SUWA D2H2 E-medicine

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

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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

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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

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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.)

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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

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t – τ ≥ 0, or τ ≤ t


Property: Stability

26


Differentiator

27


Triplets and beyond!28

𝑑𝑑𝑑𝑑𝑡𝑡



…𝑥𝑥(𝑡𝑡)𝑑𝑑𝑛𝑛𝑥𝑥 𝑡𝑡𝑑𝑑𝑡𝑡𝑛𝑛


Unit Step Response

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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


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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]

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