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信号与系统英文版教学课件：ch2 Linear Time-invariant Systems.ppt

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信号与系统英文版教学课件：ch2 Linear Time-invariant Systems.ppt

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Ch2. Linear Time-invariant Systems

INTRODUCTION

•

LTI systems possesses the superposition

property.

•

Represent signals as linear combinations of

delayed impulses .

•

Convolution sum or convolution integral.

•

linear constant-coefficient difference or

differential equations.

Ch2. Linear Time-invariant Systems

1) The Representation of Discrete-time Signals

in Terms of Impulses













knkx

nxnxnx

][ ][

]2[ ]2[ ]1[ ]1[ ][ ]0[

]1[ ]1[ ]2[ ]2[ ][









If x[n]=u[n],then









][][

knnx



Sifting Property of Unit Sample:

1. Discrete-Time LTI: Convolution Sum

Ch2. Linear Time-invariant Systems

2) The Discrete-time Unit Impulse Response of

LTI Systems

LTI

x[n]=[n] y[n]=h[n]

Unit Impulse Response h[n] :response of the

LTI system to the unit sample δ[n].

δ[n] → h[n]

Why do we need it?

Ch2. Linear Time-invariant Systems

3) The Discrete-time Response of LTI Systems to

any Input Signal: Convolution Sum

LTI

x[n] y[n]=?

Solution:

Question:

[n]  h[n]

[n-k]  h[n-k]

x[k][n-k] x[k] h[n-k]













knhkxnyknkxnx ][][][][][][



The response y[n] to x[n] is the weighted linear

combination of delayed unit sample responses.

Ch2. Linear Time-invariant Systems

Convolution Sum









knhkxny ][][][

Representing the convolution operation

symbolically as: y[n] = x[n] * h[n]

--- Convolution Sum

That is, the unit impulse response --h[n] can fully

characterize an LTI system.

Summary on calculating convolution sum

Time Inversal: h[k]  h[-k]

Time Shift: h[-k]  h[n-k]

Multiplication: x[k]h[n-k]

Summing:

[ ] [ ]

x k h n k



 





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