Copyright Monash University 2009
Signal Processing
First
Lecture5PeriodicSignals,Harmonics&TimeVaryingSinusoids
1
Copyright Monash University 2009
READING ASSIGNMENTS
ThisLecture: Chapter3,Sections32and33 Chapter3,Sections37and38
NextLecture:FourierSeriesANALYSIS Sections34,35and36
2
Copyright Monash University 2009
Problem Solving Skills
MathFormula SumofCosines Amp,Freq,Phase
RecordedSignals Speech Music Nosimpleformula
Plot&Sketches s(t)versust Spectrum
MATLAB Numerical Computation Plottinglistof
numbers
3
Copyright Monash University 2009
LECTURE OBJECTIVES
SignalswithHARMONIC Frequencies AddSinusoidswithfk =kf0
4
FREQUENCY can change vs. TIMEChirps:
Introduce Spectrogram Visualization (specgram.m) (plotspec.m)
x(t) cos(t2 )
N
kkk tkfAAtx
100 )2cos()(
Copyright Monash University 2009
SPECTRUM DIAGRAM
RecallComplexAmplitudevs.Freq
5
12 kX
0 100 250100250 f (in Hz)
3/7 je 3/7 je2/4 je 2/4 je
10
)2/)250(2cos(8)3/)100(2cos(1410)(
tttx
kjkk eAX
kX2
1
Copyright Monash University 2009
SPECTRUM for PERIODIC ?
NearlyPeriodic intheVowelRegion Periodis(Approximately)T=0.0065 sec
6
Copyright Monash University 2009
PERIODIC SIGNALS
PeriodicsignalrepeatseveryTsecs
Period:=minimumT Example:
Speechcanbequasiperiodic
7
)()( Ttxtx
)3(cos)( 2 ttx ?T3T23T
Copyright Monash University 2009
Period of Complex Exponential
8
Definition: Period is T
k = integer
tjTtj ee )(?)()(
)(txTtx
etx tj
12 kje kTe Tj 21
kkTT
k0
22
Copyright Monash University 2009
Harmonic Signal Spectrum
9
N
k
tkfjk
tkfjk
jkk
N
kkk
eXeXXtx
eAX
tkfAAtx
k
1
2212
21
0
100
00)(
)2cos()(
0:haveonly can signal Periodic fkfk
Tf 10
Copyright Monash University 2009
0 01
0 0 0
0
0
( ) cos(2 )
( 2 )
fundamental Frequency (smallest)fundamental Period (largest)
N
k kk
k
x t A A kf t
f k f f
fT
Define FUNDAMENTAL FREQ
10
00
1T
f
Copyright Monash University 2009 11
What is the fundamental frequency?
Harmonic Signal (3 Freqs)
3rd5th
10 Hz!!!
Copyright Monash University 2009
POP QUIZ: FUNDAMENTAL
Heresanotherspectrum:
12
What is the fundamental frequency?
100 Hz ? 50 Hz ?
0 100 250100250 f (in Hz)
3/7 je 3/7 je2/4 je 2/4 je
10
50 Hz !!!Greatest CommonDivisor!!!
Copyright Monash University 2009 13
SPECIAL RELATIONSHIPto get a PERIODIC SIGNAL
IRRATIONAL SPECTRUM
Copyright Monash University 2009
Harmonic Signal (3 Freqs)
14
T=0.1
Copyright Monash University 2009
NON-Harmonic Signal
15
NOTPERIODIC
Copyright Monash University 2009
FREQUENCY ANALYSIS
Now,amuchHARDERproblem Givenarecordingofasong,havethecomputerwritethemusic
16
Can a machine extract frequencies? Yes, if we COMPUTE the spectrum for x(t)
During short intervals
Copyright Monash University 2009
Time-Varying FREQUENCIES Diagram
17
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
Copyright Monash University 2009
SIMPLE TEST SIGNAL CmajorSCALE:steppedfrequencies
Frequencyisconstantforeachnote
18
IDEAL
Copyright Monash University 2009
R-rated: ADULTS ONLY
SPECTROGRAMTool MATLABfunctionisspecgram.m SPFirsthasplotspec.m&spectgr.m
ANALYSIS program Takesx(t)asinput& ProducesspectrumvaluesXk Breaksx(t)intoSHORTTIMESEGMENTS
ThenusestheFFT(FastFourierTransform)
19
Copyright Monash University 2009
SPECTROGRAM EXAMPLE TwoConstant Frequencies:Beats
20
))12(2sin())660(2cos( tt
Copyright Monash University 2009
Amplitude Modulated Radio Signal
SameasBEATNotes
21
tjtjjtjtj eeee )12(2)12(221)660(2)660(221 ))12(2sin())660(2cos( tt
ff c12600
tjtjtjtjj eeee )648(2)648(2)672(2)672(241 =))648(2cos())672(2cos( 22
122
1 tt=
Copyright Monash University 2009
SPECTRUM of AM (Beat)
4complexexponentialsinAM:
22
What is the fundamental frequency?
648 Hz ? 24 Hz ?
0 648 672 f (in Hz)672 648
2/41 je 2/41
je2/41je2/41
je
Copyright Monash University 2009
STEPPED FREQUENCIES CmajorSCALE:successivesinusoids
Frequencyisconstantforeachnote
23
IDEAL
Copyright Monash University 2009
SPECTROGRAM of C-Scale
24
ARTIFACTS at Transitions
Sinusoids ONLY
From SPECGRAMANALYSIS PROGRAM
Copyright Monash University 2009
Spectrogram of LAB SONG
25
ARTIFACTS at Transitions
Sinusoids ONLY Analysis Frame = 40ms
Copyright Monash University 2009
Time-Varying Frequency
Frequencycanchangevs.time Continuously,notstepped
FREQUENCYMODULATION(FM)
CHIRPSIGNALS LinearFrequencyModulation(LFM)
26
))(2cos()( tvtftx c VOICE
Copyright Monash University 2009
New Signal: Linear FM
CalledChirp Signals(LFM) Quadraticphase
FreqwillchangeLINEARLY vs.time ExampleofFrequencyModulation(FM) Defineinstantaneousfrequency
27
)2cos()( 02 tftAtx
QUADRATIC
Copyright Monash University 2009
INSTANTANEOUS FREQ Definition
ForSinusoid:
28
Derivativeof the Angle)()(
))(cos()(tttAtx
dtd
i
Makes sense
0
0
0
2)()(2)(
)2cos()(
ftttft
tfAtx
dtd
i
Copyright Monash University 2009
INSTANTANEOUS FREQof the Chirp
Chirp SignalshaveQuadraticphase FreqwillchangeLINEARLY vs.time
29
tttttAtx
2
2
)()cos()(
ttt dtdi 2)()(
Copyright Monash University 2009
CHIRP SPECTROGRAM
30
Copyright Monash University 2009
CHIRP WAVEFORM
31
Copyright Monash University 2009
OTHER CHIRPS
(t)canbeanything:
(t)couldbespeechormusic: FMradiobroadcast
32
))cos(cos()( tAtx)sin()()( ttt dt
di
Copyright Monash University 2009
SINE-WAVE FREQUENCY MODULATION (FM)
33
Look at CD-ROM Demos in Ch 3