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Signal Processing First

Lecture 4Spectrum Representation

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READING ASSIGNMENTS

This Lecture: Chapter 3, Section 3-1

Other Reading: Appendix A: Complex Numbers

Next Lecture: Ch 3, Sects 3-2, 3-3, 3-7 & 3-8

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LECTURE OBJECTIVES

Sinusoids with DIFFERENT Frequencies SYNTHESIZE by Adding Sinusoids

SPECTRUM Representation Graphical Form shows DIFFERENTDIFFERENT Freqs

N

kkkk tfAtx

1

)2cos()(

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FREQUENCY DIAGRAM

Plot Complex Amplitude vs. Freq

0 100 250–100–250f (in Hz)

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

10

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Another FREQ. DiagramF

req

uen

cy i

s th

e ve

rtic

al a

xis

Time is the horizontal axis

A-440

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MOTIVATION

Synthesize Complicated Signals Musical Notes

Piano uses 3 strings for many notes Chords: play several notes simultaneously

Human Speech Vowels have dominant frequencies Application: computer generated speech

Can all signals be generated this way? Sum of sinusoids?

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Fur Elise WAVEFORM

BeatNotes

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Speech Signal: BAT

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

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Euler’s Formula Reversed

Solve for cosine (or sine)

)sin()cos( tjte tj

)sin()cos( tjte tj

)sin()cos( tjte tj

)cos(2 tee tjtj

)()cos(21 tjtj eet

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INVERSE Euler’s Formula

Solve for cosine (or sine)

)()cos(21 tjtj eet

)()sin(21 tjtjj

eet

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SPECTRUM Interpretation

Cosine = sum of 2 complex exponentials:

One has a positive frequencyThe other has negative freq.Amplitude of each is half as big

tjAtjA eetA 72

72

)7cos(

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NEGATIVE FREQUENCY

Is negative frequency real? Doppler Radar provides an example

Police radar measures speed by using the Doppler shift principle

Let’s assume 400Hz 60 mph +400Hz means towards the radar -400Hz means away (opposite direction) Think of a train whistle

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SPECTRUM of SINE

Sine = sum of 2 complex exponentials:

Positive freq. has phase = -0.5 Negative freq. has phase = +0.5

tjjtjj

tjjAtj

jA

eAeeAe

eetA

75.02175.0

21

72

72

)7sin(

5.01 jj

ej

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GRAPHICAL SPECTRUMEXAMPLE of SINE

AMPLITUDE, PHASE & FREQUENCY are shown

7-7 0

tjjtjj eAeeAetA 75.02175.0

21)7sin(

5.021 )( jeA 5.0

21 )( jeA

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SPECTRUM ---> SINUSOID

Add the spectrum components:

What is the formula for the signal x(t)?

0 100 250–100–250f (in Hz)

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

10

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Gather (A,) information

Frequencies: -250 Hz -100 Hz 0 Hz 100 Hz 250 Hz

Amplitude & Phase

4 -/2 7 +/3 10 0 7 -/3 4 +/2

DC is another name for zero-freq componentDC component always has or (for real x(t) )

Note the conjugate phase

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Add Spectrum Components-1

Amplitude & Phase

4 -/2 7 +/3 10 0 7 -/3 4 +/2

Frequencies: -250 Hz -100 Hz 0 Hz 100 Hz 250 Hz

tjjtjj

tjjtjj

eeee

eeee

tx

)250(22/)250(22/

)100(23/)100(23/

44

77

10)(

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Add Spectrum Components-2

tjjtjj

tjjtjj

eeee

eeee

tx

)250(22/)250(22/

)100(23/)100(23/

44

77

10)(

0 100 250–100–250f (in Hz)

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

10

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Use Euler’s Formula to get REAL sinusoids:

Simplify Components

tjjtjj

tjjtjj

eeee

eeee

tx

)250(22/)250(22/

)100(23/)100(23/

44

77

10)(

tjjtjj eAeeAetA 21

21)cos(

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FINAL ANSWER

So, we get the general form:

N

kkkk tfAAtx

10 )2cos()(

)2/)250(2cos(8)3/)100(2cos(1410)(

tttx

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Summary: GENERAL FORM

zzze21

21}{ k

jkk

feAX k

Frequency

N

k

tfjk

keXeXtx1

20)(

N

k

tfjk

tfjk

kk eXeXXtx1

2212

21

0)(

N

kkkk tfAAtx

10 )2cos()(

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Example: Synthetic Vowel

Sum of 5 Frequency Components

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SPECTRUM of VOWEL

Note: Spectrum has 0.5Xk (except XDC)

Conjugates in negative frequency

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SPECTRUM of VOWEL (Polar Format)

k

0.5Ak

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Vowel Waveform (sum of all 5 components)