Analysis and Performance Evaluation of Convolutional Codes over Binary Symmetric Channel Using...

8/19/2019 Analysis and Performance Evaluation of Convolutional Codes over Binary Symmetric Channel Using MATLAB

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Analysis and Performance Evaluation of Convolutional Codes over BinarySymmetric Channel Using MATLAB

Moussa Hamdan 1 and Abdulati Abdullah 21,2

Electrical and Communications Department, Azzytuna University, [email protected], [email protected]

ABSTRACTThe most common concern of any communicationsystem is the data quality. There exist differentcomponents that can impact the quality of dataduring its conveying over the channel as noise,fading, etc. Forward error correcting codes (FEC)

play a major role for overcoming this noise as itadds a control bits to the original data for errordetection and correction. This paper aims atanalyzing convolutional codes with different ratesand evaluating its performance. Binary phase shiftmodulation (BPSK) scheme and binary symmetricchannel (BSC) model are used. First a convolutionencoder is presented and then additive whiteGaussian noise (AWGN) is added. The paper usesmaximum likelihood mechanism (VetribiAlgorithm) for decoding process. Simulations arecarried out using MATLAB with Simulink tools.Bit error rate (BET) is used as testing parameterand results of system behavior for both coded and

encoded are compared.

Keywords: Convolutional codes, Viterbidecoding, AWGN, Code rate and BPSK

1 INTRODUCTION The fundamental target of communicationsystems is to involve conveying the informationthrough the channel to be received with as lesserror as possible.Digital communications have been adopted to

perform such goal due to their capability of processing data faster than the conventional(analogy) communications and potentiality ofextremely less error rate. One of the majorreasons for the continuous growth in the use ofwireless communication is to increase thecapability to provide efficient communicationlinks to almost any location, at constantly

reducing cost with increasing power efficiency.For this reason, digital communication systemshave experienced quick-replacement in the areaof telecommunications [1].To appreciate the wireless version of a digitalcommunication system, firstly it is necessary toconsider what the essential components of a

digital communications system are. A general block diagram of a digital data communicationsystem is shown below:

Figure 1 Block diagram of digital communication system

In this system, it is primarily concerned withmeasuring the probability of error and themechanisms introduced to minimize it at thereceiver side [2]. The source represents anyentity that contains information to send such as,audio, image, data …etc. Whereas, the sourceencoder, provides digitization and compressionin order to remove the redundant informationthat results in reducing the bandwidth [1].Channel coding evolves adding the controlledredundancy (extra bits) to detect and correct theerrors at the receiver side for instance; linearcodes, convolution codes and turbo codes.Afterward, the baseband signal needs to bemodulated to a conceivable format that matchesthe physical medium. Multiple access is amechanism that involves more than onetransceiver to share the same medium such asTDMA, FDMA CDMA,…etc, then, the

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Proceedings of The Second International Conference on Electrical and Electronic Engineering, Telecommunication Engineering, and Mechatronics, Philippines 2016


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physical channel where the signal typicallyexperiences the distortion components, eitherAdditive White Gaussian Noise (AWGN)

process or multipath.The receiver side performs the complementary

process of the transmitter with the capability ofovercoming the interference produced throughthe channel [1] & [2].

1.1 Coding Theory OverviewCoding theory is a technique used to efficientlyand accurately transfer the information fromone point to another. This theory has beensophisticated for the purpose of severalapplications such as, minimizing noise fromcompact disc recorders, or sending of financialinformation across telephone line, data transferamong many computers in a networks or fromone location in a memory to the central

processor, and information transmission from adistance source such as a weather orcommunications satellite or the Voyagerspacecraft which sent pictures of Jupiter andSaturn to Earth. Therefore, the prior task ofcoding theory is to deal with the issue ofdetecting and correcting transmission errorsresulted from the noise that is introducedthrough the channel [3]. In practice, the controlwe have over this noise is the choice of a goodchannel to be used for transmission and the useof various noise filters to combat certain typesof interference which may be encountered [3].

1.2 Block Codes In block coding, the message is segmented into

blocks, each of which has k bits informationwhich is named dataword. Every singledataword (k) is appended an extra bits calledredundancy (m) to form codeword (v) with alength of n as shown in the following figure .

Figure 2 Datawords and codewords in block coding

Each block of message is expressed by the binary of k - tuple u = (u 1, u 2,... u 3) . In bockcoding, the symbol u is used to express a k-bitmessage instead of the entire informationsequence.

As a result, there are 2K

different possiblemessages. The encoder reforms each message uindependently into an n-tuple by addingredundancy, so that the new form will be as:v= (v 1,v2,…v k ) of discrete symbols called a

code word. Therefore, corresponding to the 2k different possible message, there are 2n different possible code words at the encoderoutput. This set of 2k code words of length n is

block code called (n, k) block code .This ratio R =k/n is called a code rate, and can

be interpreted as the number of information bitsentering the encoder per transmitted symbol[3].

1.3 Error correction mechanismsWireless technology has experienced aconsiderable enhancement in terms of theachievement of the fast deployment at lowexpenses. Whereas, the utilization of poorquality of the channel usually demandsretransmissions, that are administrated by usingAutomatic Repeat Request (ARQ) [4].Although, the ARQ is an advantageous tool tomitigate the errors of the packet, it has somedrawbacks including the increment of the

power expenditure due to retransmission andalso latency. However, since the attention wasgradually being shifted to the forward errorcorrection (FEC), it has provided severaladvantages compared to the use of ARQscheme [3]. One of its advantages is thedevelopment of the performance of the errorcontrol which is obtained by applying the errorcorrection code that is based on adding extra

bits to the original data that is known asredundancy. This redundancy is the principle offorward error correction that provides thecapability of error detection and also correctionat the destination side without the need toretransmission or waiting for acknowledgmentreturn compared to ARQ. Also, in case of

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message. In a convolution coded system, urepresents KL data bit sequence and v refers toa code word consisting of

≜ ++ (2) Where kL is the length of the informationsequence and N is the length of the codeword.The additional nm encoded symbols aregenerated after the last block of information

bits has inserted the encoder [5].It is necessary for the decoder to generate anestimate u of data sequence which is dependson the received sequence r. Equivalently,where there exists a one – to-one correspondence

between the data sequence u and the codewordv, the decoder will has capability of generatingan expectation of the code word v [8]. Cleary u= u only if (

∇ = v). The rule of decoding is

based on a mechanism of selecting an estimatedcode word for each possibly received sequencer . If the code word v was sent, a decoding errorappears if and only if ∇ ≠ v. The conditionalerror probability of the decoder is defined as () ≜∇ ≠⁄ (3)The following formula shows who to calculatethe decoding error probability which is given as

⅀ () (4)

Where P(r) is independent of the decodingruled used since r is generated prior to thedecoding [5]. As a result, the optimal decodingmust reduce () ∇≠⁄ for all r.Since to reduce P is equivalent to rise ∇ / , / (5)That is declined for a given r by selecting as thecode word v which can maximize

(6)Since,

∇

is selected to be the most likely code

word given that r is received. If all datasequences, and all codewords, are equally likely[i.e., p (v) is the same for all v], increasing ⁄ is equivalent to increasing / . Far a DMC, / П (7)Where, for a memoryless channel everyreceived symbol relies just on the

corresponding symbol sent. Consequently, amaximum likelihood decoder (MLD) isobtained when the decoder selects its estimationin order to maximize /[7]. Maximizing

/is equivalent to increasing the log-

likelihood function. ⅀ logv (8)Maximum Likelihood Decoder is then notessentially optimum, in case of the codewordsare not equally likely. Since the conditional

probabilities are / must be weighted bythe code word probabilities to specifywhich of possible codewords has capability ofmaximizing / . On the other hand, forseveral systems, the probabilities of codewordare not exactly recognized at the receiver end,resulting in an optimum decoding becomesimpossible, and an MLD then can be the bestfeasible of decoding rule [7].

2.4 Viterbi decoding of convolution codesChannel decoding is known as the technique of

recovering the received information at thedestination side once sent through the physicalchannel. Sequential decoding and MLD orViterbi decoding are the most common schemesof channel decoding for convolutoinal codes.

Viterbi decoding is used to recover the originalcodeword instead of applying the technique ofcomparing the received data sequence witheach possible sequence which requires a hugenumber of comparisons to be done. Howeverwith (n,k,m) codes, 2 (k+m-1)(L-1) paths exist overthe trellis algorithm, where L is the code framesnumber which has been considered. Thedecoding algorithm of convolution codes infersthe input information values sequence form thestream of the received distorted output symbols.

Three essential families are adopted fordecoding algorithms of convolution codeswhich are, Sequential, Viterbi and Maximum

posterior (MAP). Viterbi decoding techniquesrealizes that there is no need to consider all

paths, instead only 2 m-1 needs to be obtained[9].Viterbi is typically a method that implements

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the Maximum Likelihood decoding. The goalof the Viterbi algorithm is to find thetransmitted sequence (or codeword) that isclosest to the received sequence. As long asthe distortion is not too severe, this will be the

correct sequence [9].The Viterbi decoder block works according tothe maximum likelihood decoding. It meansfinding the most probable transmitted symbolstream from the received codeword.Moreover, The Viterbi decoder defines ametric for each path and makes a decision

based on this metric. The most commonmetric is the Hamming distance metric. Whentwo paths come together on a single node, theshortest hamming distance is kept. The

number of trellis branches is defined as trace back depth [10]. To define a convolution

decoder in MATLAB ®2014 simulation, a poly2trellis function is used to convertconvolution code to trellis description.Trellis = poly2trellis (Constraint Length, CodeGenerator) [10].

3 MODULATION SCHEMEThe modulator must select a wave format withduration of T-sec, which is conceivable for

every encoded output symbol. For binarysystem, the modulator must generate each ofsignals or to represent 0 and 1respectively [11]. Therefore, the optimumselection of signal in case of using widechannel is

sin 2πt+ 0 ≤ t ≤ T (9) sin 2πt 0 ≤ t ≤ T(10)

This is known as Binary Phase Shift-Keying(BPSK) in which the difference between 0 and1 is 180 degree as shown in figure 7. For thistype of modulation the symbol rate is the sameas bit rate.

Figure 7 BPSK modulated waveform corresponding tothe code word v = (11000)

4 AWGN & DSC CHANNELThe AWGN channel is a good model for many satellite

and deep space communication links. It is not a goodmodel for most terrestrial links because of multipath,terrain blocking, interference, etc. However, forterrestrial path modelling, AWGN is commonly used tosimulate background noise of the channel under study, inaddition to multipath, terrain blocking, interference,ground clutter and self-interference that modern radiosystems encounter in terrestrial operation. The Additive white Gaussian noise (AWGN) is

a common model of noise disturbance exists inany communication system, whereas, in case of

the channel experiences the components ofmultipath, interference or terrain blocking,AWGN does not have the capability tointroduce better performance. AWGN is themodel which has become familiar withsimulating background noise for the channelwhich is introduced in this paper. If the signalis sent as s(t)= [ or ], the receivedsignal is

(11)

Where is Gaussian random process with

one-sided power spectral density (PSD).the demodulator must generate an outputcorresponding to the received signal for everyT-second interval [11].This output may be areal number or one of a discrete set of

preselected symbols , according to the design ofthe demodulator. An optimum demodulatoroften consists of a matched filter or correlation

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detector with a sampling switch. For BPSKmodulation with coherent detection the sampledoutput is a real number,

∫ ∫ sin 2t+ . (12) Here, for memoryless channel, the symbols aretreated independently, meant the output of thedetector at an interval will rely on just thesending signal at that interval withoutdependency of the pervious transmission [11].For such case, combining M-ary inputmodulator has the physical channel (DMC). ADMC is totally prescribed by a number oftransition probabilities \ , since0 ≤ ≤1, 0 ≤ k ≤ N-1, where is denoted as aninput symbol of modulator , k represents theoutput symbols of demodulator .

\ is

interpreted as the probability to receive k ifwas sent. If a communication system in

which binary modulator is used (M=2) isconsidered as an example, the magnitude noisedistribution is symmetric, and the demodulatoroutput is quantized to Q=2 levels. In this case, aspecific simple and essential channel model,called the Binary Symmetric Channel (BSC) isused. The probability diagram of transmittingfor a BSC is shown in Figure 8, taking intoaccount that the transition probability P totally

prescribes the channel [11].

Figure 8 Transition probability diagram

When BPSK modulation is applied over anAWGN channel with optimum coherent

detection and binary quantized output, the BSCtransition probability is just the BPSK bit error

probability for equally likely signal given by

(13) Where ≜√ ∫ −∞

This is the complementary error function ofGaussian statistics. An upper bound on Q(X) is

≤ − x ≥ 0 (14)When binary coding is applied, the modulatorhas only binary input (M=2). Similarly, when

binary demodulator output quantization is used(Q=2), the decoder has only binary inputs.In this case, the demodulator is known to have

hard decisions. Majority of coded digitalcommunication systems, either block orconvolution, use binary coding with harddecision decoding which results in simplicity ofimplementation when it is compared to non-

binary systems [11].On contrast , when Q ˃ 2 (or the output is leftun-quantized), the demodulator will perform

soft decisions. Therefore, decoder must acceptmultilevel (or analogy) inputs. Although thiswill result in complexity for decoder toimplement, soft – decision decoding introducesnoticeable performance improvement.The channel is known to have memory whenthe detector has an output at the interval relieson the pervious transmitted signal.A fading channel is better example to describe achannel with memory, since in case ofmultipath transmission there is no

independency from interval to interval.If one encoded symbol is transmitted every Tseconds, then the symbol transmission rate(baud rate) is 1/T. In a coded system, if thecode rate is R= k/n, where k is information bitscorrespond to the transmission of n symbols,then data rate of transmission can be calculatedas R/T bits per second (bps) [12].All communication channels can experiencesignal distortion because of limitations in

bandwidth. To reduce the impact of this

distortion, the channel should have a bandwidthW of roughly 1/2T Hertz (Hz).In an un-coded system (R=1), the data rate is1/T= 2W, and is restricted by bandwidth ofchannel [12].In a binary-coded system, with a code rate

R˂1, the data rate is R/T = 2RW , and is

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declined by the factor R compared to an un-coded system.Hence, to keep the same data rate of un-codedsystem, the binary coded system requires anexpansion in bandwidth by a factor of 1/R. This

is one of the characteristics of binary codedsystems. They therefore need some bandwidthexpansion to keep a constant data rate. If thereis no availability for additional bandwidth, then

binary coding is not feasible, and other ways ofreliable communication must be sought [12].5 DESGIN PARAMETERS ANDMEASURMENTSBit Error Rate (BER) is considered as the mainfactor of assessing and evaluating the code

performance. It is therefore the common parameter that can assess error correctioncapability of the implemented codes in wirelesscommunications. This can be obtained bydefining the ratio of the number of the bits inerror to the total number of the transmitted bits.The bit error rate is impacted by several factorsincluding the noise introduced by the physicalchannel, noise due to a quantization process,code rate (R), the level of the transmitted power(P t) as well as the ratio energy per symbol tonoise that is introduced as ( ) and to measure

the percentage of code rate [13].The BER is demonstrated to have a forward

proportional to the code rate and an inverse proportional to energy per symbol noise ratioand transmitted power level.The process is performed as follows; firstly, theencoder encodes the data with code rate R andsends it through the noisy channel. If thetransmitter power level P t does not vary, thiscan affect the incoming energy per symbol E that reduces to R×E . Hence, the BER istypically measured at the decoder input is

bigger than the BER of the data sent withoutcoding. The increment in BER is treated byapplying a decoder which can correct errors.Minimizing the BER to more orders ofAmplitude can be achieved by selecting thespecific forward error correction code.

Coding gain is another factor can evaluate the performance of the code used. It is achieved bycalculating the difference in BER obtained byapplying error correction codes to that of un-coded transmission [14].

To assess the convolution codes performanceover the noisy channel, an Additive WhiteGaussian Noise (AWGN) channel is used.White noise is a random signal (or process)with a flat power spectral density. Gaussiannoise is the noise that statistically has a

probability density function (PDF) of thenormal distribution. In other words, the valuesthat the noise can take on are Gaussian-distributed. It is most commonly used asadditive white noise to obtain (AWGN) [7].

The addition of Gaussian noise to the encodedinformation is gained by producing Gaussianrandom numbers with wanted energy persymbol to noise ratio.The variance σ2 additive Gaussian noise whichhas the power spectrum of No/2 (Watts /Hz isgiven by 2⁄ (15)If the energy per symbol E s is set to 1,Then:

(16)As a result, the standard deviation is given by:

(17) The incoming symbols are fed to the Viterbidecoder to achieve the data bits. The decodeddata is then compared with the correspondinginput given to the encoder and then BER ismeasured [15].To measure the probability of bit error rate P b of a convolution code, the estimate ofequation is:


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distance of the code. The quantity P d is the pair wise error probability, given by:

[ ] (19)Where R is the code rate and erfc is thecomplementary error function, defined by:

√ ∫ −∞ (20)By calculating the shortest free distance ofthe code, the performance of the convolutioncode can be determined, where dfree is theminimum Hamming distance between anytwo codewords. It is also called free distance[15] & [16].

5.1 Performance AnalysisThe performance analysis generally dependson the error event. Suppose that thetransmitted code sequence is y (D) and thereceived code sequence is y’ (D). Therefore,each sequence determines a unique paththrough a minimal code trellis. These routeswill agree for long periods of time, howeverwill have disagreement over particular finiteintervals. An error event corresponds to one ofthese finite intervals. It starts at the time whenthe path y’ (D) first diverges from the path y(D), and ends when these two paths mergeagain. The error sequence is the difference(over this interval):

′ The BER is used to calculate the coding gain,which is the measurement of the difference

between signal- to-noise ratios (SNR) between

the un-coded systems and the coded system tohit the same level of BER [5]. Convolutioncoding is performed by simulations in

MATLAB ®2014 by following the steps asillustrated in Figure 9.

Figure 9 Convolution Coding Flowchart

Data generation: The information to be

sent over the channel is produced using random integer generator . It produced auniform random distribution integers inthe range of [0, M-1]

Convolution encoding: This is obtained by the recall the function known poly2trellis. It accepts a polynomialdescription of a convolution encoderand returns the corresponding trellisstructure description. The parameters

for the poly2trellis function are“constraint length” and “code genera tor

polynomial”. Adding noise to the transmitted

symbols: The AWGN function addsWhite Gaussian Noise to the channelsymbols produced by the encoder. The

parameters for this function are thecoded symbols, SNR (signal to noiseratio), the state, and the power type(whe ther in “dB” or “li near”).

Decoding: Viterbi algorithm is used toextract convolutionally encodedinformation.

Calculation of error rate: it calculates thenumber of errors, symbols and the bit error rate.

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It calculates the error rate of the receivedinformation by comparing it to a delayed copyof the data sent [14].

6 SIMULATION RESULTS

Computer simulation was applied in order tocalculate and evaluate the BER. Simulationdisplays the result of the error correction byimplementing convolution codes which are

based on Bit Error Rate (BER) performanceover a range of signal-to-noise ratios (Eb/No).The error rates of the received data werecomputed by comparing it to a delayed copy ofthe transmitted data. This comparison (as inTable 1) was done for the encoded system andothers with code rates 1/2 and 1/3 respectively.

Table 1: constraint length and generator polynomialsof 1/2 and 1/3 code rates

Table2: Simulation results

To discuss the above results, it is clearlydemonstrated that an increase in number of

symbols results in an increase in errors.The initial values of the BER for the 1/2 and1/3 codes are 0.05 and 0.0 respectively. Thesedecrease moderately to the end points of0.0408 and 0.06. As compared to the encoded

system which has a BER start point of 0.575and end point of 0.571, this is about 50% morethan the coded system. Monte-Carlo techniquewas used in draw diagrams/ graphs of bit error-rate (BER) versus signal to noise ratio (E b/No)in order to further characterize the performanceof the convolution codes. The simulationresults are shown in the Figure 10 and Figure11. As demonstrated, the vertical axisrepresents the bit error rate (BER)

performance; whereas the ratio of energy bit to

noise spectral density (E b/No) is plotted onhorizontal axis. The ( E b/No) varies between 0and 20 dB. It can be seen that the curvesdepend upon E b /No. The curves show thatBER decreases with an increase in E b/No. thesignificance of the simulation results can beillustrated as follows: With a rate 1/2 and 1/3convolution coding and a constraint length of3, a data signal can be transmitted with at least3 dB less power. This in turn contributes toreducing the transmitter or antenna expenses or

permits increased data rates for the sametransmitter power and antenna size.

Figure 10 Simulation result for1/2, 1/3, 1/4, 1convolution code using BPSK modulation

0 2 4 6 8 10 12 14 16 18 2010

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No (dB)

B E R

Performance for R=[1/2 1/3 1/4 1], K=3 Convolution Code

1/2 code1/3 code1/4 code1 code

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Figure 11 Simulation result for 1/2, 1/3 and uncodedsystem for BPSK

Higher coding rate results in a less BER at theexpense of more bits being processed andtransmitted. When more bits are being

processed, the processing time will be larger.However, a rate 1/2 and 1/3 coding results in anincrease of bandwidth by a factor of 2 and 3respectively. In general, the bandwidthexpansion factor of a convolution code issimply n/k, where k/n is the code rate which isdefined as the ratio of the number of bits in the

convolutional encoder ( k ) to the number ofchannel symbols output by the encoder ( n) in agiven encoder cycle. Slower transmission (rate


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[10] Nabeel Arshad1 and Abdul Basit ,“Implementationand analysis of convolutional codes usingMATLAB”, I nternational Journal of multidisplinarySciences engineering, VOL. 3, NO. 8, August 2012[ISSN: 2045-7057 ] www.ijmse.org 9

[11] A. J. Viterbi, “Convolutional Codes and theirPerformance in Communication Systems”, IEEETransactions on information Theory, Vol. IT-19, No.5, PP 751-772, Oct 1971.

[12] M. Divya,” Bit Error Rate Performance of BPSKModulation and OFDM-BPSK with RayleighMultipath Channel ” International Journal ofEngineeirng and Advanced Technology, vol.2, no.4,April 2013 [ISSN:2249- 8958].

[13] Ali Calhan, Celal Ceken and Ismail Ertruk,“Comparative Perf ormance Analysis of ForwardError correction Techniques used in WirelessCommunications”, Proceedings of the thirdInternational Conference on wireless and MobileCommunications,0-7695-2796-5/07.

[14] Davoud Arasteh, “ Teaching Convolutional CodingUsing MATLAB in Communications SystemsCourse” Proceedings of the 2006 ASEE Gulf-Southwest Annual Conference.

[15] Sneha Bawane and V.V. Gohokar, “ Simulation ofConvolutional Encodes”, IJRET: InternationalJournal of Research in Engineering and TechnologyISSN: 2319-1163 | pISSN: 2321-7308.

[16] Y. Yorozu, M. Hirano, K. Oka, and Y. Tagawa,“Electron spectroscopy studies on magneto -opticalmedia and plastic substrate interface,” IEEE Transl.J. Magn. Japan, vol. 2, pp. 740-741, August 1987[Digests 9th Annual Conf. Magnetics Japan, p. 301,

1982].


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