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Problem Statement

6(a) Show ${ℱ}[\underset{-\mathrm{\infty }}{\overset{t}{\int }}s(\lambda )d\lambda ]=\frac{S(f)}{j2\pi f}\text{ if }S(0)=0$.

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

Answer

a)

Remember dummy variable $\lambda =t-t_0$ Then $s(\lambda )=s(t-t_0)={ℱ}[S(f)-S(f_0)]$ and $\underset{-\mathrm{\infty }}{\overset{t}{\int }}s(\lambda )d\lambda =\underset{-\mathrm{\infty }}{\overset{t}{\int }}{ℱ}[S(f)-S(f_0)]d\lambda$

$f_0=0$ where $S(0)=S(f){|}_{f=0}=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)e^{-j2\pi ft}dt=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)dt$

$\text{ if }S(0)=0\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)dt=0$

$\underset{-\mathrm{\infty }}{\overset{t}{\int }}s(\lambda )d\lambda =\underset{-\mathrm{\infty }}{\overset{t}{\int }}{ℱ}[S(f)]dt$

${{ℱ}}^{-1}[S(f)-S(f_0)]=\underset{-\mathrm{\infty }}{\overset{t}{\int }}{e}^{j2\pi ft}dt\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}S(f)df=\frac{e^{j2\pi ft}}{j2\pi f}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}S(f)df=$

$\underset{-\mathrm{\infty }}{\overset{t}{\int }}s(\lambda )d\lambda =\underset{\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}S(f)\frac{e^{j2\pi ft}}{j2\pi f}df={{ℱ}}^{-1}\left[\frac{S(f)}{j2\pi f}\right]$

Therefore ${ℱ}[\underset{-\mathrm{\infty }}{\overset{t}{\int }}s(\lambda )d\lambda ]=\frac{S(f)}{j2\pi f}$