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Signal Processing First

Lecture 5Periodic Signals, Harmonics & Time-Varying Sinusoids

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READING ASSIGNMENTS

This Lecture: Chapter 3, Sections 3-2 and 3-3 Chapter 3, Sections 3-7 and 3-8

Next Lecture:

Fourier Series ANALYSISFourier Series ANALYSIS Sections 3-4, 3-5 and 3-6

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Problem Solving Skills

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

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LECTURE OBJECTIVES

Signals with HARMONICHARMONIC Frequencies Add Sinusoids with fk = kf0

FREQUENCY can change vs. TIMEChirps:

Introduce Spectrogram Visualization (specgram.m) (plotspec.m)

x(t) cos(t2 )

N

kkk tkfAAtx

100 )2cos()(

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SPECTRUM DIAGRAM

Recall Complex Amplitude vs. Freq

kk aX 21

0 100 250–100–250f (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

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SPECTRUM for PERIODIC ?

Nearly Periodic in the Vowel Region Period is (Approximately) T = 0.0065 sec

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PERIODIC SIGNALS

Repeat every T secs Definition

Example:

Speech can be “quasi-periodic”

)()( Ttxtx

)3(cos)( 2 ttx ?T

3T

32T

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Period of Complex Exponential

Definition: Period is T

k = integer

tjTtj ee )(

?)()()(

txTtxetx tj

12 kje

kTe Tj 21

kkTT

k0

22

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N

k

tkfjk

tkfjk

jkk

N

kkk

eXeXXtx

eAX

tkfAAtx

k

1

2212

21

0

100

00)(

)2cos()(

Harmonic Signal Spectrum

0:haveonly can signal Periodic fkfk

Tf

10

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Define FUNDAMENTAL FREQ

00

1

Tf

Periodlfundamenta Frequencylfundamenta

)2(

)2cos()(

0

0

000

100

Tf

ffkf

tkfAAtx

k

N

kkk

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What is the fundamental frequency?

Harmonic Signal (3 Freqs)

3rd5th

10 Hz

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POP QUIZ: FUNDAMENTAL

Here’s another spectrum:

What is the fundamental frequency?

100 Hz ? 50 Hz ?

0 100 250–100–250f (in Hz)

3/7 je 3/7 je2/4 je 2/4 je

10

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SPECIAL RELATIONSHIPto get a PERIODIC SIGNAL

IRRATIONAL SPECTRUM

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Harmonic Signal (3 Freqs)

T=0.1

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NON-Harmonic Signal

NOTPERIODIC

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FREQUENCY ANALYSIS

Now, a much HARDER problemNow, 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

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Time-Varying FREQUENCIES Diagram

Fre

qu

ency

is

the

vert

ical

axi

s

Time is the horizontal axis

A-440

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SIMPLE TEST SIGNAL C-major SCALE: stepped frequencies

Frequency is constant for each note

IDEAL

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R-rated: 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 Xk

Breaks x(t) into SHORT TIME SEGMENTS Then uses the FFT (Fast Fourier Transform)

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SPECTROGRAM EXAMPLE Two Constant Frequencies: Beats

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

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tjtjj

tjtj eeee )12(2)12(221)660(2)660(2

21

AM Radio Signal Same as BEAT Notes

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

))648(2cos())672(2cos( 221

221 tt

tjtjtjtjj eeee )648(2)648(2)672(2)672(2

41

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SPECTRUM of AM (Beat)

4 complex exponentials in AM:

What is the fundamental frequency?

648 Hz ? 24 Hz ?

0 648 672f (in Hz)

–672 –648

41

41

41

41

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STEPPED FREQUENCIES C-major SCALE: successive sinusoids

Frequency is constant for each note

IDEAL

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SPECTROGRAM of C-Scale

ARTIFACTS at Transitions

Sinusoids ONLY

From SPECGRAMANALYSIS PROGRAM

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Spectrogram of LAB SONG

ARTIFACTS at Transitions

Sinusoids ONLY

Analysis Frame = 40ms

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Time-Varying Frequency

Frequency can change vs. time Continuously, not stepped

FREQUENCY MODULATION (FM)FREQUENCY MODULATION (FM)

CHIRP SIGNALS Linear Frequency Modulation (LFM)

))(2cos()( tvtftx c VOICE

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)2cos()( 02 tftAtx

New Signal: Linear FM

Called Chirp Signals (LFM) Quadratic phase

Freq will change LINEARLY vs. time Example of Frequency Modulation (FM) Define “instantaneous frequency”

QUADRATIC

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INSTANTANEOUS FREQ

Definition

For Sinusoid:

Derivativeof the “Angle”)()(

))(cos()(

tt

tAtx

dtd

i

Makes sense

0

0

0

2)()(

2)()2cos()(

ftt

tfttfAtx

dtd

i

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INSTANTANEOUS FREQof the Chirp

Chirp Signals have Quadratic phase Freq will change LINEARLY vs. time

ttt

ttAtx2

2

)(

)cos()(

tttdtd

i 2)()(

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CHIRP SPECTROGRAM

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CHIRP WAVEFORM

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OTHER CHIRPS

(t) can be anything:

(t) could be speech or music: FM radio broadcast

))cos(cos()( tAtx

)sin()()( tttdtd

i

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SINE-WAVE FREQUENCY MODULATION (FM)

Look at CD-ROM Demos in Ch 3