How can you calculate the IFFT (Inverse Fast Fourier Transform) using only the forward FFT (Fast Fourier Transform)? It is easier than you think! Read on to find out how.
Cheat-Sheet: Just tell me how!
Let’s start with just saying straight out what you do. Take the input signal, let’s call it x, and take its complex conjugate. Then take the FFT of that result. Then take the complex conjugate again. Finally divide the resultant signal by N (the length of the signal). Here is the formula:
Derivation of the Method
Hopefully you are also interested in the why! To start, let’s note that the FFT and DFT (Discrete Fourier Transform), and the IFFT and IDFT (Inverse Discrete Fourier Transform), are identical. The “Fast” in FFT and IFFT just means that the algorithm used to compute the DFT and IDFT is faster than a direct approach, but it still gives the same exact results.
The first thing we will need to do is to find out how the DFT and IDFT are related. Let’s write out the definitions: