Signal and Systems
Lecture 4
Spectrum Representation
© 2003, JH McClellan & RW Schafer
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LECTURE OBJECTIVES
 Sinusoids with DIFFERENT Frequencies
 SYNTHESIZE by Adding Sinusoids
N
x(t )   Ak cos(2 f k t   k )
k 1
 SPECTRUM Representation
 Graphical Form shows DIFFERENT Freqs
© 2003, JH McClellan & RW Schafer
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FREQUENCY DIAGRAM
 Plot Complex Amplitude vs. Freq
4e j / 2
–250
7e
j / 3
–100
10
0
7e j / 3
100
4e j / 2
250
f (in Hz)
© 2003, JH McClellan & RW Schafer
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Frequency is the vertical axis
Another FREQ. Diagram
A-440
Time is the horizontal axis
© 2003, JH McClellan & RW Schafer
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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?
© 2003, JH McClellan & RW Schafer
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Fur Elise WAVEFORM
Beat
Notes
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Speech Signal: BAT
 Nearly Periodic in Vowel Region
 Period is (Approximately) T = 0.0065 sec
© 2003, JH McClellan & RW Schafer
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Euler’s Formula Reversed
 Solve for cosine (or sine)
e j t  cos( t )  j sin( t )
e j t  cos( t )  j sin( t )
e j t  cos( t )  j sin( t )
e j t  e j t  2 cos( t )
cos( t )  12 (e j t  e j t )
© 2003, JH McClellan & RW Schafer
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INVERSE Euler’s Formula
 Solve for cosine (or sine)
cos( t )  12 (e j t  e j t )
sin( t ) 
1 (e j t
2j
 e j t )
© 2003, JH McClellan & RW Schafer
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SPECTRUM Interpretation
 Cosine = sum of 2 complex exponentials:
A cos(7t ) 
A e j 7t
2
 2A e j 7t
One has a positive frequency
The other has negative freq.
Amplitude of each is half as big
© 2003, JH McClellan & RW Schafer
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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)
© 2003, JH McClellan & RW Schafer
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NEGATIVE FREQUENCY
Positive freq.
(Anti-clockwise)
Negative freq.
(Clockwise)
© 2003, JH McClellan & RW Schafer
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NEGATIVE FREQUENCY
© 2003, JH McClellan & RW Schafer
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SPECTRUM of SINE
 Sine = sum of 2 complex exponentials:
A sin(7t )  2Aj e j 7t  2Aj e  j 7t
 12 Ae  j 0.5 e j 7t  12 Ae j 0.5 e j 7t
1 
j
 Positive freq. has phase = -0.5
 Negative freq. has phase = +0.5
© 2003, JH McClellan & RW Schafer
j  e j 0.5
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GRAPHICAL SPECTRUM
EXAMPLE of SINE
A sin(7t )  12 Ae  j 0.5 e j 7t  12 Ae j 0.5 e j 7t
( 12 A)e j 0.5
( 12 A)e j 0.5
-7
0
7

AMPLITUDE, PHASE & FREQUENCY are shown
© 2003, JH McClellan & RW Schafer
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SPECTRUM ---> SINUSOID
 Add the spectrum components:
4e j / 2
–250
7e
j / 3
–100
10
0
7e j / 3
100
4e j / 2
250
f (in Hz)
What is the formula for the signal x(t)?
© 2003, JH McClellan & RW Schafer
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Gather (A,,f) information
 Frequencies:





-250 Hz
-100 Hz
0 Hz
100 Hz
250 Hz
 Amplitude & Phase





4
7
10
7
4
-/2
+/3
0
-/3
+/2
Note the conjugate phase
DC is another name for zero-freq component
DC component always has f0 or  (for real x(t) )
© 2003, JH McClellan & RW Schafer
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Add Spectrum Components-1
 Frequencies:





-250 Hz
-100 Hz
0 Hz
100 Hz
250 Hz
 Amplitude & Phase





4
7
10
7
4
-/2
+/3
0
-/3
+/2
x(t )  10 
7e  j / 3e j 2 (100)t  7e j / 3e  j 2 (100)t
4e j / 2e j 2 ( 250)t  4e  j / 2e  j 2 ( 250)t
© 2003, JH McClellan & RW Schafer
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Add Spectrum Components-2
4e j / 2
7e
–250
j / 3
–100
10
0
7e j / 3
100
4e j / 2
250
f (in Hz)
x(t )  10 
7e  j / 3e j 2 (100)t  7e j / 3e  j 2 (100)t
4e j / 2e j 2 ( 250)t  4e  j / 2e  j 2 ( 250)t
© 2003, JH McClellan & RW Schafer
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Simplify Components
x(t )  10 
7e  j / 3e j 2 (100)t  7e j / 3e  j 2 (100)t
4e j / 2e j 2 ( 250)t  4e  j / 2e  j 2 ( 250)t
Use Euler’s Formula to get REAL sinusoids:
A cos( t   )  12 Ae  j e j t  12 Ae  j e j t
© 2003, JH McClellan & RW Schafer
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FINAL ANSWER
x(t )  10  14 cos(2 (100)t   / 3)
 8 cos(2 (250)t   / 2)
So, we get the general form:
N
x(t )  A0   Ak cos(2 f k t   k )
k 1
© 2003, JH McClellan & RW Schafer
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Summary: GENERAL FORM
N
x(t )  A0   Ak cos(2 f k t   k )
k 1
N

x(t )  X 0   e X k e j 2 f k t
e{z}  12 z 
k 1
1 z
2
N
x (t )  X 0  

k 1
1
2

X k  Ak e j k
Frequency  f k
X k e j 2 f k t  12 X ke j 2 f k t
© 2003, JH McClellan & RW Schafer
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Example: Synthetic Vowel
 Sum of 5 Frequency Components
© 2003, JH McClellan & RW Schafer
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SPECTRUM of VOWEL
 Note: Spectrum has 0.5Xk (except XDC)
 Conjugates in negative frequency
© 2003, JH McClellan & RW Schafer
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SPECTRUM of VOWEL
(Polar Format)
0.5Ak
fk
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Vowel Waveform
(sum of all 5 components)
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Example
Find the spectrum of the signal:
x  t   5  2cos(100 t   / 3)  4sin  250 t   / 2
© 2003, JH McClellan & RW Schafer
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