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Random Processes

           In practical problems, we deal with time varying waveforms whose value at a time is random in nature. For example, the speech waveform recorded by a microphone, the signal received by communication receiver or the daily record of stock-market data represents random variables that change with time. How do we characterize such data? Such data are characterized as random or stochastic processes. This lecture covers the fundamentals of random processes.

Random processes

           Recall that a random variable maps each sample point in the sample space to a point in the real line. A random process maps each sample point to a waveform.

           Consider a probability space . A random process can be defined on  as an indexed family of random variables  where  is an index set, which may be discrete or continuous, usually denoting time. Thus a random process is a function of the sample point  and index variable  and may be written as .

Remark

• For a fixed      is a random variable

• For a fixed ,  is a single realization of the random process and is a deterministic function.           For a fixed  and a fixed  is a single number.            When both   and  are varying we have the random process . The random process  is normally denoted by  Figure1 illustrates a random process.

  Example 1 Consider a sinusoidal signal  where is a binary random variable with probability mass functions and 

  Clearly,  is a random process with two possible realizations and  At a particular time  is a random variable with two values  and .

   

  Continuous-time vs. Discrete-time process

            If the index set  is continuous,  is called a continuous-time process.

  Example 2 Suppose, where and are constants and  is uniformly distributed between0 and .  is an example of a continuous-time process.

•   If the index set  is a countable set,  is called a discrete-time process. Such a random process can be represented as  and called a random sequence. Sometimes the notation is used to describe a random sequence indexed by the set of positive integers.

              We can define a discrete-time random process on discrete points of time. Particularly, we can get a discrete-time random process  by sampling a continuous-time process \ at a uniform interval such that  

              The discrete-time random process is more important in practical implementations. Advanced statistical signal processing techniques have been developed to process this type of signals.

   

  Example 3 Suppose  where  is a constant and is a random variable uniformly distributed between and .Continuous-state vs. Discrete-state process

            The value of a random process  is at any time  can be described from its probabilistic model.

            The state is the value taken by at a time , and the set of all such states is called the state space. A random process is discrete-state if the state-space is finite or countable. It also means that the corresponding sample space is also finite or countable. Otherwise , the random process is called continuous state.

  Example 4 Consider the random sequence  generated by repeated tossing of a fair coin where we assign 1 to Head and 0 to Tail.

  Clearly,  can take only two values - 0 and 1. Hence  is a discrete-time two-state process.

•  

  How to describe a random process?

             As we have observed above that  at a specific time  is a random variable and can be described by itsprobability distribution function  This distribution function is called the first-order probability distribution function. 
             We can similarly define the first-order probability density function    

  To describe ,  we have to use joint distribution function of the random variables at all possible values of .  For any positive integer ,  represents  jointly distributed random variables. Thus a random process can thus be described by specifying the  joint distribution function .

  

  or th the  joint probability density function

  

  If is a discrete-state random process, then it can be also specified by the collection of joint probability mass function

  

   

  We can also define higher-order moments like The above definitions are easily extended to a random sequence  .   Examples of Random Processes   (a) Gaussian Random Process For any positive integer ,represent  jointly random variables. These  random variables define a random vector .The process  is called Gaussian if the random vector is jointly Gaussian with the joint density function given by where  and