1. Discrete-time Systems
Prof. Siripong Potisuk

2. Input-output Description
A DT system transforms DT inputs into DT outputs

3. System Interconnection
- Build more complex systems
- Modify response of a system

4. Examples of Systems
Second Difference

5. Moving Average

6. Linear Time Invariance (LTI)
A system is time-invariant if the behavior and
characteristics of the system are fixed over time.

8. Response of an LTI System

11. (Also referred to as Impulse response)

14. Properties of Convolution Sum
A discrete-time LTI system is completely characterized
by its impulse response, i.e., completely determines its
input-output behavior.
 There is only one LTI system with a given h[n]
The role of h [n] and x [n] can be interchanged
Commutative Property

15. The Distributive Property
is equivalent to

16. The Associative Property
is equivalent to

17. Causality

18. Causality for LTI Systems:
- The impulse response of a causal LTI system must be
zero before the impulse occurs.
- Causality for a linear system is equivalent to the
condition of initial rest.
Stability for LTI Systems:
A necessary and sufficient condition for an LTI system to
be BIBO stable is that the impulse response is absolutely
summable.

19. Time-domain Description of DT LTI Systems
A general Nth-order linear constant-coefficient difference
equation
Recursive equation, i.e., expresses the output at time n in
terms of previous values of the input and output

20. Solutions of LCCDE’s
- The complete solution depends on both the causal input
x[n] and the initial conditions, y[-1], y[-2],……, y[-N ].
- The solution can be decomposed into a sum of two parts:

21. Finite Impulse Response (FIR)
The equation is nonrecursive, i.e., previously computed
values of the output are not used to recursively compute
the present value of the output.
The impulse response is seen to have finite duration and
given by

22. Infinite Impulse Response (IIR)
If the system is initially at rest, the impulse response
will have infinite duration.

23. Example Solve the following difference equation

24. Example The following MATLAB routine is used to implement
a discrete-time system in terms of its 2nd order difference equation
(a and b are constants):
y1=0; y2=0; x1=0; x2=0;
for n = 1:length(x)
y(n) = x(n)+a*x1+b*x2+0.8*y1-0.16*y2;;
x2 = x1;
x1 = x(n);
y2 = y1;
y1 = y(n);
End
Determine the governing LCCDE for this system by tracing the
flow of the code in terms of changing value of n.

25. Example Given a difference equation
y[n] = x[n  1]  0.75y[n1]  0.125y[n2]
Use the MATLAB function filter() to calculate the system
response with x[n] = (0.5)nu[n] for n = 0, 1, …, 4, and
zero initial conditions.
(b) Use the MATLAB functions filter() and filtic() to calculate
the system response with x[n] = (0.5)nu[n] for n = 0, 1, …, 4,
and initial conditions x[1] = 1, y[2] = 2, and y[1] = 1.