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Linear Time-Invariant Systems Signals and SystemsLinear Time-Invariant Systems Signals and Systems

Notes Assignment

2.102.112.22 (b) (e)2.252.28 (a) (c) (e) (g)

Tutorial questions this week (Week 4)Basic Problems with Answers 2.20Basic Problems 2.29Advanced Problems 2.40, 2.43, 2.47

1


2Chapter 2 ReviewThat is ...


Unit Impulse Response

3Chapter 2 Review


Response of DT LTI Systems4

impulse

Chapter 2 Review


5

∑∞

−∞=

−=k

knhkxny ][][][

Contribution to the output signal at time n

input signal

flipped version of h[k] located at k = n

Chapter 2 Review

F S M S


Construction of the Unit-impulse function δ(t)

6Chapter 1 Review

Area=1𝛿𝛿 𝑡𝑡 = lim

Δ→0𝛿𝛿Δ(𝑡𝑡)


Representation of CT Signals

7

𝑥𝑥 𝑡𝑡 = limΔ→0

�𝑥𝑥(𝑡𝑡)


8

𝑥𝑥 −2Δ 𝛿𝛿Δ 𝑡𝑡 + 2Δ Δ

𝑥𝑥 −Δ 𝛿𝛿Δ 𝑡𝑡 + Δ Δ


Representation of CT Signals (cont.)9

?𝑥𝑥 𝑘𝑘Δ 𝛿𝛿Δ 𝑡𝑡 − 𝑘𝑘Δ Δ


10

Input: 𝑥𝑥 𝑛𝑛 = ∑𝑘𝑘=−∞+∞ 𝑥𝑥 𝑘𝑘 𝛿𝛿[𝑛𝑛 − 𝑘𝑘] 𝑥𝑥 𝑡𝑡 = ∫−∞+∞𝑥𝑥 𝜏𝜏 𝛿𝛿 𝑡𝑡 − 𝜏𝜏 𝑑𝑑𝜏𝜏

LTI: 𝛿𝛿 𝑛𝑛 → ℎ[𝑛𝑛]

Output: y 𝑛𝑛 = ∑𝑘𝑘=−∞+∞ 𝑥𝑥 𝑘𝑘 ℎ[𝑛𝑛 − 𝑘𝑘]

How to calculate the output of CT LTI systems?


Unit Impulse Response

11


Response of a CT LTI System12


Response of a CT LTI System

x(t) y(t)CT LTI

13


14

Input: 𝑥𝑥 𝑛𝑛 = ∑𝑘𝑘=−∞+∞ 𝑥𝑥 𝑘𝑘 𝛿𝛿[𝑛𝑛 − 𝑘𝑘] 𝑥𝑥 𝑡𝑡 = ∫−∞+∞𝑥𝑥 𝜏𝜏 𝛿𝛿 𝑡𝑡 − 𝜏𝜏 𝑑𝑑𝜏𝜏

LTI: 𝛿𝛿 𝑛𝑛 → ℎ[𝑛𝑛] 𝛿𝛿 𝑡𝑡 → ℎ(𝑡𝑡)

Output: y 𝑛𝑛 = ∑𝑘𝑘=−∞+∞ 𝑥𝑥 𝑘𝑘 ℎ[𝑛𝑛 − 𝑘𝑘] y 𝑡𝑡 = ∫−∞+∞𝑥𝑥 𝜏𝜏 ℎ 𝑡𝑡 − 𝜏𝜏 𝑑𝑑𝜏𝜏

Summary

= 𝑥𝑥 𝑛𝑛 ∗ ℎ 𝑛𝑛 = 𝑥𝑥 𝑡𝑡 ∗ ℎ(𝑡𝑡)


0−2𝑇𝑇h(- 𝜏𝜏)

15h(𝜏𝜏)

𝑦𝑦 𝑡𝑡 = 𝑥𝑥 𝑡𝑡 ∗ ℎ 𝑡𝑡 = �−∞

+∞

𝑥𝑥 𝜏𝜏 ℎ 𝑡𝑡 − 𝜏𝜏 𝑑𝑑𝜏𝜏

2T

𝜏𝜏2T


Flip, slide, multiply, and integrate

16


17

One Important Convolution𝑥𝑥 𝑡𝑡 ∗ 𝛿𝛿 𝑡𝑡 − 𝑎𝑎

= �−∞

+∞𝑥𝑥 𝜏𝜏 𝛿𝛿 𝑡𝑡 − 𝜏𝜏 − 𝑎𝑎 𝑑𝑑𝜏𝜏

=∫−∞+∞𝑥𝑥 𝑡𝑡 − 𝑎𝑎 𝛿𝛿 𝑡𝑡 − 𝜏𝜏 − 𝑎𝑎 𝑑𝑑𝜏𝜏

= 𝑥𝑥 𝑡𝑡 − 𝑎𝑎 �−∞

+∞𝛿𝛿 𝑡𝑡 − 𝜏𝜏 − 𝑎𝑎 𝑑𝑑𝜏𝜏

= 𝑥𝑥(𝑡𝑡 − 𝑎𝑎)

Similarly, 𝑥𝑥 𝑛𝑛 ∗ 𝛿𝛿 𝑛𝑛 − 𝑘𝑘 = 𝑥𝑥[𝑛𝑛 − 𝑘𝑘]


Property: Commutative

The role of input signal and unit impulse response is interchangeable, giving the same output signal

18

h(t)x(t)

x(t)h(t)


Property: Distributive

19


Property: Distributive (Cont.)

20

1 2 1 2[ ( ) ( )] ( ) ( ) ( ) ( ) ( )x t x t h t x t h t x t h t+ ∗ = ∗ + ∗

Problem 2.22 (b)( ) ( ) 2 ( 2) ( 5)x t u t u t u t= − − + −

2( ) (1 )th t e u t= −

0 2 5

x(t)


Properties: Associative21


Properties: Associative (Cont.)

The order in which non-linear systems are cascaded cannot be changed.

e.g.

22

y=2x y=x2

y=x2 y=2x


Property: Memory/Memoryless

23


Property: Invertibility

24


Property: Causality

This is because that the input unit impulse function δ(t)=0 at t<0

25

t – τ ≥ 0, or τ ≤ t


Property: Stability

26


Differentiator

27


Triplets and beyond!28

𝑑𝑑𝑑𝑑𝑡𝑡



…𝑥𝑥(𝑡𝑡)𝑑𝑑𝑛𝑛𝑥𝑥 𝑡𝑡𝑑𝑑𝑡𝑡𝑛𝑛


Unit Step Response

29

unit step function unit step response

The relation between unit step function and unit impulse function( )( ) '( )( )

ds th t s td t

= =

h(t)

h(t)

x(t)=δ(t)

x(t)=u(t)

h(t): unit impulse response

s(t): unit step response


Integrators30


Integrators (Cont.)

31


Notation32


33

Problem 2.391( ) ( ) / 4 ( )2

y t dy t dt x t= − +

( ) / 3 ( ) ( )dy t dt y t x t+ =

Block diagram representation - CT

+

d/dt

𝑥𝑥1(𝑡𝑡)𝑥𝑥2(𝑡𝑡)

𝑥𝑥1 𝑡𝑡 + 𝑥𝑥2(𝑡𝑡)

𝑥𝑥1(𝑡𝑡) 𝑎𝑎𝑥𝑥1(𝑡𝑡)𝑎𝑎

𝑥𝑥1(𝑡𝑡)𝑑𝑑𝑥𝑥1 𝑡𝑡𝑑𝑑𝑡𝑡


Block diagram representation - DT

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Problem 2.381 1[ ] [ 1] [ ]3 2

y n y n x n= − +

1[ ] [ 1] [ 1]3

y n y n x n= − + −

+

D

𝑥𝑥1[𝑛𝑛]𝑥𝑥2[𝑛𝑛]

𝑥𝑥1[𝑛𝑛] + 𝑥𝑥2[𝑛𝑛]

𝑥𝑥1[𝑛𝑛] 𝑎𝑎𝑥𝑥1[𝑛𝑛]𝑎𝑎

𝑥𝑥1[𝑛𝑛] 𝑥𝑥1 𝑛𝑛 − 1


From block diagram to difference equation

35

x[n] y[n]