)$$, and $$X_3(j\omega)$$ have any Dirac impulses.
It's a good idea to cross-check your result by realizing that $$x_1(t)$$ can also be represented as the convolution of two rectangular functions, i.e., its Fourier transform must be a squared sinc-function.
• Thanks! And is it possible to find $X_2(j\omega)$ from only $x_1(t)$ or $x_2(t)$ using the Fourier transform table, which is a hint from the exercise? – keanehui Apr 6 '20 at 3:49
• @keanehui: Sure, you can find $X_2(j\omega)$ from $x_2(t)$ if you know the transform of $u(t)$ and if you know the frequency domain consequence of shifting in the time domain. – Matt L. Apr 6 '20 at 7:53