1. 2. Linear Time-Invariant Systems
2.1 Discrete-time LTI system: The convolution sum
2.1.1 The Representation of Discrete-time Signals
in Terms of Impulses
If x[n]=u[n], then

2. 2 Linear Time-Invariant Systems

3. LTI 2.1.2 The Discrete-time Unit Impulse Response
2 Linear Time-Invariant Systems
2.1.2 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]

4. x[k][n-k] x[k] h[n-k]
2 Linear Time-Invariant Systems
(2) Convolution Sum of LTI System
Question:
LTI
x[n]
y[n]=?
Solution:
[n]  h[n]
[n-k]  h[n-k]
x[k][n-k] x[k] h[n-k]

5. 2 Linear Time-Invariant Systems

6. 2 Linear Time-Invariant Systems

7. (3) Calculation of Convolution Sum
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

8. 2.2 Continuous-time LTI system: The convolution integral
2 Linear Time-Invariant Systems
2.2 Continuous-time LTI system:
The convolution integral
2.2.1 The Representation of Continuous-time
Signals in Terms of Impulses
Define
We have the expression:
Therefore:

9. 2 Linear Time-Invariant Systems

10. 2 Linear Time-Invariant Systemsor

11. LTI LTI 2.2.2 The Continuous-time Unit impulse Response
2 Linear Time-Invariant Systems
2.2.2 The Continuous-time Unit impulse Response
and the convolution Integral Representation
of LTI Systems
(1) Unit Impulse Response
LTI
x(t)=(t)
y(t)=h(t)
(2) The Convolution of LTI System
LTI
x(t)
y(t)=?

12. LTI (t) h(t) x(t) y(t)=? A. Because of So,we can get
2 Linear Time-Invariant Systems
LTI
(t)
h(t)
x(t)
y(t)=? A.
Because of
So,we can get
( Convolution Integral )
or y(t) = x(t) * h(t)

13. LTI (t) h(t) (t) h(t) B. or y(t) = x(t) * h(t)
2 Linear Time-Invariant Systems
LTI
(t)
h(t)
(t)
h(t) B.
or y(t) = x(t) * h(t)
( Convolution Integral )

14. 2 Linear Time-Invariant Systems

15. (3) Computation of Convolution Integral
2 Linear Time-Invariant Systems
(3) Computation of Convolution Integral
Time Inversal: h()  h(- )
Time Shift: h(-)  h(t- )
Multiplication: x()h(t- )
Integrating:
Example

16. 2.3 Properties of Linear Time Invariant System
2 Linear Time-Invariant Systems
2.3 Properties of Linear Time Invariant System
Convolution formula:
h(t)
x(t)
y(t)=x(t)*h(t)
h[n]
x[n]
y[n]=x[n]*h[n]

17. 2.3.1 The Commutative Property
2 Linear Time-Invariant Systems
2.3.1 The Commutative Property
Discrete time: x[n]*h[n]=h[n]*x[n]
Continuous time: x(t)*h(t)=h(t)*x(t)
h(t)
x(t)
y(t)=x(t)*h(t)
y(t)=h(t)*x(t)  

18.  2.3.2 The Distributive Property Discrete time:
2 Linear Time-Invariant Systems
2.3.2 The Distributive Property
Discrete time:
x[n]*{h1[n]+h2[n]}=x[n]*h1[n]+x[n]*h2[n]
Continuous time:
x(t)*{h1(t)+h2(t)}=x(t)*h1(t)+x(t)*h2(t)
h1(t)+h2(t)
x(t)
y(t)=x(t)*{h1(t)+h2(t)}
h1(t)
x(t)
y(t)=x(t)*h1(t)+x(t)*h2(t)
h2(t) 
Example 2.10

19. y(t)=x(t)*{h1(t)*h2(t)}
2 Linear Time-Invariant Systems
2.3.3 The Associative Property
Discrete time:
x[n]*{h1[n]*h2[n]}={x[n]*h1[n]}*h2[n]
Continuous time:
x(t)*{h1(t)*h2(t)}={x(t)*h1(t)}*h2(t)
h1(t)*h2(t)
x(t)
y(t)=x(t)*{h1(t)*h2(t)}
h1(t)
x(t)
y(t)=x(t)*h1(t)*h2(t)
h2(t)

20. 2.3.4 LTI system with and without Memory
2 Linear Time-Invariant Systems
2.3.4 LTI system with and without Memory
Memoryless system:
Discrete time: y[n]=kx[n], h[n]=k[n]
Continuous time: y(t)=kx(t), h(t)=k (t)
k (t)
x(t)
y(t)=kx(t)=x(t)*k(t)
k [n]
x[n]
y[n]=kx[n]=x[n]*k[n]
Imply that: x(t)* (t)=x(t) and x[n]* [n]=x[n]

21. 2.3.5 Invertibility of LTI system
2 Linear Time-Invariant Systems
2.3.5 Invertibility of LTI system
Original system: h(t)
Reverse system: h1(t)
h(t)
x(t)
h1(t)
(t)
x(t)
x(t)*(t)=x(t)
So, for the invertible system:
h(t)*h1(t)=(t) or h[n]*h1[n]=[n]
Example

22. 2.3.6 Causality for LTI system
2 Linear Time-Invariant Systems
2.3.6 Causality for LTI system
Discrete time system satisfy the condition:
h[n]=0 for n<0
Continuous time system satisfy the condition:
h(t)=0 for t<0

23. 2.3.7 Stability for LTI system
2 Linear Time-Invariant Systems
2.3.7 Stability for LTI system
Definition of stability:
Every bounded input produces a bounded
output.
Discrete time system:
If |x[n]|<B, the condition for |y[n]|<A is

24. Continuous time system:
2 Linear Time-Invariant Systems
Continuous time system:
If |x(t)|<B, the condition for |y(t)|<A is
Example 2.13

25. 2.3.8 The Unit Step Response of LTI system
2 Linear Time-Invariant Systems
2.3.8 The Unit Step Response of LTI system
Discrete time system:
h[n]
[n]
u[n]
s[n]=u[n]*h[n]
Continuous time system:
h(t)
(t)
u(t)
s(t)=u(t)*h(t)

26. 2.4 Causal LTI Systems Described by
2 Linear Time-Invariant Systems
2.4 Causal LTI Systems Described by
Differential and Difference Equation
Discrete time system: Differential Equation
Continuous time system: Difference Equation

27. 2.4.1 Linear Constant-Coefficient Differential Equation
2 Linear Time-Invariant Systems
2.4.1 Linear Constant-Coefficient Differential Equation
A general Nth-order linear constant-coefficient
differential equation: or
and initial condition:
y(t0), y’(t0), …… , y(N-1)(t0) ( N values )

28. 2.4.2 Linear Constant-Coefficient Difference Equation
2 Linear Time-Invariant Systems
2.4.2 Linear Constant-Coefficient Difference Equation
A general Nth-order linear constant-coefficient
difference equation: or
and initial condition:
y[0], y[-1], …… , y[-(N-1)] ( N values )
Example 2.15

29. 2.4.3 Block Diagram Representations of First-order
2 Linear Time-Invariant Systems
2.4.3 Block Diagram Representations of First-order
Systems Described by Differential and Difference
Equation
(1) Dicrete time system
Basic elements:
A. An adder
B. Multiplication by a coefficient
C. An unit delay

30. 2 Linear Time-Invariant Systems
Basic elements:

31. Example: y[n]+ay[n-1]=bx[n]
2 Linear Time-Invariant Systems
Example: y[n]+ay[n-1]=bx[n]

32. (2) Continuous time system Basic elements: A. An adder
2 Linear Time-Invariant Systems
(2) Continuous time system
Basic elements:
A. An adder
B. Multiplication by a coefficient
C. An (differentiator) integrator

33. 2 Linear Time-Invariant Systems
Basic elements:

34. Example: y’(t)+ay(t)=bx(t)
2 Linear Time-Invariant Systems
Example: y’(t)+ay(t)=bx(t)

35. 2.5 Singularity Functions
2 Linear Time-Invariant Systems
2.5 Singularity Functions
The unit impulse as idealized short pulse
(1)
(2)

36. Several important formula:
2 Linear Time-Invariant Systems
Several important formula:
Problems: