Overview of the Fast Fourier Transform (FFT) #

The Fast Fourier Transform (FFT) is a widely used algorithm in signal processing that allows for efficient computation of the Discrete Fourier Transform (DFT), a fundamental operation in digital signal processing (DSP). The FFT algorithm is particularly useful for analyzing and processing signals in the frequency domain, providing insights into the spectral content of a signal. In this article, we will dive into the basics of the FFT algorithm and understand how it works mathematically.

Mathematical Background: Discrete Fourier Transform (DFT) #

The Discrete Fourier Transform (DFT) is a mathematical operation that transforms a discrete signal from the time domain into the frequency domain. It is defined as follows:

$$ X[k] = \sum_{n=0}^{N-1} x[n] \cdot e^{-j \cdot 2 \pi \cdot k \cdot n/N}$$

where \( x[n] \) is the input signal, \( X[k] \) is the output of the DFT at frequency index \(k\), and \(N\) is the number of samples in the input signal. The DFT computes the complex amplitudes of each frequency component in the signal, providing information about the signal’s spectral content.

The DFT has a computational complexity of \( O(N^2) \), which can be prohibitive for large signals. The FFT algorithm, on the other hand, reduces the computational complexity to \( O(N \log N) \), making it much faster for processing large signals.

Understanding the FFT Algorithm #

The key idea behind the FFT algorithm is to divide the input signal into even and odd parts, and then recursively compute the DFT of these smaller parts. The results are then combined to obtain the final DFT result. The algorithm can be implemented using either a radix-2 or radix-4 approach, with the radix-2 being the most common and widely used.

Steps of the FFT Algorithm: #

The FFT algorithm can be broken down into the following steps:

1. Start with an input signal \( x \) of length \( N \).

2. If the length of the input signal is not a power of 2, pad it with zeros to the nearest power of 2, denoted as \( N_p \). This is done to ensure that the algorithm works efficiently for input signals of any length.

3. Divide the input signal into even and odd-indexed parts:

$$[ x_e[n] = x[2n] \quad \text{for} \quad n = 0, 1, 2, …, \frac{N_p}{2}-1$$

$$[ x_o[n] = x[2n+1] \quad \text{for} \quad n = 0, 1, 2, …, \frac{N_p}{2}-1$$

This step essentially splits the input signal into two smaller sub-problems.

4. Recursively compute the DFT of the even and odd parts:

$$X_e[k] = \text{FFT}(x_e) \quad \text{for} \quad k = 0, 1, 2, …, \frac{N_p}{2}-1$$

$$X_o[k] = \text{FFT}(x_o) \quad \text{for} \quad k = 0, 1, 2, …, \frac{N_p}{2}-1$$

This step involves recursively applying the FFT algorithm to the even and odd parts of the input signal, until the base case of (N = 1) is reached.

5. Combine the results of the even and odd parts to obtain the final DFT result:

$$X[k] = X_e[k] + \text{exp}\left(-2j \pi \frac{k}{N_p}\right) \cdot X_o[k] \quad \text{for} \quad k = 0, 1, 2, …, N_p-1$$

6. The final result, \(X[k]\) obtained from the previous step represents the DFT of the input signal

Python Implementation of the FFT Algorithm #

import numpy as np

def fft(x):
    """
    Python FFT Code / FFT Code in Python
    Performs the Fast Fourier Transform (FFT) on an input signal x.

    Parameters:
        x (array-like): Input signal of arbitrary size.

    Returns:
        X (array-like): Complex-valued Fourier coefficients of x.
    """
    N = len(x)
    if N <= 1:
        return x
    else:
        even = fft(x[::2])
        odd = fft(x[1::2])
        t = np.exp(-2j * np.pi * np.arange(N) / N)
        return np.concatenate([even + t[:N//2] * odd, even + t[N//2:] * odd])

# Example usage:
x = np.array([1, 2, 3, 4])
X = fft(x)
print(X)

MATLAB Implementation of the FFT Algorithm #

function X = fft_arbitrary_size(x)
    % FFT code in MATLAB / MATLAB FFT code
    % Performs the Fast Fourier Transform (FFT) on an input signal x.
    %
    % Parameters:
    %   x (array-like): Input signal of arbitrary size.
    %
    % Returns:
    %   X (array-like): Complex-valued Fourier coefficients of x.
    
    N = numel(x);
    if N <= 1
        X = x;
    else
        if mod(N, 2) ~= 0
            % Pad with zeros if input length is not even
            x = [x, 0];
            N = N + 1;
        end
        even = fft_arbitrary_size(x(1:2:end));
        odd = fft_arbitrary_size(x(2:2:end));
        t = exp(-2j * pi * (0:N-1) / N);
        X = [even + t(1:N/2) .* odd, even + t(N/2+1:end) .* odd];
    end
end

% Example usage:
x = [1, 2, 3, 4];
X = fft_arbitrary_size(x);
disp(X);