Chengbin Ma UM-SJTU Joint Institute

Class#5

- Interconnections of LTI systems-Examples (2.6)

- Relations between LTI system properties and the impulse response (2.7)

Slide 1

Chengbin Ma UM-SJTU Joint Institute

Review of Previous Lecture (1)

Slide 2

Convolution Integral Convolution Sum

Convolution is used to represent

multiple impacts of the current/past

or future input signals (Memory).

Chengbin Ma UM-SJTU Joint Institute

Review of Previous Lecture (2)

Convolution integral:

Evaluation: reflect and shift (representation of

memory of a system)

Slide 3

dthxty )()()(

Why impulse response? (discuss in frequency domain)

1

frequency

Magnitude

Chengbin Ma UM-SJTU Joint Institute

Review of Previous Lecture (3)

Why reflect and shift the impulse response?

Slide 4

0

Past Future Present

Future Past Present

t

Impulse response

Input signal

Chengbin Ma UM-SJTU Joint Institute

Review of Previous Lecture (4)

Interconnections of LTI systems:

Slide 5

Distributive property

Associative property

Commutative property

Chengbin Ma UM-SJTU Joint Institute

This Class

Interconnections of Linear-Time-Invariant

Systems:

Review of the properties

Examples

Relations between LTI system properties and

the impulse response

Slide 6

Chengbin Ma UM-SJTU Joint Institute

Class#5

- Interconnections of LTI systems-Examples (2.6)

- Relations between LTI system properties and the impulse response (2.7)

Slide 7

Chengbin Ma UM-SJTU Joint Institute

Interconnections of LTI Systems

Slide 8

1 2 1 2

1 2 1 2

1 2 1 2

Distributive: ( )* ( ) ( )* ( ) ( )*{ ( ) ( )}

[ ]* [ ] [ ]* [ ] [ ]*{ [ ] [ ]}

Associative: { ( )* ( )}* ( ) ( )*{ ( )* ( )}

x t h t x t h t x t h t h t

x n h n x n h n x n h n h n

x t h t h t x t h t h t

1 2 1 2

1 2 2 1

1 2 2 1

{ [ ]* [ ]}* [ ] [ ]*{ [ ]* [ ]}

Communitative: ( )* ( ) ( )* ( )

[ ]* [ ] [ ]* [ ]

x n h n h n x n h n h n

h t h t h t h t

h n h n h n h n

Chengbin Ma UM-SJTU Joint Institute

Example 2.11, Page 130

Slide 9

][][ ],2[][

],[]2[][ ],[][

43

21

nunhnnh

nununhnunh

n

Chengbin Ma UM-SJTU Joint Institute

Another Example

Slide 10

Chengbin Ma UM-SJTU Joint Institute

Class#5

- Interconnections of LTI systems-Examples (2.6)

- Relations between LTI system properties and the impulse response (2.7)

Slide 11

Chengbin Ma UM-SJTU Joint Institute

Impulse Response and Properties

The input-output behavior (i.e., dynamics) of a

LTI system is completely characterized by its

impulse response.

The system properties, such as memory,

causality, and stability, are related to the

systems impulse response.

Slide 12

Chengbin Ma UM-SJTU Joint Institute

Memory

Memoryless LTI systems: For the system to be

memoryless, y[n] must depend only on x[n]

(For memoryless LTI systems, we dont need

convolution!)

Slide 13

[ ] [ ]* [ ] [ ]* [ ]

[ ] [ ]

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

, 0[ ] [ ] [ ] [ ]

0, 0

k

memoryless

y n x n h n h n x n

h k x n k

h x n h x n h x n

c ncx n h n h n c n

n

#Note: for continuous-time system, )()( tcth

dconvdemo.m

All memoryless LTI systems simply perform scalar multiplication on the input signal.

MichaelHighlight

MichaelRectangle

Chengbin Ma UM-SJTU Joint Institute

Causal LTI Systems

For the system to be causal, y[n] must depends

only on x[n], x[n-1], x[n-2],

Slide 14

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

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

][][

][*][][*][][

nnhnxhnxh

nxhnxhnxh

knxkh

nxnhnhnxny

causal

k

#Note:

1. negative n means future (consider the operation of flip and shift)!

2. continuous-time systems: 00)( tforth

past inputfuture input

Requiring the impulse response to be zero for negative time is equivalent to saying that the system cannot respond with an output prior to application of the input.

MichaelRectangle

MichaelRectangle

MichaelRectangle

Chengbin Ma UM-SJTU Joint Institute

Stable Systems

Bounded-Input-Bounded-Output system

Slide 15

][ then,)( If

)(][

][][][][

][][][

nykh

khMMkh

knxkhknxkh

knxkhny

k

kk

xx

k k

k

#Note:

1. continuous-time systems:

dtth )(

absolutely summable

absolutely integrable

Chengbin Ma UM-SJTU Joint Institute

Invertible LTI systems

The impulse response hinv[n] of the inverse

system for an LTI system with impulse

response h[n] must satisfy deconvolution, i.e.,

recovering x(t) from h(t)*x(t).

Slide 16

)()(*)(

)(*)(*)(

)(*)(*)()(

tthth

ththtx

ththtxtx

inv

inv

inv

#Note: the effect of the inverse system on noise also is an important

consideration in many problems.

Example2.13 in

Page 138

]1[]1[][]0[

][][][][][

nhxnhx

knhkxnhnxnyk

MichaelRectangle

Chengbin Ma UM-SJTU Joint Institute

Example 2.13, Page 138

Multipath communication channels:

Find its causal inverse system and check the

stability of the inverse system.

Slide 17

]1[][][ naxnxny

Chengbin Ma UM-SJTU Joint Institute

Homework

Problem 2.48

Problem 2.49(b)(c)(d)(e)

Due: 2:00PM, Thursday of next week

Slide 18