Homework Four

Fourier Transform Properties

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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,

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