ASN6 a,b- fixing

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Problem Statement

6(a) Show ${\mathcal {F}}\left[\int _{-\infty }^{t}s(\lambda )\,d\lambda \right]={\frac {S(f)}{j2\pi f}}{\mbox{ if }}S(0)=0$. Hint: $S(0)=S(f)\vert _{_{f=0}}=\int _{-\infty }^{\infty }s(t)e^{-j2\pi (f\rightarrow 0)t}\,dt=\int _{-\infty }^{\infty }s(t)\,dt$

6(b) If $S(0)\neq 0$ can you find ${\mathcal {F}}\left[\int _{-\infty }^{t}s(\lambda )\,d\lambda \right]$ in terms of $\displaystyle S(0)$?

Answer

a)

Remember dummy variable $\lambda =t-t_{0}\!$ Then $s(\lambda )=s(t-t_{0})={\mathcal {F}}\left[S(f)-S(f_{0})\right]\!$ and $\int _{-\infty }^{t}s(\lambda )\,d\lambda =\int _{-\infty }^{t}{\mathcal {F}}\left[S(f)-S(f_{0})\right]\,d\lambda \!$

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

${\mbox{ if }}S(0)=0\,\,\,\int _{-\infty }^{\infty }s(t)dt=0\!$

$\int _{-\infty }^{t}s(\lambda )\,d\lambda =\int _{-\infty }^{t}{\mathcal {F}}\left[S(f)\right]\,d\lambda \!$

${\mathcal {F}}\left[S(f)-S(f_{0})\right]=\int _{-\infty }^{t}e^{-j2\pi ft}\int _{-\infty }^{\infty }S(f)e^{-j2\pi ft}\,df={\frac {e^{-j2\pi ft}}{-j2\pi f}}\int _{-\infty }^{\infty }S(f)\,df$