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

**Marks:** 5M

**Year:** May 2015

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Short Note: LTI systems with stochastic input

written 8.2 years ago by | • modified 8.2 years ago |

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

**Marks:** 5M

**Year:** May 2015

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written 8.2 years ago by | • modified 8.2 years ago |

**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 $y_2(t)$ . Then the system is said to be **linear**, if for any pair of arbitrary constants $a_1$ and $a_2$ $\big[a_1x_1(t)+a_2x_2(t)\big]$ an input to T gives rise to an
output.$\big[a_1y_1(t)+a_2y_2(t)\big]$

**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

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