Fourier Methods

The Fourier transform is a commonly-used tool in mathematical physics, but Fourier analysis has now become a real-time diagnostic capability in experimental physics as well. Fourier Methods is a collection of hands-on electronic skills, and patterns of thinking, which can be used to understand the information content of signals in physics, engineering, communication, and beyond. This Immersion will develop those skills, using a combination of a Fourier-analyzer instrument (Stanford Research Systems SR770), a package of TeachSpin electronic modules, and some hardware experiments to perform and analyze. (see http://www.teachspin.com/ instruments/Fourier%20Methods/index.shtml)

Participants will experience the time-domain and frequency-domain views of various signals, ranging from simple to complex—including chaotic and noise waveforms. They will also learn the metrology of signals, and noise, in the frequency domain; the Fourier picture of recovery of weak sinusoidal signals immersed in noise will illuminate why lock-in detection works, and will also illustrate the meaning of the unit V/√Hz. Participants will acquire skills using the TeachSpin curriculum, and apply them to experiments in acoustic resonance, fluxgate magnetometry, or coupled oscillators.

Participants may bring a laptop, and are welcome (but not required) to bring along a favorite or familiar oscilloscope; the rest of the necessary equipment will be on hand.

Costs of the experiment: to replicate it as it will be used in the Immersion would cost $8k. Some, even many, of its ideas can be replicated at much lower cost, given a digital oscilloscope with FFT capability; of course, that way, lots of objects and systems would have to be improvised.

Some of the goals that participants addressed are found in LearningGoals.pdf

Two excerpts from instruction manuals introducing the language of Fourier Transforms are Ch.0 Fourier Methods Rev1.1 and CoefficientDefinition.pdf

Participants worked out the answers to these ‘homework’ exercises before the Immersion: Homework.pdf

Two examples of problems treated by Fourier Methods are found in PartialSums.pdf and ModellingTransients.pdf

The special requirements of noise waveforms are addressed in SpectralDensity.pdf , NoiseExercise.pdf , and ModelsForNoise.pdf