Signals and Systems - Electrical Engineering

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

situations, real-time processing is not required as the data can be stored and processed without the

requirements of real time. Under such circumstances causality is not necessary.

A discrete-time systemSiscausalif:

n Whenever the inputx[n]= 0 , and there are no initial conditions, the output isy[n]= 0.

n The outputy[n]does not depend on future inputs.

Causality is independent of the linearity and time-invariance properties of a system. For instance, the

system represented by the input–output equation

y[n]=x^2 [n]

wherex[n] is the input andy[n] is the output is nonlinear but time invariant. According to the above

definition it is a causal system: The output is zero whenever the input is zero, and the output depends

on the present value of the input. Likewise, an LTI system can be noncausal, as can be seen in the

following discrete-time system that computes the moving average of the input:

y[n]=

1

3

(x[n+1]+x[n]+x[n−1]).

The input–output equation indicates that at the present timento computey[n] we need a present

valuex[n], a past valuex[n−1], and a future valuex[n+1]. Thus, the system is LTI but noncausal

since it requires future values of the input.

n An LTI discrete-time system iscausalif the impulse response of the system is such that

h[n]= 0 n< 0 (8.36)

n A signalx[n]is said to becausalif

x[n]= 0 n< 0 (8.37)

n For a causal LTI discrete-time system with a causal inputx[n]its outputy[n]is given by

y[n]=

∑n

k= 0

x[k]h[n−k] n≥ 0 (8.38)

where the lower limit of the sum depends on the input causality,x[k]= 0 fork< 0 , and the upper limit

depends on the causality of the system,h[n−k]= 0 forn−k< 0 ork>n.

Thath[n]=0 forn<0 is the condition for an LTI discrete-time system to be causal is understood by

considering that when computing the impulse response, the inputδ[n] only occurs atn=0 and there

are no initial conditions, so the response forn<0 should be zero. Extending the notion of causality