class_04-convolution-continuous.pdf

Chengbin Ma UM-SJTU Joint Institute

Class#4

- The convolution integral (2.4)

- Convolution evaluation procedure (2.5)

- Interconnections of LTI systems (2.6)

Slide 1


Review of Previous Lecture (1)

Convolution sum: the output of a discrete-time LTI system.

Discrete-time signal: a time-shifted impulse with amplitude given by the value of the signal at the time

the impulse occurs.

Slide 2

[ ] [ 1] [ 1] [0] [ ] [1] [ 1]

[ ] [ ]

[ ]: the entire signal

[ ]: a specific value of the signal [ ] at time .

k

x n x n x n x n

x k n k

x n

x k x n k

Weighted sum of basis functions


Review of Previous Lecture (2)

Discrete-time LTI system: The output of an LTI system is the convolution sum of the input to the

system and the impulse response of the system.

Evaluation

1. Direct form

2. LTI form

3. Convolution table

4. Reflection and shift

Slide 3

]1[]1[][]0[

][][][][][

nhxnhx

knhkxnhnxnyk

Memory: a system is said to possess

memory if its output signal depends on

past or future values of the input signal.

Convolution is used to represent

multiple impacts of the current/past

or future input signals.


This Lecture

Convolution of continuous-time signal

Evaluation of the convolution integral

Interconnections of multiple LTI systems

Parallel connection

Distributive property

Cascade connection

Associative property

Commutative property

Slide 4


Class#4




Slide 5


LTI Systems

Slide 6


Convolution of Continuous-time Signal

Slide 7

dthx

dtHx

dtxH

txHty

tHth

)()(

)}({)(

})()({

)}({)(

)}({)(

invariance Time

Linearity

)()()( 00 txdttttx

Michael

Michael

Michael


Class#4




Slide 8


Evaluation of Convolution Integral

Reflect and shift of h() Example#1: x(t) = u(t) - u(t-3) and h(t) = u(t) - u(t-2)

Example#2: x(t) = u(t) - u(t-2) and h(t) = e-t[u(t) - u(t-

10)] (such as RC circuit)

Slide 9

- cconvdemo.m

Overlapped area with the

moving window.

Michael

The preceding example establishes a useful result for convolution with impulses: The convolution of an arbitrary signal with a time-shifted impulse simply applies the same time shift to the input signal. The analogous result holds for convolution with discrete-time impulses .

...,. Problem 2.7 Determine y(t) = e-'u(t) * {8(t + 1) - B(t) + 28(t - 2) }. Answer:

y(t) = e- (t+t>u(t + 1) - e-'u(t) + 2e-(t- 2>u(t - 2).

MichaelHighlight


Class#4




Slide 10


Parallel Connection

Slide 11

Distributive property

)}()({*)(

)}()(){(

)()()()(

)}()({*)()(*)()(*)(

)()()(

21

21

21

2121

21

ththtx

dththx

dthxdthx

ththtxthtxthtx

tytyty

dthxty )()()(


Cascade Connection

Slide 12

Associative Property

Commutative Property


Associative Properties

Slide 13

)}(*)({*)(

)(*)}(*)({)(

21

21

ththtx

ththtxty

1 2

1 2

1 2

1 2

(continued)

( ) ( ) ( )

( ) ( ) ( )

( ) ( )* ( )

( )* ( )* ( )

x h h t d d

x h h t d d

x h t h t d

x t h t h t

dthxty )()()(

1 2

1 2

1 2

1 2

( ) { ( )* ( )}* ( )

{ ( )* ( )} ( )

{ ( ) ( ) } ( )

( ) ( ) ( )

y t x t h t h t

x h h t d

x h d h t d

x h h t d d


Commutative Property

The overall system response does not depend

upon the order of the systems in cascade.

Slide 14

1 2 1 2

1 2 2 1

( )* ( ) ( ) ( )

( ) ( ) ( ) ( )t

h t h t h h t d

h t h v d h h t d

dthxty )()()(


Homework

Problem 2.33(d)(h)

Problem 2.34(a)(j)

Problem 2.38(b)

Problem 2.39(a)(b)(c)(i)

Problem 2.40(c)(m)

Problem 2.46(c)

- Due: before 2:00PM, Thursday of next week

Slide 15

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