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Transcript of Sp first l04
04/15/23 © 2003, JH McClellan & RW Schafer 1
Signal Processing First
Lecture 4Spectrum Representation
04/15/23 © 2003, JH McClellan & RW Schafer 3
READING ASSIGNMENTS
This Lecture: Chapter 3, Section 3-1
Other Reading: Appendix A: Complex Numbers
Next Lecture: Ch 3, Sects 3-2, 3-3, 3-7 & 3-8
04/15/23 © 2003, JH McClellan & RW Schafer 4
LECTURE OBJECTIVES
Sinusoids with DIFFERENT Frequencies SYNTHESIZE by Adding Sinusoids
SPECTRUM Representation Graphical Form shows DIFFERENTDIFFERENT Freqs
N
kkkk tfAtx
1
)2cos()(
04/15/23 © 2003, JH McClellan & RW Schafer 5
FREQUENCY DIAGRAM
Plot Complex Amplitude vs. Freq
0 100 250–100–250f (in Hz)
3/7 je 3/7 je2/4 je 2/4 je
10
04/15/23 © 2003, JH McClellan & RW Schafer 6
Another FREQ. DiagramF
req
uen
cy i
s th
e ve
rtic
al a
xis
Time is the horizontal axis
A-440
04/15/23 © 2003, JH McClellan & RW Schafer 7
MOTIVATION
Synthesize Complicated Signals Musical Notes
Piano uses 3 strings for many notes Chords: play several notes simultaneously
Human Speech Vowels have dominant frequencies Application: computer generated speech
Can all signals be generated this way? Sum of sinusoids?
04/15/23 © 2003, JH McClellan & RW Schafer 8
Fur Elise WAVEFORM
BeatNotes
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Speech Signal: BAT
Nearly PeriodicPeriodic in Vowel Region Period is (Approximately) T = 0.0065 sec
04/15/23 © 2003, JH McClellan & RW Schafer 10
Euler’s Formula Reversed
Solve for cosine (or sine)
)sin()cos( tjte tj
)sin()cos( tjte tj
)sin()cos( tjte tj
)cos(2 tee tjtj
)()cos(21 tjtj eet
04/15/23 © 2003, JH McClellan & RW Schafer 11
INVERSE Euler’s Formula
Solve for cosine (or sine)
)()cos(21 tjtj eet
)()sin(21 tjtjj
eet
04/15/23 © 2003, JH McClellan & RW Schafer 12
SPECTRUM Interpretation
Cosine = sum of 2 complex exponentials:
One has a positive frequencyThe other has negative freq.Amplitude of each is half as big
tjAtjA eetA 72
72
)7cos(
04/15/23 © 2003, JH McClellan & RW Schafer 13
NEGATIVE FREQUENCY
Is negative frequency real? Doppler Radar provides an example
Police radar measures speed by using the Doppler shift principle
Let’s assume 400Hz 60 mph +400Hz means towards the radar -400Hz means away (opposite direction) Think of a train whistle
04/15/23 © 2003, JH McClellan & RW Schafer 14
SPECTRUM of SINE
Sine = sum of 2 complex exponentials:
Positive freq. has phase = -0.5 Negative freq. has phase = +0.5
tjjtjj
tjjAtj
jA
eAeeAe
eetA
75.02175.0
21
72
72
)7sin(
5.01 jj
ej
04/15/23 © 2003, JH McClellan & RW Schafer 15
GRAPHICAL SPECTRUMEXAMPLE of SINE
AMPLITUDE, PHASE & FREQUENCY are shown
7-7 0
tjjtjj eAeeAetA 75.02175.0
21)7sin(
5.021 )( jeA 5.0
21 )( jeA
04/15/23 © 2003, JH McClellan & RW Schafer 16
SPECTRUM ---> SINUSOID
Add the spectrum components:
What is the formula for the signal x(t)?
0 100 250–100–250f (in Hz)
3/7 je 3/7 je2/4 je 2/4 je
10
04/15/23 © 2003, JH McClellan & RW Schafer 17
Gather (A,) information
Frequencies: -250 Hz -100 Hz 0 Hz 100 Hz 250 Hz
Amplitude & Phase
4 -/2 7 +/3 10 0 7 -/3 4 +/2
DC is another name for zero-freq componentDC component always has or (for real x(t) )
Note the conjugate phase
04/15/23 © 2003, JH McClellan & RW Schafer 18
Add Spectrum Components-1
Amplitude & Phase
4 -/2 7 +/3 10 0 7 -/3 4 +/2
Frequencies: -250 Hz -100 Hz 0 Hz 100 Hz 250 Hz
tjjtjj
tjjtjj
eeee
eeee
tx
)250(22/)250(22/
)100(23/)100(23/
44
77
10)(
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Add Spectrum Components-2
tjjtjj
tjjtjj
eeee
eeee
tx
)250(22/)250(22/
)100(23/)100(23/
44
77
10)(
0 100 250–100–250f (in Hz)
3/7 je 3/7 je2/4 je 2/4 je
10
04/15/23 © 2003, JH McClellan & RW Schafer 20
Use Euler’s Formula to get REAL sinusoids:
Simplify Components
tjjtjj
tjjtjj
eeee
eeee
tx
)250(22/)250(22/
)100(23/)100(23/
44
77
10)(
tjjtjj eAeeAetA 21
21)cos(
04/15/23 © 2003, JH McClellan & RW Schafer 21
FINAL ANSWER
So, we get the general form:
N
kkkk tfAAtx
10 )2cos()(
)2/)250(2cos(8)3/)100(2cos(1410)(
tttx
04/15/23 © 2003, JH McClellan & RW Schafer 22
Summary: GENERAL FORM
zzze21
21}{ k
jkk
feAX k
Frequency
N
k
tfjk
keXeXtx1
20)(
N
k
tfjk
tfjk
kk eXeXXtx1
2212
21
0)(
N
kkkk tfAAtx
10 )2cos()(
04/15/23 © 2003, JH McClellan & RW Schafer 23
Example: Synthetic Vowel
Sum of 5 Frequency Components
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SPECTRUM of VOWEL
Note: Spectrum has 0.5Xk (except XDC)
Conjugates in negative frequency
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SPECTRUM of VOWEL (Polar Format)
k
0.5Ak
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Vowel Waveform (sum of all 5 components)