Introduction to systems - Continuous and Discrete Time Signals and Systems - page 84

Table 2.1. Examples of CT and DT systems with and without memory

Continuous-time Discrete-time

Memoryless systems Systems with memory Memoryless systems Systems with memory

y ( t ) = 3 x ( t ) + 5 y ( t ) = x ( t − 5) y [ k ] = 3 x [ k ] + 7 y [ k ] = x [ k − 5]

y ( t ) = sin x ( t ) + 5 y ( t ) = x ( t + 2) y [ k ] = sin( x [ k ]) + 3 y [ k ] = x [ k + 3]

y ( t ) = e x ( t ) y ( t ) = x (2 t ) y [ k ] = e x [ k ] y [ k ] = x [2 k ]

y ( t ) = x 2 ( t ) y ( t ) = x ( t / 2) y [ k ] = x 2 [ k ] y [ k ] = x [ k / 2]

To compute the output voltage y ( t 0 ) at time t 0 , we require the value of the current
source for the time range ( −∞ , t 0 ], which includes the entire past. Therefore,
the electrical circuit in Fig. 2.16(b) is not a memoryless system.

In Table 2.1, we consider several examples of memoryless and dynamic systems.
The reader is encouraged to verify mathematically the classifications made in
Table 2.1.
As a side note to our discussion on memoryless systems, we consider another
class of systems with memory that require only a limited set of values of input
x ( t )in t 0 − T ≤ t ≤ t 0 to compute the value of output y ( t ). Such CT systems,
whose response y ( t ) is completely determined from the values of input x ( t ) over
the most recent past T time units, are referred to as finite-memory or Markov
systems with memory of length T time units. Likewise, a DT system is called
a finite-memory or a Markov system with memory of length M if output y [ k ]
at k = k 0 depends only on the values of input x [ k ] for k 0 − M ≤ k ≤ k 0 in the
most recent past.

2.2.4 Causal and non-causal systems

A CT system is causal if the output at time t 0 depends only on the input x ( t ) for
t ≤ t 0 . Likewise, a DT system is causal if the output at time instant k 0 depends
only on the input x [ k ] for k ≤ k 0 . A system that violates the causality condition is
called a non-causal (or anticipative ) system. Note that all memoryless systems
are causal systems because the output at any time instant depends only on
the input at that time instant. Systems with memory can either be causal or
non-causal.

Example 2.7

(i) CT time-delay system y ( t ) = x ( t − 2) ⇒ causal system;

(ii) CT time-forward system y ( t ) = x ( t + 2) ⇒ non-causal system;

y [ k ] = x [ k − 2] ⇒ causal system;

(iii) DT time-delay system

y [ k ] = x [ k + 2] ⇒ non-causal system;

(iv) DT time-advance system

(v) DT linear system y [ k ] = x [ k − 2] + x [ k + 10] ⇒ non-causal

system .