1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Double integration problem for IDSFT

  1. Feb 8, 2017 #1
    1. The problem statement, all variables and given/known data

    The 2D Discrete Space Fourier transform (DSFT) X(w1,w2) of the sequence x(n1,n2) is given by,

    $$X(w_1,w_2) = 5 + 2j sin(w_2) + cos(w_1) + 2e^{(-jw1-jw2)}$$

    determine x(n1,n2)

    2. Relevant equations

    By definition inverse DSFT is,

    $$x(n_1,n_2) = \dfrac{1}{(2π)^2} \int_{-π}^{π}\int_{-π}^{π} X(w_1,w_2) e^{(jw_1n_1+jw_2n_2)} dw_1dw_2$$

    3. The attempt at a solution

    I got zero as a final answer.

    Solving the double integral for each term I get zero when substituting pi, is it correct or did I made a mistake somewhere, what gets me confused is that when doing DSFT for some simple problem I can get cos(w1) for example but if I did the inverse DSFT I will get zero. can someone help.

    Thanks in advance.
  2. jcsd
  3. Feb 8, 2017 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Please post your working.
  4. Feb 8, 2017 #3
    Taking each term,

    $$X_1(w_1,w_2) = 5$$
    $$X_2(w_1,w_2) = 2jsin(w_2)$$
    $$X_3(w_1,w_2) = cos(w_1)$$
    $$X_4(w1,w2) = 2e^{(-jw_1-jw_2)}$$


    $$ x_1(n_1,n_2) = \dfrac{5}{(2π)^2} \int_{-π}^{π}\int_{-π}^{π} e^{jw_1n_1+jw_2n_2} dw_1dw_2$$
    $$x_1(n_1,n_2) = \dfrac{5}{(2π)^2} \int_{-π}^{π}e^{jw_2n_2}[ \int_{-π}^{π} e^{jw_1n_1} dw_1]dw_2$$
    $$x_1(n_1,n_2) = \dfrac{5}{(2π)^2} \int_{-π}^{π}e^{jw_2n_2} [\dfrac{e^{jw_1n_1}}{jn_1}]_{-π}^{π} dw_2$$
    $$x_1(n_1,n_2) = \dfrac{5}{(2π)^2} \int_{-π}^{π}e^{jw_2n_2} [\dfrac{1}{jn_1}(e^{jπn_n1} - e^{-jπn_1})] dw_2$$

    since n1 and n2 are discrete, then
    $$(e^{jπn_1} - e^{-jπn_1})] = cos(jπn_1) + jsin(jπn_1) - cos(-jπn_1) + jsin(-jπn_1) = 0$$

    I solved the remaining terms in a similar fashion, do you find anything wrong here.
  5. Feb 8, 2017 #4
    Ok I got it.

    I didn't notice that


    which is a sinc function and equals 1 at n1 = 0.
  6. Feb 8, 2017 #5


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Well done.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Threads - Double integration problem Date
Double integration problem Sep 10, 2016
Double integral problem, conceptual help. Mar 16, 2015
Double integral volume problem Nov 18, 2014
Double Integral problem Sep 2, 2014
Multivariable Double Integration Problem Apr 29, 2014