Prove that the Fourier transform of a rectangular pulse is the sinc function.

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Question statement

Consider the periodic signal $\tilde{x}(t)$ with amplitude $A=1$ , period $T =$ {T} ms and duty-cycle $\eta$ = {eta} ({eta*100}%) illustrated below.

Given that the pulse width is $w = 2t_0$ s, the duty cycle is $\eta = 2t_0/T = w/T$ ( $\eta\% = w/T\times 100$ ), and the fundametal frequency of the waveform is

$\Omega_0=\frac{2\pi}{T}\,\mathrm{rad/s}.$

Compute the following:

Give any introductory information the student needs.

			Name	Type	Generated Value

T	integer	1000
eta	number	0.25
k	integer	-1
t0	number	125

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(1000,500,200,100,50,20,10,5,2,1)

JME function reference

Description

The period of a periodic pulse chain in milliseconds.

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `integer`

1000

→ Used by:

This variable doesn't seem to be used anywhere.

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The pulse width $w = \eta T =$ {eta*T} ms.
$w = 2t_0$ so $t_0$ = {eta*T/2} ms.
$\Omega_0 = \frac{2\pi}{T} =$ {(2000/T)} $\pi$ rad/s.

$C_k = -\frac{1}{j2\pi k}\left(\exp\left(-jk \pi \frac{2t_0}{T}\right)-\exp\left(jk\pi \frac{2t_0}{T}\right)\right)$

$C_k = -\frac{1}{\pi k}\left(\frac{\exp\left(-jk \pi \frac{2t_0}{T}\right)-\exp\left(jk\pi \frac{2t_0}{T}\right)}{j2}\right)$

$C_k = \frac{1}{\pi k}\left(\frac{\exp\left(jk \pi \frac{2t_0}{T}\right)-\exp\left(-jk\pi \frac{2t_0}{T}\right)}{j2}\right)$

$C_k = \frac{1}{\pi k}\sin\left(k \pi \frac{2t_0}{T}\right)$

Converting this to the sinc function:

$C_k =\frac{T}{2t_0}\mathrm{sinc}\left(k \frac{2t_0}{T}\right)$ or

$C_k =\frac{1}{\eta}\mathrm{sinc}\left(k \eta\right)$

When $k =$ {8}, $k2t_0/T = k\eta$ = {k*eta} and

$C_k =$ {(1/eta)*sin(pi*eta)/(pi*eta)}

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Prove that the Fourier transform of a rectangular pulse is the sinc function.

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Prove that the Fourier transform of a rectangular pulse is the sinc function.

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Chris Jobling 5 years ago

Chris Jobling 5 years ago

Chris Jobling 5 years ago

Error in variable testing condition

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Generated value: integer

Parts

Number entry

Advanced settings

Generated value: `integer`