Fourier Transform Properties

Some properties to choose from if you are having difficulty....

Max Woesner

1. Find ${ℱ}[cos(w_{0}t)g(t)]$
Recall $w_0=2\pi f_0$, so ${ℱ}[cos(w_{0}t)g(t)]={ℱ}[cos(2\pi f_{0}t)g(t)]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}cos(2\pi f_{0}t)g(t)e^{-j2\pi ft}dt$
Also recall $cos(\theta )=\frac{1}{2}(e^{j\theta }+e^{-j\theta })$,so $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}cos(2\pi f_{0}t)g(t)e^{-j2\pi ft}dt=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1}{2}[e^{j2\pi f_{0}t}+e^{-j2\pi f_{0}t}]g(t)e^{-j2\pi ft}dt$
Now $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1}{2}[e^{j2\pi f_{0}t}+e^{-j2\pi f_{0}t}]g(t)e^{-j2\pi ft}dt=\frac{1}{2}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-j2\pi (f-f_0)t}g(t)dt+\frac{1}{2}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-j2\pi (f+f_0)t}g(t)dt=\frac{1}{2}G(f-f_0)+\frac{1}{2}G(f+f_0)$
So ${ℱ}[cos(w_{0}t)g(t)]=\frac{1}{2}[G(f-f_0)+G(f+f_0)]$

reviewed by Joshua Sarris

2. Find ${ℱ}[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}g(t)h^{*}(t)dt]$
Recall $g(t)={{ℱ}}^{-1}[G(f)]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}G(f)e^{j2\pi ft}df$
Similarly, $h(t)={{ℱ}}^{-1}[H(f)]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}H(f)e^{j2\pi ft}df$
So ${ℱ}[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}g(t)h^{*}(t)dt]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}G(f^{\text{'}})e^{j2\pi f^{\text{'}}t}d{f}^{\text{'}}(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}H(f^{{\text{'}}^{\prime }})e^{j2\pi f^{{\text{'}}^{\prime }}t}d{f}^{{\text{'}}^{\prime }}{)}^{*}dt$
Now $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}G(f^{\text{'}})e^{j2\pi f^{\text{'}}t}d{f}^{\text{'}}(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}H(f^{{\text{'}}^{\prime }})e^{j2\pi f^{{\text{'}}^{\prime }}t}d{f}^{{\text{'}}^{\prime }}{)}^{*}dt=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}G(f^{\text{'}})\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{H}^{*}(f^{{\text{'}}^{\prime }})\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j2\pi (f^{{\text{'}}^{\prime }}-f^{\text{'}})t}dtd{f}^{{\text{'}}^{\prime }}d{f}^{\text{'}}$

Note that $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j2\pi (f^{{\text{'}}^{\prime }}-f^{\text{'}})t}dt=\delta (f^{{\text{'}}^{\prime }}-f^{\text{'}})$

So ${ℱ}[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}g(t)h^{*}(t)dt]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}G(f)H^{*}(f)df$

Reviewed by Nick Christman

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Nick Christman

1. Find ${ℱ}[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}10^{t}g(t)e^{j2\pi f{t}_0}dt]$

To begin, we know that

${ℱ}[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}10^{t}g(t)e^{j2\pi f{t}_0}dt]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}10^{t}g(t)e^{j2\pi f{t}_0}dt)e^{-j2\pi ft}dt$

After some factoring and combinting of like terms we get:

${ℱ}[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}10^{t}g(t)e^{j2\pi f{t}_0}dt]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}10^{t}g(t)dt)e^{j2\pi f(t_0-t)}dt$

But recall that $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j2\pi f(t_0-t)}dt\equiv \delta (t_0-t)\text{ or }\delta (t-t_0)$

Because of this definition (and some "math magic") our problem has been simplified significantly:

${ℱ}[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}10^{t}g(t)e^{j2\pi f{t}_0}dt]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}10^{t}g(t)\delta (t-t_0)dt=10^{t_0}g(t_0)$

Therefore,

$mathcalF[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}10^{t}g(t)e^{j2\pi f{t}_0}dt]=10^{t_0}g(t_0)$

2.

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Joshua Sarris

Find ${ℱ}[sin(w_{0}t)g(t)]$

Recall $w_0=2\pi f_0$,

so expanding we have,

${ℱ}[sin(w_{0}t)g(t)]={ℱ}[sin(2\pi f_{0}t)g(t)]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}sin(2\pi f_{0}t)g(t)e^{-j2\pi ft}dt$

Also recall $sin(\theta )=\frac{1}{j2}(e^{j\theta }-e^{-j\theta })$,

so we can convert to exponentials.

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}sin(2\pi f_{0}t)g(t)e^{-j2\pi ft}dt=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1}{j2}[e^{j2\pi f_{0}t}-e^{-j2\pi f_{0}t}]g(t)e^{-j2\pi ft}dt$

Now integrating gives us,

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1}{j2}[e^{j2\pi f_{0}t}-e^{-j2\pi f_{0}t}]g(t)e^{-j2\pi ft}dt=\frac{1}{j2}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-j2\pi (f-f_0)t}g(t)dt-\frac{1}{2}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-j2\pi (f+f_0)t}g(t)dt=\frac{1}{j2}G(f-f_0)-\frac{1}{j2}G(f+f_0)$

So we now have the identity,

${ℱ}[sin(w_{0}t)g(t)]=\frac{1}{j2}[G(f-f_0)-G(f+f_0)]$

or rather

${ℱ}[sin(w_{0}t)g(t)]=\frac{1}{2}j[G(f+f_0)+G(f-f_0)]$

Reviewed by Max