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ECE- 2026 Spring- 2018
Lecture 6
AM, Periodic Signals &
Harmonics
29 - Jan- 2018
EE-2026 Spring 2018
General Info
HW #2 due this week (delayed by one week)
Lab #2 starting this week (delayed by one week)
Three Quizzes will count for 30 points in your grade
Written: counted 10 points each, 2/6 (M), 3/3 (F), 4/3 (M)
Final Exam will count for 30 points in your grade
2026A on 4/27 (F, 11:30am), 2026B on 5/3 (W, 8:00am)
Take all tests with your assigned section
No Quiz or Final Exam make-up
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EE-2026 Spring 2018
Quiz #1 Info (on 9 - Feb, Monday)
Covering materials in the first FOUR lectures
Chapters 1-2 and Appendices A & B
Homework: #01- 03 , Labs: 00- 02 , ITS: # 1
One-page, double-sided notes allowed
Calculators with extra batteries recommended
Graphing calculators ALLOWED, NO GRAPHICS USAGE
Review session: See T-Square Announcement
UG tutor: http://ecetutoring.gatech.edu/schedule.html
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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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EE-2026 Spring 2018
SPECTRUM DIAGRAM
Recall Complex Amplitude vs. Freq
k k
X a
2
1
- 250 – 100 0 100 250
f (in Hz)
/ 3
7
j
e
/ 3
7
j
e
/ 2
4
j
e
/ 2
4
j
e
10
8 cos( 2 ( 250 ) / 2 )
( ) 10 14 cos( 2 ( 100 ) / 3 )
t
x t t
k
j
k k
X Ae
k
X
2
1
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SPECTRUM for PERIODIC?
Nearly Periodic in the Vowel Region
Period is (Approximately) T = 0.0065 sec
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cos( 2 ( ) ) cos( 2 2 )
( )willbeperiodicif
( ) cos( 2 )
0 0 0
1
0 0
k k
N
k
k k
kf t T kft kf T
x t
x t A A kft
Harmonic Signal
0
Canonlyhave freqs:
Periodicsignal: ( ) ( )
f k f
x t x t T
k
harmonic
0
fT
9
fundamentalPeriod
/ integer,forall
fundamentalFrequency
Largest suchthat
() cos( 2 )
0
0
0
0 0 0
0
1
0 0
T
f f k
f
f kf f
f
x t A A kft
k
k
N
k
k k
EE-2026 Spring 2018
Define FUNDAMENTAL FREQ
0
0
1
T
f
Main point :
for periodic signals, all
spectral components are
integer multiples of the
fundamental frequency
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EE-2026 Spring 2018
What is the fundamental frequency?
Harmonic Spectrum (3 Freqs)
3rd
5th
10 Hz
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POP QUIZ: FUNDAMENTAL
Here’s another spectrum:
What is the fundamental frequency (unconventional GCD)?
(0.1)GCD(104,240) = (0.1)(8)=0.8 Hz
- 24 – 10.4 0 10.4 24
f (in Hz)
/ 3
7
j
e
/ 3
7
j
e
/ 2
4
j
e
/ 2
4
j
e
10
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SPECIAL RELATIONSHIP
to get a PERIODIC SIGNAL
IRRATIONAL SPECTRUM
NON-PERIODIC SIGNAL
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Harmonic Signal (3 Freqs)
T=0.
PERIODIC
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Time-Varying
FREQUENCIES Diagram
Frequency is the vertical axis
Time is the horizontal axis
A- 440
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STEPPED FREQUENCIES
C-major SCALE: successive sinusoids
Frequency is constant for each note
IDEAL
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Mixed FREQUENCIES; DTMF
or Dual Tone Multi-Frequency
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SPECTRUM of Beat Signals
1 2
( ) cos(2 ) cos(2 )
2cos(2 ) cos(2 )
c
x t f t f t
f t f t
f
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EE-2026 Spring 2018
SPECTROGRAM EXAMPLE
Two Constant Frequencies: Beats
2 cos( 2 ( 12 ))cos( 2 ( 660 ))
cos( 2 ( 672 )) cos( 2 ( 648 ))
t t
t t
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EE-2026 Spring 2018
j t j t
j
j t j t
e e e e
2 ( 12 ) 2 ( 12 )
2
1
2 ( 660 ) 2 ( 660 )
2
1
AM Radio Signal
Same as BEAT Notes
cos( 2 ( 660 ) t )sin( 2 ( 12 ) t )
cos( 2 ( 672 ) ) cos( 2 ( 648 ) )
2 2
1
2 2
1
t t
j t j t j t j t
j
e e e e
2 ( 672 ) 2 ( 672 ) 2 ( 648 ) 2 ( 648 )
4
1
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SPECTRUM of AM
(Amplitude Modulation)
SUM of 4 complex exponentials:
What is the fundamental frequency?
648 Hz? 24 Hz?
0 648 672
f (in Hz)
- 672 – 648
/ 2
4
1
j
e
/ 2
4
1
j
e
/ 2
4
1
j
e
/ 2
4
1
j
e
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EE-2026 Spring 2018
SPECTROGRAM of C-Scale
ARTIFACTS at Transitions
Sinusoids ONLY
From SPECGRAM
ANALYSIS PROGRAM
28
