| Line 10: | Line 10: | ||
---- | ---- | ||
===Answer 1=== | ===Answer 1=== | ||
| − | + | Guess: <math> X(f)=\delta (f-\frac{1}{2})</math> | |
| + | |||
| + | Proof: | ||
| + | |||
| + | <math> 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} </math> | ||
| + | |||
===Answer 2=== | ===Answer 2=== | ||
Write it here. | Write it here. | ||
Revision as of 15:56, 3 September 2011
Contents
Continuous-time Fourier transform of a complex exponential
What is the Fourier transform of $ x(t)= e^{j \pi t} $? Justify your answer.
You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!
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} $
Answer 2
Write it here.
Answer 3
write it here.
