Homework Four

Fourier Transform Properties

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Find ${ℱ}[10^{t}g(t)e^{j2\pi f{t}_0}]$

To begin, we know that

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

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

The following needs to be fixed, because the previous thing (just above this) which we just fixed wasn't an identity. Hint: $10^t$ is related to $e^t$

Because of this definition, our problem has now been simplified significantly:

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

Therefore,

${ℱ}[10^{t}g(t)e^{j2\pi f{t}_0}]=10^{t_0}g(t_0)$