Signals and Systems - Electrical Engineering

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

nExample 2.14

Consider the block diagram in Figure 2.14 with input a unit-step signal,u(t). The averager is such

that for an inputx(t)its output is

y(t)=

1

T

∫t

t−T

x(τ)dτ

Determine what the system is doing as we let the delay 1 →0. Consider that the averager and the

system with inputu(t)and outputx(t)are LTI.

FIGURE 2.14

Block diagram of the cascading of two LTI

systems, one of them being an averager.

×

+

−

u(t)

Delay Δ

Averager

x(t) y(t)

1

Δ

Solution

Since it is not clear from the given block diagram what the system is doing, using the LTI of the two

systems connected in cascade lets us reverse their order so that the averager is first (see Figure 2.15),

obtaining an equivalent block diagram.

FIGURE 2.15

Equivalent block diagram of the cascading of two

LTI systems, one of them being an averager.

×

+

−

u(t) s(t)

Delay Δ

Averager

y(t)

1

Δ

The output of the averager is

s(t)=

1

T

∫t

t−T

u(τ)dτ=







0 t< 0

t/T 0 ≤t<T

1 t≥T

as we obtained before in example 2.12. The outputy(t)of the other system is given by

y(t)=

1

1

[s(t)−s(t−1)]