Fourier Methods
In collaboration with Stanford Research Systems (SRS, Inc.), TeachSpin announces a combination of a highperformance Fourier analyzer (the SR770) and a TeachSpin ‘physics package’ of apparatus, experiments, and a selfpaced 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.
Introduction
What are the electronic instrumentation skills that physics students ought to acquire in an undergraduate advancedlab program? No doubt skills with a multimeter and oscilloscope are basic, and skills with a lockin amplifier and computer dataacquisition system are more advanced. But our ‘Fourier Methods’ offering adds an intermediatetoadvancedlevel and highlytransferable 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 handson benchtop familiarity with Fourier methods in realtime 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 16bit precision at a 256 kHz rate, and it includes antialiasing filters to permit the realtime acquisition of Fourier transforms in the 0100 kHz range. Any subrange of the spectrum can be viewed at resolutions down to milliHertz. The sensitivity and dynamic range are such that subµVolt signals can be displayed with ease, as well as Voltlevel signals with signaltonoise 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 CoupledOscillator system. Not shown is an instrumentcase 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 ‘timedomain’ and ‘frequencydomain’ views of a signal are related. One of our Electronic Modules is a voltage controlled oscillator (VCO), which can be frequencymodulated 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 50kHz center frequency, with a 1kHz modulation frequency. This spectrum shows the existence of sidebands, and the frequency ‘real estate’ required by a modulated signal; it also shows that Voltlevel signals can be detected standing >90 dB above the noise floor of the instrument.
Fig. 1: Spectrum of frequencymodulated 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 LCRcircuit that can be excited by steady sinusoids, by unitstep waveforms, or by the whitenoise 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 pointbypoint measurement of the LCRcircuit’s transfer function using sinusoids, they will be impressed to excite its timedomain transient response using a voltagestep 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.
Because frequencymixing 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 downshifted into the 0100 kHz band. Fig. 3 shows a view of part of the AMradio 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. 2: Real and the imaginary parts of the Fourier transform of an LCRresonant 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.
Fig. 3: Power spectrum arising from the downconversion of radio signals. A local oscillator, set to 1113 kHz, is mixed with signals from an antenna, revealing the downconversion of a station’s frequency of 1080 kHz to a 33kHz beat note. The sidebands to either side of the downconverted carrier reveal the program content of the AM transmission .
The SR770 includes a highgain front end making it capable of detecting very weak signals. And because it disperses those signals by frequency content, and permits timeaveraging, it is also capable of detecting weak signals that are deeply buried in noise. Our Electronic Modules include a signalundernoise 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 lockin 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 0100 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 whitenoise 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 whitenoise excitation.

fluxgate magnetometer, with a frequencydomain view into its operation, and the ability to detect micro Tesla dc and ac magnetic fields.

an electronic analogcomputer system which creates the Lorenz attractor, so students can see what chaos looks like, in the time and frequency domains.

a unique mechanical coupledoscillator 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 lineinput, audio signals, permitting students to see the realtime spectra of sounds they are hearing.
Physicists acquire Fourierthinking skills in a variety of ways, and apply these skills in many subfields of physics. Advancedlab 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.
Instrument
TeachSpin’s ‘Fourier Methods’ initiative represents the strategic combination of several elements:

SR770 FFT SpectrumAnalyzer Instrument by Stanford Research Systems

TeachSpin Electronic Modules Package

Additional ExternalHardware Experiments

Acoustic Resonator

Fluxgate Magnetometer

CoupledOscillator System


Instruction Manual with Full Curriculum
Here’s a description of what is included under each of these headings, and how together they provide an unparalleled avenue for students to learn frequencyspace thinking.
1. The Stanford Research Systems SR770 is an FFT spectrum analyzer capable of many kinds of Fourier analysis. Its capabilities extend far beyond the FFT function included in some oscilloscopes, and the TeachSpin curriculum has been designed to make full use of those capabilities:

16bit signal quantization, allowing dynamic range in excess of 90 dB (compared to <50 dB for typical 8bit ‘scope quantization)

realtime operation, with full 100kHz throughput (ie. Fouriertransforms data as fast as it is acquired)

precision analoghardware antialiasing filter (permitting 0100 kHz operation with no fear of spectral aliasing)

divisible frequency span, allowing span of 100 kHz or binary submultiples (allowing a span as small as 0.192 Hz, and centering such spans anywhere in the 0100 kHz range)

full control of display, window, and averaging modes, including vector averaging for triggered acquisitions

frontend fullscale sensitivity of 1 mV, allowing detection of signals of amplitude < 100 nV

an internal SourceOut capability, for generating sine, twotone, noise, and chirp signals of chosen frequency and amplitude

storage of setup and data files in a new USB drive

full GPIB and RS232 access to instrument control and data (for optional computer interfacing)
Simply put, this is a premier instrument, uniquely suited to the teaching, and learning, of Fourier methods. TeachSpin has created both the hardware and curriculum to exploit its full capabilities.
2. The TeachSpin 'Electronic Modules' Package  In one centrally powered box,
TeachSpin’s ‘Electronic Modules’ houses all the building blocks needed to teach
Fourier thinking.
The TeachSpin Physics Module Package Includes:

an analog Summer, and a Multiplier, to teach the difference between a
superposition and a product, and to make modulation experiments possible 
two doublebalanced Mixers, optimized for Audio and High Frequency bands,
to make demodulation experiments possible 
a WideBand Amplifier with 02 MHz coverage which can also be used with a
separate AM Antenna module, for one class of radiodetection experiments 
a Power Audio Amplifier and Speaker for various audible effects in Fourier
Methods 
a Voltage Controlled Oscillator (for experiments with frequencymodulation (FM) waveforms, and FM demodulation)

a Chaos module  an analogcomputer realization of the Lorenzattractor system for investigating deterministic chaos in time and frequencydomains

a Buried Treasure module, including choices of sinusoidal signals buried under noise for experiments in noise metrology, and signal extraction

an LRC Circuit, as an example of a linear timeinvariant system that can be used as an introduction to the transfer function, and its study by Fourier methods

an Intermodulation Distortion module to provide experiments in frequencyspace methods for detection of nonlinearity

a utility Filter module, for high, low, and bandpass filtering with various corner frequencies and Qfactors both for study via its transfer function, and as a building block in more complicated assemblies

an Audio Connections module that allows students to easily transfer signals between 3.5mm consumermarket connectors and the BNCinstrumentation world

a utility DC Voltage module that can be used with the Summer and Multiplier or Voltage Controlled Oscillator.
3. Three ExternalHardware Experiments further expand the range of investigations.
a) an Acoustic Resonator, with speaker and microphone, of cylindrical geometry with fixed diameter but variable length that is used to illustrate the simultaneous detection of a host of normal modes by noise (or chirp) excitation, and Fourier analysis
b) a Fluxgate Magnetometer, with primarycoil excitation, secondarycoil detection, and a special double solenoid for calibration to show the use of nonlinear ferromagnetic response as a tool for the detection of µTlevel magnetic fields
c) a special CoupledOscillator System, of two torsionalreed oscillators that is both magneticallydriven and magneticallycoupled to permit the systematic study of normalmode behavior in a twomode system, including the study of the avoided crossing of normal modes
4. Instrument Manual  The educational mission of TeachSpin would be incomplete without a full Instruction Manual teaching the use of all these capabilities. It includes:
Introductory Chapters on:

Learning to use a Fouriertransform spectrometer

The SourceOut capabilities of the SR770

How to produce, and detect, Amplitude Modulation

What is Heterodyning, and how does a Mixer do it?

How to produce, and detect, Frequency Modulation

Quantifying Noise waveforms, and introducing the spectral noise density

Studying the LCR system by Fourier methods

Transfer functions of Acoustic systems

Transient waveforms and their Fourier transforms
Additional Project Experiments which could be performed in any order:

Pulsemodulation and its Fourier spectrum

Downconversion and demodulation of AMradio signals

Deterministic chaos, in time and frequencydomain views

The Fluxgate Magnetometer, exploiting harmonic distortion

Frequencydomain views of audio waveforms

Signalextraction tactics for signals under noise

Coupled oscillators, in theory and experiment

Detection of nonlinearity by Fourier Methods

Demodulation of FM signals
The Manual also contains over a dozen Appendices, where various mathematical, conceptual, and computational issues are addressed. The result is an unmatched resource for teaching, and learning, Fourier methods of calculating and thinking.
Experiments
‘Fourier Methods’ is preeminently designed to teach skills in frequencydomain thinking, so our manual includes a long list of skillbuilding activities. But some of those skills have immediate application to classic experimental techniques, as noted in the projects list below.
Activities:

Learning the ropes

Superposition

Modulations

Noise

Transfer functions

Transient waveforms
Projects:

AM radio

FM communication

Fluxgate magnetometry

Couples oscillators

Chaos
Activities
1. Learning the ropes
Any FFT analyzer can be thought of as a multifrequency ac voltmeter, so a first activity is to subject the SR770 to a sinusoidal waveform of variable frequency and amplitude. Seeing a sinusoid, which is continuous in the time domain, show up in the frequency domain. as a deltafunction, a single static peak, will help students understand the way FFT analyzer presents information.
2. Superpositions
The Electronic Modules allow the easy creation of a superposition of two sinusoids, one from the 770’s internal generator, and another from any generic signal generator.
The 770 can easily resolve signals that differ by only partspermillion in frequency. To the right is the spectrum of a superposition of two sinusoidal signals: one is at 50,000.000 Hz, of 100mV amplitude; the other is at 50,000.282 Hz, of 10mV amplitude. The 770 not only reveals the frequency difference, less than 6 ppm, it also reveals that one of the oscillators involved is not quite ‘monochromatic’ at this level of frequency resolution.
The 770 can also display signals with amplitudes that differ by a huge factor. Here’s the spectrum of a superposition of frequency components at 50.12 kHz (at 100 mV amplitude) and 51.81 kHz (at 0.001 mV = 1 µV amplitude). The amplitude ratio is 105:1, or 100 dB. The spectrum displayed is the average of 64 spectra, and took less than 4 seconds to acquire.
3. Modulations
Using the Multiplier module, students can set up amplitudemodulated waveforms (using the 770’s internal source as ‘carrier’, and a generic signal generator as ‘signal’ source). They will see that the frequency spectrum of an AM wave is no longer a deltafunction, but now has sidebands. This also points out the spectral bandwidth that any informationcarrying signal must take up.
Using the VCO module, students can similarly set up frequencymodulated waveforms. Here, they will see that an FM waveform’s spectrum contains multiple sidebands, with magnitudes given by a Besselfunction expansion. They will also see that a carrier, whose frequency is modulated continuously in time, nevertheless will display a spectrum with discrete sidebands.
4. Noise
Students can use the Noise module to see noise waveforms in the time and frequency domains. They can also learn the connection between a noise signal’s rms measure (in the time domain) and its noise spectral density (in the frequency domain). They can learn the quantitative significance of the units V2/Hz and V/√Hz as appropriate units for noise.
Students can also learn how to use the 770 for the ‘blind detection’ of a weak sinusoidal signal buried under broadband noise. In the process, they’ll see how it is that a lockin amplifier can work, but with the advantage of seeing how a signal peak can be made to stand up above a ‘noise floor’.
5. Transfer functions
With noise waveforms modeled and measured as superpositions of sinusoids, students are now ready to see the usefulness of noise as an excitation source for systems. Fourier Methods includes an Acoustic Resonator, and an electronic LCR Circuit, as model linear systems. Each can have its behavior investigated by traditional onefrequencyat atime, generatorandoscilloscope methods. But each can alternatively have its entire frequency spectrum acquired ‘all at once’, in a lovely illustration of how Fourier methods are actually used in so many forms of spectroscopy.
To the right shows the spectrum of pickup from a microphone in the Acoustic Resonator, when the speaker exciting that resonator is driven by a quasinoise ‘chirp’ waveform. The entire spectrum, showing a whole collection of resonant modes, can be acquired afresh every 32 ms.
6. Transient waveforms
The SR770 allows the triggered acquisition of transient waveforms. Since this triggering establishes a t = 0 definition, it also permits the derivation of phase information in Fourier spectra. Students can learn the value of magnitude & phase, or alternatively, real & imaginary parts, in the depiction of spectra and of transfer functions. They can also learn how averaging of results from repeated pulsed excitation of transients allows powerful noisereduction techniques.
Projects
In Fourier Methods we have included some hardware apparatus especially suited to projects illustrating the conceptual value of Fourier thinking.
1. AM Radio
Using almost any available signal generator as a ‘local oscillator’, students can use the external Antenna structure included in the apparatus and a whole assembly of Electronic Modules to understand, systematically, the process of downconversion and demodulation. They will not only be able to hear live AM radio broadcasts on a speaker, but they will also understand the roles of amplification, mixing, filtering, and detection in the process.
2. FM Communication
When studying Modulation in section 3 of Activities, students learn how a signal waveform can be ‘encoded’ by its action in frequencymodulating a carrier. Then, using the Electronic Modules, including crucially the phase response of an LCR circuit, students can demodulate an FM waveform, and thereby ‘decode’ the original signal information.
3. Fluxgate magnetometry
While a Fourier analyzer is sometimes used to detect and quantify undesired ‘harmonic distortion’, there are places in physics where the harmonics generated by nonlinearity are the desired outcome. So it is with a Fluxgate Magnetometer, in which external dc (or ac!) magnetic fields, even at the µT (0.01 gauss) level, can be detected via the nonlinearity they help to create in acexcited ferromagnetic materials. Students will learn a reallife application of the triad of fields B, H, and M, and learn how to operate and calibrate a magneticfield sensor based on these properties.
4. Coupled oscillators
Many linear systems can be understood via their spectrum of normal modes, so we’ve provided a system of two mechanical oscillators which allows the systematic study of a twomode system. Our CoupledOscillator System has adjustable magnetic coupling of the two oscillators, and separatelyadjustable tuning of the two oscillators’ frequencies. Both oscillators can be excited or detected by ac electrical means, so this coupledoscillator system can be studied via its Fourier spectrum in real time. We have optimized the system to permit the detection of the ‘avoided crossing’ of two normalmode frequencies, to introduce students to this widelyencountered phenomenon.
To the right is the response to electrical stimulation of our coupledoscillator system, again obtained allatonce using chirp excitation. The two major peaks locate the system’s two normal modes, and this spectrum gives the frequencies of both modes to <0.1Hz precision. It’s those two normalmode frequencies which undergo an ‘avoided crossing’ in response to a tuning parameter.
5. Chaos
One of the revelations in physics in recent times has been to see apparent randomness arising from wholly deterministic systems. The ‘chaos revolution’ broke into physicists’ consciousness with the properties of the Lorenz attractor, derived from fluid mechanics and modeled by a set of three fullydeterministic, firstorder, but nonlinear differential equations. The electronic modules of Fourier Methods include an electronic analogcomputer realization of this Lorenz system, so students have realtime access to three chaoticallyoscillating voltages. But in the FourierMethods environment, they can also, and simultaneously, view any of these signals in the frequency domain. Because the Lorenz system can be tuned into, or out of, regimes of periodicity surrounded by chaotic behavior, observing this phenomenon in the frequency domain allows especially clear insight into the quite general feature of the ‘perioddoubling route to chaos’
While our Lorenzattractor module gives a continuous frequency spectrum when in its chaotic mode, it can also be adjusted into a regime of periodic output. In such a case, the Fourier spectrum collapses from a continuum into a series of harmonicallyrelated spectral lines.
Then a small adjustment of the Lorenz system’s rparameter will change its behavior into a perioddoubled regime. Here, the Fourier spectrum is still a harmonic series of spectral lines, but a new, and lowerfrequency, spectral line has emerged. So the fundamental frequency is about half, ie. the period is about double, compared to the previous behavior.