Signals and Systems - Electrical Engineering

478 C H A P T E R 8: Discrete-Time Signals and Systems

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FIGURE 8.7

(a) Discrete-time signal, and (b) even and (c) odd components.

8.3 Discrete-Time Systems

Just as with continuous-time systems, a discrete-time system is a transformation of a discrete-time

input signalx[n] into a discrete-time output signaly[n]—that is,

y[n]=S{x[n]} (8.26)

Just as we were when we studied the continuous-time systems, we are interested in dynamic systems

S{.}having the following properties:

n Linearity

n Time invariance

n Stability

n Causality

A discrete-time systemSis said to be

n Linear:If for inputsx[n]andv[n]and constantsaandb, it satisfies the following

n Scaling:S{ax[n]}=aS{x[n]}

n Additivity:S{x[n]+v[n]}=S{x[n]}+S{v[n]}

or equivalently ifsuperpositionapplies—that is,

S{ax[n]+bv[n]}=aS{x[n]}+bS{v[n]} (8.27)

n Time-invariant:If for an inputx[n]with a corresponding outputy[n]=S{x[n]}, the output corresponding

to a delayed or advanced version ofx[n],x[n±M], isy[n±M]=S{x[n±M]}for an integerM.