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2/9/2015 © 2003, JH McClellan & RW Schafer 1 Signal Processing First Lecture 5 Periodic Signals, Harmonics & Time-Varying Sinusoids
2/9/2015 © 2003, JH McClellan & RW Schafer 1
Signal Processing First
Lecture 5
Periodic Signals, Harmonics
& Time-Varying Sinusoids
2/9/2015 © 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 ANALYSIS Sections 3-4, 3-5 and 3-6
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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
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LECTURE OBJECTIVES
Signals with HARMONIC Frequencies
Add Sinusoids with fk = kf0
FREQUENCY can change vs. TIME
Chirps:
Introduce Spectrogram Visualization (specgram.m)
(plotspec.m)
x(t) = cos(at2)
N
k
kk tkfAAtx1
00 )2cos()(
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SPECTRUM DIAGRAM
Recall Complex Amplitude vs. Freq
kk aX 21
0 100 250 –100 –250 f (in Hz)
3/7 je3/7 je
2/4 je 2/4 je
10
)2/)250(2cos(8
)3/)100(2cos(1410)(
t
ttx
kjkk eAX
kX
21
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SPECTRUM for PERIODIC ?
Nearly Periodic in the Vowel Region
Period is (Approximately) T = 0.0065 sec
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PERIODIC SIGNALS
Repeat every T secs
Definition
Example:
Speech can be “quasi-periodic”
)()( Ttxtx
x(t) = cos(3t)?T
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Period of Complex Exponential
Definition: Period is T
k = integer
tjTtj ee )(
?)()(
)(
txTtx
etx tj
12 kje
kTe Tj 21
kkTT
k0
22
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N
k
tkfjk
tkfjk
jkk
N
k
kk
eXeXXtx
eAX
tkfAAtx
k
1
2
212
21
0
1
00
00)(
)2cos()(
Harmonic Signal Spectrum
0:haveonly can signal Periodic fkfk
Tf
10
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Define FUNDAMENTAL FREQ
00
1
Tf
x(t) = A0 + Ak cos(2pkf0t +fk )k=1
N
å
fk = kf0 (w0 = 2p f0 )
f0 = fundamental Frequency
T0 = fundamental Period
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What is the fundamental frequency?
Harmonic Signal (3 Freqs)
3rd 5th
10 Hz
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POP QUIZ: FUNDAMENTAL
Here’s another spectrum:
What is the fundamental frequency?
0 100 250 –100 –250 f (in Hz)
3/7 je3/7 je
2/4 je 2/4 je
10
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SPECIAL RELATIONSHIP
to get a PERIODIC SIGNAL
IRRATIONAL SPECTRUM
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Harmonic Signal (3 Freqs)
T=0.1
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NON-Harmonic Signal
NOT
PERIODIC
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FREQUENCY ANALYSIS
Now, 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
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Time-Varying
FREQUENCIES Diagram F
req
uen
cy
is
th
e v
ert
ical
ax
is
Time is the horizontal axis
A-440
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SIMPLE TEST SIGNAL
C-major SCALE: stepped frequencies
Frequency is constant for each note
IDEAL
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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)
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SPECTROGRAM EXAMPLE
Two Constant Frequencies: Beats
))12(2sin())660(2cos( tt
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tjtj
j
tjtj eeee )12(2)12(2
21)660(2)660(2
21
AM Radio Signal
Same as BEAT Notes
))12(2sin())660(2cos( tt
))648(2cos())672(2cos(22
122
1 tt
tjtjtjtj
jeeee )648(2)648(2)672(2)672(2
41
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SPECTRUM of AM (Beat)
4 complex exponentials in AM:
What is the fundamental frequency?
648 Hz ? 24 Hz ?
0 648 672 f (in Hz)
–672 –648
2/
41 je
2/
41 je2/
41 je2/
41 je
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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 SPECGRAM
ANALYSIS 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)
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
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INSTANTANEOUS FREQ
Definition
For Sinusoid:
Derivative
of the “Angle” )()(
))(cos()(
tt
tAtx
dtd
i
Makes sense
0
0
0
2)()(
2)(
)2cos()(
ftt
tft
tfAtx
dtd
i
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INSTANTANEOUS FREQ
of the Chirp
Chirp Signals have Quadratic phase
Freq will change LINEARLY vs. time
ttt
ttAtx2
2
)(
)cos()(
tttdtd
i 2)()(
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CHIRP SPECTROGRAM
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CHIRP WAVEFORM
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OTHER CHIRPS
(t) can be anything:
(t) could be speech or music:
FM radio broadcast
))cos(cos()( tAtx
)sin()()( tttdtd
i
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SINE-WAVE FREQUENCY
MODULATION (FM)
Look at CD-ROM Demos in Ch 3
