Contents
Question 1
$ \mathcal{F} (n^2u[n-2] - n^2 u[n+2]) = \sum^{\infty}_{n = -\infty}(n^2u[n-2] - n^2 u[n+2])e^{ j\omega n}\, $
$ =\sum^{\infty}_{n = -\infty}(n^2(u[n-2] - u[n+2])e^{ j\omega n}\, $
$ = \sum^{2}_{n = -2}(n^2 e^{ j\omega n}\, $
$ = 4e^{2 j\omega } + 4e^{-2 j\omega } + e^{ j\omega } + e^{- j\omega }\, $
$ = 8\frac{e^{2 j\omega } + e^{-2 j\omega }}{2}+ 2\frac{e^{ j\omega } + e^{- j\omega }}{2} \, $
$ = 8\cos(2\omega) + 2\cos(-\omega)\, $
Question 2
$ x(t) = \mathcal{F}^{-1} (\mathcal{X}(\omega)) \, $
$ = \frac{1}{2\pi} \pi \mathcal{F}^{-1} (\delta(\omega + \omega _o) + \delta(\omega - \omega _o)) $
$ = \frac{1}{2} \int^{\infty}_{-\infty} (\delta(\omega + \omega _o) + \delta(\omega - \omega _o)) e^{j\omega t} d\omega $
$ = \frac{1}{2} e^{j\omega _o t} \int^{\infty}_{-\infty} \delta(\omega + \omega _o) d\omega + \frac{1}{2} e^{-j\omega _o t} \int^{\infty}_{-\infty}\delta(\omega - \omega _o)) d\omega $
$ = \frac{1}{2} e^{j\omega _o t} + \frac{1}{2} e^{-j\omega _o t} d\omega $
$ = \frac{1}{2} ( e^{j\omega _o t} + e^{-j\omega _o t} ) $
$ = \cos(\omega _o t) \, $
Question 3
a
Only a Real and odd signal can be Fourier transformed into an imaginary and odd function. The Transformed output is a imaginary and even signal,thus the Fourier transform is wrong.
b
A pure imaginary signal and odd function will Fourier transform into a real and odd signal. As the function $ \frac{1}{\sin(\omega)} \, $ is a real and odd signal, Alice answer could be right.
c
As stated above, A pure imaginary signal and odd function will Fourier transform into a real and odd signal. Ths transformed signal is a real and even signal, which is transformed from a even and real signal. Thus Devin answer is wrong.
