# Inverse Discrete Fourier Transform proof help

• emnaki
In summary, the IDFT is the inverse operation of the DFT, and is written as x(n) = \frac{1}{N}\sum^{N-1}_{m=0}X(m)e^{i2\pi mn/N}. Your attempted proof has some errors, but with some corrections, you can arrive at the correct result. Keep practicing and seeking help if needed. Good luck!
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?

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

## 1. What is the Inverse Discrete Fourier Transform (IDFT)?

The Inverse Discrete Fourier Transform is a mathematical operation that converts a discrete signal from the frequency domain to the time domain. It is the inverse operation of the Discrete Fourier Transform (DFT) and is commonly used in signal processing and data analysis.

## 2. How is the IDFT different from the DFT?

The IDFT and DFT are two sides of the same coin - while the DFT converts a time-domain signal to the frequency domain, the IDFT converts the frequency-domain signal back to the time domain. This means that the IDFT is the inverse of the DFT, and the two operations can be used to transform a signal back and forth between the time and frequency domains.

## 3. What is the mathematical formula for the IDFT?

The mathematical formula for the IDFT is:

Where x(n) is the time-domain signal, N is the length of the signal, X(k) is the frequency-domain signal, and i is the imaginary unit.

## 4. How is the IDFT used in practical applications?

The IDFT has a wide range of applications in fields such as signal processing, audio and image compression, data analysis, and telecommunications. It is used to transform digital signals between the time and frequency domains, allowing for easier analysis and manipulation of the signal. For example, the IDFT is used in audio editing software to convert a recorded sound from the frequency domain to the time domain for editing and processing.

## 5. What are some common properties of the IDFT?

The IDFT shares many properties with the DFT, including linearity, convolution, and the shift theorem. It also has the property of being its own inverse - applying the IDFT twice on a signal will result in the original signal. Additionally, the IDFT is a complex-valued operation, meaning it can handle both real and imaginary signals.

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