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EE3054

Signals and Systems

Continuous Time Signals & Systems: Part I

Yao Wang

Polytechnic University

Some slides included are extracted from lecture presentations prepared by McClellan and Schafer

3/12/2008 © 2003, JH McClellan & RW Schafer 2

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2

LECTURE OBJECTIVES

� Bye bye to D-T Systems for a while

� The UNIT IMPULSE signal

� Definition

� Properties

� Continuous-time systems

� Example systems and their impulse response

�� LLinearity and TTime-IInvariant (LTI) systems

� Convolution integral

3/12/2008 © 2003, JH McClellan & RW Schafer 4

ANALOG SIGNALS x(t)

� INFINITE LENGTH� SINUSOIDS: (t = time in secs)

� PERIODIC SIGNALS

� ONE-SIDED, e.g., for t>0� UNIT STEP: u(t)

� FINITE LENGTH� SQUARE PULSE

� IMPULSE SIGNAL: δδδδ(t)

� DISCRETE-TIME: x[n] is list of numbers

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3/12/2008 © 2003, JH McClellan & RW Schafer 5

CT Signals: PERIODIC

x(t) = 10cos(200πt)Sinusoidal signal

Square Wave INFINITE DURATION

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CT Signals: ONE-SIDED

v(t) = e−tu(t)

Unit step signalu(t) =1 t > 0

0 t < 0

One-Sided

Sinusoid

“Suddenly applied”

Exponential

4

3/12/2008 © 2003, JH McClellan & RW Schafer 7

CT Signals: FINITE LENGTH

Square Pulse signal

p(t) = u(t − 2) −u(t − 4)

Sinusoid multiplied

by a square pulse

3/12/2008 © 2003, JH McClellan & RW Schafer 8

What is an Impulse?

� A signal that is “concentrated” at one point.

lim∆→0

δ∆ (t) = δ (t)δ∆ (t)dt = 1

−∞

∞

∫

5

3/12/2008 © 2003, JH McClellan & RW Schafer 9

� Assume the properties apply to the limit:

� One “INTUITIVE” definition is:

Defining the Impulse

Unit areaδ(τ )dτ−∞

∞

∫ =1

Concentrated at t=0δ(t) = 0, t ≠ 0

lim∆→0

δ∆ (t) = δ (t)

3/12/2008 © 2003, JH McClellan & RW Schafer 10

Sampling Property

f (t)δ (t) = f (0)δ (t)

f (t)δ∆ (t) ≈ f (0)δ∆ (t)

6

3/12/2008 © 2003, JH McClellan & RW Schafer 11

General Sampling Property

f (t)δ (t − t0 ) = f (t0 )δ (t − t0 )

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Properties of the Impulse

Concentrated at one time

Sampling Property

Unit area

Extract one value of f(t)

Derivative of unit step

f (t)δ(t − t0 ) = f (t0 )δ(t − t0)

δ( t − t0 )dt−∞

∞

∫ = 1

δ(t − t0 ) = 0, t ≠ t0

f (t)δ(t − t0 )dt−∞

∞

∫ = f (t0 )

du( t)

dt= δ(t)

7

Representing any signal using

impulse

∆−≈−= ∑∫ ∆

∞

∞−

)()()()()( kk txdtxtx τδτττδτ

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Continuous-Time Systems

� Examples:

� Delay

� Modulator

� Integrator

x(t) ֏ y(t)

y(t) = x(t − td )

y(t) = [A + x(t)]cosωct

y(t) = x(τ−∞

t

∫ )dτ

Input

Output

8

3/12/2008 © 2003, JH McClellan & RW Schafer 15

Impulse Response

� Output when the input is δ(t)

� Denoted by h(t)

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Ideal Delay:

� Mathematical Definition:

� To find the IMPULSE RESPONSE, h(t),let x(t) be an impulse, so

h(t) = δ (t − td )

y(t) = x(t − td )

9

3/12/2008 © 2003, JH McClellan & RW Schafer 17

Output of Ideal Delay of 1 sec

x(t) = e−tu(t)

y(t) = x(t −1) = e−(t−1)

u(t −1)

3/12/2008 © 2003, JH McClellan & RW Schafer 18

Integrator:

� Mathematical Definition:

� To find the IMPULSE RESPONSE, h(t),let x(t) be an impulse, so

y(t) = x(τ−∞

t

∫ )dτ

h(t) = δ(τ−∞

t

∫ )dτ = u(t)

Running Integral

10

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Integrator:

� Integrate the impulse

� IF t<0, we get zero

� IF t>0, we get one

� Thus we have h(t) = u(t) for the integrator

y(t) = x(τ−∞

t

∫ )dτ

δ(τ−∞

t

∫ )dτ = u(t)

3/12/2008 © 2003, JH McClellan & RW Schafer 20

Graphical Representation

δ(t) =du(t)

dt

u(t) = δ (τ )dτ =1 t > 0

0 t < 0

−∞

t

∫

11

3/12/2008 © 2003, JH McClellan & RW Schafer 21

Output of Integrator

)()(

)()(

tutx

dxty

t

∗=

= ∫∞−

ττ

)()1(25.1

0)(

00

)()(

8.0

0

8.0

8.0

tue

tdue

t

duety

t

t

t

−

−

∞−

−

−=

≥

<=

=

∫

∫

ττ

ττ

τ

τ

)()( 8.0 tuetx t−=

3/12/2008 © 2003, JH McClellan & RW Schafer 22

Differentiator:

� Mathematical Definition:

� To find h(t), let x(t) be an impulse, so

y(t) =dx(t)

dt

h(t) =dδ (t)dt

= δ (1)(t) Doublet

12

3/12/2008 © 2003, JH McClellan & RW Schafer 23

Differentiator Output: y(t) =dx(t)

dt

)1()( )1(2 −= −− tuetx t

( )

)1(1)1(2

)1()1(2

)1()(

)1(2

)1(2)1(2

)1(2

−+−−=

−+−−=

−=

−−

−−−−

−−

ttue

tetue

tuedt

dty

t

tt

t

δ

δ

Linear and Time-Invariant

(LTI) Systems

� Recall LTI property of discrete time

system

� Can be similarly defined for continuous

time systems

13

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Testing for Linearity

x1(t)

x2 (t)

y1(t)

y2 (t)

w(t)

y(t)x(t)

x2 (t)

x1(t)w(t)

y(t)

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Testing Time-Invariance

x(t) x(t − t0 )

y(t)

w(t)

y(t − t0 )

t0

w(t) y(t − t0 )

t0

14

3/12/2008 © 2003, JH McClellan & RW Schafer 27

Ideal Delay:

� Linear

� and Time-Invariant

y(t) = x(t − td )

ax1( t − td ) + bx2(t − td ) = ay1 (t) + by2 (t)

))(())(()(

))(()(

000

0

dd

d

tttxtttxtty

tttxtw

−−=−−=−

−−=

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Integrator:

� Linear

� And Time-Invariant

y(t) = x(τ−∞

t

∫ )dτ

[ax1(τ−∞

t

∫ ) + bx2 (τ )]dτ = ay1(t) + by2 (t)

w(t) = x(τ − t0−∞

t

∫ )dτ let σ = τ − t0

⇒ w( t) = x(σ )dσ−∞

t−t 0

∫ = y(t - t0 )

15

3/12/2008 © 2003, JH McClellan & RW Schafer 29

Modulator:

�� NotNot linear--obvious because

�� NotNot time-invariant

y(t) = [A + x(t)]cosωct

w(t) = [A + x(t − t0 )]cosωct ≠ y(t − t0 )

[A + ax1(t) + bx2 (t)]≠

[A + ax1(t)]+ [A + bx2 (t)]

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Continuous Time Convolution

� If a continuous-time system is both linear and

time-invariant, then the output y(t) is related to

the input x(t) by a convolution integralconvolution integral

where h(t) is the impulse responseimpulse response of the system.

y(t) = x(τ )h(t − τ )dτ = x(t)∗h(t)−∞

∞

∫

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Proof

� Representing x(t) using δ(t), using LTI property!

∆−≈−= ∑∫ ∆

∞

∞−

)()()()()( kk txdtxtx τδτττδτ

Ideal Delay:

� Recall

� Show y(t)=x(t)*h(t)

� Another important property of δ(t):� x(t)*δ(t-t0)=x(t-t0)

h(t) = δ (t − td )

y(t) = x(t − td )

17

Integrator:

� Recall

� Show: y(t)=x(t)*h(t)

y(t) = x(τ−∞

t

∫ )dτ

h(t) = δ(τ−∞

t

∫ )dτ = u(t)

READING ASSIGNMENTS

� This Lecture:

� Chapter 9, Sects 9-1 to 9-5

� Next Lecture:

� Chapter 9, Sects 9-6 to 9-8