Homework Four

Fourier Transform Properties

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 $e^{j2\pi f(t_0-t)}\equiv \delta (t_0-t)\text{ or }\delta (t-t_0)$

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,

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