Question

In this chapter, we introduced a number of general properties of systems. In particular,
a system may or may not be
(1) Memoryless
(2) Time invariant
(3) Linear
(4) Causal
(S) Stable
Determine which of these properties hold and which do not hold for each of the
following continuous-time systems. Justify your answers. In each example, y(t) denotes
the system output and x(t) is the system input.

(b) y(t) [cos(31)]x(1) (c) y() = 13, x(T)dT x(t) + x(t - 2 dx(1) x(t) < 0 x(t) 0 (f) y(t) = x(t/3) ),

Answer 1

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(1) Memory: Systems a and c are having memory because their outputs at the present instants of time depends on the past times as well.

(2) Time invariant: All systems except d and e are time invariant, because in d, even if we advance the input in time, output y is not advanced because y=0 for t<0. In e, the output y depends on the state of the input x, rather than on the time itself.

(3) All systems except d and e are linear. In d, the system output starts only after t=0, whatever the input. In e, the negative portions of x are cut out.

(4) Causal: All systems except a is causal because in a, y(0) = x(-2) + x(2), which implies that the output depends on a future value of the input.

(5) Stability: All systems except c is stable. System c is marginally stable, because the output blows up if the input is a unit step input.

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