' Gdansk Universityof Technology

Random Processes and Stochastic

Control – Part 1 of 2

Exercises

by

Maciej Niedzwiecki

Gdansk University of Technology

Departament of Automatic Control

Gdansk, Poland

Exercise 1

Using MATLAB generate and plot the sequence of 100consecutive realizations of 3 random variables: A withuniform PDF, B with Gaussian PDF and C with LaplacePDF. All random variables should have the same mean:

mA = mB = mC = 2

and the same variance:

σ2A = σ2

B = σ2C = 2

Use the same scale on the x and y axes

x ∈ [1, 100], y ∈ [−3, 7]

Exercise 2

Using MATLAB generate and plot the sequence of 100consecutive realizations of the Cauchy random variable D ∼C(2, 1).

Exercise 3

The mean and variance of a random variable X can beestimated using the following relationships

mX(N) =1N

N∑i=1

xi

σ2X(N) =

1N

N∑i=1

[xi − mX(N)]2

Estimate the mean and the variance of the random variablesA,B,C and D defined in Exercises 1 and 2, based on N =100, 1,000, 10,000 and 100,000 samples.

Exercise 4

Using MATLAB plot normalized histograms obtained for10,000 samples of random variables C and D (defined inexercises 1 and 2) and compare them with the correspondingPDF plots. In both cases divide the interval [-3, 7] (subsetof the observation space) into 20 bins of equal width 0.5.

Exercise 5

Write MATLAB procedure generating samples of a randomvariable E with the following PDF

a) pE(e) =

0.50 e ∈ [−1, 0]0.25 e ∈ (0, 2]0 elsewhere

b) pE(e) =

−e e ∈ [−1, 0]0.5 e ∈ (0, 1]0 elsewhere

Evaluate the mean, median and standard deviation of E.

∫ 0

−1

epE(e)de = 0.5e2

2

∣∣∣∣0

−1

= −14

∫ 2

0

epE(e)de = 0.25e2

2

∣∣∣∣2

0

=12

∫ 0

−1

e2pE(e)de = 0.5e3

3

∣∣∣∣0

−1

= −16

∫ 2

0

e2pE(e)de = 0.25e3

3

∣∣∣∣2

0

=23

mX =12− 1

4=

14

σ2X =

16

+23−

(14

)2

=3748

αX = 0

Exercise 6

Denote by E1, . . . , E16 the sequence of 16 independentrandom variables with PDF specified in Exercise 5. Let

F =14

16∑i=1

Ei − mE

σE

Using MATLAB plot the normalized histogram obtained for10,000 samples of F . Compare it with an appropriatelyscaled plot of a standard Gaussian PDF N (0, 1). Dividethe interval [-5, 5] (subset of the observation space) into 20bins of equal width 0.5.

Exercise 7

Let X be a random variable that is uniformly distributedover the interval (1,100). Form a new random variable Yby rounding X to the nearest integer. In MATLAB code,this could be represented by Y = round(X). Finally, formthe roundoff error variable according to Z = X − Y . UsingMATLAB generate 10,000 realizations of Z and estimateits mean and variance. Based on the available data createa normalized histogram. Find the analytical model thatseems to fit your estimated PDF.

Exercise 8

Which of the following mathematical functions could be thecumulative distribution function of some random variable?

1. FX(x) = 12 + 1

πtan−1(x)

2. FX(x) = [1 − e−x] 1(x)

3. FX(x) = e−x2

4. FX(x) = x21(x)

Note: 1(x) is the unit step function.

Exercise 9

Suppose a random variable has a cumulative distributionfunction given by FX(x) = [1 − e−x] 1(x). Find thefollowing quantities:

1. P (X > 5)

2. P (X < 5)

3. P (3 < X < 7)

4. P (X > 5|X < 7)

Exercise 10

Suppose a random variable has a cumulative distributionfunction given by

FX(x) =

0 x ≤ 00.04x 0 < x ≤ 5

0.6 + 0.04x 5 < x ≤ 101 x > 10

Find the following quantities:

1. P (X > 5)

2. P (X < 5)

3. P (3 < X < 7)

4. P (X > 5|X < 7)

Exercise 11

Which of the following are valid probability densityfunctions?

1. pX(x) = e−x1(x)

2. pX(x) = e−|x|

3. pX(x) ={

34(x

2 − 1) |x| < 20 otherwise

4. pX(x) = 2xe−x21(x)

Exercise 12

The conditional cumulative distribution function of randomvariable X , conditioned on the event A having occurred is

FX|A(x) = P (X ≤ x|A) =P (X ≤ x, A)

P (A)

(assuming that P (A) �= 0).

Suppose that X ∼ U(0, 1). Find the mathematicalexpression for FX|X<1/2(x).

Exercise 13

Consider the Laplace random variable with a PDF given by

pX(x) =b

2exp(−b|x|)

Evaluate the mean, the variance, the coefficient of skewnessand the coefficient of kurtosis of X .

Note: ∫xneaxdx =

1a

xneax − n

a

∫xn−1eaxdx

Exercise 14

Consider two independent normally distributed randomvariables

X ∼ N (0, 1), Z ∼ N (0, 1)

and letY = aX + bZ, a, b ∈ R

Determine the coefficients a and b in such a way thatY ∼ N (0, 1) and ρXY takes any prescribed value from[−1, 1].

Using MATLAB generate 100 realizations of the pair (X, Y )for ρXY = 0, ρXY = 0.5, ρXY = −0.5, ρXY = 0.9,ρXY = −0.9, ρXY = 1 and ρXY = −1.

Display the obtained results using scatter plots.

Facts about matrices

Fact 1

Given a square matrix An×n, an eigenvalue λ and itsassociated eigenvector v are a pair obeying the relation

Av = λv

The eigenvalues λ1, . . . , λn of A are the roots of thecharacteristic polynomial of A

det(A− λI)

For a positive definite matrix A it holds that λ1, . . . , λn > 0,i.e., all eigenvalues are positive real.

Fact 2

Let A be a matrix with linearly independent eigenvectorsv1, . . . ,vn. Then A can be factorized as follows

A = VΛV−1

where Λn×n = diag{λ1, . . . , λn} and Vn×n = [v1| . . . |vn].

Facts about matrices

Fact 3

Let A = AT be a real symmetric matrix. Such matrix hasn linearly independent real eigenvectors. Moreover, theseeigenvectors can be chosen such that they are orthogonalto each other and have norm one

wTi wj = 0, ∀i �= j

wTi wi = ||wi||2 = 1, ∀i

A real symmetrix matrix A can be decomposed as

A = WΛWT

where W is an orthogonal matrix: Wn×n = [w1| . . . |wn],WWT = WTW = I.

The eigenvectors wi can be obtained by normalizing theeigenvectors vi

wi =vi

||vi||2 , i = 1, . . . , n

Exercise 15

Consider a vector random variable X = [X1, X2]T with themean and covariance matrix given by

mX =[

34

], ΣX =

[5 44 5

]

1. Perform eigendecomposition of the matrix ΣX

ΣX = WΛWT

where Λ is a diagonal matrix made up of eigenvalues of ΣX

and W is an orthogonal matrix.

Sketch the covariance ellipse of X. What is the correlationcoefficient of X1 and X2?

2. Suppose that Z = [Z1, Z2]T is made up of twouncorrelated random variables with zero mean and unitvariance. Consider the following linear transformation

Y = WΛ1/2Z

where W and Λ are the matrices determined in point 1above. Show that

ΣY = ΣX.

3. Use a linear transformation, defined in point 2, togenerate two-dimensional Gaussian random variable X withthe mean and covariance matrix specified above. Create ascatter plot based on 200 realizations of X.

Exercise 16

Consider a zero-mean vector random variable X withcovariance matrix

ΣX = E[XXT] = WΛWT

where Λ is a diagonal matrix made up of eigenvalues of ΣX

and W is an orthogonal matrix.

LetY = WΛ−1/2WTX

Show thatΣY = I

i.e., Y is the vector random variable made up of uncorrelatedcomponents.

Project 1

Record and mix signals obtained from two independentspeech sources (30 second long recordings, 2 linear mixtureswith different mixing coefficients).

Perform blind source separation using the FastICAalgorithm:

1. Center the data to make its mean zero.

2. Whiten the data to obtain uncorrelated mixture signals.

3. Initialize w0 = [w1,0, w2,0]T, ||w0|| = 1 (e.g. randomly)

4. Perform an iteration of a one-source extraction algorithm

wi = avg{z(t)

[wT

i−1z(t)]3} − 3wi−1

where z(t) = [z1(t), z2(t)]T denotes the vector of whitenedmixture signals and avg(·) denotes time averaging

avg{x(t)} =1N

N∑t=1

x(t)

5. Normalize wi by dividing it by its norm

wi =wi

||wi||

6. Determine a unit-norm vector vi orthogonal to wi.

wTi vi = 0, ||vi|| = 1

7. Display and listen to the current results of sourceseparation

[x1(t)x2(t)

]=

[wT

i

vTi

] [z1(t)z2(t)

], t = 1, . . . , N

8. If wTi wi−1 is not close enough to 1, go back to step 4.

Otherwise stop.

Literature: A. Hyvarinen, E. Oja. “A fast fixed-pointalgorithm for independent component analysis”, NeuralComputation, vol. 9, pp. 1483-1492, 1997.

A special bonus will be granted for writing a procedure thatextracts n > 2 source signals from n mixtures.

Project 2

Create a noisy audio recording (add artificially generatednoise to a clean music or speech signal). The first secondof the recording should contain noise only. Then denoiserecording using the method of spectral subtraction.

Exercise 17

Consider a sinusoid with a random frequency

X(t) = cos(2πft)

where f is a random variable uniformly distributed over theinterval [0, f0].

1. Show that the theoretical mean function of X(t) is givenby

mX(t) = E[X(t)] =sin(2πf0t)

2πf0t

2. Estimate the mean function of X(t) using ensembleaveraging in the case where f0 = 0.01

mX(t) =1

1000

1000∑i=1

X(t, ξi), t = −300, . . . , 300

3. Plot the obtained estimates and compare them with thetheoretical mean function.

Exercise 18

Generate 400 samples of the first-order autoregressiveprocess governed by

Y (t) = 0.99Y (t − 1) + X(t), t = 1, 2, . . .

where Y (0) = 0 and X(t) denotes Gaussian white noisewith mean 0 and variance σ2

X = 1.

1. Depict on one plot 100 realizations of Y (t), t ∈ [1, 400].Starting from what time instant the process can beregarded as wide-sense stationary?

2. Determine analytically the mean function of Y (t) for anyinitial condition Y (0) = y0.

3. Estimate the steady-state autocorrelation function ofY (t) using time averaging

RX(τ) =1

100

300∑t=201

Y (t)Y (t + τ), τ = 0, . . . , 100

4. Estimate the steady-state autocorrelation function ofY (t) using ensemble averaging

RX(τ) =1

100

100∑i=1

Y (t, ξi)Y (t + τ, ξi)

t = 201, τ = 0, . . . , 100

5. Plot the obtained estimates and compare them with thetheoretical autocorrelation function

RX(τ) =(0.99)|τ |

1 − (0.99)2

Exercise 19

Consider random telegraph signal governed by

X(t) ∈ {−1, 1}

P (X(t + 1) = 1|X(t) = 1) = 0.99

P (X(t + 1) = −1|X(t) = 1) = 0.1

P (X(t + 1) = 1|X(t) = −1) = 0.1

P (X(t + 1) = −1|X(t) = −1) = 0.99

t = 1, . . . , 200

P (X(0) = 1) = P (X(0) = −1) = 0.5

Estimate and plot RX(τ), τ = 0, . . . , 100. To obtain theestimates use a) time averaging b) ensemble averaging.Plot and compare the obtained results.

Exercise 20

Consider a zero-mean Gausiian white noise X(t) withvariance σ2

X = 1.

1. Evaluate spectral density function SX(ω) of this process.

2. Compute and plot periodogram-based estimates ofSX(ω) using datasets consisting of N = 100, 1000 and10000 samples

PX(ωi, N) =1N

∣∣∣∣∣N∑

t=1

X(t)e−jωit

∣∣∣∣∣2

ωi = 2πi/100 , i = −50, . . . , 50

Compare the obtained results with the true spectraldensity function of X(t).

3. Compute time-averaged periodogram-based estimates

PX(ωi, N) =1M

M∑j=1

P jX(ωi, N)

ωi = 2πi/100 , i = −50, . . . , 50

where P jX(ωi, N) denotes periodogram obtained for the

j-th segment of X(t) of length N . Plot the resultsobtained for N = 100.

Exercise 21

Y (t) is a random process observed at the output of thesecond-order filter

H(z−1) =1

1 + a1z−1 + a2z−2

excited with white Gaussian noise X(t) ∼ N (0, 1) underzero initial conditions. The poles of the filter are given inthe form

z1 = rejφ, z1 = re−jφ

whereA) r = 0.9, φ = π/100B) r = 0.9, φ = π/10C) r = 0.99, φ = π/100D) r = 0.99, φ = π/10

1. Compute the coefficients a1 and a2 in each of the fourcases indicated above.

2. Plot the amplitude characteristics of H(z−1)

A(ωi) =∣∣H(e−jωi)

∣∣ , ωi = πi/1000, i = 1, . . . , 1000

What is rhe influence of r and φ on the shape of theamplitude characteristic of H(z−1) ?