| Line 23: | Line 23: | ||
:<span style="color:green">Instructor's comments: Nice and clear solution! One can also justify the answer using the shifting property directly, which would save a couple of steps.-pm </span> | :<span style="color:green">Instructor's comments: Nice and clear solution! One can also justify the answer using the shifting property directly, which would save a couple of steps.-pm </span> | ||
===Answer 2=== | ===Answer 2=== | ||
| − | + | <math> x(t) = \int_{-\infty}^{\infty} X(f)e^{j2\pi ft} df | |
| + | </math> | ||
| + | |||
| + | In order for the following to be true, | ||
| + | <math> x(t)= e^{j \pi t} </math> | ||
| + | |||
| + | <math> X(f) = \delta(f - \frac{1}{2}) </math> | ||
| + | |||
| + | because | ||
| + | |||
| + | <math> x(t) = \int_{-\infty}^{\infty} \delta(f - \frac{1}{2})e^{j2\pi ft} df = e^{j \pi t} | ||
| + | </math> with careful inspection. | ||
| + | |||
| + | |||
===Answer 3=== | ===Answer 3=== | ||
write it here. | write it here. | ||
---- | ---- | ||
[[2011_Fall_ECE_438_Boutin|Back to ECE438 Fall 2011 Prof. Boutin]] | [[2011_Fall_ECE_438_Boutin|Back to ECE438 Fall 2011 Prof. Boutin]] | ||
Revision as of 15:37, 6 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} $
using the fact that $ \delta (t-T)f(t) = \delta (t-T)f(T) $
- Instructor's comments: Nice and clear solution! One can also justify the answer using the shifting property directly, which would save a couple of steps.-pm
Answer 2
$ x(t) = \int_{-\infty}^{\infty} X(f)e^{j2\pi ft} df $
In order for the following to be true, $ x(t)= e^{j \pi t} $
$ X(f) = \delta(f - \frac{1}{2}) $
because
$ x(t) = \int_{-\infty}^{\infty} \delta(f - \frac{1}{2})e^{j2\pi ft} df = e^{j \pi t} $ with careful inspection.
Answer 3
write it here.
