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Thus, the Joint Probability Distribution function of the process will not change with any shift in time. This is called a stationary process.
Thus the white noise is a stationary process. This condition is specified by using a covariance function as. Why do we need a covariance function?
Such a process is viewed as multivariate random vector or multivariate random variable. Covariance function generalizes the notion of variance to multiple dimensions.
The above equation when represented in the matrix form gives the covariance matrix of the white noise random process. Since the random variables in the white noise process are statistically uncorrelated, the covariance function contains values only along the diagonal.
The matrix above indicates that only the auto-correlation function exists for each random variable. The diagonal elements are equal to the variance and all other elements in the matrix are zero.
Constructing the auto-correlation matrix in Matlab. This is a sufficient condition for a WSS process. Thus we can apply Weiner Khintchine Theorem.
Therefore, the power spectral density of the weakly defined white noise process is constant flat across the entire frequency spectrum.
The value of the constant is equal to the variance or power of the white noise. Plot the histogram of the generated white noise and verify the histogram by plotting against the theoretical pdf of the Gaussian random variable.
Compute the auto-correlation function of the white noise. The computed auto-correlation function has to be scaled properly. More simulation techniques available in this ebook — Simulation of Digital Communication systems using Matlab.
There are two issues here 1 The generated samples are of finite length. This is synonymous to applying truncating an infinite series of random samples.
This implies that the lags are defined over a fixed range. FFT and spectral leakage — an additional resource on this topic can be found here 2 The random number generators used in simulations are pseudo-random generators.
Due these two reasons, you will not get a flat spectrum of psd when you apply Fourier Transform over the generated auto-correlation values.
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