Introduction to systems - Continuous and Discrete Time Signals and Systems - page 83

R 2 R L

+
−

+
−

− −

v ( t ) i ( t )

R 1 y ( t ) y ( t )

C

(a) (b)

Fig. 2.16. (a) Passive electrical circuit comprising resistors R 1 and R 2 . (b) Active electrical circuit
comprising resistor R , inductor L , and capacitor C . Both inductor L and capacitor C are storage
components, and hence lead to a system with memory.

2.2.3 Systems with and without memory

A CT system is said to be without memory ( memoryless or instantaneous )ifits
output y ( t ) at time t = t 0 depends only on the values of the applied input x ( t )
at the same time t = t 0 . On the other hand, if the response of a system at t = t 0
depends on the values of the input x ( t ) in the past or in the future of time t = t 0 ,
it is called a dynamic system, or a system with memory . Likewise, a DT system
is said to be memoryless if its output y [ k ] at instant k = k 0 depends only on the

= k 0 . Otherwise, the DT system is

value of its input x [ k ] at the same instant k

said to have memory.

Example 2.6
Determine if the two electrical circuits shown in Figs. 2.16(a) and (b) are
memoryless.

Solution
The relationship between the input voltage v ( t ) and the output voltage y ( t )
across resistor R 1 in the electrical circuit of Fig. 2.16(a) is given by

R 1
R 1 + R 2 v ( τ ) .

y ( t ) = (2.46)

For time t = t 0 , the output y ( t 0 ) depends only on the value v ( t 0 ) of the input v ( t )
at t = t 0 . The electrical circuit shown in Fig. 2.16(a) is, therefore, a memoryless
system.
The relationship between the input current i ( t ) and the output voltage y ( t )in
Fig. 2.16(b) is given by

t

y ( t ) = 1
C

i ( τ )d τ. (2.47)

−∞