** An Iteration-Based Simulation Method for Getting Semi-**

Symbolic Solution of Non-coherent FSK/ASK System by

Using Computer Algebra Systems

Symbolic Solution of Non-coherent FSK/ASK System by

Using Computer Algebra Systems

**Vladimir MLADENOVIC ^{1}*, Sergey MAKOV^{2}, Viacheslav VORONIN^{2}, Miroslav LUTOVAC^{3}**

^{1 }Faculty of Technical Sciences in Cacak, University of Kragujevac,

Svetog Save 65, Cacak, 32000, Serbia

vladimir.mladenovic@ftn.kg.ac.rs

** Corresponding author*

^{2} Don State Technical University,

Shevchenko 147, Shakhty 346500, Rostov region, Russia

makovserg@yandex.ru, voroninslava@gmail.com

^{3} Singidunum University,

Belgrade, Danijelova 32, Belgrade, 11000, Serbia

lutovac@gmail.com

**Abstract: **The paper presents a development of an IBSM (Iteration-Based Simulation Method) to obtain semi-symbolic solutions of digital telecommunication non-coherent FSK/ASK system by using computer algebra systems. These solutions are applied for derivation of the symbol error probability (SEP) needed for quantitative and qualitative analysis of performances of telecommunication systems in optimization and design of low-complexity implementation into high-complexity structures. Various software tools, used in telecommunications for calculating, designing and analysing, have been developed from the viewpoint of numeric-only algorithms. But, many shortcomings cannot be neglected, such as generating of a great amount of numeric data, losing insight into the phenomenon of investigation, and numerical computation manipulates with numerical values. For these reasons, IBSM provides design, optimization, manipulation, and simplification by introducing a new parameter of iteration. The IBSM is applied to aforementioned system when the uncorrelated and correlated noise and the interferences are present in both upper and lower frequencies, respectively. The impact of number of iterations, number of bits, correlation coefficient and different values of interferences to symbol error probability are derived on semi-symbolic forms. Making use of IBSM gives an opportunity to solve very complex analyses and derive many sophisticated conclusions.

**Keywords: **Closed-form solution, Wolfram language, Rayleigh fading, Software tools, Computer algebra system.

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**CITE THIS PAPER AS**:

Vladimir MLADENOVIC^{*}, Sergey MAKOV, Viacheslav VORONIN, Miroslav LUTOVAC, **An Iteration-Based Simulation Method for Getting Semi-****Symbolic Solution of Non-coherent FSK/ASK System by ****Using Computer Algebra Systems**, *Studies in Informatics and Control*, ISSN 1220-1766, vol. 25(3), pp. 303-312, 2016. https://doi.org/10.24846/v25i3y201604

**1. Introduction
**

The rapid growth of wireless communication can be seen through the increasing number of users and the exploitation of the frequency band in which the information is exchanged. Fixed-to-fixed channels, fixed-to-vehicle channels, vehicle-to-vehicle channels, mobile-to-mobile communications, moving scatterers, vehicular communications are only some of challenges that have become the up to date subject of research in the growing mobile industry [1]. All aforementioned communication models are analysed using statistical methods. One of the phenomena is the Rayleigh fading that occurs in urban environments. It represents the propagation of electromagnetic wave when there is no a dominant component in the propagation, and received signal is sum of reflected signals. Typically, this fading is presented in places with buildings and high objects. Almost all analyses are performed using software tools that are based on the numeric-only algorithms. Using them, many graphic characteristics can be illustrated, but the behaviour of systems and processes, that describe the performance, are impossible to observe. On the other hand, the fact is ignored that numerical computation generates a large amount of data, which may sometimes lead to erroneous results [14]. They may be the result of the finite word length in the records, or errors during shortenings of numbers in fractions for example. These ways do not provide possibility to manipulate with analytical expressions.

To overcome these problems, the use of symbolic processing, which formulates a new method called the iteration-based simulation method, is introduced [10]. Symbolic processing provides advanced solutions to all the existing gaps and pushes the boundaries to new models of survey and design. The primary aspiration is speeding up in design and analysis of low-complexity implementation into high-complexity structures, which are provided by significantly increasing of processor’s power and higher storage space, so the expert-based knowledge is necessary in areas such as theory of systems, signal processing, telecommunications, and software engineering [14]. Methods of symbolic processing are known in the industry in academic fields. They are used as an aid in the design of electronic circuits and systems in the industry. In academic institutions, they are used as auxiliary tools in the classroom. Mathematical operations with errors-free concept can be performed to write simulation code for symbolic processing, even to find errors in the manual execution of published results [9]. Also, symbolic processing can be used to find expression in closed-form, and when a large number of iterations exists, or functions errors occurred. The efficiency of symbolic processing becomes more important if the systems and signals are more complex [6].

xThis paper is structured as follows. In section 2, the state-of-art is presented from viewpoint of kind of simulation and subject of researches. In section 3, the approach of knowledge programming using computer algebra system and Wolfram language is introduced. Section 4 describes the reference model and analysis of communication system. In section 5, the iteration-based simulation method is explained and Wolfram language is used to model it. Code-deriving closed-form solutions of SEP are presented in section 6 by using iteration-based simulation method. Finally, section 7 provides symbolic-based optimization procedure and convergence testing to get closed-form solutions, and transformation to semi-symbolic forms to obtain final results.

**REFERENCES**

- BORHANI, A., PATZOLD, M.,
**Modelling of Vehicle-to-Vehicle Channels in the Presence of Moving Scatterers**, Vehicular Technology Conference, 3-6 Sept. 2012, Quebec City, Canada, pp. 1-5. - BORWEIN, J. M., R. E. CRANDALL,
**Closed Forms: What They Are and Why We Care**, Notices of the American Mathematical Society, vol. 60 (1), 2013, pp. 50-65. - HOGSTAD, B. O., C. A. GUTIERREZ, M. PATZOLD, P. M. CRESPO,
**Classes of Sum-of-Cisoids Processes and Their Statistics for the Modelling and Simulation of Mobile Fading Channels**, EURASIP Journal on Wireless Communications and Networking, vol. 2013(125), 2013, pp. 1-15. - HOGSTAD, B. O., M. PATZOLD, N. YOUSSEF, V. KONTOROVITCH,
**Exact Closed-Form Expressions for the Distribution, Level-Crossing Rate, and Average Duration of Fades of the Capacity of MIMO Channels**, IEEE Transactions on Vehicular Technology, vol. 58 (2), 2008, pp. 1011-1016. - LI, W., NI, Q.,
**Joint channel parameter estimation using evolutionary algorithm**, IEEE ICC, 2010, Cape Town, South Africa, May 2010, pp. 1-6. - LUTOVAC, M., MLADENOVIC, V., LUTOVAC, M.,
**Development of Aeronautical Communication System for Air Traffic Control Using OFDM and Computer Algebra Systems**, Studies in Informatics and Control, vol. 22 (2), 2013, pp. 205-212. - MARTINEZ, W., A. MARTINEZ,
**Computational Statistics Handbook with MATLAB**, Chapman & Hall/CRC, 2002. - MLADENOVIC, V., M. LUTOVAC, M. LUTOVAC,
**Automated Proving Properties of Expectation-Maximization Algorithm Using Symbolic Tools**, Telfor Journal, vol. 4 (1), 2012, pp. 54-59. - MLADENOVIC, V., M. LUTOVAC, D. PORRAT,
**Symbolic Analysis as Universal Tool for Deriving Properties of Non-linear Algorithms–Case study of EM Algorithm**, Acta Polytechnica Hungarica, vol. 11(2), 2014, pp. 117-136. - MLADENOVIC, V., D. MILOŠEVIC,
**A Novel-Iterative Simulation Method for Performance Analysis of Non-Coherent FSK/ASK Systems Over Rice/Rayleigh Channels using the Wolfram Language**, Serbian Journal of Electrical Engineering, vol. 13 (2), 2016, pp. 157-174. - MLADENOVIC, V., M. LUTOVAC,
**Non-coherent FSK/ASK Signal Detection with Two Symbols Correlated Noise**, 2^{nd}IcETRAN 2015, Silver Lake, Serbia, EKI 1.7, June 8-11, 2015, pp. 1-4. - MLADENOVIC, V., M. LUTOVAC,
**Simulation of Symbol-Error Ratio and Energy Efficiency using Computer Algebra Tools: A Case Study of OFDM Receiver**, SINTESA Conferences, Belgrade, Serbia, April 2015, pp. 224-231. - PATZOLD, M.,
**Mobile Radio Channels**, 2^{nd}Chichester, John Wiley & Sons, 2011. - PAVLOVIC, V., V. MLADENOVIC, M. LUTOVAC,
**Telecommunications: Applications, Modern Technologies and Economic Impact**, Book chapter, Nova Science Publishers, New York, USA, 2014. - WOLFRAM, S.,
**An Elementary Introduction to the Wolfram Language**, Wolfram Media, Inc, Campaign, USA, 2015.