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Fourier Methods

Newsletter 1: "TeachSpin's Hat-Trick" - Introducing Fourier Methods, UltraSonics, Pulse Counter/Interval Timer
Newsletter 2: "Fourier Methods - More Tools!"
Newsletter 3: "Using Fourier Analysis to Diagnose Noise: A Case Study" - See Pg 2-3

Conceptual Introduction to Fourier Methods
Fourier Methods Manual - Table of Contents

Lab Topics:
Classical Mechanics
Sound & Wave

In collaboration with Stanford Research Systems (SRS, Inc.), TeachSpin announces a combination of a high-performance Fourier 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.

Fig. 1: Spectrum of frequency-modulated oscillator. Vertical scale is
logarithmic, covering 90 dB of dynamic range (an amplitude ratio of 30,000:1).

As an example of one of our Electronic Modules, let’s consider the LCR-circuit that can be excited by steady sinusoids, by unit-step waveforms, or by the white-noise generator that is built into the SR770. Exciting this one circuit, in turn, by these three signals, students can learn a great deal about the properties of resonant systems. After some point-by-point measurement of the LCR-circuit’s transfer function using sinusoids, they will be impressed to excite its time-domain transient response using a voltage-step waveform, and then seeing the Fourier transform of this transient give the entire spectrum, complete with phase characteristics, all at once in a single shot – see Fig. 2.

Fig. 2: Real and the imaginary parts of the Fourier transform of an LCR-resonant circuit, excited by a single voltage step and recorded in a single acquisition of duration 64 ms. The spectrum shows the dispersive, and the absorptive, behavior of the resonant system.

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.

Fig. 3: Power spectrum arising from the down-conversion of radio signals. A local oscillator, set to 1113 kHz, is mixed with signals from an antenna, revealing the down-conversion of a station’s frequency of 1080 kHz to a 33-kHz beat note. The sidebands to either side of the down-converted carrier reveal the program content of the AM transmission .

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.

Fig. 4: Power spectral density of white noise, showing the presence of a monochromatic signal under the noise. The noise, filtered to the 0-100 kHz bandwidth, has an rms value of 173 mV, and has a 0.83 mV sinusoidal signal contained within it. The spectrum, viewed over a band 97.5 Hz wide, and averaged for 15 seconds, reveals the signal emerging from its burial under noise, and locates it in frequency space.

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:

  • an acoustic resonator, to permit the study of acoustic modes – including finding them all at once by white-noise excitation.
  • a fluxgate magnetometer, with a frequency-domain view into its operation, and the ability to detect micro Tesla dc and ac magnetic fields.
  • an electronic analog-computer system which creates the Lorenz attractor, so students can see what chaos looks like, in the time and frequency domains.
  • a unique mechanical coupled-oscillator system, allowing the detection of two resonant modes, and a view of how their mode frequencies can be tuned through an ‘avoided crossing’.
  • inputs for bringing in microphone, and line-input, audio signals, permitting students to see the real-time spectra of sounds they are hearing.

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.