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Short Note: LTI systems with stochastic input

Mumbai University > Electronics and Telecommunication > Sem5 > Random Signal Analysis

Marks: 5 M

Year: May 2015

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Linear Time Invariant System:

Linear systems: Let T be a continuous – time system which is at rest – i.e. all of its energy storage elements are devoid of any stored energy. Let an input $x_1(t)$ to T give rise to an output $y_1(t)$ and an input $x_2(t)$ to T give rise to an output . Then the system is said to be linear, if for any pair of arbitrary constants $a_1$ and $a_2$ an input $[a_1x_1(t)+a_2x_2(t)]$ to T gives rise to an output $[a_1y_1(t)+a_2y_2(t)]$.

Time Invariant Systems: Let be the response of a continuous –time system T to an arbitrary input signal. Then the system is said to be time invariant, if for any value of the real constant $\tau$ it gives a response of $y(t-\tau)$ for an input of $x(t-\tau)$.

Mathematically a system is a functional relationship between the input x(t) and the output y(t). Relationship is written as y(t)=f (x (t)),-∞ < t <∞

If we assume that x(t) represents a sample function of a random process {X(t)}, the system produces an output or response y(t) and the ensemble of the output functions forms a random process {Y(t)}.

X(t) actually means X(s, t) where s∈S (Sample space).If the system operates only on variable t treating s as a parameter, it is called as deterministic system. If the system operates on both t and s, it is called as stochastic.

Hence we can prove that if input LTI system is WSS the output is also WSS