1. Properties of LTI Systems
Commutative Property
x[n]*h[n]=h[n]*x[n]
Distributive Property
x[n]*(h1[n]+ h2[n])= x[n]*h1[n]+x[n]* h2[n]
Associative Property
x[n]*(h1[n]* h2[n])= (x[n]*h1[n])* h2[n]
Memoryless
If h[n]=0 for n not equal 0. I.e. h[n]=Kd[n].

2. Properties of LTI Systems
Invertibility
w(t)=x(t)
x(t)
h1(t)
h(t)
y(t)
x(t)
Identity System
d(t)

3. Properties of LTI Systems
Causality
h[n]=0 for n<0
or h(t)=0 for t<0.
Stability

12. Unit Step Response of LTI System
s[n]
h[n]
The step response of a discrete-time LTI system is the convolution of the
unit step with the impulse response:-
s[n]=u[n]*h[n].
Via commutative property of convolution, s[n]=h[n]*u[n].
That means s[n] is the response to the input h[n] of a discrete-time LTI
system with unit impulse response u[n].
h[n]
s[n]
u[n]

13. Using the convolution sum:-

14. Unit Step Response of Continuous-time LTI System
Similarly, unit step response is the running integral of its impulse response.
The unit impulse response is the first
derivative of the unit step response:-

15. Causal LTI Systems Described By Differential & Difference Equations

16. Example 2.14

19. Example 2.14 with impulse input (Problem 2.56 (a) Pg 158-159 OWN)

22. General Higher N-order DE

23. Linear Constant-Coefficient Difference equations

24. Linear Constant-Coefficient Difference equations

25. Recursive case when N > or = 1.

26. Block Diagram Representations of First- Order Systems.
Provides a pictorial representation which can add to our understanding of the behavior and properties of these systems.
Simulation or implementation of the systems.
Basis for analog computer simulation of systems described by DE.
Digital simulation & Digital Hardware implementations

27. First-Order Recursive Discrete-time System.
y[n]+ay[n-1]=bx[n]
y[n]=-ay[n-1]+bx[n]
y[n] b +
x[n] D -a
y[n-1]

28. First-Order Continuous-time System Described By Differential Equation
y(t)
b/a +
x(t)
Difficult to
implement, sensitive
to errors and noise. D
-1/a

29. First-Order Continuous-time System Described By Differential Equation Alternative Block Diagram.
y(t) b +
x(t) -a