Transcript FOURIER-Analysis : Examples and Experiments Prof. Dr. R

FOURIER-Analysis : Examples and Experiments
Invited Talk, Irkutsk, June 1999
Prof. Dr. R. Lincke
Inst. für Experimentelle und Angewandte Physik der Universität Kiel
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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)
FOURIER-Theorem
FOURIER-Analysis of Some Typical Signals
A)
A) 3 complete periods of a harmonic oscillation
in the window result in a single contribution at
the 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.
B)
C)
FOURIER-Synthesis
The FOURIER coefficients determined in the
foregoing are now recombined to recover the
signal:
A)
B)
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
rectangle, but now with different phase.
C) Bk = 0.573/k2 yields a triangle.
C)
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.
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 the
Fourier spectrum contains only the contributions 1 and 2 (with frequency and amplitude).
For the right - not strictly periodic - signal there existists no FOURIER series.
Amplitude Modulation (Sine)
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]
The spectrum of an amplitude modulation thus contains the unmodified
carrier t as well as a lower an upper
side band with frequencies t - m
and t + m.
In order for the whole signal to be
strictly periodic, it was again made
sure that t und m are commensurate (compare Beats).
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.
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 = 0,322F.
This is in perfect agreement with
an independent precision
measurement!
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
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.
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.
LC Resonance : Secondary Circuits
If one couples a second LC circuit S (with nominally 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!).
2 Kreise:
PS
3 Kreise:
PST
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 rectangular 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 mathematical tricks!
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 
Resonance Curves with -Pulses
The spectrum of a delta pulse is a white conttinuum. The voltage divider (R=10 kOhm and
the impedance of the LC circuit) passes the
partial waves according to ist frequency dependent characteristic, i.e. the LC resonance curve.
The time function is a damped oscillation (or
beats with coupled secondary circuit).
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.
LS : loudspeaker
M : microphone with precision rectifier
R1 and R2 : perspex pipes with distance x.
The detailed form of the intensity curve depends strongly on
the position of the microphone.
Because of the end correction, the measured fundamental wavelength is slightly larger than 1 m corresponding to 0 = 341 Hz.
One pipe of 50 cm
Acoustic Pipes: Coupling Splits the Resonance
5 mm
20 mm
A systematic variation of the separation x
between the pipes gives the frequencies for
f+
the upper and lower maximum shown in
the right diagram.
fAt large separations, the curves converge
towards 633 Hz, for small x towards 333
Hz. (Compare the resonances of pipes of
length L and 2L).
x
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
In the lower branch the oscillations are in phase, in the upper in opposite phase
650 Hz
Acoustic Pipes: Frequency Separation
Now we excite the pipes by discharging a
capacitor through the loudspeaker (-Funktion).
f+
fs
f-
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.