w4 l1 slides
TRANSCRIPT
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AC Signals - 2
Week 4 - Lecture 1 Mark Bocko
Topics: Tone and frequency spectra
Fourier Series and Fourier Spectral Analysis
Filtering and frequency content of signals
Guitar pickups
RLC band-pass circuits
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Tone?
The quality or character of sound
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The characteristic quality or timbreof a
particular instrument or voice. The character or quality of a musical
sound or voice as distinct from its pitch
and intensity.Frequency !pitchSound pressure level !intensity (loudness)
Attack, Spectrum, Spectral evolution !tone (timbre)
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Building a Square Wave from sine waves
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S(t) = sin[2!f0t] +
1
3sin[2!(3f
0)t] +
1
5sin[2!(5f
0)t] +...
Exploring tone by building different waveforms
Building a triangle wave from sine waves
T(t) = sin[2!f0t] +1
32 sin[2!(3f0 )t] +
1
52 sin[2!(5f0 )t] +...
Notice the 1/n2coefficients!
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Tone is related to the spectrum of a waveform.
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Frequency
Amplitude
Fundamental(f0)
Overtones
The spectrum is a display of the frequency content of a waveform.
Formant (envelope)
If the overtones are at integer multiples of the fundamental theyare called harmonics, e.g., 2f0!2ndharmonic, etc.
Most musical sounds have harmonic (or nearly so) overtones.
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Fourier series
Build any periodic waveform (with period1/f0) from sine and cosine functions
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x(t) =a
0
2
+ [ancos(2!nf
0t) +b
n
n=1
!
" sin(2!nf0t)]
Starting with a function y(t), you can computethe Fourier Coefficients (the recipe!) from
an =1
!y(t) cos(2!nf0t) dt n! 0
onecycle
"
bn =
1
!
y(t) sin(2!nf0t) dt n! 1
one
cycle
"
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Meaning of Fourier Coefficient integrals
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time Amplitude
b1 (sin)
a1 (cos)
= 1
= 0
time
A
mplitude
1
-1
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Fourier Coefficient integrals are a measure of the alikeness of theoriginal function and each sine or cosine wave
time Amp
litude
b2 (sin)
a2 (cos)
= 0
= 0
time Amplitud
e
1
-1
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Spectral analysis
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timeAmplitude
T
f0 = 1/T
2f0
3f0
4f0.
.
.
Lowest frequencyin spectrum
sines b1
b2
b3
b4
Theoretically this goes on to infinite frequency
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Spectral analysis continued
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timeAmplitude
T
f0 = 1/T
2f0
3f0.
.
.
Lowest frequencyin spectrum
cosinesa1
a2
a3
Sn(f) = a
n
2+b
n
2( )1/2 Magnitude of the nth
spectral component
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Spectral Analysis Examples
Guitar
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Contrabass saxophone
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Applying a filter to alter tone
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Frequency
Amplitude
Back to circuits
Low pass
Examples
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Electrical model of a guitar pickup
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~
R
L
C
vs
RL
C 6000 turnsL "2 HenryC "120 pF
R "5,000 #(DC resistance)*
* Effective resistance at audio frequencies "10x DC resistance due to losses in the magnets
vs= voltage signalinduced in coil
from string motion
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Guitar pickup analysis: Find the output voltage, vout
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Apply KVL:
!iR! j!Li + vs!
1
j!Ci = 0
i = vs
R + j!L +1
j!C
vout
= i 1
j!C=v
s
1
j!C
R + j!L +1
j!C
vout
=vs
1
1! !2 !r
2+ j! (!
rQ)
~R vsj!L
1
j!C
- +
vout
!r
2=
1
LC
Q =1
!rRC
=
1
2"
T
RC
Resonant frequency
Q quality factor
T is the period of theresonance of the circuit
vout
= vs
1
(1! !2 !r
2 )2 + !2 (!r
2Q
2 )"# $
%
1/2
i
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Frequency response of pickups
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''=