keee313(03) signals and systems - korea...

KEEE313(03)Signals and Systems

Chang-Su Kim

Course Information

Course homepagehttp://mcl.korea.ac.kr

LecturerChang-Su KimOffice: Engineering Bldg, Rm 508E-mail: [email protected]

Tutor임경선 ([email protected])

http://mcl.korea.ac.kr/

Course Information

ObjectiveStudy fundamentals of signals and systems

Main topic: Fourier analysis

TextbookA. V. Oppenheim and A. S. Willsky, Signals & Systems, 2nd edition, Prentice Hall, 1997.

Reference M. J. Roberts, Signals and Systems, McGraw Hill, 2003.

Course Information

PrerequisiteAdvanced Engineering Mathematics

Manipulation of complex numbers

AssessmentExercises 15 %Midterm Exam 30 %Final Exam 40 %Attendance 15 %

Course Schedule

First, Chapters 1-5 of the textbook will be coveredLinear time invariant systemsFourier analysis

Then, selected topics in Chapters 6-10 will be taught, such as

FilteringSamplingModulationLaplace TransformZ-transform

Midterm exam: 19 APR 2014 (Tuesday)

Policies and Rules

ExamsScope: all materials taught Closed bookA single sheet of hint paper (both sizes)

AttendanceChecked sometimes

QuizzesPop up quizzes: not announced before# of quizzes is also variant

AssignmentsNo late submission is allowed

What Are Signals and Systems?

SignalsFunctions of one or more independent variablesContain information about the behavior or nature of some phenomenon

SystemsRespond to particular signals by producing other signals or some desired behavior

Examples of Signals and Systems (2005)


http://images.apple.com/ipodnano/gallery/images/ipodnanolarge04_20060912.jpg

http://images.apple.com/ipodnano/gallery/images/ipodnanolarge04_20060912.jpg

http://www.apple.com/ipodshuffle/gallery/

http://www.apple.com/ipodshuffle/gallery/

Audio Signals

f(t)Function of time tAcoustic pressure

S I GN AL

Image Signals

f(x,y)Function of spatial coordinates (x, y)Light intensity

Image Signals

Color imagesr(x,y)g(x,y)b(x,y)

Video Signals

Functions of space and timer(x,y,t)g(x,y,t)b(x,y,t)

원본

안정화(warping)Rolling shutter 왜곡 제거

Systems

•Input signal: left, kick, •punch, right, up, down,…

•Output signal: sound and graphic data

Systems

•Input image •Output image

•Image processing system

Scope of “Signals and Systems” is broad

Multimedia is just a tiny portion of signal classes The concept of signals and systems arise in a wide

variety of fields

•Input signal•- pressure on • accelerator • pedal

•Output signal•- velocity

Scope of “Signals and Systems” is broad

Multimedia is just a tiny portion of signal classes The concept of signals and systems arise in a wide

variety of fields

Input signal- time spenton study

Output signal- mark on midterm exam

Mathematical Framework

The objective is to develop a mathematical framework

for describing signals and systems and for analyzing them

We will deal with signals involving a single independent variable only

𝑥𝑥(𝑡𝑡) (or 𝑥𝑥[𝑛𝑛])For convenience, the independent variable 𝑡𝑡 (or 𝑛𝑛) is called time, although it may not represent actual time

It can in fact represent spatial location (e.g. in image signal)

Continuous-Time Signals 𝑥𝑥(𝑡𝑡) vs. Discrete-Time Signals 𝑥𝑥[𝑛𝑛]

𝑥𝑥[𝑛𝑛] is defined only for integer values of the independent variable n

𝑛𝑛 = ⋯ ,−2,−1, 0, 1, 2, … 𝑥𝑥[𝑛𝑛] can be obtained from sampling of CT

signals or some signals are inherently discrete

DT signalCT signal

Examples of DT Signals

Signal Energy and Power

Energy: accumulation of squared magnitudes

Power: average squared magnitudes

∫∫∞

∞−−∞→

∆

∞ == dttxdttxET

TT

22 )()(lim

2 2lim [ ] [ ]N

N n N nE x n x n

∞∆

∞ →∞=− =−∞

= =∑ ∑

21lim ( )2

T

TT

P x t dtT

∆

∞ →∞−

= ∫21lim [ ]

2 1

N

N n NP x n

N

∆

∞ →∞=−

=+ ∑

Classification of signalsCategory 1: Energy signal (𝐸𝐸∞ < ∞ and thus 𝑃𝑃∞ = 0)

e.g.

Category 2: Power signal (𝐸𝐸∞ = ∞ but 𝑃𝑃∞ < ∞)e.g.

Category 3: Remaining signals (i.e. with infinite energy and infinite power)

e.g.

≥<

= − 0,0,0

)(tet

tx t

( ) 1x t =

( )x t t=

Signal Energy and Power

Transformations of Independent Variable

Three possible time transformations1. Time Shift: 𝑥𝑥(𝑡𝑡 − 𝑎𝑎), 𝑥𝑥[𝑛𝑛 − 𝑎𝑎]

Shifts the signal to the right when 𝑎𝑎 > 0, while to the left when 𝑎𝑎 < 0.

2. Time Reversal: 𝑥𝑥 −𝑡𝑡 , 𝑥𝑥[−𝑛𝑛]Flips the signal with respect to the vertical axis.

3. Time Scale: 𝑥𝑥(𝑎𝑎𝑡𝑡), 𝑥𝑥[𝑎𝑎𝑛𝑛] for 𝑎𝑎 > 0.Compresses the signal length when 𝑎𝑎 > 1, while stretching it when 𝑎𝑎 < 1.

Time Reversal

Time Shift


21

x(-t)

t

1

-2 -1t

-3

x(t+1)1

-2 -1t

1

x(t-1)1

-2 -1t

x(t)1

Time Scaling

Combinations


-1/2t

-1

x(-2t)1

x(-t+3)

21t

43 65

1

-1/2t

-1

x(2t)1

-2 -1t

-3

x(t/2)

-4

1

-2 -1t

x(t)1

Periodic Signals

𝑥𝑥(𝑡𝑡) is periodic with period 𝑇𝑇, if𝑥𝑥(𝑡𝑡) = 𝑥𝑥(𝑡𝑡 + 𝑇𝑇) for all 𝑡𝑡

𝑥𝑥[𝑛𝑛] is periodic with period 𝑁𝑁, if𝑥𝑥[𝑛𝑛] = 𝑥𝑥[𝑛𝑛 + 𝑁𝑁] for all 𝑛𝑛

Note that 𝑁𝑁 should be an integer

Fundamental period (𝑇𝑇0 or 𝑁𝑁0): The smallest positive value of 𝑇𝑇 or 𝑁𝑁 for which the above equations hold

Periodic Signals

Periodic Signals

Is this periodic?

Even and Odd Signals 𝑥𝑥[𝑛𝑛] is even, if 𝑥𝑥[−𝑛𝑛] = 𝑥𝑥[𝑛𝑛]

𝑥𝑥[𝑛𝑛] is odd, if 𝑥𝑥[−𝑛𝑛] = −𝑥𝑥[𝑛𝑛]

Any signal 𝑥𝑥[𝑛𝑛] can be divided into even component 𝑥𝑥𝑒𝑒[𝑛𝑛] and odd component 𝑥𝑥𝑜𝑜[𝑛𝑛]

𝑥𝑥[𝑛𝑛] = 𝑥𝑥𝑒𝑒[𝑛𝑛] + 𝑥𝑥𝑂𝑂[𝑛𝑛]𝑥𝑥𝑒𝑒[𝑛𝑛] = (𝑥𝑥[𝑛𝑛] + 𝑥𝑥[−𝑛𝑛])/2𝑥𝑥𝑂𝑂[𝑛𝑛] = (𝑥𝑥[𝑛𝑛] − 𝑥𝑥[−𝑛𝑛])/2

Similar arguments can be made for continuous-time signals

Even function

Odd function

Even and Odd Signals

Even-odd decomposition

Exponential and Sinusoidal Signals

Euler’s Equation

Euler’s formula

Complex exponential functions facilitate the manipulation of sinusoidal signals.

For example, consider the straightforward extension of differentiation formula of exponential functions to complex cases.

)(21sin

)(21cos

sincos

θθ

θθ

θ

θ

θ

θθ

jj

jj

j

eej

ee

je

−

−

−=

+=

+=

Periodic Signals

Periodicity conditionx(t) = x(t+T)

If T is a period of x(t), then mT is also a period, where m=1,2,3,…

Fundamental period T0of x(t) is the smallest possible value of T.

Exercise: Find T0 for cos(ω0t+θ) and sin(ω0t+θ)

Periodicity conditionx[n] = x[n+N]

If N is a period of x[n], then mN is also a period, where m=1,2,3,…

Fundamental period N0 of x[n] is the smallest possible value of N.

CT Signal DT Signal

Sinusoidal Signals

x(t) = A cos(ωt+θ) or x[n] = A cos(ωn+θ)A is amplitudeω is radian frequency (rad/s or rad/sample)θ is the phase angle (rad)

Notice that although A cos(ω0t+θ) ≠ A cos(ω1t+θ),

it may hold that A cos[ω0n+θ] = A cos[ω1n+θ].

Do you know when?

Periodic Complex Exponential Signals

x(t) = or x[n] =A, θ and ω are real.

Is periodic?

How about the discrete case? Is periodic?

It is periodic when ω/2π is a rational number

( )j tAe ω θ+ ( )j nAe ω θ+

( ) j tz t Ae ω=

[ ] j nz n Ae ω=

23

35

22

Ex 1) [ ]

Ex 2) [ ]

Ex 3) [ ]

j n

j n

j n

z n e

z n e

z n e

π

π

π

=

=

=

Review of Sinusoidal and Periodic Complex Exponentials

CT case

These are periodic with period 2π/w.Also,

DT case

These are periodic only if w/2π is a rational number.Also,

( ) cos( ), sin( ) or j tx t wt wt e ω=

[ ] cos( ), sin( ) or j nx n wn wn e ω=

1 2

1 2

1 2 1 2

If ,cos( ) cos( ), sin( ) sin( ) and j t j t

w ww t w t w t w t e eω ω

≠

≠ ≠ ≠

1 2

1 2

1 2 1 2

If 2 ,cos( ) cos( ), sin( ) sin( ) and j n j n

w w kw n w n w n w n e eω ω

π= +

= = =

Real Exponential Signals

x(t) = A eσt or x[n] = A eσn

A and σ are real.

positive σ negative σ

General Exponential Signals( )( )

[cos( ) sin( )]

st j jw t

t

x t Xe Ae eAe wt j wt

θ σ

σ θ θ

+= =

= + + +

Real part of x(t) according to s (θ is assumed to be 0)

Impulse and Step Functions

DT Unit Impulse Function

Unit Impulse

Shifted Unit Impulse

1, 0[ ]

0, 0n

nn

δ=

= ≠

δ[n]

-1-2n

1-3 32

1

δ[n-k]

…-1n

1 k

11,[ ]

0,n k

n kn k

δ=

− = ≠

DT Unit Step Function

Unit Step

Shifted Unit Step

1, 0[ ]

0, 0n

u nn≥

= <

u[n]

-1-2n

1-3 32

1

u[n-k]

…-1n

1 k

11,[ ]

0,n k

u n kn k≥

− = <

Properties of DT Unit Impulse and Step Functions

0

0 0 0

1) [ ] [ ] [ 1]

2) [ ] [ ] [ ]

3) [ ] [ ] [0] [ ]

4) [ ] [ ] [ ] [ ]

5) [ ] [ ] [ ]

n

k k

k

n u n u n

u n k n k

x n n x n

x n n n x n n n

x n x k n k

δ

δ δ

δ δ

δ δ

δ

∞

=−∞ =

∞

=−∞

= − −

= = −

=

− = −

= −

∑ ∑

∑

CT Unit Step Function

Unit Step

Shifted Unit Step

1, 0( )

0, 0t

u tt>

= < t

1

u(t- τ)

tτ

1

u(t)

1,( )

0,t

u tt

ττ

τ>

− = <

CT Unit Step Function

Unit step is discontinuous at t=0, so is not differentiable

Approximated unit step

u∆(t) is continuous and differentiable.

0, 0

( ) , 0

1,

ttu t t

t

∆

≤= < ≤ ∆∆

> ∆

t1

u∆(t)

∆

0( ) lim ( )u t u t∆∆→

=1 , 0( )

0, otherwise

tdu tdt∆

≤ < ∆= ∆

CT Unit Impulse Function

Approximated unit impulse

Unit Impulse:

0

, 0( ) lim ( )

0, 0

( ) 1 for any 0 and 0. b

a

tt t

t

t dt a b

δ δ

δ

∆∆→

−

∞ == = ≠

= > >∫

t

δ∆(t)

∆

t

δ(t)

1 , 0( )( )0, otherwise

tdu ttdt

δ ∆∆

≤ < ∆= = ∆

1/∆

CT Unit Impulse Function

Shifted Unit Impulse

t

δ(t-τ)

τ

Properties of CT Unit Impulse and Step Functions

0

0 0 0

( )1) ( )

2) ( ) ( ) ( )

3) ( ) ( ) (0) ( )

4) ( ) ( ) ( ) ( )

5) ( ) ( ) ( )

t

du ttdt

u t d t d

x t t x t

x t t t x t t t

x t x t d

δ

δ τ τ δ τ τ

δ δ

δ δ

τ δ τ τ

∞

−∞

∞

−∞

=

= = −

=

− = −

= −

∫ ∫

∫

Comparison of DT and CT Properties

0 0

Difference becomes ( )[ ] [ ] [ 1] ( )differentiation

Summation becomes [ ] [ ] [ ] ( ) ( ) ( )

integration

Impulse functions [ ] [ ] [0] [ ] ( ) (

sample values

tn

k k

du tn u n u n tdt

u n k n k u t d t d

x n n x n x t t

δ δ

δ δ δ τ τ δ τ τ

δ δ δ

∞∞

=−∞ = −∞

= − − =

= = − = = −

=

∑ ∑ ∫ ∫

0 0 0 0 0 0

) (0) ( )

Shifted impulse [ ] [ ] [ ] [ ] ( ) ( ) ( ) ( )

functions

Sifting Property:Arbitrary functions as sum [ ] [ ] [ ] ( ) ( ) ( )or integration of delta functions k

x t

x n n n x n n n x t t t x t t t

x n x k n k x t x t d

δ

δ δ δ δ

δ τ δ τ τ∞

=−∞ −∞

=

− = − − = −

= − = −∑∞

∫

Can you represent these functions using step functions?

tc

x(t)

a b

1

y(t)

-1 1t

1

w(t)

-1 1t

2z(t)

-11 t

2

-2

x[n]

…-1n

1 N

1

y[n]

… -1n

1 4

1

-2 32 5-3 …

Can you represent these functions using step functions?

Basic System Properties

What is a System?

System is a black box that takes an input signal and converts it to an output signal.

DT System: y[n] = H[x[n]]

CT System: y(t) = H(x(t))

Hx[n] y[n]

Hx(t) y(t)

Interconnection of Systems

Series (or cascade) connection: y(t) = H2( H1( x(t) ) )

e.g. a radio receiver followed by an amplifier

Parallel connection: y(t) = H2( x(t) ) + H1( x(t) )

e.g. Carrying out a team project

H1x(t)

H1x(t) y(t)

H2y(t)

H2

+

Feedback connection: y(t) = H1( x(t)+H2( y(t) ) )

e.g. cruise control

Various combinations of connections are also possible

H1x(t) y(t)

H2

+

Interconnection of Systems

Memoryless Systems vs. Systems with Memory Memoryless Systems: The output y(t) at any instance t depends

only on the input value at the current time t, i.e. y(t) is a function of x(t)

Systems with Memory: The output y(t) at any instance t depends on the input values at past and/or future time instances as well as the current time instance

Examples:A resistor: 𝑦𝑦(𝑡𝑡) = 𝑅𝑅𝑥𝑥(𝑡𝑡)A capacitor:

𝑦𝑦 𝑡𝑡 =1𝐶𝐶 �−∞

𝑡𝑡𝑥𝑥 𝜏𝜏 𝑑𝑑𝜏𝜏

A unit delayer: 𝑦𝑦[𝑛𝑛] = 𝑥𝑥[𝑛𝑛 − 1]An accumulator:

𝑦𝑦 𝑛𝑛 = �𝑘𝑘=−∞

𝑛𝑛

𝑥𝑥[𝑘𝑘]

Causality

Causality: A system is causal if the output at any time instance depends only on the input values at the current and/or past time instances.

Examples: 𝑦𝑦[𝑛𝑛] = 𝑥𝑥[𝑛𝑛] − 𝑥𝑥[𝑛𝑛 − 1]𝑦𝑦(𝑡𝑡) = 𝑥𝑥(𝑡𝑡 + 1)Is a memoryless system causal?

Causal property is important for real-time processing. But in some applications, such as image processing,

data is often processed in a non-causal way.

image processing

Applications of Lowpass Filtering

Preprocessing before machine recognition

Removal of small gaps

Applications of Lowpass Filtering

Cosmetic processing of photos

Invertibility

Invertibility: A system is invertible if distinct inputs result in distinct outputs.

If a system is invertible, then there exists an inverse system which converts the output of the original system to the original input.

Examples:

)(41)(

)(4)(

tytw

txty

=

=

]1[][][

][][

−−=

= ∑−∞=

nynynw

kxnyn

k

System𝑥𝑥(𝑡𝑡) Inverse

System𝑤𝑤(𝑡𝑡) = 𝑥𝑥(𝑡𝑡)𝑦𝑦(𝑡𝑡)

( ) ( )

( )( )

t

y t x d

dy tw tdt

τ τ−∞

=

=

∫

Stability

Stability: A system is stable if a bounded input yields a bounded output (BIBO).

In other words, if 𝑥𝑥 𝑡𝑡 < 𝑘𝑘1 then |𝑦𝑦(𝑡𝑡)| < 𝑘𝑘2.

Examples:

0

( ) ( )t

y t x dτ τ= ∫

][100][ nxny =

Linearity A system is linear if it satisfies two properties.

Additivity: 𝑥𝑥 𝑡𝑡 = 𝑥𝑥1(𝑡𝑡) + 𝑥𝑥2(𝑡𝑡) ⇒ 𝑦𝑦(𝑡𝑡) = 𝑦𝑦1(𝑡𝑡) + 𝑦𝑦2(𝑡𝑡)Homogeneity: 𝑥𝑥 𝑡𝑡 = 𝑐𝑐 𝑥𝑥1(𝑡𝑡) ⇒ 𝑦𝑦(𝑡𝑡) = 𝑐𝑐 𝑦𝑦1(𝑡𝑡),

for any constant 𝑐𝑐

The two properties can be combined into a single property.

linearity:𝑥𝑥 𝑡𝑡 = 𝑎𝑎𝑥𝑥1(𝑡𝑡) + 𝑏𝑏𝑥𝑥2(𝑡𝑡) ⇒ 𝑦𝑦(𝑡𝑡) = 𝑎𝑎𝑦𝑦1(𝑡𝑡) + 𝑏𝑏𝑦𝑦2(𝑡𝑡)

Examples)()( 2 txty = ][][ nnxny = ( ) 2 ( ) 3y t x t= +

Time-Invariance A system is time-invariant if a delay (or a time-shift) in

the input signal causes the same amount of delay in the output.

𝑥𝑥 𝑡𝑡 = 𝑥𝑥1(𝑡𝑡 − 𝑡𝑡0) ⇒ 𝑦𝑦(𝑡𝑡) = 𝑦𝑦1(𝑡𝑡 − 𝑡𝑡0)

Examples:

][][ nnxny = )2()( txty = )(sin)( txty =

Superposition in LTI Systems

For an LTI system:Given response y(t) of the system to an input signal x(t), it is possible to figure out response of the system to any signal x1(t) that can be obtained by “scaling” or “time-shifting” the input signal x(t), because 𝑥𝑥1 𝑡𝑡 = 𝑎𝑎0 𝑥𝑥 𝑡𝑡 − 𝑡𝑡0 + 𝑎𝑎1 𝑥𝑥 𝑡𝑡 − 𝑡𝑡1 + 𝑎𝑎2𝑥𝑥 𝑡𝑡 − 𝑡𝑡2 + ⋯

⇒𝑦𝑦1 𝑡𝑡 = 𝑎𝑎0 𝑦𝑦 𝑡𝑡 − 𝑡𝑡0 + 𝑎𝑎1 𝑦𝑦 𝑡𝑡 − 𝑡𝑡1 + 𝑎𝑎2𝑦𝑦 𝑡𝑡 − 𝑡𝑡2 + ⋯

Very useful property since it becomes possible to solve a wider range of problems.

This property will be basis for many other techniques that we will cover throughout the rest of the course.

Superposition in LTI Systems

Exercise: Given response y(t) of an LTI system to the input signal x(t), find the response of that system to another input signal x1(t) shown below.

2x(t)

1t

1

y(t)

-1 1t

2x1(t)

1 t

-1

3

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