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FFT Module Overview

The numpy.fft module provides functions for computing the Discrete Fourier Transform (DFT) and its inverse using the Fast Fourier Transform (FFT) algorithm.

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What is FFT?

Fourier analysis is a method for expressing a function as a sum of periodic components, and for recovering the function from those components. The DFT has become essential in numerical computing because of the FFT algorithm, which computes it efficiently.

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Module Organization

The FFT module is organized into several categories:

Standard FFTs

1D, 2D, and N-dimensional complex FFTs

Real FFTs

Optimized transforms for real-valued input

Hermitian FFTs

Transforms for Hermitian-symmetric data

Helper Functions

Frequency bins and spectrum shifting utilities

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Mathematical Definition

The DFT is defined as: Ak=∑m=0n−1amexp⁡{−2πimkn}k=0,…,n−1A_k = \sum_{m=0}^{n-1} a_m \exp\left\{-2\pi i\frac{mk}{n}\right\} \qquad k = 0,\ldots,n-1 The inverse DFT is: am=1n∑k=0n−1Akexp⁡{2πimkn}m=0,…,n−1a_m = \frac{1}{n}\sum_{k=0}^{n-1}A_k\exp\left\{2\pi i\frac{mk}{n}\right\} \qquad m = 0,\ldots,n-1

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Normalization Modes

All FFT functions accept a norm parameter with three options:
  • "backward" (default): Forward transform unscaled, inverse scaled by 1/n
  • "ortho": Both transforms scaled by 1/√n (unitary transforms)
  • "forward": Forward transform scaled by 1/n, inverse unscaled

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Quick Example

import numpy as np

# Create a simple signal
t = np.linspace(0, 1, 500)
signal = np.sin(2 * np.pi * 50 * t) + 0.5 * np.sin(2 * np.pi * 120 * t)

# Compute FFT
fft_result = np.fft.fft(signal)
freqs = np.fft.fftfreq(len(signal), t[1] - t[0])

# Get power spectrum
power = np.abs(fft_result) ** 2

print(f"Peak frequencies: {freqs[np.argsort(power)[-3:]]}")

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Type Promotion

numpy.fft promotes float32 and complex64 arrays to float64 and complex128 respectively. For an FFT implementation that preserves input precision, see scipy.fftpack.

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Common Use Cases

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Signal Processing

Analyze frequency content of time-domain signals (audio, sensor data)

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Image Processing

Apply frequency-domain filters, compression (JPEG uses DCT, similar to FFT)

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Fast Convolution

Multiply in frequency domain instead of convolving in time domain

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Differential Equations

Solve PDEs using spectral methods
  • SciPy FFT: The scipy.fft module is a more comprehensive superset of numpy.fft
  • Performance: FFT is most efficient when the array size is a power of 2
  • References: Based on Cooley-Tukey algorithm (1965)

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See Also

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