Random Processes and Statistical Data Analysis Random Processes

Properties: The sine signal is a deterministic and periodic signal with π2 period. Therefore the autocorrelation function must be periodic too and has maximum ...
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Random Processes and Statistical Data Analysis

Random Processes By: Nabil Haddad

Nabil Haddad

Random processes By: Nabil Haddad

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Introduction: This report render an account of the methodology used to reach the goals defined by the wording of the two labs, as well as the problems. For each of theses labs the corresponding solution will be detailed. All the files used to reach the final design can be found in the annex following this report. Therefore, it contains the m-Matlab files. The aim of the first lab is to analyze the properties of some random and deterministic signals and to examine the relationship between them. The methods that were studed during the course’s lectures will be used to analyze the different results. The aim of the second lab is to give a close look to real experiment and to examine each variable in the experiment and to find out the correlation between them. .

Nabil Haddad

2 of 7

MA 412S

Random Processes and Statistical Data Analysis

I – Random & Deterministic Signals

1) Objective: The objective of this lab is to analysis some signals’ properties and to exam the relationship among them. In particular interests are the following signals: • White Noise Signal • Sine Signal • Mix of Sine and white Noise signal • Pseudo Random Binary Signal (PRBS)

2) White Noise Signal: •

Definition: White noise signal is a random signal witch takes its values randomly on a sample space. Its power spectral density is constant and its mean is equal to zero. The upper part of figure 1 shows a white noise signal centered at zero with range values between -4 to +4.



Properties: The autocorrelation of white noise has maximum value at zero which equals to the signal’s variance and it formed according to:

ϕ •

xx

(0) =

σ

2 x

Analysis: To imitate white noise behavior, I used Matlab to generate a random vector of length 31416 centered at zero. By calculating the mean, autocorrelation, standard Figure 1: White noise signal and its autocorrelation deviation and the variance of this signal it can be inferred that it has the same properties of a white noise signal as expected. The lower part of figure 1 shows the autocorrelation function of this signal and its maximum value at:

ϕ

Nabil Haddad

xx

(0 ) ≅ 1

3 of 7

MA 412S

Random Processes and Statistical Data Analysis

3) Sine Signal: •

Definition: A sinusoidal signal, that is, one moving in simple harmonic motion according to the function: f ( t ) = A × sin( 2 π ft )

Where A is the amplitude f is the frequency and t is time. Figure 2 shows a sinusoidal signal. •

Properties: The sine signal is a deterministic and periodic signal with 2π period. Therefore the autocorrelation function must be periodic too and has maximum value at zero. The autocorrelation function for periodic sine signal represented by:

ϕ •

Figure 2: Sinusoidal signal

xx

(τ ) =

1

π



∫π sin(

t ) × sin( t + τ ) dt



Analysis: Using the sine function provided in Matlab, I generate a long vector, which compromises the values of the sine function. Then calculating the autocorrelation function of the sine signal. The upper part of Figure 3 plot is the sine function and the lower part shows the autocorrelation. It is observed that the autocorrelation function satisfies the periodical requirement, and has a maximum value at zero.

ϕ

xx

( 0 ) ≅ 0 .5

It also observed that the symmetric property of the autocorrelation around the Y-axis is conserved.

Figure 3: Sine signal and its autocorrelation

4) Mix of Sine and white Noise signal: •

Definition: This is a sine signal excited by with noise signal to form a new random signal. This signal is shown in the upper part of figure 4.

Nabil Haddad

4 of 7

MA 412S

Random Processes and Statistical Data Analysis



Properties: The mix signal is a random signal. Its autocorrelation function should be equal to the addition of the autocorrelation of sine and white noise signals.



Analysis: The autocorrelation function of the mix signal is shown in the figure 4. It has a maximum value at zero, and it is equal to the addition of compromised autocorrelation functions as:

ϕ

Mixsignal

(0) = ϕ

Figure 4: Mix signal and its autocorrelation

WhiteNoise

(0 ) + ϕ

Sine

( 0 ) ≅ 1 .5

To check if there is correlation between the sine signal and the mix signal the calculation of the inter correlation function of these two signals is needed. Figure 5 plot the inter correlation function of these two signals. Because the maximum value of the inter correlation is less that 0.87 it induced that there is no correlation between the signals. By examining the inter correlation function in figure 5, it is observed that the symmetrical property is not conserved and the function doesn’t have maximum at zero.

Figure 5: Sine and mix signal intercorrelation

Nabil Haddad

5 of 7

MA 412S

Random Processes and Statistical Data Analysis

5) Pseudo Random Binary Signal (PRBS): •

Definition: This is a sequence of rectangular signal between two levels A1 and A2 with a period L, and a frequency F. The nature of this signal is deterministic but its behavior seems to be random.



Properties: The autocorrelation function seems to be as the autocorrelation function of white noise when L is large.



Analysis: The implementation of this signal is shown in figure 6. The upper and lower levels are ±1 and the period is 1000. The autocorrelation has maximum value at zero:

ϕ

xx

Figure 6: PRBS signal and its autocorrelation

(0 ) ≅ 1

It has been noted that the autocorrelation’s value decrease at zero when the signals’ period decrease and increase elsewhere. The lower part of figure 6 shows several periods of the autocorrelation function of PRBS. The attached Matlab file Lab1.m explains step by step all the procedures that were used during this lab.

Nabil Haddad

6 of 7

MA 412S

Random Processes and Statistical Data Analysis

II – Distillation column experiment

TI 3106

FI 2301C

PC 3102 FI 3110

AI 3102T

cold DAD 301

FC 3101

PI 3103

FAD 306

1 LC 3007

TI 3019

DAD 302 12

TI 3114

FI 3003

TC 3101

Vapour 4 bars

TV 3101

PC 3105

AI 3102C

DAD 303

LC 3102 LC 4009

TI 3104

18

feed

TI 3101

EAD 314

FI 3107C

34

TI 3115

DPI 3101 D A D 2 0 2

40 TI 3109-03

LC 3101

EAD 315 A/B

TI 3102 TI 3113

Nabil Haddad

AI 3402T convers i

AI 2203T AT 2203A

TC 6201 PC 6205

48

FC 3103

D A D 3 0 4

AC 3402A

? ?

Variable used for identification

? ?

Controlled output

? ?

Control variable

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