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Continuous-time Fourier transform of a complex exponential
What is the Fourier transform of $ x(t)= e^{j \pi t} $? Justify your answer.
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Answer 1
Guess: $ X(f)=\delta (f-\frac{1}{2}) $
Proof:
$ x(t)=\int_{-\infty}^{\infty} X(f)e^{j2\pi ft} df = \int_{-\infty}^{\infty} \delta (f-\frac{1}{2})e^{j2\pi ft} df = \int_{-\infty}^{\infty} \delta (f-\frac{1}{2})e^{j\pi t} df = e^{j\pi t} \int_{-\infty}^{\infty} \delta (f-\frac{1}{2}) df = e^{j\pi t} $
using the fact that $ \delta (t-x)f(t) = \delta (t-x)f(x) $
Answer 2
Write it here.
Answer 3
write it here.
