P To understand the concepts of Multiple Random Variables and operations that may be performed on Multiple Random variables. Learning Outcomes: A student will able to determine the temporal and spectral characteristics of random signal response of a given linear system. Unequal Distribution, Equal Distributions.

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The random processes are also called as stochastic processes which deal with randomly varying time wave forms such as any message signals and noise.

They are described statistically since the complete knowledge about their origin is not known. So statistical measures are used. Probability distribution and probability density functions give the complete statistical characteristics of random signals. A random process is a function of both sample space and time variables. Deterministic and Non-deterministic processes: In general a random process may be deterministic or non deterministic.

A process is called as deterministic random process if future values of any sample function can be predicted from its past values. If future values of a sample function cannot be detected from observed past values, the process is called non-deterministic process. Classification of random process: Random processes are mainly classified into four types based on the time and random variable X as follows. Continuous Random Process: A random process is said to be continuous if both the random variable X and time t are continuous.

The below figure shows a continuous random process. The fluctuations of noise voltage in any network is a continuous random process. Discrete Random Process: In discrete random process, the random variable X has only discrete values while time, t is continuous.

The below figure shows a discrete random process. A digital encoded signal has only two discrete values a positive level and a negative level but time is continuous. So it is a discrete random process. Page Continuous Random Sequence: A random process for which the random variable X is continuous but t has discrete values is called continuous random sequence. A continuous random signal is defined only at discrete sample time intervals.

Discrete Random Sequence: In discrete random sequence both random variable X and time t are discrete. It can be obtained by sampling and quantizing a random signal. This is called the random process and is mostly used in digital signal processing applications. The amplitude of the sequence can be quantized into two levels or multi levels as shown in below figure s d and e.

Joint distribution functions of random process: Consider a random process X t. The function FX x1;t1 is known as the first order distribution function of X t. Joint density functions of random process:: Joint density functions of a random process can be obtained from the derivatives of the distribution functions.

Independent random processes: Consider a random process X t. If the random process X t is statistically independent, then the Nth order joint density.

Similarly the joint distribution will be the product of the. Statistical properties of Random Processes: The following are the statistical properties of random processes. Stationary Processes: A random process is said to be stationary if all its statistical properties such as mea n, moments, variances etcâ€¦ do not change with time.

The stationarity which depends on the density functions has different levels or orders. First order stationary process: A random process is said to be stationary to order one or first. Therefore the condition for a process to be a first order stationary random process is that its mean value must be constant at any time instant. Second order stationary process: A random process is said to be stationary to order two or. It is a function of time difference t2, t1 and not absolute time t.

Note that a second order stationary process is also a first order stationary process. The condition for a process to be a second order stationary is that its autocorrelation should depend only on time differences and not on absolute time.

The condition for a wide sense stationary process are 1. Joint wide sense stationary process:. Consider two random processes X t and Y t. Therefore the conditions for a process to be joint wide sense stationary are 1. A p rocess that is stationary to all orders. But the reverse is not true. N is called strict sense stationary process.

Ergodic Theorem and Ergodic Process: The Ergodic theorem states that for any random process X t , all time averages of sample functions of x t are equal to the corresponding statistical or ensemble averages of X t.

Random processes that satisfy the Ergodic theorem are called Ergodic processes. Joint Ergodic Process: Let X t and Y t be two random processes with sample functions x t and y t respectively. The two random processes are said to be jointly Ergodic if they are individually Ergodic and their time cross correlation functions are equal to their respective statistical cross correlation functions.

Mean Ergodic Random Process: A random process X t is said to be mean Ergodic if time average of any sample function x t is equal to its statistical average, which is constant and the probability of all other sample functions is equal to one. Autocorrelation Ergodic Process: A stationary random process X t is said to be Autocorrelation Ergodic if and only if the time autocorrelation function of any sample function x t is equal to the statistical autocorrelation function of X t.

Cross Correlation Ergodic Process: Two stationary random processes X t and Y t are said to be cross correlation Ergodic if and only if its time cross correlation function of sample functions x t and y t is equal to the statistical cross correlation function of X t and Y t. Then the following are the properties of the autocorrelation function of X t.

Then the following are the properties of cross correlation function. The auto covariance function can be expressed as.

The autocorrelation coefficient of the random process, X t is defined as. The cross correlation coefficient of random processes X t and Y t is defined as. Gaussian Random Process: Consider a continuous random process X t. Let N random variables. If random tN. Then the random. The Gaussian density function is given as. And at any time instants. And the probability density function is. Learn more about Scribd Membership Home.

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