
Contents
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11.1 Quick Fast Fourier Transform Example 11.1 Quick Fast Fourier Transform Example
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11.2 Fourier Transform 11.2 Fourier Transform
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11.3 Discrete Sampling and the Nyquist Frequency 11.3 Discrete Sampling and the Nyquist Frequency
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11.4 Discrete Fourier Transform 11.4 Discrete Fourier Transform
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11.5 Fast Fourier Transform 11.5 Fast Fourier Transform
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11.6 FFT of Sinusoids 11.6 FFT of Sinusoids
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11.7 Applying FFT to Various Sinusoidal Inputs 11.7 Applying FFT to Various Sinusoidal Inputs
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11.8 Magnitude of Complex-Amplitude 11.8 Magnitude of Complex-Amplitude
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11.9 Observing Spectral Leakage 11.9 Observing Spectral Leakage
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11.10 Windowing 11.10 Windowing
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11.11 Estimating Frequency and Amplitude 11.11 Estimating Frequency and Amplitude
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11.12 Aliasing 11.12 Aliasing
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Do It Yourself Do It Yourself
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Use It! Use It!
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Problems Problems
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11 Data Analysis: Fast Fourier Transform
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Published:November 2024
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Abstract
Using SciPy functions, readers write Python programs to investigate the proper use of the fast Fourier transform (FFT). After a quick introduction to FFT properties, the theory of the discrete Fourier transform and discrete inverse Fourier transform is presented, including the mathematics behind how these quantities are used to produce discrete complex-amplitude and frequency spectra from a given input waveform. Then, readers write a program in which SciPy’s complex Fourier transform function is applied to various sinusoidal inputs, demonstrating that this function produces the expected complex-amplitude spectra. Finally, a program is written that simply plots the magnitude of the complex-amplitude as a function of frequency, and the effects of spectral leakage, windowing, and aliasing for input frequencies above the Nyquist frequency are explored. By applying this understanding of the FFT, along with the use of a DAQ device, readers construct a computer-based spectrum analyzer as a Do It Yourself project. A Use It! example investigates another of SciPy’s frequency-related signal-processing capabilities—digital filtering.
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