1. Communication Theory Lab (1)
Prepared by:
Eng.Lamiaa Daoud

2. Agenda MATLAB Revision Random Variable Definition
Types of Random Variables
Types and Features
Task

3. Variables: ∞ π Syntax Description X=6 Integer Y=[1 2 3 4] Vector
Z=[ ; ]
Matrix
L=3+4i
Complex number
M=‘hello’
String
R=Inf ∞
S=pi π

4. Exponential Functions:
Syntax
Description
a^b ab
exp(a) ea
log10(a)
log10 a
log2(a)
log2 a
log(a)
ln a
sqrt(a)

5. Vectors: Description Output Syntax X=[1 3 5 7 9] X=[x1 x2 x3 x4 x5]
X=start: step: end
Y=X(2)
Y=2nd element in X
Y=3
X(3)=4
Set the 3rd element in X to 4
X=[ ]
Y=X(2:1:3)
Y=2nd to 3rd element in X
Y=[3 4]
Y=X(1:2:end)
Y=1st to last element in X with a step of 2
Y=[1 4 9]
Y=X(5:-2:2)
Y=5th to 2nd element in X with a step of -2
Y=[9 4 ]
Y=X([2 3 5])
Y=2nd 3rd and 5th elements in X
Y=[3 4 9]

6. Y= Conjugate transpose of X in case of complex numbers
Vectors:
Syntax
Description
Output
Y=X.’
Y=transpose of X
Y=[1 3 4 7 9]
Y=X’
Y= Conjugate transpose of X
in case of complex numbers
Y=X.^2
Y=[x12 x22 x32 x42 x52]
Y=[ ]

7. Vectors: Description Output Syntax Y=length(X)
Y=number of elements in X
Y=5
Y=sum(X)
Y=sum of elements of X
Y=24
Y=mean(X)
Y=sum(X)/length(X)
Y=24/5
=4.8
Y=find(X==4)
Y=location of element 5 in X
Y=3
Y=find(X>5)
Y=location of elements >5 in X
Y=[4 5]
Y=find(X>5 & X<9)
Y=location of elements >5 and <9 in X
Y=[4]
Y=find(X<3 |X>5)
Y=location of elements <3and >9 in X
Y=[1 4 5 ]

8. Plotting Description X=[x1 x2 x3 x4 x5…………xn] Y=[y1 y2 y3 y4 y5…………yn]
Syntax
Description
X=[x1 x2 x3 x4 x5…………xn]
Y=[y1 y2 y3 y4 y5…………yn]
plot(X,Y)
Define both vectors then plot
X=-pi:0.01*pi:pi;
Y=sin(X);
Define one vector and define the other as a function of it
xlabel(‘x’)
label the x axis
ylabel(‘sin(x)’)
label the y axis
title(‘Sine Wave’)
Give a title
hold on /hold off
Plot more than one graph in the same figure
figure
Create a new figure
legend(‘str1’,’str2’,…..)
Display a legend

9. Help:

10. Help:
Type the functionality you are searching for

11. Notes:
Matlab is case sensetive which means that X is not equivalent to x.
Typing a ; at the end of every command means not to display the values in the command window after executing the command.
Typing a % before a command means it is a comment and the program will not execute it.

12. Random Experiment
It is an experiment whose outcome cannot be predicted with certainty
Examples:
Tossing a Coin
Rolling a Die

13. Random Variable Definition
A variable that results due to a random process
Example:
Rolling a dice
Tossing a coin

14. Types of Random variables
Discrete Random Variables
The random variable can only take a finite number of values (ex: tossing a coin)
Continuous Random Variables
The random variable can take a continous of values (phase of the carrier)

15. Mean of a Random Variable
Discrete Random Variable
Continuous Random Variable

16. Mean of Discrete Random Variable
Probability mass function P[X=x]
Mean: Weighted average

17. Probability Density Functions: (Continuous random variables)
A probability density function: gives the probability that a random variable x takes on values within a range.
Probability density f(x)
Properties of probability density function:
0 a b 17

18. Variance of the random variable
Variance is a measure of the randomness of the random variable i.e it is a measure of the amount of variation within the values of the random variable.

19. Variance of the random variable
Discrete Random Variable:
Continuous Random Variable:

20. Example: Gaussian Random Variable
Probability density function:
Where
is the mean of the random variable x.
is the variance of x.

21. Gaussian (Normal) RV
m: mean
: variance

22. Task 1
It is required to generate 1000 normally distributed samples with mean 3 and variance 2
Add the previous samples to another 1000 normally distributed samples with mean 4 and variance 3
Get the mean and the variance of the resulted signal
What do you notice ?

23. Variancetotal = Variance1 + Variance2
meantotal = mean1 + mean2

24. Task 2
Plot the probability density functions of the previous example on the same graph
Notice the mean of the plotted functions from the graph

25. Useful Commands
normrnd(mean, standard deviation, number of rows, number of columns)