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Inverse Discrete Fourier Transform proof help

  1. Jan 28, 2009 #1
    I'm reading the text Understanding Digital Signal Processing Second Edition and in the text they give the IDFT without any proof and so I tried to do a quick proof, but I have not been able do it here is my attempted steps:
    [tex]
    X(m)=\sum^{N-1}_{n=0}x(n)e^{-i2\pi mn/N} \\
    [/tex]
    Considering x(1),
    [tex]
    x(1)e^{-i2\pi m/N}=X(m)-\sum^{N-1}_{n=0,n\neq 1}x(n)e^{-i2\pi mn/N} \\
    [/tex]
    [tex]
    x(1)=X(m)e^{i2\pi m/N}-\sum^{N-1}_{n=0,n\neq 1}x(n)e^{-i2\pi mn/N} e^{i2\pi m/N} \\
    [/tex]
    [tex]
    x(1)=X(m)e^{i2\pi m/N}-\sum^{N-1}_{n=0,n\neq 1}x(n)e^{i2\pi m(1-n)/N} \\
    [/tex]
    [tex]
    \mbox{Add up for all m} \\
    [/tex]
    [tex]
    Nx(1)=\sum^N_{m=1}X(m)e^{i2\pi m/N}-\sum^N_{m=1}\sum^{N-1}_{n=0,n\neq 1}x(n)e^{i2\pi m(1-n)/N} \\
    [/tex]
    [tex]
    x(1)=\frac{1}{N}(\sum^N_{m=1}X(m)e^{i2\pi m/N}-\sum^N_{m=1}\sum^{N-1}_{n=0,n\neq 1}x(n)e^{i2\pi m(1-n)/N}) \\
    [/tex]

    So it looks like I got the first term, but how do I remove the second term with the double summation? Is my working wrong? Does the second term actually cancel out?
     
  2. jcsd
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