ECE600 F13 Stochastic Processes mhossain - Rhea

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Stochastic Processes

We have already seen discrete-time random processes, but we will now formalize the concept of random process, including both discrete-time and continuous time.

'Definition $ \qquad $ a stochastic process, or random process, defines on (S,F,P) is a family of random variables {X(t), t ∈ T} indexed by a set T.

Fig 1: The mapping from the sample space to the reals under X$ _j $.

Each waveform is referred to as a sample realization. Note that T can be uncountable, as shown above, or countable.

Note that

• X(t,</math>\omega</math>) (or simply X(t)) is a random process.
• X(t$ _0 $,</math>\omega</math>) is a random variable for fixed t$ _0 $.
• X(t,</math>\omega_0</math>) is a real-valued function of t for fixed $ \omega_0 $.
• X(t$ _0 $,</math>\omega_0</math>) is a real number for fixed t$ _0<math> and <math>\omega_0 $.

There are four types or random processes we will consider

1. T ⊂ R uncountable, X(t) a discrete random variable for every t ∈ T is a continuous-time discrete random process.
2. T ⊂ R uncountable, X(t) a continuous random variable for every t ∈ T is a continuous time continuous random process.
3. T ⊂ R countable, X(t) a discrete random variable for every t ∈ T is a discrete-time discrete random process.
4. T ⊂ R countable, X(t) a continuous random variable for every t ∈ T is a discrete-time continuous random process.

Example $ \qquad $ if T = N = {1,2,3,...}, then X(t) is a discrete time random process, usually written as X$ <_1 $,X$ _2 $

Example $ \qquad $ a binary waveform with random transition times

Fig 2: A binary waveform with random transition times.

Example $ \qquad $ A sinusoid with random frequency

where $ \Omega $ is a random variable.

Probabilistic Description of a Random Process

We can use joint pdfs of pmfs, but often we use the first and second order moments instead.