- •Contents
- •Introduction
- •General direction to carrying out of laboratory work and technical safety rules
- •A rules of laboratory work decoration
- •Laboratory work 1 Study of the determined signals characteristics
- •1.The Procedure.
- •Laboratory work 2 Spectral representation of periodic and acyclic signals
- •Short theoretical data
- •3. Approbation of the received results
- •Test questions
- •List of references
- •2.The procedure.
- •3.Test Questions
- •List of References
- •Laboratory work 3 Characteristics of random processes (signals, noises)
- •Short theoretical data.
- •The Procedure.
- •Approbation of the received results
- •4.Test Questions
- •Laboratory work 4 Methods of quantization and discretization of signals
- •1. Short theoretical data
- •1.1. Discretization
- •1.2. Fourier Fast transform.
- •1.3. Quantization
- •2.Study of processes of discretization and quantization
- •2.1. Discretization.
- •2.2. Quantization
- •3. Approbation of the received results
- •4.Test questions
- •5.List of References
- •Laboratory work 5
- •Information characteristics of sources of messages and communication channels
- •1.Short theoretical data
- •2.Solution of standard examples
- •3. Tasks for independent work
- •4.Approbation of the received results
- •5.Test questions
- •Laboratory work 6 Effective coding of messages. Codes of Shannon – Fano, Haffman.
- •1.Short theoretical data.
- •2. The procedure.
- •3. Approbation of the received results
- •4. Test questions
- •Laboratory work 7 Noiseproof coding. HAmming Code
- •2. The procedure.
- •2. Approbation of the received results
- •3.Test questions
- •Laboratory work 8 Cyclic codes
- •1.Short theoretical data
- •Coding with the using of cyclic codes
- •2.The Procedure.
- •2.1. Research of data transform with binary symmetric channel and bch code with fixing length of coding combination..
- •2.2. Research of a data transmission system with the bch codes when using the Gaussian channel
- •3.Approbation of the received results
- •4.Test questions
- •Laboratory practical work for the discipline “the applied theory of information”
3.Test questions
1. A characteristic of binary symmetric communication channel.
2. a principle of distortion probable valuation of coding sequence in the channels with independent errors.
3. Order of parameter task information source of modeling.
4. Order parameter task of modeling coder (decoder ).
5. Order parameter task of communication channel.
6. Order error appearance in modeling communication channel.
7. Order of set of virtual oscillograph.
8. A principle of analysis and installation of time modeling
9. Principle of element synchronization of imitative modeling.
List of References
1. Лазарев Ю. MatLAB 5.X / Ю. Лазарев. – Киев:, 2000.– 383 с.
2. Васильев К. К. Основы помехоустойчивых кодов / К. К. Васильев, Л. Я. Новосельцев, В. Н. Смирнов. – Ульяновск: УлГТУ, 2000.
S.M.Moser, Po – Ning Chen. A Student’s Guide to Coding and Information. Theory. Cambridge University Press, 2012.
Laboratory work 8 Cyclic codes
The Purpose: Study of cyclic codes on an example of BCH codes.
1.Short theoretical data
The name of the codes came from their property contained in that that every coding combination can be received by cyclic transposition of symbols of combination belonged to this code. It means if for example combination is allowed a0 a1 a2 … an - 1 is allowed, then combination an - 1a0 a1 a2 … an - 2 is also allowed. Cyclic codes usually have polynomial representation, knowing, that binary coding elements of word a0 a1 a2 … an - 1 is interpreted as coefficients of polonium of same fictive variable х: f(x)=an - 1xn - 1 + an - 2xn - 2 +…+ a2x2+a1x + a0
For example password 11010 is presented in the form of polynomial x4+x3+x.
The maximum degree х in composed with non zero coefficient is called polynomial degree. Therefore in polynomial n - 1 degree the highest coefficient an - 1 is equal to 1.
Actions on code combinations resolve into actions on polynomial. At that the operations of resolving is carried out by module 2. A lot of such polynomials and actions on them form a Gallus field of order 2 (means GF(2)).
Cyclic move of polynomial coefficient is received by multiplying of polynomial on x with simultaneous composition with binomial xn+1. In fact let
f(x)=1•xn - 1+an - 2xn - 2 +…+a2x2+a1x+a0.
Then f(x)•x + (xn+1) = xn+an - 2xn - 1+ … +a2x3+a1x2+a0x + xn+1 =
= an - 2xn - 1+ … +a2x3+a1x2+a0x +1
Consequently, if you take some polynomials as input g(x), then a process of permitted code combinations can be presented as following
u1(x) = g(x)
u2(x) = g(x)∙x + (xn+1)
u3(x) = u2(x)∙x + (xn+1) = g(x)∙x2 + (xn+1)(x+1)
…
un(x) = g(x)∙xn - 1 + (xn+1)(xn - 2+xn - 3+…+1)
A construction by such method polynomial is called generating. If to require that polynomial g(x) was divider of binomial (xn+1), that all permitted combinations attain a property of divisibility on g(x). It follows from that it is easy to check if the combination is permitted. It is enough to check its divisibility on g(x).
The main properties of cyclic codes are:
Ve .In cyclic (n,k) - code every code polynomial must have no more degree than n - 1.
2.It is existed polynomial g(x) of degree (n - k) form
g(x)=xn - k+gn - k - 1xn - k - 1+…+g2x2+g1x+1,
called generating polynomial of code.
Every polynomial f code u(x) is short является кратным g(x) tuple, ie u(x) = m(x)•g(x).