Example of implementing FFT fast Fourier transform algorithm in python

Introduction to FFT Fast Fourier Transform

FFT (Fast Fourier Transform) is an efficient algorithm for calculating discrete Fourier Transform (DFT) and its inverse transformation.

DFT is a mathematical tool for converting signals from the time domain to the frequency domain, and FFT accelerates this process by reducing the amount of computation.

The basic idea of FFT

FFT utilizes symmetry and periodicity in DFT, decomposing DFT into smaller DFTs through a divide-and-conquer strategy, thereby significantly reducing computational complexity.
For sequences of length N, the complexity of the DFT is directly calculated as O(N^2), while the complexity of the FFT can be reduced to O(N log N).

Algorithm steps of FFT (taking the Cooley-Tukey algorithm as an example)

Selection Decomposition: First, decompose N (sequence length) into two smaller factors, usually even factors that select N. The most common decomposition is to decompose N into two equal factors (i.e. N is a power of 2).
Reorder (bit inversion): Sorting the input sequence bit inversion, which is to rearrange the input sequence into a sequence that facilitates subsequent processing.
Butterfly operation: The core of the FFT algorithm is butterfly operation. In butterfly operations, two inputs (usually from a specific position in the sequence) are combined into one output by a series of multiplication and addition operations. This process is repeated at different levels of the algorithm.
Layer-by-layer calculation: The FFT algorithm calculates DFT by applying butterfly operations layer by layer (from the lowest layer to the highest layer). Each layer is based on the output of the previous layer, and the calculations of each layer are closer to the final DFT result.

Python FFT Fast Fourier Transform

In Python, an easy way to implement fast Fourier transform (FFT) is to use the fft module in the numpy library. numpy provides efficient FFT algorithm implementation, capable of handling one-dimensional or multi-dimensional arrays.

Here is a simple example showing how to calculate the FFT of a one-dimensional array using functions in numpy.

First, make sure you have the numpy library installed. If not installed, you can install it via pip:

pip install numpy

Then you can use the following Python code to calculate the FFT:

import numpy as np
import  as plt

# Create a sample signal, such as a composite signal containing sine waves and cosine wavesFs = 1000  # Sampling frequencyT = 1/Fs  #Sampling cycleL = 1500  # Signal lengtht = (0, L-1, L)*T  # Time vector
# Create a composite signalS = 0.7*(2**50*t) + (2**120*t)

# Use numpy's fft function to calculate FFTY = (S)

# Get the FFT frequency axisP2 = (Y/L)
P1 = P2[:L//2+1]
P1[1:-1] = 2*P1[1:-1]  # Remove symmetryf = Fs*(0,(L//2+1))/L

# Draw the amplitude spectrum of FFT()
(f, P1) 
('Single-Sided Amplitude Spectrum of S(t)')
('f (Hz)')
('|P1(f)|')
()

In this example, we first generate a composite signal S containing two sine waves of different frequencies. Then, we calculated the FFT of the signal using the function. To obtain the amplitude spectrum of FFT, we calculate the absolute value of Y and divide it by the signal length L to obtain the normalized amplitude. Since the results of FFT are symmetric (for real inputs), we usually focus on only half of the frequency range and adjust the amplitude accordingly (double the amplitude in half of the frequency range except for the first and last points). Finally, we plot the amplitude spectrum of FFT using the matplotlib library.

Notice:

This example is only used to demonstrate how to use the numpy library for FFT calculations in Python.
In practical applications, you may need to adjust the way signal generation, FFT calculation and result analysis according to your specific needs.

FFT (Fast Fourier Transform) is a fast algorithm for calculating discrete Fourier Transform (DFT) that converts signals from the time domain to the frequency domain.

In Python, the NumPy library can be used to implement FFT.

FFT python example

Here is an example code to implement FFT using NumPy:

import numpy as np

def fft(x):
    N = len(x)
    if N &lt;= 1:
        return x
    even = fft(x[0::2])
    odd = fft(x[1::2])
    T = [(-2j *  * k / N) * odd[k] for k in range(N // 2)]
    return [even[k] + T[k] for k in range(N // 2)] + [even[k] - T[k] for k in range(N // 2)]

# Example input signalx = [0, 1, 2, 3, 4, 5, 6, 7]
# Execute FFTX = fft(x)
# Output FFT resultsprint(X)

Output result:

[(28+0j), (-4+9.65685424949238j), (-4+4j), (-4+1.6568542494923806j), (-4+0j), (-4-1.6568542494923806j), (-4-4j), (-4-9.65685424949238j)]

In the above codefftThe function uses recursion to divide the input signal into two parts of an even and an odd index, and then merges the two parts. The return value of the function is the result of the FFT transformation.

Notice:

The above code is a simplified FFT implementation, regardless of performance optimization and handling of exceptions.
In practical applications, you can use the NumPy libraryFunctions perform FFT calculations, which provide a more efficient and stable implementation.

Summarize

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