EE360: SIGNALS AND SYSTEMS - UNLV

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http://www.ee.unlv.edu/~b1morris/ee360 EE360: SIGNALS AND SYSTEMS CH3: FOURIER SERIES 1

Transcript of EE360: SIGNALS AND SYSTEMS - UNLV

http://www.ee.unlv.edu/~b1morris/ee360

EE360: SIGNALS AND SYSTEMS

CH3: FOURIER SERIES

1

FOURIER SERIES OVERVIEW, MOTIVATION, AND HIGHLIGHTSCHAPTER 3.1-3.2

2

BIG IDEA: TRANSFORM ANALYSIS

Make use of properties of LTI systems to simplify analysis

Represent signals as a linear combination of basic signals with two properties

Simple response: easy to characterize LTI system response to basic signal

Representation power: the set of basic signals can be use to construct a broad/useful class of signals

3

When plucking a string, length is divided into integer divisions or harmonics

Frequency of each harmonic is an integer multiple of a โ€œfundamental frequencyโ€

Also known as the normal modes

Any string deflection could be built out of a linear combination of โ€œmodesโ€

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NORMAL MODES OF VIBRATING STRING

When plucking a string, length is divided into integer divisions or harmonics

Frequency of each harmonic is an integer multiple of a โ€œfundamental frequencyโ€

Also known as the normal modes

Any string deflection could be built out of a linear combination of โ€œmodesโ€

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NORMAL MODES OF VIBRATING STRING

Caution: turn your sound down

https://youtu.be/BSIw5SgUirg

Fourier argued that periodic signals (like the single period from a plucked string) were actually useful

Represent complex periodic signals

Examples of basic periodic signals

Sinusoid: ๐‘ฅ ๐‘ก = ๐‘๐‘œ๐‘ ๐œ”0๐‘ก

Complex exponential: ๐‘ฅ ๐‘ก = ๐‘’๐‘—๐œ”0t

Fundamental frequency: ๐œ”0

Fundamental period: ๐‘‡ =2๐œ‹

๐œ”0

Harmonically related period signals form family

Integer multiple of fundamental frequency

๐œ™๐‘˜ ๐‘ก = ๐‘’๐‘—๐‘˜๐œ”0๐‘ก for ๐‘˜ = 0,ยฑ1,ยฑ2,โ€ฆ

Fourier Series is a way to represent a periodic signal as a linear combination of harmonics

๐‘ฅ ๐‘ก = ฯƒ๐‘˜=โˆ’โˆžโˆž ๐‘Ž๐‘˜๐‘’

๐‘—๐‘˜๐œ”0๐‘ก

๐‘Ž๐‘˜ coefficient gives the contribution of a harmonic (periodic signal of ๐‘˜times frequency)

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FOURIER SERIES 1 SLIDE OVERVIEW

SAWTOOTH EXAMPLE

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signalHarmonics: height given by coefficient

Animation showing approximation as more harmonics added

๐‘Ž1

๐‘Ž2

๐‘Ž3๐‘Ž4โ€ฆ

SQUARE WAVE EXAMPLE

Better approximation of square wave with more coefficients

Aligned approximations

Animation of FS

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1

2

3

4

Note: ๐‘†(๐‘“) ~ ๐‘Ž๐‘˜

#๐‘Ž๐‘˜

coef

fici

ents

ARBITRARY EXAMPLES

Interactive examples [flash (dated)][html]

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RESPONSE OF LTI SYSTEMS TO COMPLEX EXPONENTIALS CHAPTER 3.2

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TRANSFORM ANALYSIS OBJECTIVE

Need family of signals ๐‘ฅ๐‘˜ ๐‘ก that have 1) simple response and 2) represent a broad (useful) class of signals

1. Family of signals Simple response โ€“ every signal in family pass through LTI system with scale change

2. โ€œAnyโ€ signal can be represented as a linear combination of signals in the family

Results in an output generated by input ๐‘ฅ(๐‘ก)

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๐‘ฅ๐‘˜(๐‘ก) โŸถ ๐œ†๐‘˜๐‘ฅ๐‘˜(๐‘ก)

๐‘ฅ ๐‘ก =

๐‘˜=โˆ’โˆž

โˆž

๐‘Ž๐‘˜๐‘ฅ๐‘˜(๐‘ก)

๐‘ฅ ๐‘ก โŸถ

๐‘˜=โˆ’โˆž

โˆž

๐‘Ž๐‘˜๐œ†๐‘˜๐‘ฅ๐‘˜(๐‘ก)

IMPULSE AS BASIC SIGNAL

Previously (Ch2), we used shifted and scaled deltas

๐›ฟ ๐‘ก โˆ’ ๐‘ก0 โŸน ๐‘ฅ ๐‘ก = โˆซ ๐‘ฅ ๐œ ๐›ฟ ๐‘ก โˆ’ ๐œ ๐‘‘๐œ โŸถ ๐‘ฆ ๐‘ก = โˆซ ๐‘ฅ ๐œ โ„Ž ๐‘ก โˆ’ ๐œ ๐‘‘๐œ

Thanks to Jean Baptiste Joseph Fourier in the early 1800s we got Fourier analysis

Consider signal family of complex exponentials

๐‘ฅ ๐‘ก = ๐‘’๐‘ ๐‘ก or ๐‘ฅ ๐‘› = ๐‘ง๐‘›, ๐‘ , ๐‘ง โˆˆ โ„‚

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Using the convolution

๐‘’๐‘ ๐‘ก โŸถ๐ป ๐‘  ๐‘’๐‘ ๐‘ก

๐‘ง๐‘› โŸถ๐ป ๐‘ง ๐‘ง๐‘›

Notice the eigenvalue ๐ป ๐‘ depends on the value of โ„Ž(๐‘ก)and ๐‘ 

Transfer function of LTI system

Laplace transform of impulse response

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COMPLEX EXPONENTIAL AS EIGENSIGNAL

TRANSFORM OBJECTIVE

Simple response

๐‘ฅ ๐‘ก = ๐‘’๐‘ ๐‘ก โŸถ ๐‘ฆ ๐‘ก = ๐ป ๐‘  ๐‘ฅ ๐‘ก

Useful representation?

๐‘ฅ ๐‘ก = ฯƒ๐‘Ž๐‘˜๐‘’๐‘ ๐‘˜๐‘ก โŸถ ๐‘ฆ ๐‘ก = ฯƒ๐‘Ž๐‘˜๐ป ๐‘ ๐‘˜ ๐‘’๐‘ ๐‘˜๐‘ก

Input linear combination of complex exponentials leads to output linear combination of complex exponentials

Fourier suggested limiting to subclass of period complex exponentials ๐‘’๐‘—๐‘˜๐œ”0๐‘ก , ๐‘˜ โˆˆ โ„ค,๐œ”0 โˆˆ โ„

๐‘ฅ ๐‘ก = ฯƒ๐‘Ž๐‘˜๐‘’๐‘—๐‘˜๐œ”0๐‘ก โŸถ ๐‘ฆ ๐‘ก = ฯƒ๐‘Ž๐‘˜๐ป ๐‘—๐‘˜๐œ”0 ๐‘’๐‘ ๐‘˜๐‘ก

Periodic input leads to periodic output.

๐ป ๐‘—๐œ” = ๐ป ๐‘  ศ๐‘ =๐‘—๐œ” is the frequency response of the system

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CONTINUOUS TIME FOURIER SERIESCHAPTER 3.3-3.8

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CTFS TRANSFORM PAIR

Suppose ๐‘ฅ(๐‘ก) can be expressed as a linear combination of harmonic complex exponentials

๐‘ฅ ๐‘ก = ฯƒ๐‘˜=โˆ’โˆžโˆž ๐‘Ž๐‘˜๐‘’

๐‘—๐‘˜๐œ”0๐‘ก synthesis equation

Then the FS coefficients {๐‘Ž๐‘˜} can be found as

๐‘Ž๐‘˜ =1

๐‘‡โˆซ๐‘‡ ๐‘ฅ(๐‘ก) ๐‘’โˆ’๐‘—๐‘˜๐œ”0๐‘ก๐‘‘๐‘ก analysis equation

๐œ”0 - fundamental frequency

๐‘‡ = 2๐œ‹/๐œ”0 - fundamental period

๐‘Ž๐‘˜ known as FS coefficients or spectral coefficients

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CTFS PROOF

While we can prove this, it is not well suited for slides.

Key observation from proof: Complex exponentials are orthogonal

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VECTOR SPACE OF PERIODIC SIGNALS

All signals

Periodic signals, ๐œ”0

Each of the harmonic exponentials are orthogonal to each other and span the space of periodic signals

The projection of ๐‘ฅ(๐‘ก) onto a particular harmonic (๐‘Ž๐‘˜) gives the contribution of that complex exponential to building ๐‘ฅ ๐‘ก

๐‘Ž๐‘˜ is how much of each harmonic is required to construct the periodic signal ๐‘ฅ(๐‘ก)

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VECTOR SPACE OF PERIODIC SIGNALS

Periodic signals, ๐œ”0

๐‘ฅ(๐‘ก)

๐‘’๐‘—0๐‘ก = 1๐‘Ž0

๐‘’๐‘—(โˆ’๐œ”0)๐‘ก

๐‘’๐‘—๐œ”0๐‘ก

๐‘’๐‘—2๐œ”0๐‘ก

๐‘’๐‘—๐‘˜๐œ”0๐‘ก

๐‘Žโˆ’1

๐‘Ž1

๐‘Ž2

๐‘Ž๐‘˜

HARMONICS

๐‘˜ = ยฑ1 โ‡’ fundamental component (first harmonic)

Frequency ๐œ”0, period ๐‘‡ = 2๐œ‹/๐œ”0

๐‘˜ = ยฑ2 โ‡’ second harmonic

Frequency ๐œ”2 = 2๐œ”0, period ๐‘‡2 = ๐‘‡/2 (half period)

โ€ฆ

๐‘˜ = ยฑ๐‘ โ‡’ Nth harmonic

Frequency ๐œ”๐‘ = ๐‘๐œ”0, period ๐‘‡๐‘ = ๐‘‡/๐‘ (1/N period)

๐‘˜ = 0 โ‡’ ๐‘Ž0 =1

๐‘‡โˆซ๐‘‡ ๐‘ฅ ๐‘ก ๐‘‘๐‘ก, DC, constant component, average

over a single period

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HOW TO FIND FS REPRESENTATION

Will use important examples to demonstrate common techniques

Sinusoidal signals โ€“ Eulerโ€™s relationship

Direct FS integral evaluation

FS properties table and transform pairs

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๐‘ฅ ๐‘ก = 1 +1

2cos 2๐œ‹๐‘ก + sin 3๐œ‹๐‘ก

First find the period

Constant 1 has arbitrary period

cos 2๐œ‹๐‘ก has period ๐‘‡1 = 1

sin 3๐œ‹๐‘ก has period ๐‘‡2 = 2/3

๐‘‡ = 2, ๐œ”0 = 2๐œ‹/๐‘‡ = ๐œ‹

Rewrite ๐‘ฅ ๐‘ก using Eulerโ€™s and read off ๐‘Ž๐‘˜coefficients by inspection

๐‘ฅ ๐‘ก = 1 +1

4๐‘’๐‘—2๐œ”0๐‘ก + ๐‘’โˆ’๐‘—2๐œ”0๐‘ก +

1

2๐‘—๐‘’๐‘—3๐œ”0๐‘ก โˆ’ ๐‘’โˆ’๐‘—3๐œ”0๐‘ก

Read off coeff. directly

๐‘Ž0 = 1

๐‘Ž1 = ๐‘Žโˆ’1 = 0

๐‘Ž2 = ๐‘Žโˆ’2 = 1/4

๐‘Ž3 = 1/2๐‘—, ๐‘Žโˆ’3 = โˆ’1/2๐‘—

๐‘Ž๐‘˜ = 0, else

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SINUSOIDAL SIGNAL

๐‘ฅ ๐‘ก = แ‰1 ๐‘ก < ๐‘‡1

0 ๐‘‡1 < ๐‘ก <๐‘‡

2

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PERIODIC RECTANGLE WAVE

Important signal/function in DSP and communication

sinc ๐‘ฅ =sin ๐œ‹๐‘ฅ

๐œ‹๐‘ฅnormalized

sinc ๐‘ฅ =sin ๐‘ฅ

๐‘ฅunnormalized

Modulated sine function

Amplitude follows 1/x

Must use Lโ€™Hopitalโ€™s rule to get ๐‘ฅ = 0 time

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SINC FUNCTION

Consider different โ€œduty cycleโ€ for the rectangle wave

๐‘‡ = 4๐‘‡1 50% (square wave)

๐‘‡ = 8๐‘‡1 25%

๐‘‡ = 16๐‘‡1 12.5%

Note all plots are still a sincshape

Difference is how the sync is sampled

Longer in time (larger T) smaller spacing in frequency more samples between zero crossings

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RECTANGLE WAVE COEFFICIENTS

Special case of rectangle wave with ๐‘‡ = 4๐‘‡1

One sample between zero-crossing

๐‘Ž๐‘˜ = แ‰1/2 ๐‘˜ = 0

sin(๐‘˜๐œ‹/2)

๐‘˜๐œ‹๐‘’๐‘™๐‘ ๐‘’

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SQUARE WAVE

๐‘ฅ ๐‘ก = ฯƒ๐‘˜=โˆ’โˆžโˆž ๐›ฟ(๐‘ก โˆ’ ๐‘˜๐‘‡)

Using FS integral

Notice only one impulse in the interval

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PERIODIC IMPULSE TRAIN

PROPERTIES OF CTFS

Since these are very similar between CT and DT, will save until after DT

Properties are used to avoid direct evaluation of FS integral

Be sure to bookmark properties in Table 3.1 on page 206

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DISCRETE TIME FOURIER SERIESCHAPTER 3.6

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DTFS VS CTFS DIFFERENCES

While quite similar to the CT case,

DTFS is a finite series, ๐‘Ž๐‘˜ , k < K

Does not have convergence issues

Good News: motivation and intuition from CT applies for DT case

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DTFS TRANSFORM PAIR

Consider the discrete time periodic signal ๐‘ฅ ๐‘› = ๐‘ฅ ๐‘› + ๐‘

๐‘ฅ ๐‘› = ฯƒ๐‘˜=<๐‘> ๐‘Ž๐‘˜๐‘’๐‘—๐‘˜๐œ”0๐‘› synthesis equation

๐‘Ž๐‘˜ =1

๐‘ฯƒ๐‘›=<๐‘> ๐‘ฅ ๐‘› ๐‘’โˆ’๐‘—๐‘˜๐œ”0๐‘› analysis equation

๐‘ โ€“ fundamental period (smallest value such that periodicity constraint holds)

๐œ”0 = 2๐œ‹/๐‘ โ€“ fundamental frequency

ฯƒ๐‘›=<๐‘> indicates summation over a period (๐‘ samples)

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DTFS REMARKS

DTFS representation is a finite sum, so there is always pointwise convergence

FS coefficients are periodic with period N

32

DTFS PROOF

Proof for the DTFS pair is similar to the CT case

Relies on orthogonality of harmonically related DT period complex exponentials

Will not show in class

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HOW TO FIND DTFS REPRESENTATION

Like CTFS, will use important examples to demonstrate common techniques

Sinusoidal signals โ€“ Eulerโ€™s relationship

Direct FS summation evaluation โ€“ periodic rectangular wave and impulse train

FS properties table and transform pairs

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๐‘ฅ[๐‘›] = 1 +1

2cos

2๐œ‹

๐‘๐‘› + sin

4๐œ‹

๐‘๐‘›

First find the period

Rewrite ๐‘ฅ[๐‘›] using Eulerโ€™s and read off ๐‘Ž๐‘˜ coefficients by inspection

Shortcut here

๐‘Ž0 = 1, ๐‘Žยฑ1 =1

4, ๐‘Ž2 = ๐‘Žโˆ’2

โˆ— =1

2๐‘—

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SINUSOIDAL SIGNAL

๐‘ฅ(๐‘ก) = cos๐œ”0๐‘ก

๐‘Ž๐‘˜ = แ‰Š1/2 ๐‘˜ = ยฑ10 ๐‘’๐‘™๐‘ ๐‘’

๐‘ฅ ๐‘› = cos๐œ”0๐‘›

๐‘Ž๐‘˜ = แ‰Š1/2 ๐‘˜ = ยฑ10 ๐‘’๐‘™๐‘ ๐‘’

Over a single period must specify period with period N

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SINUSOIDAL COMPARISON

Type equation here.

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PERIODIC RECTANGLE WAVE

Consider different โ€œduty cycleโ€ for the rectangle wave

50% (square wave)

25%

12.5%

Note all plots are still a sincshaped, but periodic

Difference is how the sync is sampled

Longer in time (larger N) smaller spacing in frequency more samples between zero crossings

38

RECTANGLE WAVE COEFFICIENTS

๐‘ฅ[๐‘›] = ฯƒ๐‘˜=โˆ’โˆžโˆž ๐›ฟ[๐‘› โˆ’ ๐‘˜๐‘]

Using FS integral

Notice only one impulse in the interval

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PERIODIC IMPULSE TRAIN

PROPERTIES OF FOURIER SERIESCHAPTER 3.5, 3.7

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PROPERTIES OF FOURIER SERIES

See Table 3.1 pg. 206 (CT) and Table 3.2 pg. 221 (DT)

In the following slides, suppose:

Most times, will only show proof for one of CT or DT

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CT

๐ด๐‘ฅ ๐‘ก + ๐ต๐‘ฆ ๐‘ก โŸท ๐ด๐‘Ž๐‘˜ + ๐ต๐‘๐‘˜

DT

๐ด๐‘ฅ ๐‘› + ๐ต๐‘ฆ ๐‘› โŸท ๐ด๐‘Ž๐‘˜ + ๐ต๐‘๐‘˜

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LINEARITY

CT

๐‘ฅ(๐‘ก โˆ’ ๐‘ก0) โŸท ๐‘Ž๐‘˜๐‘’โˆ’๐‘—๐‘˜๐œ”0๐‘ก0

Proof

Let ๐‘ฆ ๐‘ก = ๐‘ฅ(๐‘ก โˆ’ ๐‘ก0)

DT

๐‘ฅ[๐‘› โˆ’ ๐‘›0] โŸท ๐‘Ž๐‘˜๐‘’โˆ’๐‘—๐‘˜๐œ”0๐‘›0

43

TIME-SHIFT

CT

๐‘’๐‘—๐‘€๐œ”0๐‘ก๐‘ฅ ๐‘ก โŸท ๐‘Ž๐‘˜โˆ’๐‘€

DT

๐‘’๐‘—๐‘€๐œ”0๐‘›๐‘ฅ ๐‘› โŸท ๐‘Ž๐‘˜โˆ’๐‘€

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FREQUENCY SHIFT

Note: Similar relationship with Time Shift (dualilty). Multiplication by exponential in time is a shift in frequency. Shift in time is a multiplication by exponential in frequency.

CT

๐‘ฅ(โˆ’๐‘ก) โŸท ๐‘Žโˆ’๐‘˜

DT

๐‘ฅ[โˆ’๐‘›] โŸท ๐‘Žโˆ’๐‘˜

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TIME REVERSAL

CT

โˆซ๐‘‡ ๐‘ฅ ๐œ ๐‘ฆ ๐‘ก โˆ’ ๐œ ๐‘‘๐œ โŸท ๐‘‡๐‘Ž๐‘˜๐‘๐‘˜

DT

ฯƒ๐‘Ÿ=<๐‘> ๐‘ฅ ๐‘Ÿ ๐‘ฆ[๐‘› โˆ’ ๐‘Ÿ] โŸท ๐‘๐‘Ž๐‘˜๐‘๐‘˜

46

PERIODIC CONVOLUTION

CT

๐‘ฅ ๐‘ก ๐‘ฆ(๐‘ก) โŸท ฯƒ๐‘™=โˆ’โˆžโˆž ๐‘Ž๐‘™๐‘๐‘˜โˆ’๐‘™ = ๐‘Ž๐‘˜ โˆ— ๐‘๐‘˜

DT

๐‘ฅ ๐‘› ๐‘ฆ[๐‘›] โŸท ฯƒ๐‘™=<๐‘>๐‘Ž๐‘™๐‘๐‘˜โˆ’๐‘™ = ๐‘Ž๐‘˜ โˆ— ๐‘๐‘˜

Convolution over a single period (DT FS is periodic)

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MULTIPLICATION

Note: Similar relationship with Convolution (dualilty). Convolution in time results in multiplication in frequency domain. Multiplication in time results in convolution in frequency domain.

CT

1

๐‘‡โˆซ๐‘‡ ๐‘ฅ ๐‘ก 2๐‘‘๐‘ก = ฯƒ๐‘˜=โˆ’โˆž

โˆž ๐‘Ž๐‘˜2

DT

1

๐‘ฯƒ๐‘›=<๐‘> ๐‘ฅ ๐‘› 2 = ฯƒ๐‘˜=<๐‘> ๐‘Ž๐‘˜

2

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PARSEVALโ€™S RELATION

Note: Total average power in a periodic signal equals the sum of the average power in all its harmonic components

1

Tเถฑ๐‘‡

๐‘Ž๐‘˜๐‘’๐‘—๐‘˜๐œ”0๐‘ก

2๐‘‘๐‘ก =

1

๐‘‡เถฑ๐‘‡

๐‘Ž๐‘˜2๐‘‘๐‘ก = ๐‘Ž๐‘˜

2

Average power in the ๐‘˜th harmonic

CT

๐‘ฅ(๐›ผ๐‘ก) โŸท ๐‘Ž๐‘˜

๐›ผ > 0

Periodic with period ๐‘‡/๐›ผ

DT

๐‘ฅ(๐‘š) ๐‘› = แ‰Š๐‘ฅ[๐‘›/๐‘š] ๐‘› multiple of ๐‘š

0 ๐‘’๐‘™๐‘ ๐‘’

Periodic with period ๐‘š๐‘

๐‘ฅ(๐‘š)[๐‘›] โŸท1

๐‘š๐‘Ž๐‘˜

Periodic with period ๐‘š๐‘

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TIME SCALING

Note: Not all properties are exactly the same. Must be careful due to constraints on periodicity for DT signal.

FOURIER SERIES AND LTI SYSTEMSCHAPTER 3.8

50

EIGENSIGNAL REMINDER

๐‘ฅ ๐‘ก = ๐‘’๐‘ ๐‘ก โŸท ๐‘ฆ ๐‘ก = ๐ป ๐‘  ๐‘’๐‘ ๐‘ก ๐‘ฅ ๐‘› = ๐‘ง๐‘› โŸท ๐‘ฆ ๐‘› = ๐ป ๐‘ง ๐‘ง๐‘›

๐ป ๐‘  = โˆซโˆ’โˆžโˆžโ„Ž ๐‘ก ๐‘’โˆ’๐‘ ๐‘ก๐‘‘๐‘ก ๐ป ๐‘ง = ฯƒ๐‘›=โˆ’โˆž

โˆž โ„Ž ๐‘› ๐‘งโˆ’๐‘˜

๐ป ๐‘  ,๐ป ๐‘ง known as system function (๐‘ , ๐‘ง โˆˆ โ„‚)

For Fourier Analysis (e.g. FS)

Let ๐‘  = ๐‘—๐œ” and ๐‘ง = ๐‘’๐‘—๐œ”

Frequency response (system response to particular input frequency)

๐ป ๐‘—๐œ” = ๐ป ๐‘  ศ๐‘ =๐‘—๐œ” = โˆซโˆ’โˆžโˆžโ„Ž ๐‘ก ๐‘’โˆ’๐‘—๐œ”๐‘ก๐‘‘๐‘ก

๐ป ๐‘’๐‘—๐œ” = ๐ป ๐‘ง ศ๐‘ง=๐‘’๐‘—๐œ” = ฯƒ๐‘›=โˆ’โˆžโˆž โ„Ž ๐‘› ๐‘’โˆ’๐‘—๐œ”๐‘›

51

FOURIER SERIES AND LTI SYSTEMS I

Consider now a FS representation of a periodic signals

๐‘ฅ ๐‘ก = ฯƒ๐‘˜ ๐‘Ž๐‘˜๐‘’๐‘—๐‘˜๐œ”0๐‘ก

โ†’ ๐‘ฆ ๐‘ก = ฯƒ๐‘˜ ๐‘Ž๐‘˜๐ป ๐‘—๐‘˜๐œ”0 ๐‘’๐‘—๐‘˜๐œ”0๐‘ก

Due to superposition (LTI system)

Each harmonic in results in harmonic out with eigenvalue

๐‘ฆ(๐‘ก) periodic with same fundamental frequency as ๐‘ฅ(๐‘ก) โ‡’ ๐œ”0

๐‘‡ =2๐œ‹

๐œ”0- fundamental period

FS coefficients for ๐‘ฆ ๐‘ก

๐‘๐‘˜ = ๐‘Ž๐‘˜๐ป(๐‘—๐‘˜๐œ”0)

๐‘๐‘˜ is the FS coefficient ๐‘Ž๐‘˜ multiplied/affected by frequency response at ๐‘˜๐œ”0

52

FOURIER SERIES AND LTI SYSTEMS III

System block diagram

53

๐‘Žโˆ’๐‘˜๐‘’โˆ’๐‘—๐‘˜๐œ”0๐‘ก ๐ป(๐‘—๐œ”)

๐‘ฅ ๐‘ก =

๐‘ฆ(๐‘ก)๐‘Ž0๐‘’๐‘—(0)๐œ”0๐‘ก ๐ป(๐‘—๐œ”)

๐‘Ž๐‘˜๐‘’๐‘—๐‘˜๐œ”0๐‘ก ๐ป(๐‘—๐œ”)

โ‹ฎ

โ‹ฎ

๐‘Žโˆ’๐‘˜๐ป(โˆ’๐‘—๐‘˜๐œ”0)๐‘’โˆ’๐‘—๐‘˜๐œ”0๐‘ก

๐‘Ž๐‘˜๐ป(๐‘—๐‘˜๐œ”0)๐‘’โˆ’๐‘—๐‘˜๐œ”0๐‘ก

๐‘Ž0๐ป(โˆ’๐‘—0)

DTFS AND LTI SYSTEMS

๐‘ฅ ๐‘› = ฯƒ๐‘˜=<๐‘>๐‘Ž๐‘˜๐‘’๐‘—๐‘˜2๐œ‹/๐‘๐‘› โ†’

๐‘ฆ ๐‘› =

๐‘˜=<๐‘>

๐‘Ž๐‘˜๐ป(๐‘’๐‘—2๐œ‹๐‘๐‘˜)๐‘’๐‘—๐‘˜2๐œ‹/๐‘๐‘›

Same idea as in the continuous case

Each harmonic is modified by the Frequency Response at the harmonic frequency

54

LTI system with

โ„Ž ๐‘› = ๐›ผ๐‘›๐‘ข ๐‘› ,โˆ’1 < ๐›ผ < 1

Find FS of ๐‘ฆ[๐‘›] given input

๐‘ฅ ๐‘› = cos2๐œ‹๐‘›

๐‘

Find FS representation of ๐‘ฅ[๐‘›]

๐œ”0 = 2๐œ‹/๐‘

๐‘ฅ ๐‘› =1

2๐‘’๐‘—2๐œ‹/๐‘๐‘› +

1

2๐‘’โˆ’๐‘—2๐œ‹/๐‘๐‘›

๐‘Ž๐‘˜ = เต1

2๐‘˜ = ยฑ1,ยฑ ๐‘ + 1 ,โ€ฆ

0 else

Find frequency response

๐ป ๐‘’๐‘—๐œ” = ฯƒ๐‘› โ„Ž ๐‘› ๐‘’โˆ’๐‘—๐œ”๐‘›

๐ป ๐‘’๐‘—๐œ” = ฯƒ๐‘›๐›ผ๐‘›๐‘ข[๐‘›]๐‘’โˆ’๐‘—๐œ”๐‘›

55

EXAMPLE 1

๐ป ๐‘—๐œ” =

๐‘›=0

โˆž

๐›ผ๐‘›๐‘’โˆ’๐‘—๐œ”๐‘›

๐ป ๐‘—๐œ” =

๐‘›=0

โˆž

๐›ผ๐‘’โˆ’๐‘—๐œ”๐‘›

Let ๐›ฝ = ๐›ผ๐‘’โˆ’๐‘—๐œ”

๐ป ๐‘—๐œ” =1

1 โˆ’ ๐›ฝ

๐ป ๐‘—๐œ” =1

1 โˆ’ ๐›ผ๐‘’โˆ’๐‘—๐œ”

EXAMPLE 1 II

Use FS LTI relationship to find output

๐‘ฆ ๐‘› = ฯƒ๐‘˜=<๐‘> ๐‘Ž๐‘˜๐ป(๐‘’๐‘—๐‘˜๐œ”0) ๐‘’๐‘—๐‘˜๐œ”0๐‘›

๐‘ฆ ๐‘› =1

2๐ป ๐‘’๐‘—1

2๐œ‹

๐‘๐‘› ๐‘’๐‘—1

2๐œ‹

๐‘๐‘› +

1

2๐ป ๐‘’โˆ’๐‘—1

2๐œ‹

๐‘๐‘› ๐‘’โˆ’๐‘—1

2๐œ‹

๐‘๐‘›

๐‘ฆ ๐‘› =1

2

1

1โˆ’๐›ผ๐‘’โˆ’๐‘—๐‘˜2๐œ‹/๐‘๐‘’๐‘—

2๐œ‹

๐‘๐‘› +

1

2

1

1โˆ’๐›ผ๐‘’๐‘—๐‘˜2๐œ‹/๐‘๐‘’โˆ’๐‘—

2๐œ‹

๐‘๐‘›

Output FS coefficients

๐‘๐‘˜ = เต1

2

1

1โˆ’๐›ผ๐‘’โˆ’๐‘—๐‘˜2๐œ‹/๐‘๐‘˜ = ยฑ1

0 ๐‘’๐‘™๐‘ ๐‘’

56

Periodic with period ๐‘

๐‘ฅ(๐‘ก) has fundamental period ๐‘‡and FS ๐‘Ž๐‘˜

Sometimes direct calculation of ๐‘Ž๐‘˜ is difficult, at times easier to calculate transformation

๐‘๐‘˜ โ†” ๐‘” ๐‘ก =๐‘‘๐‘ฅ ๐‘ก

๐‘‘๐‘ก

Find ๐‘Ž๐‘˜ in terms of ๐‘๐‘˜ and ๐‘‡, given

โˆซ๐‘‡2๐‘‡๐‘ฅ ๐‘ก ๐‘‘๐‘ก = 2

๐‘Ž0 =1

๐‘‡โˆซ๐‘‡ ๐‘ฅ ๐‘ก ๐‘’โˆ’๐‘— 0 ๐œ”0๐‘ก๐‘‘๐‘ก =

1

๐‘‡โˆซ๐‘‡ ๐‘ฅ ๐‘ก ๐‘‘๐‘ก โ‡’

2

๐‘‡

From Table 3.1 pg 206

๐‘๐‘˜ โ†” ๐‘—๐‘˜2๐œ‹

๐‘‡๐‘Ž๐‘˜ โ‡’ ๐‘Ž๐‘˜ =

๐‘๐‘˜

๐‘—๐‘˜2๐œ‹/๐‘‡

๐‘Ž๐‘˜ = แ‰2/๐‘‡ ๐‘˜ = 0๐‘๐‘˜

๐‘—๐‘˜2๐œ‹/๐‘‡๐‘˜ โ‰  0

57

EXAMPLE PROBLEM 3.7

EXAMPLE PROBLEM 3.7 II

Find FS of periodic sawtooth wave

Take derivative of sawtooth

Results in sum of rectangular waves

FS coefficients of rectangular waves from Table 3.2 to get ๐‘๐‘˜ โ†” ๐‘”(๐‘ก)

Then use previous result to find ๐‘Ž๐‘˜ โ†” ๐‘ฅ(๐‘ก)

See examples 3.6, 3.7 for similar treatment

58

CHAPTER 3.9FILTERING

59

FILTERING

Important process in many applications

The goal is to change the relative amplitudes of frequency components in a signal

In EE480: DSP you can learn how to design a filter with desired properties/specifications

60

LTI FILTERS

Frequency-shaping filters โ€“ general LTI systems

Frequency-selective filters โ€“ pass some frequencies and eliminate others

Common examples include low-pass (LP), high-pass (HP), bandpass (BP), and bandstop (BS) [notch]

61

MOTIVATION: AUDIO EQUALIZER

Basic equalizer gives user ability to adjust sound from to match taste โ€“ e.g. bass (low freq) and treble (high freq)

62

Log-log plot to show larger range of frequencies and response

dB = 20 log10 ๐ป ๐‘—๐œ”

Magnitude response matches are intuition Boost low and high frequencies

but attenuate mid frequencies

EXAMPLE: DERIVATIVE FILTER

๐‘ฆ ๐‘ก =๐‘‘

๐‘‘๐‘ก๐‘ฅ ๐‘ก โŸท ๐ป ๐‘—๐œ” = ๐‘—๐œ”

High-pass filter used for โ€œedgeโ€ detection

63

EXAMPLE: AVERAGE FILTER

๐‘ฆ ๐‘› =1

2๐‘ฅ ๐‘› + ๐‘ฅ ๐‘› โˆ’ 1

Low-pass filter used for smoothing

64

MATLAB FOR FILTERS

Very helpful to visualize filters

65

Continuous Case

๐‘ฅ ๐‘ก = ฯƒ๐‘˜ ๐‘Ž๐‘˜๐‘’๐‘—๐‘˜๐œ”0๐‘ก

๐‘Ž๐‘˜ =1

๐‘‡โˆซ๐‘‡ ๐‘ฅ ๐‘ก ๐‘’โˆ’๐‘—๐‘˜๐œ”0๐‘ก๐‘‘๐‘ก

Fundamental frequency ๐œ”0

Fundamental period ๐‘‡ =2๐œ‹

๐œ”0

Discrete Case

๐‘ฅ[๐‘›] = ฯƒ๐‘˜=<๐‘>๐‘Ž๐‘˜๐‘’๐‘—๐‘˜๐œ”0๐‘›

๐‘Ž๐‘˜ =1

๐‘ฯƒ๐‘›=<๐‘> ๐‘ฅ ๐‘› ๐‘’โˆ’๐‘—๐‘˜๐œ”0๐‘›

Fundamental frequency ๐œ”0

Fundamental period ๐‘ =2๐œ‹

๐œ”0

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FOURIER SERIES SUMMARY