Signal and Systems Chapter 2: LTI Systems Representation of DT - - PowerPoint PPT Presentation

Signal and Systems

Chapter 2: LTI Systems

1)

Representation of DT signals in terms of shifted unit samples System properties and examples

2)

Convolution sum representation of DT LTI systems

3)

Examples

4)

The unit sample response and properties of DT LTI systems

5)

Representation of CT Signals in terms of shifted unit impulses

6)

Convolution integral representation of CT LTI systems

7)

Properties and Examples

8)

The unit impulse as an idealized pulse that is “short enough”: The operational definition of δ(t)

Exploiting Superposition and Time- Invariance

 𝑦[𝑜] =

𝑙

𝑏𝑙𝑦𝑙[𝑜]

𝑀𝑗𝑜𝑓𝑏𝑠𝑇𝑧𝑡𝑢𝑓𝑛 𝑧[𝑜] = 𝑙

] 𝑏𝑙𝑧𝑙[𝑜

 Question: Are there sets of “basic” signals so that:

 We can represent rich classes of signals as linear

combinations of these building block signals.

 The response of LTI Systems to these basic signals are both

simple and insightful.

 Fact: For LTI Systems (CT or DT) there are two natural

choices for these building blocks

 Focus for now: DT Shifted unit samples

CT Shifted unit impulses

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 2

Representation of DT Signals Using Unit Samples

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 3

 That is …

𝑦[𝑜] =. . . +𝑦[−2]𝜀[𝑜 + 2] + 𝑦[−1]𝜀[𝑜 + 1] + 𝑦[0]𝜀[𝑜] + 𝑦[1]𝜀[𝑜 − 1]+. . . => 𝑦[𝑜] =

𝑙=−∞ ∞

] 𝑦[𝑙]𝜀[𝑜 − 𝑙 The Shifting Property of the Unit Sample

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 4

Coefficients Basic Signals

 Suppose the system is linear, and define

] ℎ𝑙[𝑜 as the response to ] 𝜀[𝑜 − 𝑙 : ] 𝜀[𝑜 − 𝑙] → ℎ𝑙[𝑜

 From superposition:

𝑦[𝑜] =

𝑙→−∞ ∞

𝑦[𝑙]𝜀[𝑜 − 𝑙] → 𝑧[𝑜] =

𝑙→−∞ ∞

] 𝑦[𝑙]ℎ𝑙[𝑜

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 5

 Now suppose the system is LTI, and define the unit

sample response ] ℎ[𝑜 : ] 𝜀[𝑜] → ℎ[𝑜 From TI: ] 𝜀[𝑜 − 𝑙] → ℎ[𝑜 − 𝑙 From LTI: 𝑦[𝑜] =

𝑙→−∞ ∞

𝑦[𝑙]𝜀[𝑜 − 𝑙] → 𝑧[𝑜] =

𝑙→−∞ ∞

] 𝑦[𝑙]ℎ[𝑜 − 𝑙 convolution sum

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 6

Convolution Sum Representation of Response of LTI Systems

𝑧[𝑜] = 𝑦[𝑜] ∗ ℎ[𝑜] =

𝑙→−∞ ∞

] 𝑦[𝑙]ℎ[𝑜 − 𝑙 Interpretation:

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 7

Visualizing the calculation of ] 𝑧[𝑜] = 𝑦[𝑜] ∗ ℎ[𝑜

 Choose value of n and consider it fixed

𝑧[𝑜] =

𝑙→−∞ ∞

] 𝑦[𝑙]ℎ[𝑜 − 𝑙

View as functions of k with n fixed

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 8

prod of

• verlap for

prod of

• verlap for

Calculating Successive Values: Shift, Multiply, Sum

𝑧[𝑜] = 0 𝑜 < −1 𝑧[−1] = 1 × 1 = 1 𝑧[0] = 0 × 1 + 1 × 2 = 2 𝑧[1] = (−1) × 1 + 0 × 2 + 1 × (−1) = −2 𝑧[2] = (−1) × 2 + 0 × (−1) + 1 × (−1) = −3 𝑧[3] = (−1) × (−1) + 0 × (−1) = 1 𝑧[4] = (−1) × (−1) = 1 𝑧[𝑜] = 0 𝑜 > 4

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 9

Properties of Convolution and DT LTI Systems

 A DT LTI System is completely characterized by its unit

sample response ]

• Ex. 1: ℎ[𝑜] = 𝜀[𝑜 − 𝑜0

There are many systems with this response to ] 𝜀[𝑜 There is only one LTI System with this response to ] 𝜀[𝑜 ] 𝑧[𝑜] = 𝑦[𝑜 − 𝑜0] ⇒ 𝑦[𝑜] ∗ 𝜀[𝑜 − 𝑜0] = 𝑦[𝑜 − 𝑜0

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 10

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 11

• An Accumulator

Unit Sample response

[ ] [ ] [ ]

n k

h n k u n 



 



[ ]* [ ] [ ]

n k

x n u n x k



 

Example 2:

𝑧 𝑜 =

𝑙=−∞ 𝑜

𝑦[𝑙]

The Commutativity Property

] 𝑧[𝑜] = 𝑦[𝑜] ∗ ℎ[𝑜] = ℎ[𝑜] ∗ 𝑦[𝑜 Ex: Step response ] 𝑡[𝑜 of an LTI system ] 𝑡[𝑜] = 𝑣[𝑜] ∗ ℎ[𝑜] = ℎ[𝑜] ∗ 𝑣[𝑜 ⇒ 𝑡[𝑜] =

𝑙→−∞ 𝑜

] ℎ[𝑙

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 12

The Distributivity Property

] 𝑦[𝑜] ∗ (ℎ1[𝑜] + ℎ2[𝑜]) = 𝑦[𝑜] ∗ ℎ1[𝑜] + 𝑦[𝑜] ∗ ℎ2[𝑜 Interpretation:

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 13

The Associativity Property

] 𝑦[𝑜] ∗ (ℎ1[𝑜] ∗ ℎ2[𝑜]) = (𝑦[𝑜] ∗ ℎ1[𝑜]) ∗ ℎ2[𝑜 ⇕ Commutativity ] 𝑦[𝑜] ∗ (ℎ2[𝑜] ∗ ℎ1[𝑜]) = (𝑦[𝑜] ∗ ℎ2[𝑜]) ∗ ℎ1[𝑜

 Implication (Very special to LTI Systems):

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 14

Properties of LTI Systems

Causality [ ] h n  

 n

a) Sufficient condition: Causality ⇒ ℎ 𝑜 = 0, 𝑜 < 0 b) Necessity: Proof If h[n]=0 for n<0, 𝑧 𝑜 =

𝑙=−∞ ∞

𝑦 𝑙 ℎ[𝑜 − 𝑙] Which is equivalent to: Meaning that the output at n depends only on previous inputs

[ ] [ ] [ ]

k

y n h k x n k

 

 



Properties of LTI Systems

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 16

Stability | [ ]|

k

h k

 

  



a) sufficiency: If 𝑦 𝑜 < 𝐶 𝑔𝑝𝑠 𝑏𝑚𝑚 𝑜 𝑧 𝑜 = |

𝑙=−∞ +∞

ℎ 𝑙 𝑦 𝑜 − 𝑙 | 𝑧 𝑜 ≤

𝑙=−∞ +∞

ℎ 𝑙 |𝑦 𝑜 − 𝑙 | 𝑧 𝑜 ≤ 𝐶

𝑙=−∞ +∞

ℎ 𝑙 𝑔𝑝𝑠 𝑏𝑚𝑚 𝑜 So we can conclude that if the impulse response is absolutely summable, that is, if:

𝑙=−∞ +∞

ℎ 𝑙 < ∞ Then, y[n] is bounded and hence, the system is stable.

 b) necessity:  Assume we have a stable system.  Suppose the input to the system is:

𝑦 𝑜 = 0, 𝑗𝑔 ℎ −𝑜 = 0 ℎ[−𝑜] |ℎ −𝑜 | 𝑗𝑔 ℎ[−𝑜] ≠ 0 This is a bounded input, 𝑦 𝑜 < 1 𝑔𝑝𝑠 𝑏𝑚𝑚 𝑜 The output at n=0 is: 𝑧 0 =

𝑙=−∞ ∞

𝑦 𝑙 ℎ[−𝑙] 𝑧 0 = 𝑙=−∞

∞ ℎ2[−𝑙] |ℎ −𝑙 | = 𝑙=−∞ ∞ ℎ2[𝑙] |ℎ 𝑙 | = 𝑙=−∞ ∞

|ℎ 𝑙 | < ∞ since the system is assumed to be stable.

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 17

Properties of LTI Systems

Representation of CT Signals

Book Chapter#2: Section# Computer Engineering Department, Signal and Systems 18

 Approximate any input x(t) as a sum of shifted, scaled

pulses

^

( ) ( ) ( 1) x t x k k t k       

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 19

has unit area

) (t





    



) ( ) ( k t k x 

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 20

The Shifting Property of the Unit Impulse

( ) ( ) ( )

k

x t x k t k 

  

    



limit as

 

( ) ( ) ( ) x t x t d    

 

 



Response of CT LTI system

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 21

Impulse response : Taking limits

) ( ) ( t h t

 

 

 

       

          

k k

k t h k x t y k t k x t x ) ( ) ( ) ( ) ( ) ( ) ( 

) ( ) ( t h t  

 

Convolution Integral

       d t h x t y d t x t x ) ( ) ( ) ( ) ( ) ( ) (     

 

     

Operation of CT Convolution

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 22



  

      d t h x t h t x t y ) ( ) ( ) ( * ) ( ) (

( ) h 

Flip

( ) h   ( ) h t  

Slide

( ) ( ) x h t   

Multiply Integrate

( ) ( ) x h t d   

 





PROPERTIES AND EXAMPLES Commutativity Shifting property Example: An integrator Step response:

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 23

( )* ( ) ( )* ( ) x t h t h t x t 

) ( ) ( * ) ( ), ( ) ( * ) ( t x t t x t t x t t t x      

So if input

 





t

d x t y   ) ( ) (

) ( ) ( t t x  

) ( ) ( t h t y 

) ( ) ( ) ( t u d t h

t

   



  

• utput

 



  

t

d x t u t x t h t x t y   ) ( ) ( * ) ( ) ( * ) ( ) (

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

t

s t u t h t h t u t h d  



   

DISTRIBUTIVITY

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 24

)] ( ) ( [ * ) ( ) (

2 1

t h t h t x t y  

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

2 1

t h t x t h t x t y  

ASSOCIATIVITY

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 25

) ( * ] ) ( * ) ( [ ) (

2 1

t h t h t x t y  )] ( * ) ( [ * ) ( ) (

2 1

t h t h t x t y  )] ( * ) ( [ * ) ( ) (

1 2

t h t h t x t y  ) ( * ] ) ( * ) ( [ ) (

1 2

t h t h t x t y 

Causality and Stability

Computer Engineering Department, Signal and Systems Book Chapter#: Section# 26



  

     d h | ) ( | , ) (    t t h

The impulse as an idealized “short” pulse

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 27

Consider response from initial rest to pulses of different shapes and durations, but with unit area. As the duration decreases, the responses become similar for different pulse shapes.

) ( 1 ) ( 1 ) ( t x RC t y RC dt t dy  

The Operational Definition of the Unit Impulse δ(t)

 δ(t) —idealization of a unit-area pulse that is so short that,

for any physical systems of interest to us, the system responds only to the area of the pulse and is insensitive to its duration

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 28

Operationally: The unit impulse is the signal which when applied to any LTI system results in an output equal to the impulse response of the system. That is,

δ(t) is defined by what it does under convolution.

) ( ) ( * ) ( t h t h t  

for all h(t)

The Unit Doublet —Differentiator

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 29

Impulse response = unit doublet

The operational definition of the unit doublet:

dt t dx t y ) ( ) ( 

dt t d t u ) ( ) (

1

 

dt t dx t u t x ) ( ) ( * ) (

1



Triplets and beyond!

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 30

 n

n is number of differentiations

) ( * ... * ) ( ) (

1 1

t u t u t un 

n times

) ( ) ( ) ( * ) (   n dt t x d t u t x

n n n

Operational definitions

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 31

“-1 derivatives" = integral ⇒I.R.= unit step

Impulse response:

Operational definition: Cascade of n integrators :

) ( ) (

1

t u t u 



 

 



t

d x t u t x   ) ( ) ( * ) (

1

) ( ), ( * ... * ) ( ) (

1 1

 

  

n t u t u t u n

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 32

Integrators (continued)

the unit ramp

 

     

 

t t

d u d u t u     ) ( ) ( ) (

1 2

) ( . t u t 

) ( )! 1 ( ) (

1

t u n t t u

n n

 

 

More generally, for n>0

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 33

Then E.g. Define

) ( ) ( t t u   ) ( ) ( * ) ( t u t u t u

m n m n 



) ( ) ( * ) (

1 1

t u t u t u 



) ( ) ( t dt t du        

n and m can be positive or negative

Sometimes Useful Tricks

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 34

Differentiate first, then convolve, then integrate

) ( * ) ( * ) ( ) ( * ) ( t h t t x t h t x   ) ( * ) ( * ) ( * ) (

1 1

t h t u t u t x



 ) ( * )} ( * )] ( * ) ( {[

1 1

t u t h t u t x





Example

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 35

) 2 ( ) 1 ( 2 ) 1 ( ) (       t t t dt t dx   

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 36

Example (continued)

) 2 ( ) 1 ( 2 ) 1 ( ) ( * ) (       t h t h t h t h dt t dx

    d h d dx t h t x

t

 



       ) ( * ) ( ) ( * ) (