Homework Four

Fourier Transform Properties

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Nick Christman

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

This is a fairly straightforward property and is known as complex modulation

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

Combining terms, we get:

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

Now let's make the following substitution $\displaystyle \theta =f-f_{0}$

This now gives us a surprisingly familiar function:

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

This looks just like $\displaystyle G(\theta )$!

We can now conclude that:

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

Looks good - Kevin