Homework Four

Fourier Transform Properties

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Nick Christman

1. Find ${ℱ}[g(t)e^{j2\pi f_{0}t}]$

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

${ℱ}[g(t)e^{j2\pi f_{0}t}]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}[g(t)e^{j2\pi f_{0}t}]e^{-j2\pi ft}dt$

Combining terms, we get:

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}[g(t)e^{j2\pi f_{0}t}]e^{-j2\pi ft}dt=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}g(t)e^{-j2\pi (f-f_0)t}dt$

Now let's make the following substitution $\theta =f-f_0$

This now gives us a surprisingly familiar function:

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}g(t)e^{-j2\pi (f-f_0)t}dt=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}g(t)e^{-j2\pi \theta t}dt$

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

We can now conclude that:

${ℱ}[g(t)e^{j2\pi f_{0}t}]=G(\theta )=G(f-f_0)$

Looks good - Kevin