Transformation of Random Variables & noise concepts using

MATLAB

PREPARED BY-DARSHAN BHATT

Content• Introduction

• Transformation of RV

• Monotone transformation

• WSS Random process

• ACF and PSD for WSS random process

• Gaussian White Noise (GWN)

• Estimated ACF and PDF of GWN

• Conclusion

• References

Introduction• What is MATLAB ?

MATLAB is a technical computing environment developed by the Math-Works Inc. It is used in many fields of science and engineering like statistics, matrix algebra, signal processing, image processing etc.

• Reasons for choosing MATLAB for statistics:MATLAB is very simple to use and provides powerful computation and excellent graphics capabilities.MATLAB has a powerful facility for statistics called Statistics Toolbox with interactive inbuilt commands for computation of statistical signals and probability models.

Transformations of RV using MATLAB• MATLAB based programs for transformation of RV in context of its parameters,

mean value and variance and PDF’s.

• Monotone Transformation

(1 input RV – 1 output RV)

• Non-Monotone Transformation

(2 input RV – 1 output RV)

• Linear Transform ation

• Non-linear Transformation

Transformations of RV

• Fig.1 shows the transformation of input RV ‘X’. • Transformed RV is ‘Y’.

• We will discuss two types of transformation for input RV ‘X’ that

is monotone type and Non-monotone type.

Monotone Transformations of RV

• In a monotone transformation the output fy(y) and input fx(x) PDF’s are

related as:

• In monotone transformation, we consider two cases i.e linear and non-linear

transformation.

• In linear transformation, the output – input RV are related by –

Y = aX + b

Monotone Transformations of RV • Consider a linear transformation for two different values of ‘a’

and one value of ‘b’.

MATLAB based program for Auto-correlation function & noise concepts

Random process• MATLAB has an attractive toolbox called ‘Statistic toolbox’ by which

analysis of statistical problems can be done efficiently.

• We will discuss about ACF for Wide sense stationary (WSS) random process.

• An indexed family of random variables { X(t , ζ), t ϵ I } is called a random

process or stochastic random process.

• A discrete time or continuous time random process X(t) is stationary if the

joint distribution of any set of samples does not depend on the placement of

time origin.

Wide Sense Stationary Random process• A discrete time or continues time random process X(t) is called WSS random

process if it satisfies two conditions:

1. mx(t) = m for all t.

2. Cx(t1 , t2) = Cx(t1 - t2) for all t1 and t2

• ACF for WSS random process is given by –

RX(τ) = E[ X(t+τ) X(t) ] , E = expectation parameter

ACF and PSD• A fourier transform of an ACF is called power spectral density (PSD) : SX(f)

• For GWN, PSD is given by –

• For GWN, ACF is given by -

Analysis of ACF using MATLAB• STEP 1 : Two signals Y1 and Y2 are generated randomly for the first 250

values.

• Generate a random signal using MATLAB :-

x = randn(1,250);

plot(x)

Analysis of ACF using MATLAB• STEP 2 : Estimate PDF’s for both random signals

Analysis of ACF using MATLAB• STEP 3 : Estimate ACF’s for both random signals

Analysis of ACF using MATLAB• Signals & its corresponding ACF’s

Gaussian White Noise (GWN)• In signal processing, white noise is a random signal with a constant power

spectral density.

• If each sample has a normal distribution with zero mean, the signal is said to

be Gaussian White Noise (GWN).

• ACF for GWN, WSS random process is given by -

Gaussian White Noise (GWN)

ACF of Gaussian White Noise (GWN)

Conclusion• First part of this presentation shows about transformation of RV like monotone &

non-monotone. In that transformation, how the transformation affects on mean and

variance of the linear and non-linear function is shown.

• Second part is about WSS random process with important characteristics like mean,

variance, ACF and PSD. From that, we conclude that PDF’s are same for two

different random signals if they are zero mean Gaussian signals with same variance.

And we conclude that fast process will have smaller values of ACF.

References [1]. G.J.Dolecek, “MATLAB based program for teaching auto-correlation function and noise concepts”, IEEE Transaction on education, vol.55, no.3, August 2012. [2]. G.J.Dolecek, “Using MATLAB for teaching transformation of random variables”, IEEE Transactions on education, ICACCI, 2013.

[3]. Book – Leon Garcia, “Probability & random processes for electrical engineer”, 2nd edition, PE, India.

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