ch2 Linear Time-Invariant Systems.ppt

2 Linear Time-Invariant Systems
Discrete-time LTI system: The convolution sum
The Representation of Discrete-time Signals
in Terms of Impulses
2. Linear Time-Invariant Systems
If x[n]=u[n], then
2 Linear Time-Invariant Systems
2 Linear Time-Invariant Systems
The Discrete-time Unit Impulse Response
and the Convolution Sum Representation
of LTI Systems
(1) Unit Impulse(Sample) Response
LTI
x[n]=[n]
y[n]=h[n]
Unit Impulse Response: h[n]
2 Linear Time-Invariant Systems
(2) Convolution Sum of LTI System
LTI
x[n]
y[n]=?
Solution:
Question:
[n]  h[n]
[n-k]  h[n-k]
x[k][n-k] x[k] h[n-k]
2 Linear Time-Invariant Systems
( Convolution Sum )
So
or y[n] = x[n] * h[n]
(3) Calculation of Convolution Sum
Time Inversal: h[k]  h[-k]
Time Shift: h[-k]  h[n-k]
Multiplication: x[k]h[n-k]
Summing:
Example
2 Linear Time-Invariant Systems
Continuous-time LTI system:
The convolution integral
The Representation of Continuous-time
Signals in Terms of Impulses
Define
We have the expression:
Therefore:
2 Linear Time-Invariant Systems
or