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Practice Question on "Digital Signal Processing"

Topic: Continuous-space Fourier transform of a 2D "rect" function


Question

Compute the Continuous-space Fourier transform (CSFT) of

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

Justify your answer.


Share your answers below

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

$ x[n] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-2j \pi (ux +vy) }dxdy $

$ = \frac{1}{2j\pi(u)}\frac{1}{2j\pi(v)}[-e^{-j \pi (u)} + e^{j \pi (u)}][-e^{-j \pi (v)} + e^{j \pi (v)}] $

Instructor's comment:
a) You could perhaps simplify your answer a bit. (It's actually a sinc!).
b) The parenthesis around the u and v in the denominator of the answer are a bit confusing, but you would not lose any point for that of course.
c) You should add a few extra steps before writing the answer. If you do the steps in your head, there is a high likelihood of making a mistake. If you make a mistake in the answer and your wrote no intermediate steps, then you would get very little partial credit.
d) Actually, these is a slight problem with your answer at u=0 or v=0. So technically, you should split your solution into three cases: "u=0", "v=0", and "neither u nor v equal to zero".
pm

Answer 2

$ F(u,v) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-2j \pi (ux +vy) }dxdy = \int_{-\frac{1}{2}}^{\frac{1}{2}}\int_{-\frac{1}{2}}^{\frac{1}{2}} e^{-2j \pi (ux +vy) }dxdy $ $ = \frac{(e^{j \pi u}-e^{-j \pi u} )(e^{j \pi v}-e^{-j \pi v})}{(2j\pi u)(2j\pi v)} $

$ = \frac{sin(u)sin(v)}{(\pi u)(\pi v)} = sinc(u)sinc(v)= sinc(u,v) $ --Xiao1 23:26, 12 November 2011 (UTC)


Instructor's comment:
a) This is a good amount of intermediate steps: not too much, not too little, altough adding an extra step to do the actual integration would not hurt.
b) Technically, your reasoning is not valid at u=0 or v=0 (because you would be dividing by zero).
pm

Answer 3

Write it here.


Back to ECE438 Fall 2011 Prof. Boutin

Alumni Liaison

Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

Buyue Zhang