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White Noise Process

A random process (or signal for your visualization) with a constant power spectral density (PSD) function is a white noise process.

Power Spectral Density

Power Spectral Density function shows how much power is contained in each of the spectral component. For example, for a sine wave of fixed frequency, the PSD plot will contain only one spectral component present at the given frequency. PSD is an even function and so the frequency components will be mirrored across the Y-axis when plotted. Thus for a sine wave of fixed frequency, the double sided plot of PSD will have two components – one at +ve frequency and another at –ve frequency of the sine wave. (How to plot PSD/FFT in Matlab)

Gaussian and Uniform White Noise:

A white noise signal (process) is constituted by a set of independent and identically distributed (i.i.d) random variables. In discrete sense, the white noise signal constitutes a series of samples that are independent and generated from the same probability distribution. For example, you can generate a white noise signal using a random number generator in which all the samples follow a given Gaussian distribution. This is called White Gaussian Noise (WGN) or Gaussian White Noise. Similarly, a white noise signal generated from a Uniform distribution is called Uniform White Noise. White Gaussian Noise and Uniform White Noise are frequently used in system modelling.

In modelling/simulation, a white noise can be generated using an appropriate random generator. White Gaussian Noise can be generated using “randn” function in Matlab which generates random numbers that follow a Gaussian distribution. Similarly, “rand” function can be used to generate Uniform White Noise in Matlab that follows a uniform distribution. When the random number generators are used, it generates a series of random numbers from the given distribution. Let’s take the example of generating a White Gaussian Noise of length 10 using “randn” function in Matlab – with zero mean and standard deviation=1.

noise =   -1.5121    0.7321   -0.1621    0.4651    1.4284    1.0955   -0.5586    1.4362   -0.8026    0.0949

What is i.i.d ?

This simply generates 10 random numbers from the standard normal distribution. As we know that a white process is seen as a random process composing several random variables following the same Probability Distribution Function (PDF). The 10 random numbers above are generated from the same PDF (standard normal distribution). This condition is called “identically distributed” condition. The individual samples given above are “independent” of each other. Furthermore, each sample can be viewed as a realization of one random variable. In effect, we have generated a random process that is composed of realizations of 10 random variables. Thus, the process above is constituted from “independent identically distributed” (i.i.d) random variables.

Strictly and weakly defined White noise:

Since the white noise process is constructed from i.i.d random variable/samples, all the samples follow the same underlying probability distribution function (PDF). 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. As with a stationary process which can be classified as Strict Sense Stationary (SSS) and Wide Sense Stationary (WSS) processes, we can have white noise that is SSS and white noise that is WSS. Correspondingly they can be called “strictly white noise signal” and “weakly white noise signal”.

What’s with Covariance Function/Matrix ?

A white noise signal, denoted by x(t), is defined in weak sense is a more practical condition. Here, the samples are statistically uncorrelated and identically distributed with some variance equal to σ2. This condition is specified by using a covariance function as

covariance function of white noise

Why do we need a covariance function? Because, we are dealing with a random process which composes “n” random variables(10 variables in the modelling example above). Such a process is viewed as multivariate random vector or multivariate random variable. For multivariate random variables, Covariance function specified how each of the “n” variables in the given random process behaves with respect to each other. 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.

covariance matrix of white noiseSince 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 cross-correlation values are zero (samples/variables are statistically uncorrelated with respect to each other).

The diagonal elements are equal to the variance and all other elements in the matrix are zero.The ensemble auto-correlation function of the weakly defined white noise is given by

auto correlation function of white noise

This indicates that the auto-correlation function of weakly defined white noise process is zero everywhere except at lag τ=0

Frequency Domain Characteristics:

Wiener-Khintchine Theorem states that for Wide Sense Stationary Process (WSS), the power spectral density function \(S_{xx}(f)\) of the random process can be obtained by Fourier Transform of auto-correlation function of the random process. In continuous time domain, this is represented as

Wiener Khinchin Theorem equation

For the weakly defined white noise process, we find that the mean is a constant and its covariance does not vary with respect to time. This is a sufficient condition for a WSS process. Thus we can apply Weiner Khintchine Theorem.

Wiener Khinchin Theorem equation for white noise

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.

Wiener Khinchin Theorem

Testing the characteristics of White Gaussian Noise in Matlab:

Generate a Gaussian white noise signal of length L=100,000 using the randn function in Matlab and plot it. Here the underlying pdf is a Gaussian pdf with mean μ=0 and standard deviation σ=2. Thus the variance of the Gaussian pdf is σ2=4.

White noise in Matlab

Plot the histogram of the generated white noise and verify the histogram by plotting against the theoretical pdf of the Gaussian random variable.

PDF of Gaussian random variable

White Noise Histogram in Matlab

Compute the auto-correlation function of the white noise. The computed auto-correlation function has to be scaled properly. If the ‘xcorr’ function (inbuilt in Matlab) is used for computing the auto-correlation function, use the ‘biased’ argument in the function to scale it properly.

Autocorrelation of White noise in Matlab

Simulating the PSD of the white noise:

Simulating the Power Spectral Density (PSD) of the white noise is a little tricky business. 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.The wavering effect of the psd can be minimized by generating sufficiently long random signal and averaging the psd over several realizations of the random signal.

Simulating Gaussian White Noise as a Multivariate Gaussian Random Vector:

To verify the power spectral density of the white noise, we will use the approach of envisaging the white noise as a composite of ‘N’ Gaussian random variables. We want to average the psd over L such realizations.

Since there are ‘N’ Gaussian random variables (N individual samples) per realization, the covariance matrix Cxx will be of dimension NxN. The vector of mean for this multivariate case will be of dimension 1xN. Cholesky decomposition of covariance matrix gives the equivalent standard deviation for the multivariate case. Cholesky decomposition can be viewed as square root operation. Matlab’s randn function is used here to generate the multi-dimensional Gaussian random process with the given mean matrix and covariance matrix.

Compute PSD of the above generated multi-dimensional process and average it to get a smooth plot.

PSD power spectral density of White noise in Matlab

The PSD plot of a white noise gives almost fixed power in all the frequencies. In other words, for a white noise signal, the PSD is constant (flat) across all the frequencies (-∞ to + ∞). The y-axis in the above plot is expressed in dB/Hz unit. We can see from the plot that the constant power = 10*log10(σ2)=10*log10(4)=6 dB.

References:

[1] Robert Grover Brown, Introduction to Random Signal Analysis and Kalman Filtering. John Wiley and Sons, 1983

[2] Athanasios Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. WCB/McGraw-Hill, 1991

Recommended Books on Probability and Random Process: