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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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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
3/12/2008 © 2003, JH McClellan & RW Schafer 6
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
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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
−∞
∞
∫
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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)
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Sampling Property
f (t)δ (t) = f (0)δ (t)
f (t)δ∆ (t) ≈ f (0)δ∆ (t)
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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)
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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
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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 )
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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)
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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
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3/12/2008 © 2003, JH McClellan & RW Schafer 19
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
∫
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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−=
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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
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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
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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
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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 )
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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)]
3/12/2008 © 2003, JH McClellan & RW Schafer 30
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 )
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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