Determine which of these properties hold or don\'t hold for the following contin

Question

Determine which of these properties hold or don't hold for the following continuous time systems. Justify your answers. IN each example, y(t) denotes the system output and x(t) denotes the system input. Properties: 1) Memoryless 2) Time invariant 3) Linear 4) Causal 5) Stable d) y(t)= 0 t<0    x(t)+x(t-2) t>=0 Determine which of these properties hold or don't hold for the following continuous time systems. Justify your answers. IN each example, y(t) denotes the system output and x(t) denotes the system input. Properties: 1) Memoryless 2) Time invariant 3) Linear 4) Causal 5) Stable d) y(t)= 0 t<0    x(t)+x(t-2) t>=0

Explanation / Answer

Y(t) = 0 , ty1(t) = x1(t)+x1(t-2) x2(t)->y2(t) = x2(t)+x2(t-2) Let x3(t) be the linear combination of x1(t)and x2(t), that is x3(t) = ax1(t)+bx2(t) Where a and b are arbitrary scalars. If x3(t) is the input to the system y(t) then the corresponding output can be expressed as y3(t)=x3(t)+x3(t-2) = ax1(t)+bx2(t)+ ax1(t-2)+bx2(t-2) = a(x1(t)+x1(t-2))+b(bx2(t)+x2(t-2)) = a(x1(t)+x1(t-2))+b(bx2(t)+x2(t-2)) = ay1(t)+by2(t) So the given system is a linear system. 4. A system is said to be causal if the output at any time depends only on values of the input at the present time and in the past. Such a system is often referred to as being non-anticipative, as the system output does not anticipate future values of the input. The given system y(t) depends only on the present values[x(t)] and past values[x(t-2)] of x(t), so it is a causal system. 5. A stable system is one in which small inputs lead to responses that do not diverge. In the given system, a constant input [x(t)+x(t-2)] leads a bounded output, so it is a stable system.

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