Signals and Systems - Electrical Engineering

144 C H A P T E R 2: Continuous-Time Systems

which can be written as

y(t)=

1

2 T

∫t

t−T

x(τ)dτ+

1

2 T

∫t+T

t

x(τ)dτ

At the present time t,y(t)consists of the average of a past and present values in[t−T,t]of the input, and

of the average of future values of the signal (i.e., the average of values x(t)for[t,t+T]). Thus, this system is

not causal.

An LTI system represented by its impulse responseh(t)iscausalif

h(t)= 0 fort< 0 (2.21)

The output of a causal LTI system with a causal inputx(t)(i.e.,x(t)= 0 fort< 0 ) is

y(t)=

∫t

0

x(τ)h(t−τ)dτ (2.22)

One can understand the above results by considering the following:

n The choice of the starting time ast=0 is for convenience. It is purely arbitrary as the system being

considered is time invariant, so that similar results are obtained for any other starting time.

n When computing the impulse responseh(t), the inputδ(t)only occurs att=0 and there are no

initial conditions. Thus,h(t)should be zero fort<0 since fort<0 there is no input and there

are no initial conditions.

n A causal LTI system is represented by the convolution integral

y(t)=

∫∞

−∞

x(τ)h(t−τ)dτ

=

∫t

−∞

x(τ)h(t−τ)dτ+

∫∞

t

x(τ)h(t−τ)dτ

where the second integral is zero according to the causality of the system (h(t−τ)=0 whenτ >t

since the argument ofh(.)becomes negative). Thus, we obtain

y(t)=

∫t

−∞

x(τ)h(t−τ)dτ