- #1

emnaki

- 5

- 0

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X(m)=\sum^{N-1}_{n=0}x(n)e^{-i2\pi mn/N} \\

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Considering x(1),

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x(1)e^{-i2\pi m/N}=X(m)-\sum^{N-1}_{n=0,n\neq 1}x(n)e^{-i2\pi mn/N} \\

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[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} \\

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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} \\

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\mbox{Add up for all m} \\

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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} \\

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[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}) \\

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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?