Discrete-time Systems Prof. Siripong Potisuk

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

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

Examples of Systems Second Difference

Moving Average

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

Response of an LTI System

(Also referred to as Impulse response)

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

The Distributive Property is equivalent to

The Associative Property is equivalent to

Causality

- 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. Causality for LTI Systems:

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

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:

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

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

Example Consider the difference equation