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06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 1
[p. 3] Analog signal definition[p. 4] Periodic signal[p. 5] One-sided signal[p. 6] Finite length signal[p. 7] Impulse function[p. 9] Sampling property[p.11] Impulse properties[p.17] Continuous time system building blocks
[p. 18] Delay block[p. 20] Integrator block[p. 25] Differentiator block
[p. 26] LTI system[p. 26] Definition[p. 27] Testing for linearity[p. 30] Testing for time invariance
[p. 32] Evaluating convolution[p.37] Convolving unit steps[p.38] Convolving impulses[p. 39] Convolution properties[p. 44] Combining LTI systems[p. 47] System stability[p. 51] System causality[p. 53] Examples
IV. Continuous-Time Signals & LTI Systems
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 3
Analog signal x(t)INFINITE LENGTH
SINUSOIDS: (t = time in secs)PERIODIC SIGNALS
ONE-SIDED, e.g., for t>0UNIT STEP: u(t)
FINITE LENGTHSQUARE PULSE
Processing of continuous time Signals
ANALOGSystem
x ( t ) y ( t )
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 4
CT Signals: PERIODIC
x (t ) = 10 cos( 200 π t ) Sinusoidal signal
Square Wave
Signals of INFINITE DURATION
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 5
CT Signals: ONE-SIDED
v(t ) = e − t u(t )
Unit step signal1 0( )
0 0t
u tt
≥⎧= ⎨ <⎩
One-SidedSinusoid
“Suddenly applied”Exponential
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 6
CT Signals: FINITE LENGTH
Square Pulse signal
p(t) =u(t −2)−u(t −4)
Sinusoid multipliedby a square pulse
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 7
What is an Impulse?
A signal that is “concentrated” at one point.
0( ) lim ( )t tδ δ ΔΔ →
=
δ Δ ( t ) dt = 1−∞
∞
∫where
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 8
One “INTUITIVE” definition is:
Defining the Impulse
δ ( τ ) d τ−∞
∞
∫ = 1
Concentrated at t=0 with unit area
δ(t) =0, t≠0
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 9
δ(t) Sampling Property
f ( t )δ ( t ) = f ( 0 )δ ( t )
f (t)δ Δ (t) ≈ f (0)δ Δ (t )
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 10
General Sampling Property
f (t)δ (t − t0 ) = f (t0 )δ (t − t0 )
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 11
Properties of the ImpulseConcentrated at one time
Sampling Property
Of Unit area
Extract one value of f(t)sifting property
Derivative of unit step
f (t)δ(t −t0 ) = f (t0 )δ(t − t0)
δ ( t − t 0 )dt−∞
∞
∫ = 1
δ ( t − t0 ) = 0 , t ≠ t0
f ( t ) δ ( t − t 0 ) dt−∞
∞
∫ = f ( t 0 )
du ( t )dt
= δ (t )
1( ) ( )at ta
δ δ= Scaling Property
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 12
Example:2( 1)( ) ( 1)
Compute and plot ( ) /
tx t e u tdx t dt
− −= −
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 13
Example:
{ }
1
3
1
11
0
Compute:
( 3)
( 1)
( 3)
sin( ) ( 3)
t
t
t
t
jt
A e d
dB e u tdt
C d
D t e d
τδ τ τ
δ τ τ
δ τ τ
−
−∞
−
+
−
−
= +
= −
= +
= +
∫
∫
∫
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 14
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 15
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 16
Example:Assume x(t)=u(t)-u(t-5), sketch: 1) dx(t)/dt, 2) x(2-t)
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 17
CT Building Blocks
DELAY by to
INTEGRATOR (CIRCUITS)
DIFFERENTIATOR
MULTIPLIER & ADDER
Others…
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 18
Ideal Delay:
Mathematical Definition:
To find the IMPULSE RESPONSE of a system, h(t), let x(t) be an impulse, so
( ) ?h t =
y ( t ) = x ( t − t d )
System Sx(t) y(t)
System Sx(t) y(t)
x(t)=δ(t) y(t)=h(t)
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 19
Output of Ideal Delay of 1 sec
y(t) = x(t −1) = e−(t−1)u(t −1)
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 20
Integrator:
Mathematical Definition:
To find the IMPULSE RESPONSE, h(t), let x(t) be an impulse, so
y ( t ) = x (τ−∞
t∫ )d τ
h(t) = δ (τ−∞
t
∫ )dτ = u(t)
Running Integral
System Sx(t) y(t)
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 21
Integrator:
Integrate the impulse
IF t<0, we get zeroIF t>0, we get one
Thus we have h(t) = u(t) for the integrator
y ( t ) = x (τ−∞
t∫ ) d τ
δ (τ−∞
t
∫ )d τ = u ( t )
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 22
Graphical Representation
δ ( t ) =du ( t )
dt
1 0( ) ( )
0 0
t tu t d
tδ τ τ
−∞
≥⎧= = ⎨ <⎩
∫
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 23
Output of Integrator
( ) ( )t
y t x dτ τ− ∞
= ∫
y(t)=
System Sx(t) y(t)
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 24
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 25
Differentiator Output:y ( t ) =
dx ( t )dt
y(t)=
Differentiator:System S
x(t) y(t)
Example:
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 26
Linear and Time-Invariant (LTI) Systems
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.
System Sx(t)
y(t)
x(t)=δ(t) h(t)=y(t)
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 27
Testing for Linearity
Identical process to discrete signal case
{ }{ } { }
1 2
1 2
( ) ( )
( ) ( )
S x t x t
S x t S x t
α β
α β
+ =
+
System Sx(t) y(t)
Examples: y(t)=2x(t)y(t)=2x(t)+1y(t)=x(t2)y(t)=x(t-D)
( ) ( )t
y t x dτ τ−∞
= ∫
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 28
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 29
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 30
Testing Time-Invariance
A system is time-invariant if a shift in the input produces the same shift in the output
System Sx(t) y(t)
Examples: y(t)=2x(t)y(t)=x(2t)y(t)=x(t-D)
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 31
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 32
Evaluating a Convolution
∫∞
∞−
∗=−= )()()()()( txthdtxhty τττ
)()( tueth t−=)1()( −= tutx
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 33
Solutiony(t)=
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 34
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 35
Example
( ) ( ), ( ) ( )at btx t e u t h t e u t− −= =
y(t)=
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 36
Special Case: h(t)=u(t)
( ) ( ), 0, ( ) ( )atx t e u t a h t u t−= ≠ =
y(t)=
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 37
Convolving Unit Steps
( ) ( ), ( ) ( )x t u t h t u t= =
y(t)=
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 39
Convolution Properties
1) Commutative
2) Associative
3) Distributive
4) Derivative of convolution
[ ] [ ]
[ ]
( ) ( ) ( ) ( )
( ) ( )
d dx t y t x t y tdt dt
dx t y tdt
∗ = ∗
= ∗
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 40
Convolution Properties
5) Time invariance
0 0
( ) ( ) ( )( )* ( ) ( )
y t x t h tx t t h t y t t
= ∗⇒ − = −
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 41
Proofs:
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 42
Examples: Compute
( )
( 1)*( ( 2) 2 )
sin(5 ) ( 1/ 2)
tA t t edB t u tdt
δ δ −= − + +
= −
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 43
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 44
Cascade of LTI Systems
h(t) = h1(t)∗ h2 (t) = h2(t) ∗h1(t)
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 47
Stability
A system is stable if every bounded input produces a bounded output.
A continuous-time LTI system is stable if and only if
h ( t ) dt < ∞−∞
∞∫
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 48
Stability
Examples:
0
( )
( ) 3 ( ), ( ) ( )
( )( ) , ( ) ( )
( )
t
x t
y t x t y t x d
dx ty t y t tx tdt
y t e
τ τ= =
= =
=
∫
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 49
Note: 1) Finding a counter-example is OK. 2) You can’t prove a property with an example
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 50
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 51
Causal SystemsA system is causal if and only if y(t0) depends only on x(τ) for τ< t0 .
An LTI system is causal if and only if0for 0)( <= tth
Example:Are the LTI systems with the following imput/output relationships or impulse responses causal?
12
2
3
( ) ( )
( ) ( )( ) ( 3) 2 ( 3)
y t x t
y t x th t u t u t
= −
== + − −
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 52
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 54
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 55
Example:LTI systemh1
(t)=δ(t+2)
LTI systemh2
(t)=δ(t-2)
LTI systemh3
(t)=u(t−Τ)-+
1)
Compute the impulse response to the overall system and plot it2)
How do you need to pick T so that the overall system is causal3)
Are systems 1, 2, 3 causal, is the overall system causal?
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 56
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 57
Example:
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 58
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 59
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 60