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===Answer 1=== | ===Answer 1=== | ||

− | + | Fourier Transform of rect(t): | |

+ | |||

+ | <math> X(f)=\int_{-\infty}^{\infty} x(t)e^{-j2\pi ft} dx | ||

+ | =\int_{\frac{-1}{2}}^{\frac{1}{2}} e^{-j2\pi ft} dx | ||

+ | =\frac{e^{-j2\pi ft}}{-j2\pi f} </math> from t=-1/2 to t=1/2 | ||

+ | |||

+ | <math> =\frac{e^{-j\pi f}-e^{j\pi f}}{-j2\pi f} | ||

+ | =\frac{sin(\pi f)}{\pi f} </math> | ||

+ | |||

+ | |||

+ | Fourier Transform of <math>\frac{sin(\pi t)}{\pi t}</math>: | ||

+ | |||

+ | Guess: <math> X(f)=rect(t) </math> | ||

+ | |||

+ | Proof: | ||

+ | |||

+ | <math> x(t)=\int_{-\infty}^{\infty} X(f)e^{j2\pi ft} df | ||

+ | =\int_{\frac{-1}{2}}^{\frac{1}{2}} e^{j2\pi ft} df | ||

+ | =\frac{e^{j2\pi ft}}{j2\pi t} </math> from f=-1/2 to f=1/2 | ||

+ | |||

+ | <math> =\frac{e^{j\pi t}-e^{j\pi t}}{j2\pi t} | ||

+ | =\frac{sin(\pi t)}{\pi t} </math> | ||

+ | |||

===Answer 2=== | ===Answer 2=== | ||

Write it here. | Write it here. |

## Revision as of 16:49, 3 September 2011

## Contents

# Continuous-time Fourier transform computation (in terms of frequency f in hertz)

Compute the Continuous-time Fourier transform of the two following functions:

$ x(t)= \text{rect}(t) = \left\{ \begin{array}{ll} 1, & \text{ if } |t|<\frac{1}{2}\\ 0, & \text{ else} \end{array} \right. $

$ y(t)= \frac{ \sin ( \pi t )}{\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

Fourier Transform of rect(t):

$ X(f)=\int_{-\infty}^{\infty} x(t)e^{-j2\pi ft} dx =\int_{\frac{-1}{2}}^{\frac{1}{2}} e^{-j2\pi ft} dx =\frac{e^{-j2\pi ft}}{-j2\pi f} $ from t=-1/2 to t=1/2

$ =\frac{e^{-j\pi f}-e^{j\pi f}}{-j2\pi f} =\frac{sin(\pi f)}{\pi f} $

Fourier Transform of $ \frac{sin(\pi t)}{\pi t} $:

Guess: $ X(f)=rect(t) $

Proof:

$ x(t)=\int_{-\infty}^{\infty} X(f)e^{j2\pi ft} df =\int_{\frac{-1}{2}}^{\frac{1}{2}} e^{j2\pi ft} df =\frac{e^{j2\pi ft}}{j2\pi t} $ from f=-1/2 to f=1/2

$ =\frac{e^{j\pi t}-e^{j\pi t}}{j2\pi t} =\frac{sin(\pi t)}{\pi t} $

### Answer 2

Write it here.

### Answer 3

write it here.