Inverse Discrete Fourier Transform proof help

[emnaki]

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:
$$X(m)=\sum^{N-1}_{n=0}x(n)e^{-i2\pi mn/N} \\$$
Considering x(1),
$$x(1)e^{-i2\pi m/N}=X(m)-\sum^{N-1}_{n=0,n\neq 1}x(n)e^{-i2\pi mn/N} \\$$
$$x(1)=X(m)e^{i2\pi m/N}-\sum^{N-1}_{n=0,n\neq 1}x(n)e^{-i2\pi mn/N} e^{i2\pi m/N} \\$$
$$x(1)=X(m)e^{i2\pi m/N}-\sum^{N-1}_{n=0,n\neq 1}x(n)e^{i2\pi m(1-n)/N} \\$$
$$\mbox{Add up for all m} \\$$
$$Nx(1)=\sum^N_{m=1}X(m)e^{i2\pi m/N}-\sum^N_{m=1}\sum^{N-1}_{n=0,n\neq 1}x(n)e^{i2\pi m(1-n)/N} \\$$
$$x(1)=\frac{1}{N}(\sum^N_{m=1}X(m)e^{i2\pi m/N}-\sum^N_{m=1}\sum^{N-1}_{n=0,n\neq 1}x(n)e^{i2\pi m(1-n)/N}) \\$$

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?

 

[mighty2000]

Thank you for sharing your attempted proof. It seems like you are on the right track, but there are a few errors in your working. Let me try to help clarify some of the steps for you.

First of all, the expression X(m) that you have written is actually the DFT (Discrete Fourier Transform) of the sequence x(n). The IDFT (Inverse Discrete Fourier Transform) is the inverse operation, which is written as x(n) = \frac{1}{N}\sum^{N-1}_{m=0}X(m)e^{i2\pi mn/N}. This is an important distinction to make, as the two operations are inverses of each other.

Now, moving on to your attempt at the proof, the first error that I noticed is when you write x(1)e^{-i2\pi m/N}=X(m)-\sum^{N-1}_{n=0,n\neq 1}x(n)e^{-i2\pi mn/N}. This is not correct. The correct equation should be X(1) = \sum^{N-1}_{n=0}x(n)e^{-i2\pi n/N}. This is because the DFT is a sum of the sequence x(n) multiplied by complex sinusoids at different frequencies, but it does not involve any subtraction.

Next, when you try to isolate x(1) in your working, you should use the fact that e^{i2\pi m/N} is a periodic function with period N, so e^{i2\pi m/N} = e^{i2\pi (m + kN)/N} for any integer k. Therefore, the second term in your equation should be written as \sum^{N-1}_{n=0,n\neq 1}x(n)e^{i2\pi m(1-n)/N} = \sum^{N-1}_{n=0,n\neq 1}x(n)e^{i2\pi (m + kN)(1-n)/N} for any integer k. This will allow you to cancel out the second term when you add up for all m.

I hope this helps to clarify some of the steps for you. Keep in mind that the proof for the IDFT can be quite involved and may require some advanced mathematical concepts. If you are struggling with it, I would suggest consulting other resources or