Understanding the Inverse DFT (FFT)

Question

I was reading this article to understand the fft - JACK SCHAEDLER - A WALKTHROUGH OF THE DFT - STEP BY STEP.

I understand that:

1. the fft takes a block (256 or 512) of the signal, creates a sinusoidal and cosineidal wave of frequency "f" and multiplies each sample for each sample of the signal to decompose,
2. add the value of all the samples of the cosine and sine wave product, if it gives greater than 0 it means that the frequency wave "f" is present in the signal
3. the result of the sum of each product wave gives a complex number

now what I want to know what is the next formula to rebuild the signal, ie an inverse fft

Answer 1

If you know Linear Algebra then the process can be explained using Linear Algebra in a very simple way.

Let's say you have Orthonormal Basis for $ \mathbb{R}^{k} $ given by the Columns of $ A \in \mathbb{R}^{k \times k} $.

Now imagine having a vector $ x \in \mathbb{C}^{k} $.
How would represent the vector $ x $ in the set of coordinate of matrix $ A $, namely using the Orthonormal Basis?

Well, you'd calculate the projection (Inner Product) between any column of $ A $ to the vector $ x $ by $ {x}_{A} = {A}^{H} x $.
How would you recreate $ x $ from the columns of $ A $ using $ {x}_{A} $?
Well, you can see that $ x = A {x}_{A} $ since $ A $ is Unitary Matrix.
Basically you sum the columns of $ A $ using the weights of $ {x}_{A} $.

So, Now just replace $ A $ with the DFT Matrix and you have the whole process of DFT and Inverse DFT.

In practice, the Inverse DFT and the DFT are very similar, multiplication and summation.
The only difference is on one direction (Forward) we multiply by $ {e}^{- \frac{2 \pi j}{N} k n} $ the signal and on the backward we multiply the DFT coefficients by $ {e}^{\frac{2 \pi j}{N} k n} $.