Digital Signal Processing
Prof. Nizamettin AYDIN
[email protected]
http://www.yildiz.edu.tr/~naydin
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Digital Signal Processing
Lecture 4
Spectrum Representation
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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
4
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
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FREQUENCY DIAGRAM
• Plot Complex Amplitude vs. Freq
4e
 j / 2
–250
7e
j / 3
10
7e
 j / 3
4e
–100
0
100
j / 2
250
f (in Hz)
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Frequency is the vertical axis
Another FREQ. Diagram
A-440
Time is the horizontal axis
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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?
8
Fur Elise WAVEFORM
Beat
Notes
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Speech Signal: BAT
• Nearly Periodic in Vowel Region
– Period is (Approximately) T = 0.0065 sec
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Euler’s Formula Reversed
• Solve for cosine (or sine)
e
e
e
j t
 cos( t )  j sin(  t )
 j t
 cos(  t )  j sin(  t )
 j t
 cos( t )  j sin(  t )
e
j t
e
 j t
cos( t ) 
 2 cos( t )
1 (e j t
2
e
 j t
)
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INVERSE Euler’s Formula
• Solve for cosine (or sine)
cos( t ) 
1 (e j t
2
sin(  t ) 
1 ( e j t
2j
e
 j t
e
)
 j t
)
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SPECTRUM Interpretation
• Cosine = sum of 2 complex exponentials:
A cos(7t ) 
A e j 7t
2

A e  j 7t
2
One has a positive frequency
The other has negative freq.
Amplitude of each is half as big
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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)
– Think of a train whistle
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SPECTRUM of SINE
• Sine = sum of 2 complex exponentials:
j 7t
 j 7t
A
A
A sin( 7t )  2 j e  2 j e
 j 0.5 j 7 t
j 0.5  j 7 t
1
1
 2 Ae
e  2 Ae
e
1
j
 je
j 0.5
– Positive freq. has phase = -0.5
– Negative freq. has phase = +0.5
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GRAPHICAL SPECTRUM
EXAMPLE of SINE
Asin( 7t )  Ae
1
2
1
(2
A)e
-7
 j 0.5 j 7t
e
j 0.5
 Ae
1
2
1
(2
0
j 0.5  j 7t
e
A)e
 j 0.5
7

AMPLITUDE, PHASE & FREQUENCY are shown
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SPECTRUM ---> SINUSOID
• Add the spectrum components:
4e
 j / 2
–250
7e
j / 3
10
7e
 j / 3
4e
–100
0
100
j / 2
250
f (in Hz)
What is the formula for the signal x(t)?
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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) )
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Add Spectrum Components-1
• Frequencies:
–
–
–
–
–
• Amplitude & Phase
–
–
–
–
–
-250 Hz
-100 Hz
0 Hz
100 Hz
250 Hz
4
7
10
7
4
-/2
+/3
0
-/3
+/2
x (t )  10 
7e
4e
 j / 3 j 2 (100) t
e
j / 2 j 2 ( 250) t
e
 7e
 4e
j / 3  j 2 (100) t
e
 j / 2  j 2 ( 250) t
e
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Add Spectrum Components-2
4e
 j / 2
7e
j / 3
10
7e
 j / 3
4e
–250
–100
0
j / 2
100
250
 7e
j / 3  j 2 (100) t
f (in Hz)
x (t )  10 
7e
4e
 j / 3 j 2 (100) t
e
j / 2 j 2 ( 250) t
e
 4e
e
 j / 2  j 2 ( 250) t
e
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Simplify Components
x (t )  10 
7e
4e
 j / 3 j 2 (100) t
e
j / 2 j 2 ( 250) t
e
 7e
 4e
j / 3  j 2 (100) t
e
 j / 2  j 2 ( 250) t
e
Use Euler’s Formula to get REAL sinusoids:
A cos( t   )  Ae
1
2
 j j t
e
 Ae
1
2
 j  j t
e
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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
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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
e{z}  12 z 
k 1
1 z
2
N
x (t )  X 0  

k 1
1
2
Xke
j 2 f k t
j 2 f k t


j k
X k  Ak e
Frequency  f k
1
2
  j 2 f k t
Xke

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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)
0.5Ak
fk
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Vowel Wavefor (sum of all 5 components)
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