FOURIER-Analysis : Examples and Experiments Invited Talk, Irkutsk, June 1999 Prof. Dr. R. Lincke...
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Transcript of FOURIER-Analysis : Examples and Experiments Invited Talk, Irkutsk, June 1999 Prof. Dr. R. Lincke...
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FOURIER-Analysis : Examples and ExperimentsInvited Talk, Irkutsk, June 1999
Prof. Dr. R. LinckeInst. für Experimentelle und Angewandte Physik der Universität Kiel
• Definition of the FOURIER-Series
• FOURIER-Series of Sine, Rectangle and Triangle
• FOURIER-Synthesis and Phase
• Analysis of Vowels, Beats and Amplitude Modulation
• Resonance Curves of LC-Circuits (Generator: Sine and Rectangle)
• FOURIER-Spectra of Rectangles and -Function
• Resonance Curves excited by a -Pulse
• Coupled Acoustic Tubes (Repeating the above Topics)
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FOURIER-Theorem
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FOURIER-Analysis of Some Typical Signals
A) 3 complete periods of a harmonic oscillation in the window result in a single contribution atthe 3rd harmonic.B) A rectangular signal with the on/off ratio 1:1 contains only the odd harmonics i = 1, 3, 5 .. with Ai=1/i.C) A triangular signal also contains only odd harmonics, but now with Ai=1/i2.A)
B) C)
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FOURIER-Synthesis
The FOURIER coefficients determined in the foregoing are now recombined to recover the signal:
A) Ak = 0.9/k with k = 1, 3, 5, ··· yields a rectangle. B) Bk = 0.9/k with k = 1,-3, 5, -7 again produces a rectangle, but now with different phase. C) Bk = 0.573/k2 yields a triangle.
A)
B) C)
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FOURIER-Analysis of Vowels
These are real microphone recordings. Left: the vowel A with many harmonics. Right: the vowel U, which is nearly harmonic and contains only two important partial waves. The period of the acoustic signal was always chosen as window for the analysis.
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FOURIER-Analysis of Beats
Definition of beats : Ys = y1 ·cos(1·t) + y2 ·cos(2·t)Beats are strictly periodic only if the frequencies 1 and 2 are commensurable, i.e. if 1 = n ·(2- 1 ) with n = 1,2,3,4,···. In the left recording, this condition is fulfilled, and theFourier spectrum contains only the contributions 1 and 2 (with frequency and amplitude).For the right - not strictly periodic - signal there existists no FOURIER series.
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Amplitude Modulation (Sine)
The spectrum of an amplitude modu-lation thus contains the unmodifiedcarrier t as well as a lower an upperside band with frequencies t - m
and t + m.
In order for the whole signal to bestrictly periodic, it was again madesure that t und m are commensu-rate (compare Beats).
Definition: YAM = [yt + ym·sin(m·t)]·sin(t·t) (t=carrier, m=modulation)
Rearranging : YAM = yt·sin(t·t)] + ½·ym ·cos[(t - m)·t] - ½·ym ·cos[(t + m)·t]
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Amplitude Modulation (Rectangle)
Here we use a rectangular signal
(50:50) for the amplitude modulation.
This rectangle contains the odd
FOURIER components An ~ 1/n² with
n = 1, 3, 5 ··. They reappear as lower
and upper side bands in the FOURIER
spectrum of the amplitude modulation.
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Measuring C from the Discharge Time
An important application for the FOURIER analysis will be the electromagnetic LC-circuit. For this we measure the capacity using the time constant of discharge: Three time constants R·C correspond to a faktor e-3 = 0,0498, i.e. the voltage must decay from 2000 to a value of 99,6. Here we get 3 R·C = 966 ms. Using the input impedance of UNIMESS
R = 1 M , we get C = 0,322F. This is in perfect agreement with an independent precision measurement!
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Measuring L from the Decay Time
Here we measure the inductance from the decay time of the current through the coil: The time constant T = L/R corresponds to a factor e-1 = 0,368, i.e. the voltage over the shunt must decay from 1900 to 699. We measure L/R = 2,82 ms. With the ohmic resistance of the
coil equal to 11,26 and the shunt equal to 2,74 , there results L = 39,5 mH. This is 5% larger than obtained from a precision measurement.
8,2
2,74
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Damped LC Oscillations
After charging the capacitor to 12V, it is being discharged by the coil: The program triggers on the falling slope and records the damped oscillation. The 6 measured periods correspond to 4.12 ms or = 1456 Hz. With L = 37.75 mH and C = 0.323 F we expected = 1/2· L·C = 1441 Hz. From the two marked amplitudes one calculates the damping constant = ln(1748/776)/(4.77-0.65) ·1000 = 197. From theory one expects = R/2L. With R = 11.26 and L = 37.75 mH this yields = 149. The reason for this error of 32% is not clear.
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LC Resonance Curve with Sinusoidal Driver
The program sweeps the frequency of the generator FD4E and measures and plots the rectified and averaged voltage.
With the FD4E set to a sinusoidal driving voltage, the spectrum contains only the resonance at = 1/2· L·C.
With L = 37.75 mH and C = 0.323 F it should lie at 1441 Hz.
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LC Resonance : Secondary Circuits
If one couples a second LC circuit S (with nomi-nally equal L and C) inductively to the primary circuit P, then the resonance splits up. If one couples a 3rd circuit T onto S one obtains a 3rd maximum in the curve (PST). If, however, one adds the 3rd circuit symmetrically to the other side of the primäry circuit (SPS), it behaves like a 2nd secondary circuit, and the curve has two maxima (now the phases are equal!).
3 Kreise: PST
2 Kreise: PS
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LC Resonance Curve with Rectangular Driver The program sweeps the frequency of the function generator FD4E and records the rectified signal across R.
With the function generator set to rectan-gular output, each harmonic (Ak=1/k2) produces its resonance at k times the frequency (see below). The partial waves thus are physically real, not just mathema-tical tricks!
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FOURIER-Analysis of Rectangles
Sequence of rectangular signals with decreasing on/off ratios :
• rectangle 50% Ai = 1/i, only odd harmonics
• rectangle 10% the 10th harmonic is missing
• rectangle 1% the 100th harmonic is missing
• -function white continuum to
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The spectrum of a delta pulse is a white cont- tinuum. The voltage divider (R=10 kOhm and the impedance of the LC circuit) passes the partial waves according to ist frequency depen-dent characteristic, i.e. the LC resonance curve. The time function is a damped oscillation (or beats with coupled secondary circuit).
Resonance Curves with -Pulses
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Acoustic Pipes: an Alternative to LC Circuits
Coupled acoustic pipes, excited with a loudspeaker, show many of the features discussed in connection with LC circuits.
The detailed form of the inten- sity curve depends strongly on the position of the microphone.
One pipe of 50 cm
LS : loudspeaker M : microphone with precision rectifier R1 and R2 : perspex pipes with distance x.
Because of the end correction, the measured fundamental wave- length is slightly larger than 1 m corresponding to 0 = 341 Hz.
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Acoustic Pipes: Coupling Splits the Resonance
x
f+
f-
A systematic variation of the separation x between the pipes gives the frequencies for the upper and lower maximum shown in the right diagram.
At large separations, the curves converge towards 633 Hz, for small x towards 333 Hz. (Compare the resonances of pipes of length L and 2L).
5 mm 20 mm
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Acoustic Pipes: Phases in Maxima
These are 2-channel recordings made at the upper and lower maximum of the resonance crve (x =3 mm) :
569 Hz 650 Hz
In the lower branch the oscillations are in phase, in the upper in opposite phase
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Acoustic Pipes: Frequency Separation
Now we excite the pipes by discharging a capacitor through the loudspeaker (-Funktion).
The time signal shows damped beats. If one measures the beat frequency fs one obtains the difference between the upper and the lower frequency f+ - f - in the resonance curve.
A FOURIER analysis of the time function recovers the resonance curve.
f+
f -
fs