SPFirst-L05 EV1

Copyright Monash University 2009

Signal Processing

First

Lecture5PeriodicSignals,Harmonics&TimeVaryingSinusoids

1


READING ASSIGNMENTS

ThisLecture: Chapter3,Sections32and33 Chapter3,Sections37and38

NextLecture:FourierSeriesANALYSIS Sections34,35and36

2


Problem Solving Skills

MathFormula SumofCosines Amp,Freq,Phase

RecordedSignals Speech Music Nosimpleformula

Plot&Sketches s(t)versust Spectrum

MATLAB Numerical Computation Plottinglistof

numbers

3


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()(


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


SPECTRUM for PERIODIC ?

NearlyPeriodic intheVowelRegion Periodis(Approximately)T=0.0065 sec

6


PERIODIC SIGNALS

PeriodicsignalrepeatseveryTsecs

Period:=minimumT Example:

Speechcanbequasiperiodic

7

)()( Ttxtx

)3(cos)( 2 ttx ?T3T23T


Period of Complex Exponential

8

Definition: Period is T

k = integer

tjTtj ee )(?)()(

)(txTtx

etx tj

12 kje kTe Tj 21

kkTT

k0

22


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


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!!!


POP QUIZ: FUNDAMENTAL

Heresanotherspectrum:

12


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


Harmonic Signal (3 Freqs)

14

T=0.1


NON-Harmonic Signal

15

NOTPERIODIC


FREQUENCY ANALYSIS

Now,amuchHARDERproblem Givenarecordingofasong,havethecomputerwritethemusic

16

Can a machine extract frequencies? Yes, if we COMPUTE the spectrum for x(t)

During short intervals


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


SIMPLE TEST SIGNAL CmajorSCALE:steppedfrequencies

Frequencyisconstantforeachnote

18

IDEAL


R-rated: ADULTS ONLY

SPECTROGRAMTool MATLABfunctionisspecgram.m SPFirsthasplotspec.m&spectgr.m

ANALYSIS program Takesx(t)asinput& ProducesspectrumvaluesXk Breaksx(t)intoSHORTTIMESEGMENTS

ThenusestheFFT(FastFourierTransform)

19


SPECTROGRAM EXAMPLE TwoConstant Frequencies:Beats

20

))12(2sin())660(2cos( tt


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=


SPECTRUM of AM (Beat)

4complexexponentialsinAM:

22


648 Hz ? 24 Hz ?

0 648 672 f (in Hz)672 648

2/41 je 2/41

je2/41je2/41

je


STEPPED FREQUENCIES CmajorSCALE:successivesinusoids

Frequencyisconstantforeachnote

23

IDEAL


SPECTROGRAM of C-Scale

24

ARTIFACTS at Transitions

Sinusoids ONLY

From SPECGRAMANALYSIS PROGRAM


Spectrogram of LAB SONG

25

ARTIFACTS at Transitions

Sinusoids ONLY Analysis Frame = 40ms


Time-Varying Frequency

Frequencycanchangevs.time Continuously,notstepped

FREQUENCYMODULATION(FM)

CHIRPSIGNALS LinearFrequencyModulation(LFM)

26

))(2cos()( tvtftx c VOICE


New Signal: Linear FM

CalledChirp Signals(LFM) Quadraticphase

FreqwillchangeLINEARLY vs.time ExampleofFrequencyModulation(FM) Defineinstantaneousfrequency

27

)2cos()( 02 tftAtx

QUADRATIC


INSTANTANEOUS FREQ Definition

ForSinusoid:

28

Derivativeof the Angle)()(

))(cos()(tttAtx

dtd

i

Makes sense

0

0

0

2)()(2)(

)2cos()(

ftttft

tfAtx

dtd

i


INSTANTANEOUS FREQof the Chirp

Chirp SignalshaveQuadraticphase FreqwillchangeLINEARLY vs.time

29

tttttAtx

2

2

)()cos()(

ttt dtdi 2)()(


CHIRP SPECTROGRAM

30


CHIRP WAVEFORM

31


OTHER CHIRPS

(t)canbeanything:

(t)couldbespeechormusic: FMradiobroadcast

32

))cos(cos()( tAtx)sin()()( ttt dt

di


SINE-WAVE FREQUENCY MODULATION (FM)

33

Look at CD-ROM Demos in Ch 3

Documents

SPFirst-L05 EV1