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04/15/23 © 2003, JH McClellan & RW Schafer 1
Signal Processing First
Lecture 5Periodic Signals, Harmonics & Time-Varying Sinusoids
04/15/23 © 2003, JH McClellan & RW Schafer 3
READING ASSIGNMENTS
This Lecture: Chapter 3, Sections 3-2 and 3-3 Chapter 3, Sections 3-7 and 3-8
Next Lecture:
Fourier Series ANALYSISFourier Series ANALYSIS Sections 3-4, 3-5 and 3-6
04/15/23 © 2003, JH McClellan & RW Schafer 4
Problem Solving Skills
Math Formula Sum of Cosines Amp, Freq, Phase
Recorded Signals Speech Music No simple formula
Plot & Sketches S(t) versus t Spectrum
MATLAB Numerical Computation Plotting list of
numbers
04/15/23 © 2003, JH McClellan & RW Schafer 5
LECTURE OBJECTIVES
Signals with HARMONICHARMONIC Frequencies Add Sinusoids with fk = kf0
FREQUENCY can change vs. TIMEChirps:
Introduce Spectrogram Visualization (specgram.m) (plotspec.m)
x(t) cos(t2 )
N
kkk tkfAAtx
100 )2cos()(
04/15/23 © 2003, JH McClellan & RW Schafer 6
SPECTRUM DIAGRAM
Recall Complex Amplitude vs. Freq
kk aX 21
0 100 250–100–250f (in Hz)
3/7 je 3/7 je2/4 je 2/4 je
10
)2/)250(2cos(8)3/)100(2cos(1410)(
tttx
kjkk eAX
kX2
1
04/15/23 © 2003, JH McClellan & RW Schafer 7
SPECTRUM for PERIODIC ?
Nearly Periodic in the Vowel Region Period is (Approximately) T = 0.0065 sec
04/15/23 © 2003, JH McClellan & RW Schafer 8
PERIODIC SIGNALS
Repeat every T secs Definition
Example:
Speech can be “quasi-periodic”
)()( Ttxtx
)3(cos)( 2 ttx ?T
3T
32T
04/15/23 © 2003, JH McClellan & RW Schafer 9
Period of Complex Exponential
Definition: Period is T
k = integer
tjTtj ee )(
?)()()(
txTtxetx tj
12 kje
kTe Tj 21
kkTT
k0
22
04/15/23 © 2003, JH McClellan & RW Schafer 10
N
k
tkfjk
tkfjk
jkk
N
kkk
eXeXXtx
eAX
tkfAAtx
k
1
2212
21
0
100
00)(
)2cos()(
Harmonic Signal Spectrum
0:haveonly can signal Periodic fkfk
Tf
10
04/15/23 © 2003, JH McClellan & RW Schafer 11
Define FUNDAMENTAL FREQ
00
1
Tf
Periodlfundamenta Frequencylfundamenta
)2(
)2cos()(
0
0
000
100
Tf
ffkf
tkfAAtx
k
N
kkk
04/15/23 © 2003, JH McClellan & RW Schafer 12
What is the fundamental frequency?
Harmonic Signal (3 Freqs)
3rd5th
10 Hz
04/15/23 © 2003, JH McClellan & RW Schafer 13
POP QUIZ: FUNDAMENTAL
Here’s another spectrum:
What is the fundamental frequency?
100 Hz ? 50 Hz ?
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 14
SPECIAL RELATIONSHIPto get a PERIODIC SIGNAL
IRRATIONAL SPECTRUM
04/15/23 © 2003, JH McClellan & RW Schafer 17
FREQUENCY ANALYSIS
Now, a much HARDER problemNow, a much HARDER problem Given a recording of a song, have the
computer write the music
Can a machine extract frequencies? Yes, if we COMPUTE the spectrum for x(t)
During short intervals
04/15/23 © 2003, JH McClellan & RW Schafer 18
Time-Varying FREQUENCIES Diagram
Fre
qu
ency
is
the
vert
ical
axi
s
Time is the horizontal axis
A-440
04/15/23 © 2003, JH McClellan & RW Schafer 19
SIMPLE TEST SIGNAL C-major SCALE: stepped frequencies
Frequency is constant for each note
IDEAL
04/15/23 © 2003, JH McClellan & RW Schafer 20
R-rated: ADULTS ONLY
SPECTROGRAM Tool MATLAB function is specgram.m SP-First has plotspec.m & spectgr.m
ANALYSIS program Takes x(t) as input & Produces spectrum values Xk
Breaks x(t) into SHORT TIME SEGMENTS Then uses the FFT (Fast Fourier Transform)
04/15/23 © 2003, JH McClellan & RW Schafer 21
SPECTROGRAM EXAMPLE Two Constant Frequencies: Beats
))12(2sin())660(2cos( tt
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tjtjj
tjtj eeee )12(2)12(221)660(2)660(2
21
AM Radio Signal Same as BEAT Notes
))12(2sin())660(2cos( tt
))648(2cos())672(2cos( 221
221 tt
tjtjtjtjj eeee )648(2)648(2)672(2)672(2
41
04/15/23 © 2003, JH McClellan & RW Schafer 23
SPECTRUM of AM (Beat)
4 complex exponentials in AM:
What is the fundamental frequency?
648 Hz ? 24 Hz ?
0 648 672f (in Hz)
–672 –648
41
41
41
41
04/15/23 © 2003, JH McClellan & RW Schafer 24
STEPPED FREQUENCIES C-major SCALE: successive sinusoids
Frequency is constant for each note
IDEAL
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SPECTROGRAM of C-Scale
ARTIFACTS at Transitions
Sinusoids ONLY
From SPECGRAMANALYSIS PROGRAM
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Spectrogram of LAB SONG
ARTIFACTS at Transitions
Sinusoids ONLY
Analysis Frame = 40ms
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Time-Varying Frequency
Frequency can change vs. time Continuously, not stepped
FREQUENCY MODULATION (FM)FREQUENCY MODULATION (FM)
CHIRP SIGNALS Linear Frequency Modulation (LFM)
))(2cos()( tvtftx c VOICE
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)2cos()( 02 tftAtx
New Signal: Linear FM
Called Chirp Signals (LFM) Quadratic phase
Freq will change LINEARLY vs. time Example of Frequency Modulation (FM) Define “instantaneous frequency”
QUADRATIC
04/15/23 © 2003, JH McClellan & RW Schafer 29
INSTANTANEOUS FREQ
Definition
For Sinusoid:
Derivativeof the “Angle”)()(
))(cos()(
tt
tAtx
dtd
i
Makes sense
0
0
0
2)()(
2)()2cos()(
ftt
tfttfAtx
dtd
i
04/15/23 © 2003, JH McClellan & RW Schafer 30
INSTANTANEOUS FREQof the Chirp
Chirp Signals have Quadratic phase Freq will change LINEARLY vs. time
ttt
ttAtx2
2
)(
)cos()(
tttdtd
i 2)()(
04/15/23 © 2003, JH McClellan & RW Schafer 33
OTHER CHIRPS
(t) can be anything:
(t) could be speech or music: FM radio broadcast
))cos(cos()( tAtx
)sin()()( tttdtd
i