Unit4_LTI_TimeDomain - EE3210 Signals and Systems Unit 4...

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EE3210 Signals and Systems Unit 4 Linear Time-Invariant (LTI) Systems (Time-Domain Analysis)
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Unit 4: LTI Systems (Time Domain) 2 Outline 1. Unit Impulse Function and Unit Step Function 2. Convolution Sum for Discrete-Time Systems 3. Convolution Integral for Continuous-Time Systems 4. Properties of Convolution and Characterization of LTI Systems 5. Systems described by Differential & Difference Equations
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Unit 4.1 Unit Impulse Function and Unit Step Function After studying this unit, you will be able to 1. describe the impulse function and the step functions. 2. describe their properties.
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Unit 4: LTI Systems (Time Domain) 4 Two Useful Signal Models Two useful signal models: 1. Unit Step Function u(t) 2. Unit Impulse Function δ (t) Why are they useful? 1. They can be used to represent other signals. 2. They can simplify the analysis of many systems.
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Unit 4: LTI Systems (Time Domain) 5 1. Unit Step Function Continuous-Time: u(t) = 0, t<0 1, t>0 Note : u(t) is discontinuous at t=0, and u(0) is not defined. Note : Sometimes, u(0) is defined as 1, or defined as 0.5. Discrete-Time: u[n] = 0, n<0 1, n 0 1 t u(t) u[ n ] n -3 -2 -1 0 1 2 3 1 ... ...
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Unit 4: LTI Systems (Time Domain) 6 Example How to represent a one-sided signal using u(t)? Example: Answer: 1 t 2 π y ( t )
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Unit 4: LTI Systems (Time Domain) 7 Example How to represent a rectangular pulse using u(t)? Example: Answer: 2 t x ( t ) 37
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Unit 4: LTI Systems (Time Domain) 8 2. Unit Impulse Function Continuous-Time: Discrete-Time: δ[ n ] = 0, n 0 1, n=0 δ [ n ] n -3 -2 -1 0 1 2 3 1 ... ... 1 t δ (t) = = 1 ) ( 0 0 ) ( dt t t t δ
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Unit 4: LTI Systems (Time Domain) 9 Example How to represent a discrete-time signal by δ[ n ] ? Answer: x [ n ] n -3 -2 -1 0 1 2 3 4 5 2 3
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Unit 4: LTI Systems (Time Domain) 10 Representation of D.T. Signals (…, 2, 2, -1, 3, 1, -3, 2,…) 0 ... | | 0 + x [-1] δ [ n +1] 0 + x [0] [ n ] 0 + x [2] [ n -2] + ... 0 x [-2] [ n +2] ... 0 x [1] [ n -1] +
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Unit 4: LTI Systems (Time Domain) 11 Representation of D.T. Signals In general, we have This is called the sifting property of δ [n]. −∞ = = + + + + + + + + = k k n k x n x n x n x n x n x n x ] [ ] [ ] 2 [ ] 2 [ ] 1 [ ] 1 [ ] [ ] 0 [ ] 1 [ ] 1 [ ] 2 [ ] 2 [ ] [ δ L L
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Unit 4: LTI Systems (Time Domain) 12 Representation of C.T. Signals How about the continuous case? Recall that What does the integral mean? 1 t δ (t) = = 1 ) ( 0 0 ) ( dt t t t δ
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Unit 4: LTI Systems (Time Domain) 13 Impulse Approximation A continuous-time unit impulse can be visualized as a tall, narrow, rectangular pulse of unit area: 1/ Δ t δ Δ (t) Δ Area = 1 1 t δ (t)
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Unit 4: LTI Systems (Time Domain) 14 Example What is k δ (t) ? The function k δ (t) is an impulse whose area is k. k t k δ (t) Area = k
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Unit 4: LTI Systems (Time Domain) 15 Properties of Unit Impulse We consider two of its properties: 1. Being the derivative of unit step function 2. Sampling/Sifting property
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Unit 4: LTI Systems (Time Domain) 16 i) Derivative of Step Function 1 t u Δ (t) Δ 1/ Δ t δ Δ (t) Δ δ Δ (t) = du Δ (t) / dt δ (t) = lim δ Δ (t) Δ 0 1 t δ (t) u(t) = lim u Δ (t) Δ 0 1 t u(t)
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Unit 4: LTI Systems (Time Domain) 17 ii) Sampling Property Consider a function φ (t), which is continuous at t=0.
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This note was uploaded on 11/22/2009 for the course ELECTRONIC EE3210 taught by Professor Sungchiwan during the Spring '09 term at École Normale Supérieure.

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Unit4_LTI_TimeDomain - EE3210 Signals and Systems Unit 4...

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