Problem 3: (a) Is the following statement true or false? The series interconnect

Question

Problem 3: (a) Is the following statement true or false? The series interconnection of two linear time-invariant systems is itself a linear, time-invaiant system. Justify your answer. (b) Is the following statement true or false? The series interconnection of two nonlinear systems is itself nonlinear Justify your answer (c) Consider three systems with the following input-output relationships x[n2].n even n odd System 1: y[a] = System 2: n-x(n] + Systen 3:yin] [2n) x(n-11+ n-2] Suppose that these systems are connected in series as depicted in Figure P1.42. Find the input-output relationship for the overall interconnected system. Is this system linear? Is it time invariant? xin) System System 2 System 3 vIn) Figure PT.42

Explanation / Answer

Ans a ) True. As both the LTI system will follow associative, distributive and commulative properties which leads the interconnection to follow the same properties.

Ans b ). False. We use a counter-example. Consider two systems S1 : x(t) x(t)+1 and S2 : x(t) x(t)1; both are clearly nonlinear. Now, the series interconnection of S1 and S2 gives S1S2 : x(t) x(t), which is linear. Hence, the statement is false.

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