Propeties of the System

Shift-invariance: If the input unit step response is shifted by t0 output is also shifted by t0. Hence the system is shift invariant.




Causality: A system is causal if the output at any time depends only on inputs at present time and the past, i.e. the system output does not anticipate the future values of the input. Here the system is non-anticipative, hence it is causal.

Stability: A system is stable when a bounded input gives a bounded output.

For 0 < x(t) < Mx < infinity
we have y(t) As derived earlier, transfer function H(s) = k/(s+a).Hence |H(s)| < k/a and Y(s) is bounded for bounded inputs.

Linearity: Output response function for input step response x(t)=c is given by
y(t) = 0 t < T
= c [1-exp (-at)] t > T
x1(t) = c1
y1(t) = 0 t < T
= c1 [1-exp (-at)] t > T
x2(t) = c2
y2(t) = 0 t < T
= c2 [1-exp (-a’t)] t > T

Additivity:
x3(t) = x1(t) + x2(t) = c1 + c2
y3(t) = 0 t < T
= c3 [1-exp (-a’’t)] t > T

Homogeneity:
x4(t) = k x1(t)
y4(t) = 0 t < T
= c4 [1-exp (-a’’’t)] t > T
If rate of exponential response (a = a’=a’’=a’’’) is constant then the system is additive and homogenous. Hence it will be linear with c3 = c1 + c2 and c4= kc1.

Else if the rate of exponential response a is a function of x(t) system will be neither additive nor homogenous. Hence it will be nonlinear. n Memory: The system is memoryless if the output for each value of the independent variable depends only on the input at the same time. For system under consideration input step response x(t)=a,

Output response function is given by
y(t) = 0 t < T =a[1-exp(-kt)] t > T
Here the output depends on x(t) as well as t hence the system has memory.

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