Some applications of Fourier's great discovery for beginners

Abstract

Nearly two centuries ago, Fourier discovered that any periodic function of period T can be presented as a sum of sine waveforms of frequencies equal to an integer times the fundamental frequency ω = 2π/T (Fourier's series). It is impossible to overestimate the importance of Fourier's discovery, and all physics or engineering students should be familiar with this subject. A suitable device for demonstrating spectra of electrical signals is a digital storage oscilloscope. Spectra of various waveforms and of AM and FM signals are demonstrated, as well as AM signals from a broadcasting station. Changes in the signals filtered by frequency-selective circuits are seen by comparing the spectra of the input and output voltages. All the experiments are suitable for undergraduate laboratories and usable as classroom demonstrations.

Signal Frequency Spectra with Audacity®

“…4) because the square wave is a weighted sum of odd multiples of the fundamental waveform frequency. 8 During rising or falling square-wave edges, the signal changes very quickly from low to high or high to low. Fast changing parts of a signal give the signal higher frequencies than the 1000-Hz base frequency, resulting in the peaks at higher frequencies.…”

“…If students have access to a digital storage oscilloscope, they can view the time domain and frequency spectrum of a signal on the same screen and experiment with the effects of low-pass, band-pass, and highpass filtering. 8 This work presents the use of Audacity to allow students to first experiment with the frequency spectrum of simple waveforms (sine wave, square wave, and sawtooth wave) before looking at the frequency spectrum of a more complicated waveform (a voice recording). Next, students manufacture a noisy voice recording and attempt to filter out the unwanted noise while preserving the voice signal as much as possible.…”

Exploring digital signal processing using an interactive Jupyter notebook and smartphone accelerometer data

“…Physics lab tasks utilizing the DFT and its algorithm implementation fast Fourier transform (FFT) have existed for decades [3,4]. Reference [5] showcases several examples of electric circuits. Reference [6] uses the FFT in a mechanics context to analyze the eigenfrequencies of coupled harmonic oscillators from force transducer data.…”

Applications of sinusoidal phase modulation in temporal optics to highlight some properties of the Fourier transform

“…Therefore, we want to form an idea of the corollary that the application of a phase modulation in the time domain deeply affects the signal in the frequency domain. We wish here to highlight the possibilities offered by Fourier analysis [7,8], which plays a major role in countless branches of physics and engineering [9] such as electronics, acoustics, as well as optics [10][11][12][13]. While master students are used to manipulating this concept in the case of real initial signals (for example, in the case of Fraunhofer diffraction caused by an aperture), we found that they had much more difficulties in managing phase-modulated signals.…”