Fourier Analysis

Fourier Analysis is a mathematical framework for representing complex, time-varying signals as combinations of simpler sinusoidal components, revealing their frequency content. Using Fourier series for periodic signals and the Fourier transform for aperiodic or sampled data, it converts information between the time and frequency domains, where amplitude and phase describe each frequency component. In engineering, this approach supports filtering, noise reduction, vibration analysis, communications, image processing, and system characterization. By identifying dominant frequencies and how systems respond to them, Fourier Analysis helps engineers diagnose faults, design reliable technologies, and interpret measurements across electrical, mechanical, and aerospace applications.

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Research

JoVE Journal - Engineering
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Fourier-Based Diffraction Analysis of Live Caenorhabditis elegans

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Cited by 4 •

2017

This manuscript describes how to distinguish different nematodes using far-field diffraction signatures. We compare the locomotion of 139 wild type and 108 "Roller" C. elegans by averaging frequencies associated with the temporal Fraunhofer diffraction signature at a single location using a continuous wave laser.

Research

JoVE Journal - Engineering

A Multimodal Wide-Field Fourier-Transform Raman Microscope

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2025

A wide-field Fourier-transform microscope, based on a compact and ultra-stable birefringent interferometer, allows the parallel acquisition of spectra for all pixels of a 2D detector. The time-domain approach enables the disentanglement of photoluminescence and Raman signals, and allows rapid Raman mapping (~5 ms/pixel) with ~1-µm spatial and 23-cm-1 spectral resolution.

A Fourier Transform Infrared Spectroscopy Technique to Study Peptide Self-Assembly

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2025

This video demonstrates the self-assembly of amyloid peptides into a beta sheet-rich supramolecular structure using Fourier transform infrared spectroscopy. An infrared beam is passed through a crystal with a high refractive index, and energy absorption across the interface of the crystal and the sample is recorded. The secondary structures of the sample proteins are identified by analyzing their characteristic absorption peaks at specific regions of the infrared spectrum, which aids in...

Education

JoVE Core - Electrical Engineering

Exponential Fourier series

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2025

In audio signal processing, the exponential Fourier series plays a crucial role in sound synthesis, allowing complex sounds to be broken down into simpler sinusoidal components. This decomposition process is fundamental in analyzing and reconstructing musical notes and other audio signals. The exponential Fourier series expresses periodic signals as the sum of complex exponentials at both positive and negative harmonic frequencies, providing a powerful tool for signal analysis. Euler's identity...

Properties of Fourier series I

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2024

The Fourier series is a powerful tool in signal processing and communications, allowing periodic signals to be expressed as sums of sine and cosine functions. A foundational property of the Fourier series is linearity. If we consider two periodic signals, their linear combination results in a new signal whose Fourier coefficients are simply the corresponding linear combinations of the original signals' coefficients. This property is crucial in applications like frequency modulation (FM) radio,...

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