Homework Four

Fourier Transform Properties

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Find ${\mathcal {F}}[10^{t}g(t)e^{j2\pi ft_{0}}]$

To begin, we know that

${\mathcal {F}}[10^{t}g(t)e^{j2\pi ft_{0}}]=\int _{-\infty }^{\infty }10^{t}g(t)e^{j2\pi ft_{0}}e^{-j2\pi ft}\,dt=\int _{-\infty }^{\infty }10^{t}g(t)e^{j2\pi f(t_{0}-t)}\,dt$

But recall that $\int _{-\infty }^{\infty }e^{j2\pi f(t_{0}-t)}df\equiv \delta (t_{0}-t){\mbox{ or }}\delta (t-t_{0})$

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:

${\mathcal {F}}[10^{t}g(t)e^{j2\pi ft_{0}}]=\int _{-\infty }^{\infty }10^{t}g(t)\delta (t-t_{0})\,dt=10^{t_{0}}g(t_{0})$

Therefore,

${\mathcal {F}}[10^{t}g(t)e^{j2\pi ft_{0}}]=10^{t_{0}}g(t_{0})$