Signal
Processing
First
Copyright Monash University 2009
Lecture5
PeriodicSignals,
Harmonics&
TimeVarying
Sinusoids
READING ASSIGNMENTS
ThisLecture:
Chapter3,Sections32and33
Chapter3,Sections37and38
NextLecture:
FourierSeriesANALYSIS
Sections34,35and36
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Problem Solving Skills
MathFormula
SumofCosines
Amp,Freq,Phase
RecordedSignals
Speech
Music
Nosimpleformula
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Plot&Sketches
s(t)versust
Spectrum
MATLAB
Numerical
Computation
Plottinglistof
numbers
3
LECTURE OBJECTIVES
SignalswithHARMONIC Frequencies
AddSinusoidswithfk =kf0
N
x (t ) A0 Ak cos(2 kf 0t k )
k 1
FREQUENCY can change vs. TIME
Chirps:
2
x(t) cos( t )
Introduce Spectrogram Visualization (specgram.m)
(plotspec.m)
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SPECTRUM DIAGRAM
RecallComplexAmplitudevs.Freq
1
2
Xk
4e
j / 2
7e
250
j / 3
10
7e
j / 3
4e
X k Ak e j k
100
1
2
100
j / 2
250
x (t ) 10 14 cos(2 (100)t / 3)
8 cos(2 ( 250)t / 2)
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Xk
f (in Hz)
SPECTRUM for PERIODIC ?
NearlyPeriodic intheVowelRegion
Periodis(Approximately)T=0.0065 sec
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PERIODIC SIGNALS
PeriodicsignalrepeatseveryTsecs
x (t ) x (t T )
Period:=minimumT
Example:
x (t ) cos (3t )
2
T ?
T 2
3
T
3
Speechcanbequasiperiodic
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Period of Complex Exponential
j t
x (t ) e
Definition: Period is T
x (t T ) x (t ) ?
j ( t T )
j t
e
e
e j 2 k 1
jT
e
1 T 2 k
2 k 2
k = integer
k 0k
T
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Harmonic Signal Spectrum
Periodic signal can only have : f k k f 0
N
x (t ) A0 Ak cos(2 kf 0t k )
k 1
X k Ak e
N
x (t ) X 0
k 1
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1
2
j k
X ke
j 2 kf 0t
1
2
1
f0
T
j 2 kf 0t
X ke
Define FUNDAMENTAL FREQ
N
x(t ) A0 Ak cos(2 kf 0t k )
k 1
fk k f0
(0 2 f 0 )
1
f0
T0
f 0 fundamental Frequency (smallest)
T0 fundamental Period (largest)
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10
Harmonic Signal (3 Freqs)
3rd
What is the fundamental frequency?
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5th
10 Hz!!!
11
POP QUIZ: FUNDAMENTAL
Heresanotherspectrum:
4e
j / 2
7e
250
j / 3
10
7e
j / 3
4e
100
100
j / 2
250
f (in Hz)
What is the fundamental frequency?
100 Hz ?
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50 Hz ?
50 Hz !!!
Greatest
Common
Divisor!!!
12
IRRATIONAL SPECTRUM
SPECIAL RELATIONSHIP
to get a PERIODIC SIGNAL
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13
Harmonic Signal (3 Freqs)
T=0.1
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14
NON-Harmonic Signal
NOT
PERIODIC
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15
FREQUENCY ANALYSIS
Now,amuchHARDERproblem
Givenarecordingofasong,havethe
computerwritethemusic
Can a machine extract frequencies?
Yes, if we COMPUTE the spectrum for x(t)
During short intervals
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16
Frequency is the vertical axis
Time-Varying FREQUENCIES
Diagram
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A-440
Time is the horizontal axis
17
SIMPLE TEST SIGNAL
CmajorSCALE:steppedfrequencies
Frequencyisconstantforeachnote
IDEAL
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18
R-rated: ADULTS ONLY
SPECTROGRAMTool
MATLABfunctionisspecgram.m
SPFirsthasplotspec.m &spectgr.m
ANALYSIS program
Takesx(t)asinput&
ProducesspectrumvaluesXk
Breaksx(t)intoSHORTTIMESEGMENTS
ThenusestheFFT(FastFourierTransform)
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19
SPECTROGRAM EXAMPLE
TwoConstant Frequencies:Beats
cos(2 (660)t ) sin(2 (12)t )
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20
Amplitude Modulated
Radio Signal
SameasBEATNotes
1
2
1
4j
1
2
cos(2 (660)t ) sin(2 (12)t )
j 2 ( 660 ) t
j 2 ( 672 ) t
j 2 ( 660 ) t
e
1
2j
j 2 (12 ) t
j 2 (12 ) t
e j 2 ( 672 ) t e j 2 ( 648) t e j 2 ( 648) t
cos(2 (672)t 2 ) 12 cos(2 (648)t 2 )
600 12 f c f
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21
SPECTRUM of AM (Beat)
4complexexponentialsinAM:
1
4
e j / 2
672
1
4
e j / 2
648
1
4
j / 2
648
1
4
e j / 2
672
f (in Hz)
What is the fundamental frequency?
648 Hz ?
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24 Hz ?
22
STEPPED FREQUENCIES
CmajorSCALE:successivesinusoids
Frequencyisconstantforeachnote
IDEAL
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23
SPECTROGRAM of C-Scale
Sinusoids ONLY
From SPECGRAM
ANALYSIS PROGRAM
ARTIFACTS at Transitions
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24
Spectrogram of LAB SONG
Sinusoids ONLY
Analysis Frame = 40ms
ARTIFACTS at Transitions
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25
Time-Varying Frequency
Frequencycanchangevs.time
Continuously,notstepped
FREQUENCYMODULATION(FM)
x (t ) cos(2 f c t v (t ))
VOICE
CHIRPSIGNALS
LinearFrequencyModulation(LFM)
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26
New Signal: Linear FM
Called Chirp Signals(LFM)
QUADRATIC
Quadraticphase
x (t ) A cos( t 2 f 0 t )
2
FreqwillchangeLINEARLY vs.time
ExampleofFrequencyModulation(FM)
Defineinstantaneousfrequency
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27
INSTANTANEOUS FREQ
Definition
x (t ) A cos( (t ))
d (t )
i (t ) dt
Derivative
of the Angle
ForSinusoid:
x (t ) A cos(2 f 0t )
(t ) 2 f 0t
i ( t )
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d (t )
dt
Makes sense
2 f 0
28
INSTANTANEOUS FREQ
of the Chirp
Chirp SignalshaveQuadraticphase
FreqwillchangeLINEARLY vs.time
x (t ) A cos( t t )
2
(t ) t t
2
i ( t )
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d (t )
dt
2 t
29
CHIRP SPECTROGRAM
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30
CHIRP WAVEFORM
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31
OTHER CHIRPS
(t)canbeanything:
x (t ) A cos( cos( t ) )
i (t )
(
t
)
dt
sin( t )
(t)couldbespeechormusic:
FMradiobroadcast
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32
SINE-WAVE FREQUENCY
MODULATION (FM)
Look at CD-ROM Demos in Ch 3
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33
