1/28/2005 2003, J H McClellan &RW Schafer 1

Si gnal Pr oc essi ng Fi r st
Lec t ur e 5
Per i odi c Si gnal s, Har moni c s
& Ti me-Var yi ng Si nusoi ds
1/28/2005 2003, J H McClellan &RW Schafer 3
READI NG ASSI GNMENTS
This Lecture:
Chapter 3, Sections 3-2 and 3-3
Chapter 3, Sections 3-7 and 3-8
Next Lecture:
Fourier Series ANALYSIS Fourier Series ANALYSIS
Sections 3-4, 3-5 and 3-6
1/28/2005 2003, J H McClellan &RW Schafer 4
Pr obl em Sol vi ng Sk i l l s
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
1/28/2005 2003, J H McClellan &RW Schafer 5
LECTURE OBJ ECTI VES
Signals with HARMONIC HARMONIC Frequencies
Add Sinusoids with f
k
=kf
0
FREQUENCY can change vs. TIME
Chirps:
Introduce Spectrogram Visualization (specgram.m)
(plotspec.m)
x(t)= cos(t
2
)

=
+ + =
N
k
k k
t kf A A t x
1
0 0
) 2 cos( ) (
1/28/2005 2003, J H McClellan &RW Schafer 6
SPECTRUM DI AGRAM
Recall Complex Amplitude vs. Freq
k k
a X =
2
1
0 100 250 100 250
f (in Hz)
3 /
7
j
e
3 /
7
j
e

2 /
4
j
e
2 /
4
j
e
10
) 2 / ) 250 ( 2 cos( 8
) 3 / ) 100 ( 2 cos( 14 10 ) (


+ +
+ =
t
t t x
k
j
k k
e A X

=

k
X
2
1
1/28/2005 2003, J H McClellan &RW Schafer 7
SPECTRUM f or PERI ODI C ?
Nearly Periodic in the Vowel Region
Period is (Approximately) T =0.0065 sec
1/28/2005 2003, J H McClellan &RW Schafer 8
PERI ODI C SI GNALS
Repeat every T secs
Definition
Example:
Speech can be quasi-periodic
) ( ) ( T t x t x + =
) 3 ( cos ) (
2
t t x =
? = T
3

= T
3
2
= T
1/28/2005 2003, J H McClellan &RW Schafer 9
Per i od of Compl ex Ex ponent i al
Definition: Period is T
k =integer
t j T t j
e e

=
+ ) (
? ) ( ) (
) (
t x T t x
e t x
t j
= +
=

1
2
=
k j
e

k T e
T j

2 1 = =
k k
T T
k
0
2 2

= =
1/28/2005 2003, J H McClellan &RW Schafer 10
{ }

=

=
+ + =
=
+ + =
N
k
t kf j
k
t kf j
k
j
k k
N
k
k k
e X e X X t x
e A X
t kf A A t x
k
1
2
2
1
2
2
1
0
1
0 0
0 0
) (
) 2 cos( ) (

Har moni c Si gnal Spec t r um
0
: have only can signal Periodic f k f
k
=
T
f
1
0
=
1/28/2005 2003, J H McClellan &RW Schafer 11
Def i ne FUNDAMENTAL FREQ
0
0
1
T
f =
(shortest) Period l fundamenta
(largest) Frequency l fundamenta
) 2 (
) 2 cos( ) (
0
0
0 0 0
1
0 0
=
=
= =
+ + =

=
T
f
f f k f
t kf A A t x
k
N
k
k k


1/28/2005 2003, J H McClellan &RW Schafer 12
What is the fundamental frequency?
Har moni c Si gnal (3 Fr eqs)
3rd
5th
10 Hz
1/28/2005 2003, J H McClellan &RW Schafer 13
POP QUI Z: FUNDAMENTAL
Heres another spectrum:
What is the fundamental frequency?
100 Hz ? 50 Hz ?
0 100 250 100 250
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
I RRATI ONAL SPECTRUM
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Har moni c Si gnal (3 Fr eqs)
T=0.1
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NON-Har moni c Si gnal
NOT
PERIODIC
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FREQUENCY ANALYSI S
Now, a much HARDER problem 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
1/28/2005 2003, J H McClellan &RW Schafer 18
Ti me-Var yi ng
FREQUENCI ES Di agr am
F
r
e
q
u
e
n
c
y

i
s

t
h
e

v
e
r
t
i
c
a
l

a
x
i
s
Time is the horizontal axis
A-440
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SI MPLE TEST SI GNAL
C-major SCALE: stepped frequencies
Frequency is constant for each note
IDEAL
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R-r at ed: 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 X
k
Breaks x(t) into SHORT TIME SEGMENTS
Then uses the FFT (Fast Fourier Transform)
1/28/2005 2003, J H McClellan &RW Schafer 21
SPECTROGRAM EXAMPLE
Two Constant Frequencies: Beats
) ) 12 ( 2 sin( ) ) 660 ( 2 cos( t t
1/28/2005 2003, J H McClellan &RW Schafer 22
( ) ( )
t j t j
j
t j t j
e e e e
) 12 ( 2 ) 12 ( 2
2
1
) 660 ( 2 ) 660 ( 2
2
1

+
AM Radi o Si gnal
Same as BEAT Notes
) ) 12 ( 2 sin( ) ) 660 ( 2 cos( t t
) ) 648 ( 2 cos( ) ) 672 ( 2 cos(
2 2
1
2 2
1
+ + t t
( )
t j t j t j t j
j
e e e e
) 648 ( 2 ) 648 ( 2 ) 672 ( 2 ) 672 ( 2
4
1

+
1/28/2005 2003, J H McClellan &RW Schafer 23
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 /
4
1
j
e
2 /
4
1
j
e
2 /
4
1
j
e
2 /
4
1
j
e
1/28/2005 2003, J H McClellan &RW Schafer 24
STEPPED FREQUENCI ES
C-major SCALE: successive sinusoids
Frequency is constant for each note
IDEAL
1/28/2005 2003, J H McClellan &RW Schafer 25
SPECTROGRAM of C-Sc al e
ARTIFACTS at Transitions
Sinusoids ONLY
From SPECGRAM
ANALYSIS PROGRAM
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Spec t r ogr am of LAB SONG
ARTIFACTS at Transitions
Sinusoids ONLY
Analysis Frame = 40ms
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Ti me-Var yi ng Fr equenc y
Frequency can change vs. time
Continuously, not stepped
FREQUENCY MODULATION (FM) FREQUENCY MODULATION (FM)
CHIRP SIGNALS
Linear Frequency Modulation (LFM)
)) ( 2 cos( ) ( t v t f t x
c
+ =
VOICE
1/28/2005 2003, J H McClellan &RW Schafer 28
) 2 cos( ) (
0
2
+ + = t f t A t x
New Si gnal : Li near FM
Called Chirp Signals (LFM)
Quadratic phase
Freq will change LINEARLY vs. time
Example of Frequency Modulation (FM)
Define instantaneous frequency
QUADRATIC
1/28/2005 2003, J H McClellan &RW Schafer 29
I NSTANTANEOUS FREQ
Definition
For Sinusoid:
Derivative
of the Angle
) ( ) (
)) ( cos( ) (
t t
t A t x
dt
d
i


=
=
Makes sense
0
0
0
2 ) ( ) (
2 ) (
) 2 cos( ) (
f t t
t f t
t f A t x
dt
d
i



= =
+ =
+ =
1/28/2005 2003, J H McClellan &RW Schafer 30
I NSTANTANEOUS FREQ
of t he Chi r p
Chirp Signals have Quadratic phase
Freq will change LINEARLY vs. time


+ + =
+ + =
t t t
t t A t x
2
2
) (
) cos( ) (
+ = = t t t
dt
d
i
2 ) ( ) (
1/28/2005 2003, J H McClellan &RW Schafer 31
CHI RP SPECTROGRAM
1/28/2005 2003, J H McClellan &RW Schafer 32
CHI RP WAVEFORM
1/28/2005 2003, J H McClellan &RW Schafer 33
OTHER CHI RPS
(t) can be anything:
(t) could be speech or music:
FM radio broadcast
) ) cos( cos( ) ( + = t A t x
) sin( ) ( ) ( t t t
dt
d
i
= =
1/28/2005 2003, J H McClellan &RW Schafer 34
SI NE-WAVE FREQUENCY
MODULATI ON (FM)
Look at CD-ROM Demos in Ch 3