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

Marks: 10M

Year: Dec 2015

Deterministic System: A deterministic system is one in which the occurrence of all events is known with certainty. If the description of the system state at a particular point of time of its operation is given, the next state can be perfectly predicted.

Stochastic system: A random process, also called a stochastic process, is a family of random variables, indexed by a parameter t from an indexing set T . For each experiment outcome ω ∈Ω, we assign a function X that depends on t X(t, ω) t ∈ T , ω∈Ω –t is typically time, but can also be a spatial dimension –t can be discrete or continuous – The range of t can be finite, but more often is infinite, which means the process contains an infinite number of random variables.

Memoryless system: A system is memoryless if and only if the output y(t)at any time $t_0$ depends only on the input x(t) at that same time: x($t_0$)

i.e. P$(X_{(n+1)})$=j│$X_0$=$i_0$,$X_1$=$i_1$,$X_2$=$i_2$……….$X_{(n-1)}$=$i_{(n-1)}$,$X_n$=i

=P($X_{n+1}$)=j|$X_n$=i)

In the above expression $X_0$,$X_1$,$X_2$,………$X_{(n-1)}$is the ‘past’ ,$X_n$ is the ‘present’ and $X_{(n+1)}$ is the ‘future’. The statement states that the probability of the ‘future’ events $X_{(n+1)}$ depends only on the ‘present’ event {$X_n$} and not on the past events

{$X_0$,$X_1$,$X_2$,………$X_{(n-1)}$

Such a chain is called memoryless.