U NIVERSITY of V AASA Communications and Systems Engineering Group TLTE.3120 Computer Simulation in Communication and Systems (5 ECTS) timan/TLTE3120

UNIVERSITY of VAASA Communications and Systems Engineering Group

TLTE.3120 Computer Simulation in Communication and Systems (5 ECTS)

http://www.uva.fi/~timan/TLTE3120

Lecture 1 – 9.9.2015

Timo Mantere

Professor, Embedded systemsUniversity of Vaasa

http://www.uva.fi/[email protected]

1

http://www.uva.fi/~timan

mailto:[email protected]


Lectures & Excercises

Lectures Wednesdays 14-17 in Technobotnia class TF4104

Week 38 Thursday 17.9. 8-11 … Ends week 44

Exercises Thursdays 15-18 TF4104

Week 38 Thursday 17.9. 13-16 … Ends week 45

Participating is not obligatory

Due duties in Seinäjoki some classes might have to be rescheduled

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Lecture 1

Information about Course arrangements & schedule What is Matlab What is Simulink What is Simulation Basics of modelling

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Course Website

Follow the page continuously – every piece of course material will be added there Lectures Exercises Homeworks Important announcements Course results (announcement is e-mailed over Oodi, when

results are available)

http://www.uva.fi/~timan/TLTE3120

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General

Credits: 5 ECTS Prerequisites: one can take this course without prior

knowledge of Matlab or Simulink

Passing the course Passing the mandatory exercises, homeworks and final exam

50% Final homework ( “Project”) 50% Final examination 0% Homeworks 0% Class exercises

Grading Usual 1-5 or failed Based on points collected from mandatory exercises,

homeworks and final exam

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Course Contents

Getting started with MATLAB and SIMULINK Writing simple programs with Matlab Design simple models with Simulink Matlab and Simulink toolboxes Matlab as a part of the simulation environment Use of Matlab in Control and in Communications Possibly other simulation environments (Labview, Comsol,

etc.) are briefly reviewed Combining Matlab and C (Optional, if course time is

sufficient)

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Matlab-Imitating Freeware/Shareware Policy

The use of any kind of Matlab-imitating freeware/shareware such as Octave etc. is NOT allowed in this course

The use of any other licensed mathematical software, such as Maple, Mathematica or Mathcad etc., is NOT allowed in this course

Exercises or homeworks solved by using some other software than Matlab & Simulink are automatically ignored

We have an up-to-date Matlab license in the University of Vaasa, and the computer classes in Technobothnia are available

7


Why Matlab?

It is not a toy The most common program for mathematical computation and

simulations in professional use

A huge amount of toolboxes

Compatibility – the program is de facto standard in professional and in research use File exchange Huge amount of Matlab-software including additional toolboxes

available (one must be careful, though)

8


What is Matlab?

MATLAB (short for matrix laboratory) is a company run by the company MathWorks

It is computer software meant for numerical computations, it contains also it’s own programming language.

MATLAB provides opportunities for easy and efficient handling of matrices, functions, and visualization of data, implementation of algorithms, creation of user interfaces and interaction with the programs created with other languages.

In addition to the basic configuration of the software, there is available a number of additional tools (toolboxes), which allows specific areas of functionality, for example, or the possibility of symbolic computation.

9


What is Simulink?

Simulink is a additional program for MATLAB that can be used to model, analyze and simulate dynamic systems in various technical areas.

Systems can be linear or non-linear and can be time dependent. The time dependence can be discrete, continuous, or a mixed

type. Simulink is a graphical user interface that allows the model

to be created using block diagrams, which consist of blocks and signal lines.

Simulink block diagrams may contain a variety of mathematical operations, MATLAB code and the inputs and outputs.

Block diagrams are hierarchical, allowing both top-down and a bottom-up approach, when the model is created.

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What is Simulink?

Common Simulink toolboxes include e.g. Communication System Toolbox Control System toolbox DSP System Toolbox Fuzzy logic toolbox Neural Network Toolbox 3D Animation Toolbox System Identification Toolbox Etc.

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Learn Matlab by yourself

There are Matlab training courses offered by Mathworks

https://se.mathworks.com/

https://matlabacademy.mathworks.com/R2015a/?s_tid=hp_mi_mlacademy_0515

https://se.mathworks.com/programs/nrd/buy-matlab-student.html?ref=ggl&s_eid=ppc_15474673282&q=%2Bmatlab%20%2Bstudent

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https://se.mathworks.com/







Matlab

Many Matlab codes and Simulink systems are created all around the world and many have given their codes and models for the free use of others via Matlab file exchange (and other webpages) http://www.mathworks.se/matlabcentral/fileexchange/

Matlab examples are available in http://www.mathworks.se/products/matlab/examples.html http://www.mathworks.se/help/matlab/examples.html

Matlab tutorial videos http://www.mathworks.se/products/matlab/videos.html https://www.youtube.com/results?search_query=matlab

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http://www.mathworks.se/matlabcentral/fileexchange/

http://www.mathworks.se/products/matlab/examples.html

http://www.mathworks.se/help/matlab/examples.html

http://www.mathworks.se/products/matlab/videos.html

https://www.youtube.com/results?search_query=matlab


Simulation

Simulation or simulating is an imitation of reality. The reality is the world around you. Man simulate reality in their

own world of imagination. An important and growing area of simulation is computer

simulation. Computer simulations are used by building an artificial reality in the computer that tries to mimic the real reality, as far as it can.

The simulation of the reality include, child play, many games and playing, imagination and the simulation of reality in physical or other models and their calculation (solar eclipses, exsosolar planet finding models etc.). In humans and mammals, a significant imitation of reality is a

game. Simulation is often cheaper than the reality (flight simulator

training), safer than the real situations (for a nuclear power plant simulator, playing war, a war game), and is often the only way to get information (atmosphere simulator to the greenhouse effect).

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The basics of modeling

Modeling of the systems may have started a long time ago, at least from the Isaac Newton’s differential equations which he used to model physical systems

Usually computer simulations require coal to which the simulation aims or physical reality that the simulation tries to mimic

Typically we need to think about To what the model is used for The profitability of the modeling Users of the model What is modelled and calculated and how accurately How detailed model is needed

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Usage of the model Model can be used for several purposes, e.g. to

predict the system behavior (weather, economy, population growth, climate change etc.)

Basic process model uses are To improve the existing systems Changing how the system is used Improve control system To design a new process To design a new manufacturing system or equipment

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Profitability of the model In industry the profitability of the modeling and

simulation have to be justified Often simulation results are abstract and the

company lead requires proofs that simulation brings economic benefits and cost savings

Simulation is often used for decreasing the bugs in the system in advance

Profits of the model may include Direct benefits To increase the knowledge about the process Better control increases output Teaching the users to use system with the help of simulator

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The users of the model In industry models are used by designers and

development managers Simulations could be used to introduce the idea for

management And for teaching staff how to use a new system To introduce the system to the customers

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How detailed model is needed?

If we try to define the values for different components in the system we may need very accurate model E.g. fluid flow systems, building structure strenght

etc. (often static models)

When planning control systems models may be less accurate, but dynamic.

Models may contain transfer functions and state models and differential equations

Control systems usually require feedback loops

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Modeling project consist of Preplanning

Delimit, time table, resources, experiments Selecting the variables

What model is used for, theory, measurements needed Designing the model structure

Linear, nonlinear, real-time, state model, discrete, continuous

Designing the test plan (scenarios) Doing the experiments Analyzing the results To improve the model and testing

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Numeric calculus

Matlab is an Numerical calculation system Numerical analysis is the branch of mathematics that seeks

to find the approximate solution of mathematical problems that can not or need not be solved accurately.

Numerical methods are often applied in the needs of physics, for example, to solve the differential equations. Numerical methods such as equations, numerical solving of function, approximation and numerical integration.

An example of numerical analysis methods is forking which aims to find a continuous function zeros. Forking algorithm is as follows: Select number interval within which to examine the function,

and that the end points of the function has a different sign values . (* if search zeros)

Shortening of the interval at either end, so that the function gets a head to tail values of the endpoints of the interval.

Repeat until the desired accuracy is achieved.

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Numerical and Symbolic Computation

Example:

Symbolic computation computation in closed form by using mathematical symbols values of the variables are irrelevant (no rounding errors) great influence in development of mathematical theories programs supporting symbolic calculus can handle algebraic

expressions and calculate e.g. integrals and derivatives using symbols

sin 0.70714

Numerical computation:1

sin4 2

Symbolic computation:

1

0

1.7183xe dx 1

0

1xe dx e

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Numerical and Symbolic Calculus

Numerical calculus: e.g. pocket calculator: results are always approximate values

depending on the calculator’s internal computation accuracy risk: is the computation accuracy sufficient? numerical methods are efficient in solving complex problems it’s not even possible to solve all problems in closed form -

numerical calculus is needed (sometimes iterative) many integrals are difficult to solve in symbolic form whereas

numerical method would be more straightforward and faster therefore many practical applications (mathematical modeling,

simulation) usually exploit numerical calculus

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The development of computer programs started at 1970’s the first programs were developed for

numerical calculus due to limited computing power

In the beginning it was difficult to use these programs operation based on programming e.g. Fortran (calculation of derivative

using parable interpolation) Due to increased computing power

graphical features and user-friendliness have become more popular

subroutine deriv1(h, y, dy, n)integer nreal h, hp, y(n), dy(n)hp = 0.5 / hC endpointsdy(1) = hp * (-3*y(1) + 4*y(2) - y(3))dy(n) = hp * (y(n-2) - 4*y(n-1) + 3 y(n))C midpointsdo i = 2,n-1dy(i) = hp * (y(i+1) - y(i-1))end doreturnend

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It may not be possible to use numerical calculus program in real time computation in some process to be controlled reason: program intended for numerical computation is not

efficient enough in real time solution (if needed): implementation using C-programming

language programs are efficient in development because some of the

needed functions are already programmed

Some programs intended for symbolic calculus also support numerical calculus and vice versa

Symbolic calculus: Mathematica Maple MathCad

Numerical calculus: Matlab Excel Octave MATRIXx

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Lecture 2

Basics of Matlab Number representation Matrix operations Matlab data types Complex numbers in Matlab

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Number Representation,Computational Accuracy

27


Bit and Byte

Computer’s memory handles numbers in bits and bytes

How are bits and bytes interpreted? e.g. integer data type

7

0

2iii

d b

0 1 2 3 4 5 6 71 2 1 2 0 2 0 2 0 2 0 2 1 2 0 2

1 2 0 0 0 0 64 0 67.

d

1000010 1

Byte = 8 bits

Bit

7 6 5 4 3 2 1 0

1000010 1

Byte = 8 bits

Bit

7 6 5 4 3 2 1 0

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Fixed and Floating Point Numbers

Fixed point numbers fixed amount of digits / number every number reserves memory space like an integer data type

(1-4 bytes) using four digits one can handle range [0, 9999] dot defines the accuracy of the number (12.81 versus 912.6)

Floating point numbers data type single represented in picture(4 bytes, range about

±10^38) exponent defines the order of magnitude and mantissa the

decimal part huge range can be represented in a very compact form (vs.

fixed point numbers) 012345678910111213141516171819202122232425262728293031

s Exponent n Mantissa m

s = sign bit

012345678910111213141516171819202122232425262728293031

s Exponent n Mantissa m

s = sign bit

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Matlab GUI

User Interface (UI) Editor

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Finite Computation Accuracy

Handling numbers as floating point numbers can cause rounding errors in practice

Matlab:

» 1 5 0.2

ans =

0

» 1 0.2 0.2 0.2 0.2 0.2

ans =

5.5511e-017.

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Data Types in Matlab

char numeric

single (4 bytes) double (8 bytes, default data type) uint8 (1 byte) int64 (8 bytes) ...

struct cell

Addition information using command: help datatypes

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Matrices and Character Arrays

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Matrix

Matrix is an array of numbers consisting of horizontal rows and vertical columns

Individual numbers in matrix are called elements: for example element aij refers to the matrix element in row i and column j. Matrixes consisting of only one row or column are called row or column vectors accordingly.

1

2 1 - dimension

(column)vector

n

a

a na

a

11 12 1, 1 1,

12 22 2, 1 2,

,1 ,2 , 1 ,

row

column

matrix

m m

m m

n n n m n m

a a a a

a a a aa

a a a a

n m

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Matrix in Matlab

Matrix is created by feeding the matrix elements one by one inside brackets ( [ ] )

Elements on the same row are separated by white space Rows are separated from each other with a semi-colon (;)

Matlab prints the return values to the command line if the row is not ended with a semi-colon

1 2 3

4 5 6A

>> A = [1 2 3; 4 5 6];

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Matrix in Matlab

Creating vectors in different ways 1st Way

2nd Way

vector = minimum value:step:maximum value 3rd Way

vector = linspace(minimum value, maximum value, number of elements)

a = [0 1 2 3 4 5] >> a = 0:5;

b = [1 3 5 7 9 11] >> b = 1:2:11;

c = [6 4.5 3 1.5 0] >> c = 6:-1.5:0;

a = [0 1 2 3 4 5] >> a = linspace(0, 5, 6);

a = [0 1 2 3 4 5] >> a = [0 1 2 3 4 5];

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Basic Matrix Arithmetic's

Transpose

Summation and subtraction

Multiplication with a scalar value

2 33 2

1 41 2 3

2 54 5 6

3 6

TA A

2 3

1 2 3

4 5 6A

2 3

7 8 9

10 11 12B

2 3

8 10 12

14 16 18A B

1 2 3

4 5 6A

2 3

4 5 6A

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Matrix product The product of matrixes A and B is defined only if it the

following holds for matrix dimensions (dim)

dim , dim , dim A n x p B p x k C n x k

Esimerkki: MATLAB:

2 33 2

1 21 2 3 1 1 2 ( 1) 3 ( 2) 1 2 2 ( 1) 3 0 7 0

1 14 5 6 4 1 5 ( 1) 6 ( 2) 4 2 5 ( 1) 6 0 13 3

2 0

BA

C

» A = [1 2 3; 4 5 6]; » B = [1 2; -1 -1; -2 0]; » C = A*B

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Matrix product is not commutative in general

Product’s zero-rule does not hold for matrix product

AB BA

1 2 1 1 5 1 2 2, ,

3 4 2 0 11 3 2 4A B AB BA

1 2 2 2 2 4, 0,

2 4 1 1 1 2A B AB BA

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Basic Matrix Arithmetic's in Matlab

Transpose using apostrophe (’) Summation and subtraction using plus- (+) and minus (-)

signs

But:

Two signs for product in Matlab: * and .* * is the matrix product .* is a product where matrix elements in the same row and

same column are multiplied with each other (element-wise product)

Same applies for division and involution / and ./ ^ and .^

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*

1 2

3 4

5 6

7 8

19 22

43 50

23 34

31 46

A

B

AB

BA

>> A = [1 2; 3 4];>> B = [5 6; 7 8];

>> A * B;>> B * A;

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*

1 2 3 , 4 5 6

4

1 2 3 5 32

6

1 4 5 6

2 4 5 6 8 10 12

3 12 15 18

T

T

a b

ab

ab

a b

Not possible

>> a * b’;

>> a’ * b;

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.*

1 2 5 6,

3 4 7 8

1 5 2 6 5 12 .*

3 7 4 8 21 32

1 2 3 , 4 5 6

.* 1 4 2 5 3 6 4 10 18

A B

A B

a b

a b

>> A .* B;

>> a .* b;

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Coupling of Matrices

Matrices can be coupled if the dimensions match

1 2 3

1 1 1

2 2 2

3 3 3

1 2, 5 6

3 4

1 2 1

3 4 2

5 6 3

Tx

X x x x

A b

AY x

b

>> x = [1 2 3]’;>> X = [x x x];

>> A = [1 2; 3 4];>> b = [5 6];>> Y = [[A; b] x];

44


The Determinant of a Matrix

The determinant is an important concept in calculation of an inverse of a matrix

The determinant of a matrix is either real or complex number (scalar)

The determinant of a 2 x 2 –matrix is by definition

11 1211 22 12 21

21 22

, det a a

A A A a a a aa a

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11 12 1

21 22 2

1 2

1 det (cofactors)

n

n

n n nn

n n

i j

ij ij

a a a

a a aA

a a a

c A

Inverse of a Matrix

Matrix B is the inverse of matrix A , if The inverse of matrix A is denoted by Calculation rules

AB BA I 1A

11

1 1

1 1 1

1 1

1 ( scalar)

TT

A A

A A

AB B A

A A

1 adj ,

det when det 0

AA

AA

adj TA C

46


/, \, ^-1 and inv

Esimerkki: MATLAB: 1 2

3 4A

» A = [1 2; 3 4];

5 6

7 8B

» B = [5 6; 7 8];

1 3 2

2 1AB

» A/B

tai » A*inv(B)

1 3 4

4 5A B

» A\B

tai » inv(A)*B

1 1 2

2 3BA

» B/A

tai » B*inv(A)

1 5 4

4 3B A

» B\A

tai » inv(B)*A 47


Referring to Matrix Elements

Matrix is the basic data type in Matlab Data to be processed is usually stored in matrices Sometimes it’s of interest to manipulate only part of the data

stored in matrix. This interesting data has to be “picked out” of the matrix Plotting Analysis Data handling etc.

There may be need to remanipulate the data in matrix. In that case assignment is used, e.g. so that the part of the matrix data is replaced with new information

One has to know how to refer to desired matrix elements to be able to manipulate data properly

48



Matrix elements are referred using parentheses first row(s) and then column(s)

The following can be used for referring: number vector end-command colon (:)

1 11 111 1111 11111

2 22 222 2222 22222

3 33 333 3333 33333

4 44 444 4444 44444

5 55 555 5555 55555

A

>> A(2,3)ans = 222

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Referring to individual elements using numbers – eg. A(2,3) It’s possible to refer to many rows/columns using vector

1 11 111 1111 11111

2 22 222 2222 22222

3 33 333 3333 33333

4 44 444 4444 44444

5 55 555 5555 55555

A

>> A(2,[1 3])ans = 2 222

>> A(2:4, 2)ans = 22 33 44

>> A(4, 1:2:5)ans = 4 444 44444

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The end-command is useful when referring to the last matrix element

1 2 3 4

1 11 111 1111 11111

2 22 222 2222 22222

3 33 333 3333 33333

4 44 444 4444 44444

5 55 555 5555 55555

x

A

>> x(end)ans = 4

>> x(end-1)ans = 3

>> A(4, 3:end)ans = 444 4444 44444

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Entire row or column can be referred using colon (:)


1 11 111 1111 11111

2 22 222 2222 22222

3 33 333 3333 33333

4 44 444 4444 44444

5 55 555 5555 55555

A

>> A(:,4)ans = 1111 2222 3333 4444 5555

>> A(2,:)ans = 2 22 222 2222 22222

52



All the above situations can be combined with insertion

>> A(1,:) = [1 2 3 4 5];

>> A(4,4:end) = [8 9];

>> A(1:2:end,2) = 7;

1 2 3 4 5

2 22 222 2222 22222

3 33 333 3333 33333

4 44 444 4444 44444

5 55 555 5555 55555

A

1 2 3 4 5

2 22 222 2222 22222

3 33 333 3333 33333

4 44 444 8 9

5 55 555 5555 55555

A

1 7 3 4 5

2 22 222 2222 22222

3 7 333 3333 33333

4 44 444 8 9

5 7 555 5555 55555

A

53


zeros & ones

Command zeros creates a matrix consisting of nothing but zeros This command can also be used for creating a vector

zeros(3)

zeros(3, 4)

zeros(1,2)

Same applies to command ones with the exception that the matrix contains only number ones

0 0 0

0 0 0

0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0

54


eye

Command eye creates an identity matrix

eye(3)

eye(3,4)

1 0 0

0 1 0

0 0 1

1 0 0 0

0 1 0 0

0 0 1 0

55


Useful Commands

56


Useful Commands

Command Description

min(X) Returns the value (and the index) of the smallest element in vector X. If X happens to be a matrix, then min(X) returns the smallest values of every column (in a row vector).

max(X) Returns the value (and the index) of the greatest element in vector X. If X is matrix, then the greatest values of each column are returned as a row vector.

length(X) Returns the length of the vector (or the matrix).

size(X) Returns the amount of rows and columns in the matrix.

find(condition)

Returns the indices of those elements in which given condition is satisfied.

sort(X) Returns a matrix X in which elements in every column are sorted in ascending order. The indices are also returned in another matrix so the original matrix can be reformed from the return values.

57


Useful Commands

Command Description

clc Clear screen, removes all writings from the command line

clear Clears all the variables currently defined in workspace

clear all Same as previous

clear A Clears the variable A

clear a b Clears variable a and b

who Tells what variables you have in your workspace

58


min & max

A = [1 2 3 4 5 6]; min(A)

ans =

1 max(A)

ans =

6

1 2 3 4 5 6A

1 2 3

4 5 6

1 5 10

B

B = [A(:,1:3); A(:,4:6); [-1 5 10]]; min(B)

ans =

-1 2 3 [m, index] = max(B)

m =

4 5 10

index =

2 2 3

59


length & size

>>A = [5 -1 3; 4 5 6; 8 -3 -5; 9 10 -11];>>length(A)

ans = 4

>>[rows, cols] = size(A)rows =

4cols =

3

5 1 3

4 5 6

8 3 5

9 10 11

A

for row_index = 1:rowsfor col_index = 1:cols

B(row_index, col_index) = row_index*col_index;end

end

1 2 3

2 4 6

3 6 9

4 8 12

B

60


find & sort

A = [5 -1 3 4 5 6 8 -3 -5 9 10 -11]; indext = find(A < 0 | rem(A,3) == 0)

indext = 2 3 6 8 9 10 12

5 1 3 4 5 6 8 3 5 9 10 11A

[B, index] = sort(A)B =

-11 -5 -3 -1 3 4 5 5 6 8 9 10 index =

12 9 8 2 3 4 1 5 6 7 10 11 C = A(index)

C =-11 -5 -3 -1 3 4 5 5 6 8 9 10

61


Solving a system of equations

System of equations

can be presented in matrix form

Solve the unknowns

2 3 6

7 15 8

x y

x y

2 3 6

7 15 8X YA

x

y

1 1 1

I

AX Y A A X A Y IX X A Y

3815 3 6 114 2.241 1 177 2 8 26 26 0.51det 51

51

XA

62


Strings

Strings are char-arrays A string is created by writing the characters between

apostrophes (’)

Multiple strings can be concatenated by using brackets ([ ]) (compare to the concatenaton of matrixes)

Note the addition of a space (’ ’) between string1- and string2-variables

Compare with:

>> string3 = [string1 ’ ’ string2]string3 = Hello world! How do you do?

>> string1 = ’Hello world!’;>> string2 = ’How do you do?’;

>> string3 = [’string1’ ’ ’ ’string’]string3 = string1 string2

63


Strings

Adding a number to the string The number must be transformed into string format by num2str-

function (number to string)

Or with shorter version (note the brackets, separate strings have to be concatenated with each other by using the brackets before they can be given as a parameter to the disp-function)

>> number = 200;>> message = [’I had ’ num2str(number) ’ rats in my car!’];>> disp(message);I had 200 rats in my car!

>> number = 200;>> disp([’I had ’ num2str(number) ’ rats in my car!’]);I had 200 rats in my car!

64


Lecture 3

Basics of Matlab Complex numbers Basics of programming Basics of plotting

65


Complex numbers

A set of complex numbers C is a natural extension of the real numbers. The complex number z is of the form

z = x + yi, where x and y are real numbers and i is the imaginary unit,

which can be labeled i2 = -1. The x of complex number is called a real part (Re (z)) and a number y, respectively, is the imaginary part (Im (z)).

Set of real numbers is a subset of the complex numbers obtained by setting the imaginary part of a complex number to zero: y = 0.

If x = 0, the complex number is called purely imaginary. Originally complex numbers were developed partly because

of the need to get a larger number of the polynomial equations to be solved. For example, the equation x2 + 1 = 0 has no real roots, because

x2 is positive for all real values of x. With complex numbers, however, we can found solutions x = + i and x = -i.

66


Complex numbers

Complex numbers form a set of complex numbers , the numbers belonging to this group are of form

where x and y are real numbers

and

(real part) (imaginary

part)

z

Re

Im

y

x

z x yi

2 1i

Re

Im

z x

z y

67


Complex numbers

Every complex number has

complement

conjugate

magnitude

inverse

Polar angle (argument)

z x yi

z x yi

z x yi

2 2z x y

12 2

1 z x yiz

z zz x y

arg z

z

Re

Im

y

x

φ

-y

-x

φ -y

x

φ

z

z

z

68


The polar coordinate form

Complex numbers can be presented in polar coordinate form

, where

The relations between the polar coordinates (r, φ) and the Cartesian coordinates (x, y) of a complex number are

Euler’s formula

cos sinz r i

ja arg , tan , 0y

r z z xx

cos

sin

x r

y r

cos sin i ir i re z e

69


Complex Matrix

The elements of the matrix can be complex

When dealing with complex matrix hermitting is used instead of transposing. The hermiting is defined as:

So the hermitted matrix is achieved by taking the complex conjugate and transpose

1 1

2 3 4 ( on imaginary unit)

3 1 2

i i

C i i i

i

H TC C

1 2 3 3

1 4 1 2H i iC

i i i

70


Complex matrixes in Matlab

The elements of the matrix can be either real or complex numbers

The imaginary unit is either i or j When giving complex numbers as input note that the real

and imaginary parts must be written without space between them, otherwise they will be interpreted as different numbers.

a = [2+i 3-2i 4i] >> a = [2+i 3-2*i 4*i];

b = [2 i 3-2i 4i] >> b = [2 +i 3-2*i 4*i];

71


Complex matrixes in Matlab

In Matlab matrix C is hermitted with command C’ and transposed with command C.’

>> C = [1+i 1-i; 2-3*i 4*i; 3 -1+2*i]C = 1 + i 1 - i 2 - 3i 0 + 4i 3 -1 + 2i>> C'ans = 1 - i 2 + 3i 3 1 + i 0 - 4i -1 - 2i>> C.'ans = 1 + i 2 - 3i 3 1 - i 0 + 4i -1 + 2i

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Programming and Command String Files (or scripts)

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Introduction

Matlab was first used by entering simple commands from the command line

This is not practical when implementing complex things requiring tens of commands or multiple repetitions of the same task. Reasons for this are: errors are easily made all the commands have to be entered once again when

detecting an error in the beginning of the command series entering commands by hand is very slow there will be no documentation of the entered commands (well,

Matlab command history remembers them for a while)

Programming helps!

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Branching

It’s often useful to make a choise between one or multiple alternatives

% marks a comment

if condition % do something...else % do something else...end

if (x > y) bigger = x;else bigger = y;end

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Creating Conditional Clauses

Relational operators in Matlab:

Example:

Operator Description < Smaller than > Bigger than <= Smaller than or equal to >= Bigger than or equal to == Equal to ~= Not equal to

» 1 < 2ans = 1

» 1 > 2ans = 0

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Note!

These operators are different: == “is equal to” = “is signed a value”

Character strings can not be compared using operator ==. Use strcmp-function instead

» 'cats' == 'dogs'ans = 0 0 0 1

» strcmp('cats', 'dogs')ans = 0» strcmp('cats', 'cats')ans = 1

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Logical operators in Matlab:

Examples: (true and false with small letters are predefined in Matlab)

Operator Description Priority ~ Logical NOT Greatest & Element-wise logical AND | Element-wise logical OR && AND || OR Smallest

» TRUE = 1; FALSE = 0;» TRUE & TRUEans = 1» TRUE & FALSEans = 0

» TRUE | FALSEans = 1» TRUE | TRUEans = 1

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& or &&? Right hand side of the clause is not evaluated if the left hand

side already gives the result of the clause when using &&-operator

Imagine a situation where b = 0 and you shoul calculate a/b (division by zero!)» a = 1;

» b = 0;» x = (b ~= 0) & (a/b > 18.5)Warning: Divide by zero.x = 0» x = (b ~= 0) && (a/b > 18.5)x = 0

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for-loop

for variable = initialValue:step:finalValue % do something...end

for index = -7:2:0 indexend

index = -7index = -5index = -3index= -1

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for-loop

[M, N] = size(X);sum = 0;for column = 1:N for row = 1:M sum = sum + X(row, column); endend

It’s possible to go through every matrix element by using two internal for-loops

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while-loop

while condition % do something...end

element = 1;while (element <= 10) x(element) = element; element = element + 1;end

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for vs. while

for-loop is used when it’s exactly known a priori how many repetitions are to be made E.g. going through every row and column in a matrix number of repetitions does not necessarily have to be a

constant for instance, the size of the matrix to go through can change from

one execution to another

while-loop is used when the number of repetitions is unknown in the beginning of the loop for example, the user feeds an unknown amount of numbers

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Question

How much is x?

Don’t do this at home! If you do CTRL C will help (terminate the run by user)

x = 0;while (2) x = x + 1;end

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break and continue

break-command exits the loop and program execution continues from the line after the loop

command continue forces the loop to start immediately from the beginning

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break and continue

index = 1;while (index < 5) if (index == 3) break; end; index index = index + 1;end;

index = 1;while (index < 5) if (index == 3) index = index + 1; continue; end; index index = index + 1;end;

index = 1index = 2

index = 1index = 2index = 4

Note write index to the comand line, what is the value? Why?

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Good Programming Practice

Program code should be easily read and understood easier to debug the errors easier to remember what the code was intended for after weeks

or months other people can more easily utilize the code in their programs

Code is easily read when: variables have descriptive names (do NOT use scandinavian

letters å, ä, ö) indentation and empty space is used properly it is well commented

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Good Programming Practice

% Initialize the variablesmatrixSize = 5;X = zeros(matrixSize);

% Let’s create a matrix in which diagonal elements% are 1 and elements over diagonal are 2 for row = 1:matrixSize for column = 1:matrixSize

%Checking if we’re on main diagonal if (row == column) % Yes – on main diagonal X(row, column) = 1; elseif (row < column) % Above main diagonal X(row, column) = 2; end endend

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Bad Programming Practice

x=5;X=zeros(x);for y=1:xfor z=1:xif y==zX(y,z)=1;elseif y>zX(y,z)=2;end,end,end

x=5;X=zeros(x);for y=1:x,for z=1:x,if y==z,X(y,z)=1;elseif y>z,X(y,z)=2;end,end,end

These are BAD examples

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Command String Files

Matlab command string file (script, macro) is a text file consisting of consecutive Matlab-function calls (commands). These files are of type *.m and they are called Matlab m-files.

Matlab executes the commands when the name of the command string file is written to the command line. These files utilize the existing variables in the workspace and store new variables created during execution to the workspace. These new variables can be used from the command line after the script execution has stopped.

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Plotting

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Basic Commands

Command Description

plot Plots the function/functions according to the parameters

loglog Otherwise the same as plot with the exception that the scales are logarithmic.

semilogx Logarithmic x-axis, normal y-axis.

semilogy Logarithmic y-axis, normal x-axis.

polar Polar coordinate representation.

Plotyy Same as plot but with a possibility to define two y-axis with different scales.

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Plotting Basic command: plot

First parameter: values in the x-axis

Second parameter: values in the y-axis

Additional parameters to customize the line type and colors.

Empty figure window: figure>> t = (0:0.1:10)';>> y = sin(2*pi*0.2*t);>> figure>> plot(t,y,'r.')>> hold on>> line([0 10],[0 0])>> xlabel(’Time (s)')>> ylabel('\it{sin(2\pift)}')>> title(’Graph')>> grid on

0 1 2 3 4 5 6 7 8 9 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

sin(2

ft)

Graph

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Documents

U NIVERSITY of V AASA Communications and Systems Engineering Group TLTE.3120 Computer Simulation in Communication and Systems (5 ECTS) timan/TLTE3120