6 - Fourier Transform 2

(a) Show ${ℱ}[\underset{-\mathrm{\infty }}{\overset{t}{\int }}s(\lambda )d\lambda ]=\frac{S(f)}{j2\pi f}\text{ if }S(0)=0$. Hint: $S(0)=S(f){|}_{{f=0}_{}}=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)e^{-j2\pi (f\to 0)t}dt=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)dt$

(b) If $S(0)\ne 0$ can you find ${ℱ}[\underset{-\mathrm{\infty }}{\overset{t}{\int }}s(\lambda )d\lambda ]$ in terms of $S(0)$?

(c) Do another property on the Wiki and review a second property

Find ${ℱ}[e^{j2\pi f_{0}t}s(t)]$
First ${ℱ}[e^{j2\pi f_{0}t}s(t)]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j2\pi f_{0}t}s(t)e^{-j2\pi ft}$
or rearranging we get $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j2\pi f_{0}t}s(t)e^{-j2\pi ft}dt=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)e^{j2\pi t(f_0-f)}dt$
Which leads to $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)e^{j2\pi t(f_0-f)}dt=S(f-f_0)$
So ${ℱ}[e^{j2\pi f_{0}t}s(t)]=S(f-f_0)$

Reviewed Nicks 2nd Fourier transform made comment about one possible error other than that looked good