Homework Six

Perform the following tasks:

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

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

(c) Do another property on the Wiki and get it reviewed (i.e. review a second property) -- Fourier Transform Properties

Find ${\mathcal {F}}\left[g(t-t_{0})e^{j2\pi f_{0}t}\right]$

-- Using the above definition of complex modulation and the definition from class of a time delay (a.k.a "the slacker function"), I will attempt to show a hybrid of the two...

By definition we know that:

${\mathcal {F}}\left[g(t-t_{0})e^{j2\pi f_{0}t}\right]=\int _{-\infty }^{\infty }\left[g(t-t_{0})e^{j2\pi f_{0}t}\right]e^{-j2\pi ft}\,dt$

Rearranging terms we get:

$\int _{-\infty }^{\infty }\left[g(t-t_{0})e^{j2\pi f_{0}t}\right]e^{-j2\pi ft}\,dt=\int _{-\infty }^{\infty }g(t-t_{0})e^{-j2\pi (f-f_{0})t}\,dt$

Now lets make the substitution $\lambda =t-t_{0}\rightarrow t=\lambda +t_{0}$.
This leads us to:

$\int _{-\infty }^{\infty }g(t-t_{0})e^{-j2\pi (f-f_{0})t}\,dt=\int _{-\infty }^{\infty }g(\lambda )e^{-j2\pi (f-f_{0})(\lambda +t_{0})}\,dt$

After some simplification and rearranging terms, we get:

$\int _{-\infty }^{\infty }g(\lambda )e^{-j2\pi (f-f_{0})(\lambda +t_{0})}\,dt=\int _{-\infty }^{\infty }g(\lambda )e^{-j2\pi (f-f_{0})\lambda }e^{-j2\pi (f-f_{0})t_{0}}\,dt$

Rearranging the terms yet again, we get:

$\int _{-\infty }^{\infty }g(\lambda )e^{-j2\pi (f-f_{0})\lambda }e^{-j2\pi (f-f_{0})t_{0}}\,dt=e^{-j2\pi (f-f_{0})t_{0}}\left[\int _{-\infty }^{\infty }g(\lambda )e^{-j2\pi (f-f_{0})\lambda }\,dt\right]$

We know that the exponential in terms of $\displaystyle t_{0}$ is simply a constant and because of the Fourier Property of complex modualtion, we finally get:

${\mathcal {F}}\left[g(t)e^{j2\pi f_{0}t}\right]=G(f-f_{0})e^{-j2\pi (f-f_{0})t_{0}}$