IV. Continuous-Time Signals & LTI Systemsfaculty.nps.edu/fargues/teaching/ec2410/Section4MPF-SuFY... · 2010. 7. 4. · Continuous-Time Signals & LTI Systems. 06/12/10 2003rws/jMc-

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


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 )


CT Signals: PERIODIC

x (t ) = 10 cos( 200 π t ) Sinusoidal signal

Square Wave

Signals of INFINITE DURATION


CT Signals: ONE-SIDED

v(t ) = e − t u(t )

Unit step signal1 0( )

0 0t

u tt

≥⎧= ⎨ <⎩

One-SidedSinusoid

“Suddenly applied”Exponential


CT Signals: FINITE LENGTH

Square Pulse signal

p(t) =u(t −2)−u(t −4)

Sinusoid multipliedby a square pulse


What is an Impulse?

A signal that is “concentrated” at one point.

0( ) lim ( )t tδ δ ΔΔ →

=

δ Δ ( t ) dt = 1−∞

∞

∫where


One “INTUITIVE” definition is:

Defining the Impulse

δ ( τ ) d τ−∞

∞

∫ = 1

Concentrated at t=0 with unit area

δ(t) =0, t≠0


δ(t) Sampling Property

f ( t )δ ( t ) = f ( 0 )δ ( t )

f (t)δ Δ (t) ≈ f (0)δ Δ (t )


General Sampling Property

f (t)δ (t − t0 ) = f (t0 )δ (t − t0 )


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


Example:2( 1)( ) ( 1)

Compute and plot ( ) /

tx t e u tdx t dt

− −= −


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

−

−∞

−

+

−

−

= +

= −

= +

= +

∫

∫

∫




Example:Assume x(t)=u(t)-u(t-5), sketch: 1) dx(t)/dt, 2) x(2-t)


CT Building Blocks

DELAY by to

INTEGRATOR (CIRCUITS)

DIFFERENTIATOR

MULTIPLIER & ADDER

Others…


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)


Output of Ideal Delay of 1 sec

y(t) = x(t −1) = e−(t−1)u(t −1)


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)


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 )


Graphical Representation

δ ( t ) =du ( t )

dt

1 0( ) ( )

0 0

t tu t d

tδ τ τ

−∞

≥⎧= = ⎨ <⎩

∫


Output of Integrator

( ) ( )t

y t x dτ τ− ∞

= ∫

y(t)=

System Sx(t) y(t)



Differentiator Output:y ( t ) =

dx ( t )dt

y(t)=

Differentiator:System S

x(t) y(t)

Example:


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)


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−∞

= ∫




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)



Evaluating a Convolution

∫∞

∞−

∗=−= )()()()()( txthdtxhty τττ

)()( tueth t−=)1()( −= tutx


Solutiony(t)=



Example

( ) ( ), ( ) ( )at btx t e u t h t e u t− −= =

y(t)=


Special Case: h(t)=u(t)

( ) ( ), 0, ( ) ( )atx t e u t a h t u t−= ≠ =

y(t)=


Convolving Unit Steps

( ) ( ), ( ) ( )x t u t h t u t= =

y(t)=


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

∗ = ∗

= ∗


Convolution Properties

5) Time invariance

0 0

( ) ( ) ( )( )* ( ) ( )

y t x t h tx t t h t y t t

= ∗⇒ − = −


Proofs:


Examples: Compute

( )

( 1)*( ( 2) 2 )

sin(5 ) ( 1/ 2)

tA t t edB t u tdt

δ δ −= − + +

= −



Cascade of LTI Systems

h(t) = h1(t)∗ h2 (t) = h2(t) ∗h1(t)


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 < ∞−∞

∞∫


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= =

= =

=

∫


Note: 1) Finding a counter-example is OK. 2) You can’t prove a property with an example



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

= −

== + − −




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?



Example:




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IV. Continuous-Time Signals & LTI Systemsfaculty.nps.edu/fargues/teaching/ec2410/Section4MPF-SuFY... · 2010. 7. 4. · Continuous-Time Signals & LTI Systems. 06/12/10 2003rws/jMc-