Fourier Methods - NEW
"TeachSpin's Hat-Trick" - Introducing Fourier Methods, UltraSonics, Pulse Counter/Interval Timer
Conceptual Introduction to Fourier Methods
Fourier Methods Manual - Table of Contents
analyzer (the SR770) and a TeachSpin ‘physics package’ of apparatus, experiments, and a self-paced curriculum. Together, they form an ideal system for students to use in learning about ‘Fourier thinking’ as an alternative way to analyze physical systems. This whole suite of electronic modules and physics experiments is designed to show off the power of Fourier transforms as tools for picturing and understanding physical systems.
What are the electronic instrumentation skills that physics students ought to acquire in an undergraduate advanced-lab program? No doubt skills with a multimeter and oscilloscope are basic, and skills with a lock-in amplifier and computer data-acquisition system are more advanced. But our ‘Fourier Methods’ offering adds an intermediate-to-advanced-level and highly-transferable skill set to students’ capabilities. Using it, they can go beyond a passing encounter with the Fourier transform as a mathematical tool in theory courses, to a hands-on benchtop familiarity with Fourier methods in real-time electronic experiments. It represents a skill set that will serve them well in any kind of theoretical or experimental science they might encounter.
The SR770 wave analyzer (shown in the photo) digitizes input voltage signals with 16-bit precision at a 256 kHz rate, and it includes anti-aliasing filters to permit the real-time acquisition of Fourier transforms in the 0-100 kHz range. Any sub-range of the spectrum can be viewed at resolutions down to milli-Hertz. The sensitivity and dynamic range are such that sub-µVolt signals can be displayed with ease, as well as Volt-level signals with signal-to-noise ratio over 30,000:1.
The only additional instruments required to perform these experiments are a digital oscilloscope and any ordinary signal generator. The photo above also shows three ‘hardware’ experiments from TeachSpin: a cylindrical Acoustic Resonator, the Fluxgate Magnetometer in its solenoid, and the mechanical Coupled-Oscillator system. Not shown is an instrument-case full of our ‘Electronic Modules’, which are devised to make possible a host of investigations on the Fourier content of signals.We are confident that the simultaneous use of a ‘scope and the FFT analyzer, viewing the same signal, is the best way to give students intuition for how ‘time-domain’ and ‘frequency-domain’ views of a signal are related. One of our Electronic Modules is a voltage- controlled oscillator (VCO), which can be frequency-modulated by an external voltage. Fig. 1 shows the 770’s view of the spectrum of this VCO’s output, when it is set for a 50-kHz center frequency, with a 1-kHz modulation frequency. This spectrum shows the existence of sidebands, and the frequency ‘real estate’ required by a modulated signal; it also shows that Volt-level signals can be detected standing >90 dB above the noise floor of the instrument.
Because frequency-mixing technologies are so important across the board in experimental physics, our Electronic Modules include an electronic multiplier, as well as two kinds of mixers. When combined with a ‘local oscillator’ from a signal generator, a signal in any frequency range can be down-shifted into the 0-100 kHz band. Fig. 3 shows a view of part of the AM-radio spectrum, as received in Buffalo, NY. Our modules include all the parts, and all the instructions, to make the audio content of this AM transmission audible through a speaker.
The SR770 includes a high-gain front end making it capable of detecting very weak signals. And because it disperses those signals by frequency content, and permits time-averaging, it is also capable of detecting weak signals that are deeply buried in noise. Our Electronic Modules include a signal-under-noise experiment, in which weak sinusoidal signals are overlaid with analog white noise. Fig.4 shows how such weak signals can be detected by spectral resolution, without the need for a ‘reference signal’ that a lock-in amplifier would require.
The noise source in the Modules, and the noise source within the SR770, can both be quantified for spectral noise density, so students will finally be able to use an instrument whose output is calibrated in those mysterious units, Volts/√Hz. They’ll be able to see that the units for measuring the amplitude of spectral peaks (in Volts) and the level of noise floors (in V/√Hz) are incommensurate, and also see that spending more acquisition time will enhance the degree to which a monochromatic signal stands up above the white-noise floor.
Because ‘Fourier methods’ are a set of mental skills transferable to many areas of physics and technology, we have included a set of experiments and projects which showcase the applicability of Fourier analysis:
Physicists acquire Fourier-thinking skills in a variety of ways, and apply these skills in many sub-fields of physics. Advanced-lab instructors might want to share, with their theorist colleagues as well as those teaching mechanics, waves & optics and mathematical physics, the capabilities of this Fourier Methods package so that they too can see, and demonstrate for their students, how Fourier analysis works in action.