2.3 LTI Continuous-Time Systems 143
where we used the sifting property of the impulse and that its area is unity. If we letT=T 0 and
let the unit step beu( 0 )=0.5, the signalyT 0 (t)=1 for−∞<t<∞. WhenT= 2 T 0 , the signal
y 2 T 0 (t)is a periodic train of rectangular pulses of period 2T 0. See Figure 2.10.
y 2 T 0 (t)
−T 0 T 0 2 T 0 3 T 0
t
−T 0 T 0 2 T 0 3 T 0
t
yT 0 (t)
1
1
FIGURE 2.10
Convolution with a sequence of unit impulses as input. Notice the result is the superposition of the input signal
shifted by the time-shift kT of the impulses. ForT=T 0 ,yT(t)= 1 and forT= 2 T 0 is a sequence of pulses. n
2.3.6 Causality
Causality relates to the conditions under which processing of a signal can be performed in real time—
when it is necessary to process the signal as it comes into the system. For real-time processing the
system needs to be causal. In many situations the data can be stored and processed without the
requirements of real-time processing; under such circumstances causality is not necessary.
A continuous-time systemSis calledcausalif:
n Whenever the inputx(t)= 0 and there are no initial conditions, the output isy(t)= 0.
n The outputy(t)does not depend on future inputs.
For a valueτ >0, when considering causality it is helpful to think of
n The timet(the time at which the outputy(t)is being computed) as thepresent.
n Timest−τas thepast.
n Timest+τas thefuture.
Remarks
Causality is independent of the linearity and the time-invariance properties of a system. For instance, the
system represented by the input–output equation
y(t)=x^2 (t)
where x(t)is the input and y(t)the output, is nonlinear but time invariant, and according to the above
definition is a causal system. Likewise, an LTI system can be noncausal. Consider the following averager:
y(t)=
1
2 T
∫t+T
t−T
x(τ)dτ
