**Mumbai University > Electronics and Telecommunication Engineering > Sem 5 > Random Signal Analysis

Marks: 10M

Year: May 2016

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 $a_2$ an input [$a_1$$x_1$(t) + $a_2$$x_2$(t)] to T gives rise to an output $a_1$$y_1$(t) + $a_2$$y_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 it gives a response of y(t-τ) for an input of x(t-τ). We have the output y(t)=h(t)* x(t) = $\int_{-\infty}^\infty h(τ) x(t-τ)dτ$ Hence, E(y(t))= $\int_{-\infty}^\infty h(τ) E (X(t-τ))dτ$ =$u_x$ $\int_{-\infty}^\infty h(τ) dτ$ [Since {X(t)} is WSS]

$\implies$ E(y(t)) is a constant. ........(i)

Now, the autocorrelation of the output process is given by

$R_{YY}$($t_1$,$t_2$) = E{Y($t_1$ Y($t_2$)}

=E $\int_{-\infty}^\infty$$\int_{-\infty}^\infty$ h($τ_1$)h($τ_2$)X($t_1$-$τ_1$)X($t_2$-$τ_2$)d$τ_1$d$τ_2$

=E $\int_{-\infty}^\infty$$\int_{-\infty}^\infty$ h($τ_1$)h($τ_2$)$R_{XX}$($t_1$-$τ_1$,$t_2$-$τ_2$)d$τ_1$d$τ_2$

=E $\int_{-\infty}^\infty$$\int_{-\infty}^\infty$ h($τ_1$)h($τ_2$)$R_{XX}$($t_1$-$t_2$-($τ_1$-$τ_2$))d$τ_1$d$τ_2$

Since{X(t)} is WSS.

Now the RHS is a function of ($t_1$-$t_2$) .Therefore the LHS is also a function only of ($t_1$-$t_2$). Hence $R_{YY}$($t_1$,$t_2$) is a function of only ($t_1$-$t_2$) Hence the output {Y(t)} is also WSS.

Ergodic Processes

A random process is said to be ergodic if the time averages of the process tend to the appropriate ensemble averages. This definition implies that with probability 1, any ensemble average of {X(t)} can be determined from a single sample function of {X(t)}.

Clearly, for a process to be ergodic, it has to necessarily be stationary. But not all stationary processes are ergodic.