EE361: SIGNALS AND SYSTEMS II CH3: FOURIER SERIES HIGHLIGHTS - - PowerPoint PPT Presentation

http://www.ee.unlv.edu/~b1morris/ee361

EE361: SIGNALS AND SYSTEMS II

CH3: FOURIER SERIES HIGHLIGHTS

1

FOURIER SERIES OVERVIEW AND MOTIVATION

2

BIG IDEA: TRANSFORM ANALYSIS

 Make use of properties of LTI system 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

• r harmonics

 Frequency of each harmonic is an

integer multiple of a “fundamental frequency”

 Also known as the normal modes

 Any string deflection could be

built out of a linear combination of “modes”

4

NORMAL MODES OF VIBRATING STRING

 When plucking a string, length

is divided into integer divisions

• r harmonics

 Frequency of each harmonic is an

integer multiple of a “fundamental frequency”

 Also known as the normal modes

 Any string deflection could be

built out of a linear combination of “modes”

5

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: 𝑦 𝑢 = 𝑑𝑝𝑡𝜕0𝑢  Complex exponential: 𝑦 𝑢 = 𝑓𝑘𝜕0t  Fundamental frequency: 𝜕0  Fundamental period: 𝑈 = 2𝜌

𝜕0

 Harmonically related period

signals form family

 Integer multiple of fundamental

frequency

 𝜚𝑙 𝑢 = 𝑓𝑘𝑙𝜕0𝑢 for 𝑙 = 0, ±1, ±2, …

 Fourier Series is a way to

represent a periodic signal as a linear combination of harmonics

 𝑦 𝑢 = σ𝑙=−∞

∞

𝑏𝑙𝑓𝑘𝑙𝜕0𝑢

 𝑏𝑙 coefficient gives the contribution

• f a harmonic (periodic signal of 𝑙

times frequency)

6

FOURIER SERIES 1 SLIDE OVERVIEW

SAWTOOTH EXAMPLE

7 signal Harmonics: height given by coefficient Animation showing approximation as more harmonics added 𝑏1 𝑏2 𝑏3 𝑏4 …

SQUARE WAVE EXAMPLE

 Better approximation of square

wave with more coefficients

 Aligned approximations  Animation of FS

8 1 2 3 4 Note: 𝑇(𝑔) ~ 𝑏𝑙 #𝑏𝑙 coefficients

ARBITRARY EXAMPLES

 Interactive examples [flash (dated)][html]

9

RESPONSE OF LTI SYSTEMS TO COMPLEX EXPONENTIALS

CHAPTER 3.2 10

TRANSFORM ANALYSIS OBJECTIVE

 Need family of signals 𝑦𝑙 𝑢

that have 1) simple response and 2) represent a broad (useful) class of signals

1.

Family of signals Simple response – every signal in family pass through LTI system with scale change

2.

“Any” signal can be represented as a linear combination of signals in the family

 Results in an output generated by input 𝑦(𝑢)

11

𝑦𝑙(𝑢) ⟶ 𝜇𝑙𝑦𝑙(𝑢)

𝑦 𝑢 = ෍

𝑙=−∞ ∞

𝑏𝑙𝑦𝑙(𝑢) 𝑦 𝑢 ⟶ ෍

𝑙=−∞ ∞

𝑏𝑙𝜇𝑙𝑦𝑙(𝑢)

IMPULSE AS BASIC SIGNAL

 Previously (Ch2), we used shifted and scaled deltas



𝜀 𝑢 − 𝑢0 ⟹ 𝑦 𝑢 = ∫ 𝑦 𝜐 𝜀 𝑢 − 𝜐 𝑒𝜐 ⟶ 𝑧 𝑢 = ∫ 𝑦 𝜐 ℎ 𝑢 − 𝜐 𝑒𝜐

 Thanks to Jean Baptiste Joseph Fourier in the early

1800s we got Fourier analysis

 Consider signal family of complex exponentials

 𝑦 𝑢 = 𝑓𝑡𝑢 or 𝑦 𝑜 = 𝑨𝑜, 𝑡, 𝑨 ∈ ℂ

12

 Using the convolution

 𝑓𝑡𝑢 ⟶ 𝐼 𝑡 𝑓𝑡𝑢  𝑨𝑜 ⟶ 𝐼 𝑨 𝑨𝑜

 Notice the eigenvalue 𝐼 𝑡

depends on the value of ℎ(𝑢) and 𝑡

 Transfer function of LTI system  Laplace transform of impulse

response

13

COMPLEX EXPONENTIAL AS EIGENSIGNAL

TRANSFORM OBJECTIVE

 Simple response

 𝑦 𝑢 = 𝑓𝑡𝑢 ⟶ 𝑧 𝑢 = 𝐼 𝑡 𝑦 𝑢

 Useful representation?

 𝑦 𝑢 = σ𝑏𝑙𝑓𝑡𝑙𝑢 ⟶ 𝑧 𝑢 = σ𝑏𝑙𝐼 𝑡𝑙 𝑓𝑡𝑙𝑢

 Input linear combination of complex exponentials leads to output linear

combination of complex exponentials

 Fourier suggested limiting to subclass of period complex exponentials

𝑓𝑘𝑙𝜕0𝑢, 𝑙 ∈ ℤ, 𝜕0 ∈ ℝ

 𝑦 𝑢 = σ𝑏𝑙𝑓𝑘𝑙𝜕0𝑢 ⟶ 𝑧 𝑢 = σ𝑏𝑙𝐼 𝑘𝑙𝜕0 𝑓𝑡𝑙𝑢

 Periodic input leads to periodic output.  𝐼 𝑘𝜕 = 𝐼 𝑡 ȁ𝑡=𝑘𝜕 is the frequency response of the system

14