Understanding and Implementing the Fast Fourier Transform (FFT) Algorithm in C#

Mahmoud Alyosify

·

Follow

2 min read

·

Jan 26, 2023

--

Listen

Share

Fast Fourier Transform (FFT) is an efficient algorithm for computing the discrete Fourier transform (DFT) of a sequence. The DFT is a representation of a sequence in the frequency domain, which is useful for analyzing and manipulating signals. The FFT algorithm uses a divide-and-conquer approach to efficiently compute the DFT, reducing the number of calculations required compared to a direct computation of the DFT.

The basic idea behind the FFT is to decompose the DFT of a sequence into two smaller DFTs of roughly half the length. This is achieved by rearranging the terms of the DFT into a new sequence and then applying the DFT to each of these smaller sequences. This process is repeated recursively until we have reduced the problem to computing the DFT of a sequence of length 2, which can be done in a single step.

Here is an example of C# code that computes the FFT of a sequence:

using System;
using System.Numerics;
// This code is written by Mahmoud Alyosify
class FFTExample {
    static void Main() {
        // Define the sequence to be transformed
        double[] x = {1.0, 2.0, 3.0, 4.0};

        // Compute the FFT of the sequence
        Complex[] y = FFT(x);

        // Print the results
        for (int i = 0; i < y.Length; i++) {
            Console.WriteLine("y[{0}] = {1}", i, y[i]);
        }
    }

    static Complex[] FFT(double[] x) {
        int N = x.Length;
        Complex[] y = new Complex[N];

        // Base case: N = 2
        if (N == 2) {
            y[0] = x[0] + x[1];
            y[1] = x[0] - x[1];
            return y;
        }

        // Split the input into two parts
        double[] x_even = new double[N/2];
        double[] x_odd = new double[N/2];
        for (int i = 0; i < N/2; i++) {
            x_even[i] = x[2*i];
            x_odd[i] = x[2*i + 1];
        }

        // Recursively compute the FFT of each part
        Complex[] y_even = FFT(x_even);
        Complex[] y_odd = FFT(x_odd);

        // Combine the results
        for (int k = 0; k < N/2; k++) {
            double angle = -2 * k * Math.PI / N;
            Complex w = Complex.FromPolarCoordinates(1, angle);
            y[k] = y_even[k] + w * y_odd[k];
            y[k + N/2] = y_even[k] - w * y_odd[k];
        }

        return y;
    }
}

This example computes the FFT of the sequence [1, 2, 3, 4], and prints the results to the console.

It’s worth to mention that, this example is a simple implementation of the FFT algorithm. It does not handle edge cases such as sequences of length that is not power of 2, and also it’s not optimized for performance.