ee235-lectures1-4_TP(1)

Continuous-Time Linear System

Pham Van Tuan, DUT

Gernot Kubin, TUGraz

Akshay Sharma, UW

Goals: To study signal analysis, linear systems, and frequency analysis. Tobegin learning and using MATLAB for signal analysis in the time andfrequency domains Grading policies:

Grading policiesHW (20%) + Lab (30%) + Midterm (20%) + Final exam (30%)

Textbooks:

Signals Systems and Transforms by Philips, Parr, Riskin, 4th edition

(Chapters 1-7)

References:

• Interactive notes

http://www.ee.washington.edu/class/235dl/

• Textbook webpage:

http://www.ee.washington.edu/class/SST_textbook/textbook.html

Acknowledgement: courses materials are supported by EE, UW

Course Administration

http://www.ee.washington.edu/class/235dl/

http://www.ee.washington.edu/class/SST_textbook/textbook.html

Policy & Schedule Homeworks

– A HW will be due every week

– I will accept HWs up to 24 hours late. (Penalty 20%?)

– Collaboration is good. Copying verbatim BAD!

Please turn in your own work.

Labs―You must have a lab partner

― You must read the lab beforehand

― During each lab session: teamwork but each person

must program yourself, discussion is encouraged

― Ending lab session: both show your works to TA,

answer TA’s questions (not always that you & your

lab partner will get the same grade)

Lecture Overview

• A Gentle Introduction to Signal Processing

Applications of Signal Processing

What is a Signal

What is a System

How Signals and Systems are Ubiquitous

Applications of Signal Processing

Communications

Multimedia

Medical

What is a Signal?

Signals

• A signal is a function of one or more

independent variables (usually time or space).

• Signals often carry information.

Example: Velocity of a car

t (seconds)

v(t)

velocity

(mph)

v(t)

30

1 60

v(1) = 30

v(60) = 0

Systems

A system is a relationship between input and

output signal(s).

Example: Airbag deployment system

What is a System?

t (seconds)

v(t)

t (seconds)

y

n

y(t)Deploy

Airbag?velocity

v(t) y(t)A

A

y(t) =

½y if d

dtv(t) < a

n otherwise

Many signals are oscillatory

Signals due to

vibrations

Signals with cycles

speech signal

Examples

• Geological: temperature, rainfall

• Electromagnetic: AM/FM radio

• Sound: music, speech.

Case study: A simple sound

Air pressure

t (seconds)

Frequency

Tf0=1

T

s(t) = sin(2¼ f0t)

The Violin

G3

D4

A4

E5

196 Hz

293.66 Hz

440 Hz

659.26 Hz

Signals in the Frequency Domain

Each signal can be represented by its frequency content

196

g(t) G(f)

g(t) = sin(2¼ f0t)

f0= 196Hz

t (seconds) f (Hz)

Note on Convention

• Time Domain signals are represented as lower case.

• Frequency Domain signals are represented with upper

case.

Signals in the Frequency Domain

196

293.66

440

659.26

Each signal can be represented by its frequency content

g(t)

d(t)

e(t)

G(f)

D(f)

A(f)

E(f)

t (seconds)f (Hz)

a(t)

Combining Signals

+

g(t)

d(t)

a(t)

e(t)

y(t)

Frequency (Fourier) Analysis

196 293.66 440 659.26

y(t)Y (f)

t (seconds) f (Hz)

Trumpet signals in time

g(t)

f(t)

p(t)

Trumpet signals in frequency

P (f)

F (f)

G(f)

Other Example Time Signals• Charge on a capacitor over time

• Power consumed over time by a household

• Signal transmitted by a radio station

• Sound of a heartbeat over time

• Solar power hitting a space shuttle along its

flight path.

• Others? Class?

2D signals

Some signals are defined over space, generally called

“images”

hyperspectral image:

224 spectral bands, 2D

Concepts so far

• Signals as functions over time or space

• Signals carry information

• Signals in the frequency (Fourier) domain

What about Systems?

Systems


output signal(s).


What is a System?

t (seconds)

v(t)

t (seconds)

y

n

y(t)Deploy

Airbag?velocity

v(t) y(t)A

A

y(t) =

½y if d

dtv(t) < a

n otherwise

Systems

A system inputs and/or outputs signal(s)

Player outputs signal

via mouth trumpet

sound wave signal leaves the trumpet

microphone

electrical signal speakers

sound

wave

signal

systems

in blue

Some systems are filters

• Filters decrease or increase certain

frequencies (through multiplication)

Hz

LOW PASS Filter

Low freq High freq

1

1 .33

0

Some systems are filters

Filters decrease or increase certain

frequencies (through multiplication)

Hz200 400

Hz200 400

ORIGINAL

SIGNAL

LOW PASS SIGNAL

time

time

multipl

y

Cochlea

Unrolled

View of the

Cochlea

Audio Processing in the Brain

Art from Scientific American http://128.200.122.84/weinberger/publications/Weinberger,%202004a.pdf

The Brain as a Filter bank

Groups of brain

cells respond to

frequencies

differently. Each

cell filters out a

particular range.

Art from Scientific American http://128.200.122.84/weinberger/publications/Weinberger,%202004a.pdf

Band Pass

Filter!

Begin Lecture 2

Lecture 2-3 Overview

Signals and Signals…then more Signals

• Types of signals

• Some useful signals

• Common signal operations

What is a Signal?

Signals

A signal is a function of one or more

independent variables (usually time or space).

Signals often carry information.

Example: Velocity of a car

Systems


output signal(s).


Types of signals

Continuous time

vs

Discrete time

(an X-axis relationship)

types of signals

continuous-time vs. discrete-time

• A Continuous Time signal is specified at all values of

time (all Real numbers ).

Must be a functional form

• A Discrete Time signal is specified for only discrete

values of (only on Integers ).

Can be functional OR specified at a set of values

R

Z

f¡1; 0; 1; 7;¡586; 358; : : :gf1; 2:5; ¼; e;

p10; 3; : : :g

types of signals

• A Continuous Time signal

• A Discrete Time signal

f [n] = [0; 1; ¼; 0]

STEM PLOT

0 1 2 3

¼

1

NOTE: Discrete Time but Analog signal!

types of signals

continuous-time vs. discrete-time

1) 88.5 FM radio transmitted signal (continuous)

2) stock’s daily close price (discrete)

3) a capacitor’s charge over time (continuous)

4) local news broadcast to your tv (continuous)

5) picture taken by a digital camera (discrete)

Types of signals

Analog

vs

Digital

(a Y-axis relationship)

analog vs. digital

analog vs. digital

or or

where is a set of discrete values.

• An Analog signal whose amplitude can take any value in

a continuous interval (all Real numbers ).

• A Digital signal whose amplitude can take only a

discrete set of values of (from some arbitrary set ).

R

G

types of signals

• An Analog signal

• A Digital signal

analog vs. digital

or

1) 88.5 FM transmitted signal (analog)

2) stock market daily close price (digital)

3) a capacitor’s charge over time (analog)

4) local news broadcast to your tv (analog)

5) picture taken with a digital camera (digital)

6) Comedy Central’s the Daily Show on your hard drive (digital)

7) your voice (analog)

or

where is a finite set of values.

analog vs. digital

signals to know

constant signal

t0

a

unit step signal

t0

1

unit ramp signal

t0

1

or: r(t) = f{u(t)}

Find function f ?

Signal operations

signal arithmetic: add, subtract, multiply, divide signals pointwise

Delay: x(t-n) is x(t) delayed by n time units

Amplify: Ax(t) is x(t) amplified by some constant A

Speed-up: x(at) is x(t) sped up by a factor of a

Slow-down: x(t/a) slows x(t) down by a factor of a

Begin Lecture 3

playing with signals

signal arithmetic: add, subtract, multiply, divide signals pointwise

t0

a

t

1

Given

looks like: t

and

then

a+1

a

playing with signals0

unit pulse signal

t0

1

1

unit step signal

t

1

Describe p(t) in terms of u(t). Hint: use signal arithmetic and delays.

Hey, I didn’t

get an intro

slide!

playing with signals

unit pulse signal

t0

1

1

What does look like?

t p(t-3)

t

1-5 0

0 0

3 1

4 1

4.1 0

Delay: x(t-n) is x(t) delayed by n time units


3 4

Time Scaling /

Time Shifting

Time Scaling

• Speed-up

• Slow-Down

Time Shifting

• Delay

• Advance

10

p(t)

p(at)

01

a

a0

p(t=a)

y(t) = p(at)

y(t) = p(t=a)

b0

-b 0

p(t¡ b)

p(t+ b)

y(t) = p(t¡ b)

y(t) = p(t+ b)

Combining Time Scaling and

Time Shifting

So how do we deal with: f(t) = p(at – b) ?

Convert to form f(t) = p(a(t – b/a))

10

p(t)

Scale/Shift

0

x(t) = p(at)

1a 0

f1(t) = x(t¡ b

a)

ba

¡ba+ 1

a

¢

In words

Mathematically analyze


Time Shifting

So how do we deal with: f(t) = p(at – b) ?

Another solution: x(t) = p(t – b), f(t) = x(at) ?

10

p(t)

Scale/Shift

In words


Time Shifting

Sketch: f(t) = p(3t + 8) ?

By any methods you want !


Time Shifting – Dealing with Speech

Let s(t) be a speech signal

Describe in words: y(t) = 2s(5t - 4) ?

Apply both methods for your explanation

Time reversal

t

1


2

1

-2

This is also a form of Time Scaling!

Only by a negative number!

= w((-1)t)

playing with time

Describe z(t) in terms of w(t)

t

1 1

-2 12 3 t

playing with time


t

1 1

-2 12 3 t

Double check:

Pick out features to match

1) Short Side of Triangle 2) Tall Side of Triangle

playing with time, example 2


t

1 1

-2 12 5

playing with time, example 2


t

1 1

-2 12 5

Doublecheck:

1) Short Side of Triangle 2) Tall Side of Triangle

more fun with signals

t

8

3 5

in terms of unit pulse p(t)

y(t) =

8<:0 if t < 3

1 if 3 · t · 50 if 5 < t

One definition (but not in terms of p(t)):

more fun with signals

t

8

3 5

in terms of unit pulse p(t)

t

8

2

first step:

t

8

2nd step:

3 5

3rd step:

replace t by t-3:

Double-check!pulse starts:

pulse ends:

Signals You Need to Know

The Dirac delta

The Dirac delta

The Dirac delta is denoted:

•also known as the unit impulse or impulse

•it is a signal we will use a lot.

•it is not a function, so do not treat it as a function.

•but we do have rules for dealing with it.

“a spike of signal at time 0”

0

The Dirac delta

0

1²

²

1²

²

1²

²

lim²!0

The Dirac delta

0

1²

²

1²

²

1²

²

The Dirac delta

Dirac delta properties:

0

0 t0

±(t¡ t0)

Shifted to time instant t0:

Dirac delta example 1

Evaluate:


Evaluate:

x(t)±(t¡ t0) = x(t

0)±(t¡ t

0)Hint:


Evaluate:


Evaluate:

Begin Lecture 4

Dirac Properties

R1¡1 x(t)±(t¡ t

0)dt = x(t

0)

Sift-ing Property

0

Scaling the Dirac is weirdConsider

R1¡1 ±(ax)dx for some constant a

Scaling the Dirac is weirdZ 1

¡1±(ax)dx =

1

jaj

Z 1

¡1±(x)dx

Ex :

Z10

t=0

±(5t¡ 2)dt

Important Types of Signals

Exponential signals

General Form: Ceat; where a = ¾ + j!

Equivalently: Ce¾tej!t

Euler’s Relationej!t = cos(!t) + j sin(!t)

In Parts:

amplitude

Increasing

Constant

Decaying

Exponentialt

¾ > 0

¾ < 0

¾ = 0

sinusoidal w/

angular freq. !

Ce¾tej!t

Exponential signals

t

1

-1

1 2

t

1

-1

1 2

f(t) = Ce¾tej!t

where C = 1, ¾ = ¡1, ! = 2¼

Reff(t)g Imff(t)g

Periodic Signals I

)(tx is periodic if there exists a T such that x ( t)

repeats itself every T seconds.

Fundamental Frequency:

t (seconds)

T

f0=1

THz !

0=2¼

Trad !

0= 2¼f

0

x(t) = x(t+ nT ) for all integers n

Periodic Signals: Caused by

Oscillation/Vibration

Periodicity of (sums of periodic signals)

x1(t) = x

1(t+ nT

1) for all integers n

z(t) = x1(t) + x

2(t)

x2(t) = x

2(t+mT

2) for all integers m

Given:

Find T s.t. z(t) = z(t+ rT ) for all integers r


-1 0 1 2 3 4 5 6 7 8 9 10 11 12

1

2

1

1

T1

T2

Ex: x1(t) has period T

1= 2 and x

2(t) has period T

2= 3.

What is the Period T of z(t) = x1(t) + x

2(t)?


-1 0 1 2 3 4 5 6 7 8 9 10 11 12

1

2

1

1

T1

T2

Ex: x1(t) has period T

1= 2 and x

2(t) has period T

2= 3.

What is the Period T of z(t) = x1(t) + x

2(t)?

4

Periodicity of (sums of periodic signals)x1(t) = x

1(t+ nT

1) for all integers n

z(t) = x1(t) + x

2(t)

x2(t) = x

2(t+mT

2) for all integers m

Given:

Find T s.t. z(t) = z(t+ rT ) for all integers r

Solution: T = LCM of T1; T2.

Least

Common

Multiple

Periodicity of sum of periodic signalsTry it: Find the period of z(t) = sin(3t) + cos(5t)

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ee235-lectures1-4_TP(1)