Analysis of Error Correction Codes in Unique Word OFDM · sion, the linear convolution with the channel impulse response becomes a circular convolution and also inter symbol interferences

AuthorBernhard Hiptmair

SubmissionInstitute of SignalProcessing

Thesis SupervisorUniv.-Prof. Dr. MarioHuemer

RefereeDI (FH) ChristianHofbauer

January 2016

JOHANNES KEPLERUNIVERSITAT LINZAltenbergerstraße 694040 Linz, Osterreichwww.jku.atDVR 0093696

Analysis of Error CorrectionCodes in Unique WordOFDM

Bachelor’s Thesis

to confer the academic degree of

Bachelor of Science

in the Bachelor’s Program

Information Electronics

2

Abstract

Unique word orthogonal frequency division multiplexing (UW-OFDM) is an im-proved version of the well known modulation method cyclic prefix orthogonal fre-quency division multiplexing (CP-OFDM). The implementation of a known se-quence, the so called unique word (UW) instead of the cyclic prefix improves theperformance of the transmission. The reason is that due to the UW, a certain re-dundancy is introduced in frequency domain, which can be beneficially exploited.In this thesis the impact of different error correction codes in a UW-OFDM sim-ulation chain is analysed and compared against a CP-OFDM system.

For the analysis, three error correction codes were used, a convolution code, a lowdensity parity check (LDPC) code and a Reed Solomon (RS) Code . To evaluatethe impact, these codes have been implemented in existing Matlab frameworks forUW-OFDM and CP-OFDM. For several channel models the bit error ratio (BER)performance was simulated and compared.

Kurzfassung

Unique Word Orthogonal Frequency Division Multiplexing (UW-OFDM) ist eineverbesserte Version des bekannten Modulationsverfahren Cyclic Prefix OrthogonalFrequency Division Multiplexing (CP-OFDM). Die Verwendung einer bekanntenSequenz, genannt Unique Word (UW) anstatt des cyklischen Prafixes verbessertdie Leistung des Verfahrens. Der Grund dafur ist, dass durch die Verwendung desUW eine gewisse Redundanz im Frequenzbereich hinzugefugt wird, die vorteilhaftgenutzt werden kann. In dieser Arbeit wird die Auswirkung von verschiedenenFehlerkorrekturverfahren in einer UW-OFDM Simulation untersucht und mit CP-OFDM verglichen.

Fur die Analyse wurden drei Fehlerkorrekturverfahren verwendet, ein Faltungscode,ein Low Density Parity Check (LDPC) Code und ein Reed Solomon (RS) Code.Um die Auswirkung zu untersuchen wurden diese Fehlerkorrekturverfahren inbestehende Matlab Simulationen fur UW-OFDM und CP-OFDM eingebunden.Fur verschiedene Ubertragungskanal Modelle wurden die Bitfehlerraten (BER)ermittelt und verglichen.

CONTENTS 3

Contents1 Introduction 4

2 Orthogonal Frequency Division Multiplexing 52.1 Principles of OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Cyclic Prefix OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Unique Word OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Error Correction Coding 83.1 Convolutional Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Block Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2.1 Low Density Parity Check Code . . . . . . . . . . . . . . . . 103.2.2 Reed Solomon Code . . . . . . . . . . . . . . . . . . . . . . 11

4 Simulation Results 144.1 Additive White Gaussian Noise Channel . . . . . . . . . . . . . . . 164.2 Frequency Selective Indoor Environment - Channel A . . . . . . . . 174.3 Frequency Selective Indoor Environment - Channel B . . . . . . . . 194.4 Frequency Selective Indoor Environment - Multiple Channels . . . . 20

5 Conclusion 22

1 INTRODUCTION 4

1 Introduction

Data communication is getting more and more important nowadays, because ev-eryone is permanently down- and uploading data with their cell phones, laptops,tablets and so on. So the amount of data which has to be transmitted increasescontinuously.

One significant part of data communication are wireless local area networks (WLAN),with the physical layer defined by the IEEE standard 802.11 [1]. This standardincludes the usage of Orthogonal Frequency Division Multiplexing (OFDM), wherea cyclic prefix is added to every transmitted data block. Due to the cyclic exten-sion, the linear convolution with the channel impulse response becomes a circularconvolution and also inter symbol interferences (ISI) are eliminated. But the cyclicextension with random data can not be used to increase the transmission perfor-mance.

To improve this technology, this cyclic prefix is replaced by a deterministic se-quence called unique word (UW). Due to this known extension, several parame-ters can be estimated and also synchronisation behaviour can be improved at thereceiver [2].

To improve the quality of transmission, error correction coding are normally usedin a digital communication system. Although convolutional codes are well knownand already quite powerful, even more powerful block codes and non binary codeslike low density parity check codes (LDPC) and Reed-Solomon (RS) codes becamemore attractive, because of the rising processing power of chips and the latestresearch.

The goal of this thesis is to analyse the impact of different error correction codesto the improvement of the unique word implementation in Orthogonal FrequencyDivision Multiplexing systems.

2 ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING 5

2 Orthogonal Frequency Division Multiplexing

2.1 Principles of OFDM

In frequency division multiplexing (FDM) the available frequency band is splitup in several sub-carriers (SC). To avoid inter carrier interference, the SCs areseparated, but this is not very efficient. By using orthogonal carriers, the SCs areallowed to overlap by 50 % without any crosstalk between the SCs, leading to ahigh spectral efficiency. This concept is known as orthogonal frequency divisionmultiplexing (OFDM). By introducing a guard interval (GI), the orthogonalitycan be maintained even over dispersive channels. OFDM is an effective paralleldata transmission scheme, which is robust against narrowband interferences, butsensitive to frequency offset and phase offset.

Figure 1 shows a simple OFDM transmission scheme. The scheme can be splitup in three groups, where only the first part is relevant for this thesis. The firstpart covers symbol mapping and forward error correction coding, which is furtherexplained in section 3. The second part is the actual OFDM part including modu-lation, where the data given in frequency domain is transformed into time domainusing inverse discrete Fourier transform (IDFT). That part also includes the guardinterval (GI) insertion at the transmitter side, GI removal and demodulation, us-ing discrete Fourier transform (DFT) at the receiver side. The third part is theRF-modulation/demodulation and transmission over the communication channel[3].

Figure 1: Simple OFDM transmission block diagram.


2.2 Cyclic Prefix OFDM

By extending the OFDM symbol by a guard interval, inter symbol interference(ISI) can be eliminated. When the guard interval time is chosen to be longer thanthe channel impulse response, the symbols can not interfere with each other. Ifthe guard interval in empty, the inter carrier interference (ICI) can still arise andthe subcarriers are not orthogonal any more, therefore a cyclic extension is used[4]. In CP-OFDM the tail of the OFDM symbol is used as the cyclic extension, asshown in Figure 2.

2.3 Unique Word OFDM

In UW-OFDM, known sequences (unique words) are inserted instead of a cyclicprefix. In Figure 2, the transmit data structures of CP- and UW-OFDM are shown.The main difference between these two structures is that the guard interval (GI)is part of the DFT interval in UW-OFDM, but in CP-OFDM it is not. Due tothat the symbol duration in UW-OFDM reduces from TDF T + TGI to TDF T . [2]

CP1 Data CP1 CP2 Data CP2 CP3 ...

TGI TDF T TGI TDF T

TGI TDF T TDF T

UW Data UW Data UW ...

Figure 2: Transmit data structure using CPs (above) or UWs (below).

Unique Word Implementation

NotationLower-case bold face variables indicate vectors, whereas upper-case bold face vari-ables indicate matrices. To distinguish between time and frequency domain vari-ables, a tilde is used to express frequency domain vectors and matrices. FN donatesthe N-point DFT.

To generate a UW-OFDM symbol in time domain, given as x′ = [xTd xT

u ]T withthe unique word forming the tail, two steps are performed.


The first step is to generate a zero UW such as x = [xTd 0T]T in time domain

and relation x = F−1N x to frequency domain. This step, performed in frequency

domain, includes the insertion of zero subcarriers and redundant subcarriers (r).The introduction of zero subcarriers can be described by the matrix B whichconsists of zero-rows at the position of the zero subcarriers. For the generationof the redundant subcarriers, a permutation matrix P is introduced, thereforethe data in frequency domain, can be written as x = BP

[dT rT

]T. Thus the

redundant subcarriers depend on the data vector, where the expression can berewritten as x = BP

[IT TT

]Td = BGd. G can be interpreted as a generation

matrix for the non zero part of the OFDM symbol.

The second step contains the addition of the unique word xu = FN[0T xT

u

]T.

The complete OFDM symbol is then given by x = F−1N (BGd + xu).

Figure 3: Time- and frequency-domain view of an OFDM symbol in UW-OFDM.

3 ERROR CORRECTION CODING 8

3 Error Correction Coding

During transmission of data over a communication channel, errors will naturallyoccur. By appending redundancy, these errors can be detected and/or correctedupon the received data. The addition of redundancy decreases the data rate, butincreases the transmission quality [5] .

3.1 Convolutional Code

A convolutional encoder can be described by shift registers and modulo 2 adders.The content of the shift registers determines the state of the encoder. The codingrate of convolutional codes is given by R = k/n, where n represents the numberof input bits and k the number of output bits.The common encoders are either systematic, or non-systematic or recursive convo-lutional encoders. Figure 4(a) shows a systematic encoder, where the coded outputconsists of the input data and a modulo 2 sum of states. Figure 4(b) displays anon-systematic encoder, where the input data is no longer visible in the codewordand is replaced by a modulo 2 sum of states. The encoder in figure 4(c) though issystematic, but the state of the encoder is a result of the previous state and theinput data, which is called a recursive encoder.

A convolutional code can also be seen as a finite state machine or described aspolynomial, but the most common representation is the trellis diagram. Eachpoint represents a state and each state transition is represented by a line, which islabelled by the output of the encoder. Figure 5 shows an exemplary trellis diagramof a convolutional encoder with an initial state (000).

The most common algorithm for decoding convolutional codes, is the Viterbi al-gorithm. This algorithm is based on the search of the codeword which has theshortest Hamming distance to the received word.

Puncturing

Puncturing is used to obtain different coding rates from one code. It involvesusing an encoder with low coding rate, but transmitting only parts of a codeword.With this technique it is possible to vary the coding rate without changing theencoder and is therefore easy to implement. At the decoder, the bits which arenot transmitted, have to be filled by neutral elements. For example by usingbinary detection (+1 for logical ’0’ and -1 for logical ’1’), the null value is takenas neutral element. For the decoder this neutral element contains no information


(a)

(b)

(c)

Figure 4: Example of a (a) systematic, (b) non-systematic and (c) recursive con-volutional encoder.

about the transmitted bit. Of course, by not transmitting parts of the codeword,the correction capability of the code decreases.

3.2 Block Codes

In block codes, a data block d of k symbols, gets mapped to a codeword c of nsymbols. The ratio k/n is called the coding rate. The dataword as well as thecodeword often take their values out of a Galoise field GFq with q elements. Alinear block code, where the mapping is a linear function, has the feature that the


Figure 5: Trellis diagram of a convolutional encoder.

sum of two codewords form a new codeword and the null word is also a codeword.

Block codes with binary symbols take their elements out of the Galois field GF2and the generation of the codeword can be described by a simple multiplication.Therefore we introduce a code generation matrix G with the size k × n, leadingto cT = dTG. The permutation of the rows or columns of the generation matrixproduces the same set of codewords, which means that the generation matrix isnot unique. As a consequence the generator matrix can be brought in a formG = [Ik P], where Ik is the k × k identity matrix. The code is then systematicand the dataword is directly visible in the codeword.

Now we define a dual linear block code HT, which satisfies that any codeword of thedual code is orthogonal to any codeword of the original code. This is given, whentheir scalar product result to null. This condition leads to 0 = cTHT = dTGHT,hence GHT = 0. That feature is used in the receiver to check if a received datais a valid codeword. The matrix H is called the parity check matrix [5].

3.2.1 Low Density Parity Check Code

Low density parity check (LDPC) codes are powerful codes, which are capableof closely approaching the channel capacity. Their performance is comparable toturbo codes. Due to their iterative decoding algorithms, they are easy to imple-ment. The coding rate is selectable by specifying the shape of the parity checkmatrix. LDPC codes have very sparse parity check matrices and are linear block


codes. In this thesis, I focus on binary LDPC codes. A LDPC code is regular, ifthe column and the row weight, which is equivalent with the number of non zeroelements of them, is constant. Irregular LDPC codes are more difficult to design,but can be optimised more efficient, due to the higher amount of degrees of freedom.

LDPC codes can be represented as a bipartite graph called Tanner graph. Figure6 shows an exemplary schematic of a Tanner graph. A Tanner graph has twodifferent types of nodes, variable nodes ci and parity nodes ei. The n variablenodes represent the bits of the codeword (columns of H), the m parity nodesrepresent the parity constraints (rows of H). Two nodes are connected, if theparity check matrix H contains a 1 on the corresponding position. Bipartite graphscontain cycles, defined by the capability of leafing and returning to the same node,without passing a link twice. This cycles have major influence of the decodingperformance.

Figure 6: Exemplary schematic diagram of a Tanner graph.

For decoding an LDPC code, the belief propagation algorithm is used. In everyiteration, the a priori information is sent from the variable nodes to the paritynodes. Further, the parity nodes compute and return the extrinsic information.This steps are performed till all parity equations are satisfied or the maximumnumber of iterations is reached. For soft decoding, log likelihood ratios are usedas a priori information. The performance of the belief propagation algorithmincreases with the length of the smallest cycle of the graph and gets optimal for acycle free graph [5].

3.2.2 Reed Solomon Code

Reed-Solomon (RS) codes are block codes with non-binary symbols. The coeffi-cients of the datawords and codewords take there values in a Galois field GFq withq = 2m, where each dataword is encoded by m binary symbols. The addition of


two codewords form another codeword and also the cyclic rotation of a codewordleads to another valid codeword, therefore RS codes are called linear and cyclic.RS codes are maximum distance codes with the minimal distance d=n-k+1, whichequals their constriction distance. The length of the codeword of such codes isn=q-1, the length of the dataword is given by k = n-2t with t, the number oferrors, which can be corrected. The generator polynomial is defined by his d-1roots αj, αj+1, αj+2, ..., αj+d−2 and can be written as product of minimal polyno-mials g(x) = ∏d−2

i=0 (x + αj+i). The parameter j is either set to 0 or 1, like forBose Chaudhuri Hocquenghem (BCH) codes. For encoding, the k data symbolsare represented as polynomial D(x) = ∑m−1

i=0 cixi, multiplied by xn−k and divided

by the generator polynomyal.

D(x) · xn−k

g(x) = q(x) + r(x)g(x)

The codeword C(x) is than found by C(x) = M(x)·xn−k +r(x), with the remainderr(x) of the division.

Figure 7 shows a schematic diagram of a RS encoder with generation polynomialg(x) = x4 + α3x3 + α6x2 + α3x+ α10.

Figure 7: Exemplary schematic diagram of a RS encoder.

For decoding RS codes, syndroms Si are calculated. Thus the reminder was addedin the encoding process, a valid codeword is divisible by the generation polynomialand so by any factor of it, without a reminder. If no errors occur, the fallowingequation leads to no syndrom, otherwise the error pattern can be determined outof the syndrom. The following equation shows the mathematical background,R(x) are the received symbols, (x + αi) a minimal polynomial of the generatorpolynomial.


R(x)(x+ αi) = Q(x) + Si(x)

x+ αi

The syndroms do not depend on the data, but on the error pattern, which relievesthe error correction. For more than t = (n-k)/2 errors, the code is exceeded andthe errors can not be corrected [5].

4 SIMULATION RESULTS 14

4 Simulation Results

As mentioned is section 2, we consider two approaches, the cyclic prefix OFDMand the unique word OFDM. In CP-OFDM the symbols were cyclic extended toavoid inter carrier interferences. As reference for the classical CP-OFDM system,the IEEE 802.11a WLAN standard is used. In UW-OFDM the cyclic extensionis replaced by a known sequence, the unique word, which can be used to improvetransmission behaviour. In table 1 the most important parameters of the usedframeworks are confronted. The difference at the number of data subcarriers isdue to the difference in additional subcarriers. In CP-OFDM 4 pilot subcarriersare used for synchronisation, in UW-OFDM synchronisation is done with the re-dundant subcarriers, which also defines the unique word. The second difference isthe symbol duration, hence the guard interval is part of the DFT/IDFT intervalin UW-OFDM. As a result of the unique word, there is also a gain in perfor-mance. We compare three error correcting codes, presented in section 3, aboutthe preservation of this gain.

We take the convolutional code, with generator polynomial (133, 171)oct, as rever-ence and compare a LDPC and a RS code to it. For the LDPC code a codewordlength of 648 bits is used, being aware that LDPC codes become more powerfulthe longer they get. The codeword length of the used RS code is 255 bits. For allsimulations binary phase-shift keying (BPSK) is applied for modulation and coderates of 1/2 and 3/4 are used. For the RS code hard decision decoding is used,whereas for the other code soft decision decoding is applied. For visualisation thebit error rate (BER) over the bit-energy to noise ratio (Eb/N0) is plotted.

Table 1: Main physical parameters of the IEEE 802.11a and UW-OFDM frame-work.

Data IEEE 802.11a UW-OFDMNumber of total subcarriers 64 64Number of zero subcarriers 12 12Additional subcarriers 4 (pilot) 16 (redundant)Number of data subcarriers 48 36Guard interval duration 800ns 800nsFFT/IFFT duration 3.2µs 3.2µsOFDM symbol duration 4µs 3.2µssample frequency 20 MHz 20 MHzModulation BPSK BPSKCode rate 1/2, 3/4 1/2, 3/4


At the transmission of data over a channel multiple effects occur. For examplethere is always a superposition of Additive White Gaussian Noise (AWGN). Thiseffect can mathematically be written as simple addition of a noise vector n withthe same length as the sent data vector s. The received signal r is than given byr = s + n. The noise vector for AWGN is assumed to be a zero-mean complexGaussian random variable with variance σ2

n.

Another major effect is the multipath propagation, which is modelled as a FiniteImpulse Response (FIR) filter with the Channel Impulse Response (CIR) h =[h0 h1 h2 ... hNk−1]T . The length Nk has to be smaller or equal than the length ofthe guard interval to eliminate inter symbol interferences. Each entry of the CIRhas uniformly distributed phase, Rayleigh distributed magnitude and the poweris decreasing exponentially. Figure 8 shows two channel snapshots featuring andelay spread of 100ns. Channel A which involves two deep fading holes within thesystem bandwidth, whereas channel B does not involve such fading holes.

Figure 8: Frequency-domain representation of two multipath channel snapshots(channel A, channel B).


4.1 Additive White Gaussian Noise Channel

The first simulation setup is an AWGN channel, where just noise is added to thesent data, no further multipath disorders are considered. For all simulations BPSKmodulation is used.

Comparison of the Error Correction Codes

Figure 9 shows the BER behaviour of the different error correction codes for theCP-OFDM and the UW-OFDM system. For both cases, three different codes areused, a convolutional, a LDPC and a RS code. The profit of the higher coding rateR=1/2 toward the coding rate R=3/4 is the same for CP-OFDM and UW-OFDM.LDPC codes are characterised by their abrupt fall after a certain Eb/N0 level, whichis the reason why they outperform the convolutional code. A disadvantage of theLDPC codes is the bad behaviour at low levels of Eb/N0, where they do not workproperly. That behaviour leads to a worse BER for the higher coding rate, till aEb/N0 level of 2 dB. The RS code also shows the behaviour of the LDPC code.Hence, hard decision demapping is used instead of log likelihood ratios, the RScode drops 3 dB after the LDPC code.

(a) (b)

Figure 9: AWGN - BER comparison between convolutional, LDPC and RS codefor (a) CP-OFDM, (b) UW-OFDM system.

Comparison of the UW-OFDM and the CP-OFDM system

Figure 10 shows a direct comparison of the CP-OFDM and the UW-OFDM system.In sub-figure 10(a) the comparison for coding rate R=1/2 is presented, coding rate


R=3/4 is presented in sub-figure 10(b). In both cases, the introduction of the UWenhances the BER behaviour of approximately 1 dB for a BER value of 10−4. InAWGN environment all simulated error correction codes show the same amountof gain, due to the unique word.

(a) (b)

Figure 10: AWGN - BER comparison between UW-OFDM and CP-OFDM atdifferent coding rates, (a) coding rate 1/2, (b) coding rate 3/4.

4.2 Frequency Selective Indoor Environment - Channel A

In this simulation, we determine a frequency selective environment. Channel A isa multipath channel involving two deep fading holes in the frequency response anda delay spread of 100ns. A snapshot of the channel is shown in figure 8.


Figure 11 shows the BER behaviour of the different error correction codes for theCP-OFDM and the UW-OFDM system. In both cases, again three different codesare used, a convolutional, a LDPC and a RS code. Due to the difference, the profitof the coding rates is significant, although there is a slight difference for the CP-and the UW-OFDM system, for the convolutional code. The characteristic dropof the LDPC code and the drop of the RS code 3 dB afterwards, is clearly evident.Therefore the LDPC code outperforms the convolutional code.


(a) (b)

Figure 11: Channel A - BER comparison between convolutional, LDPC and RScode for (a) CP-OFDM, (b) UW-OFDM system.


Figure 12 shows a direct comparison of the CP-OFDM and the UW-OFDM system.In sub-figure 12(a) the comparison for coding rate R=1/2 is presented, codingrate R=3/4 is presented in sub-figure 12(b). In both cases, the introduction ofthe UW enhances the BER behaviour of approximately 1 dB for a BER value of10−4, convolutional code and for the LDPC code. At the coding rate of R=3/4the gain of the convolutional code increases with rising Eb/N0 level and also thegain for the RS code slightly differs.

(a) (b)

Figure 12: Channel A - BER comparison between UW-OFDM and CP-OFDM atdifferent coding rates, (a) coding rate 1/2, (b) coding rate 3/4.


4.3 Frequency Selective Indoor Environment - Channel B

In this simulation we determine another frequency selective environment. ChannelB is a multipath channel involving no deep fading holes in the frequency responseand a delay spread of 100ns. A snapshot of the channel is shown in figure 8.


Figure 13 shows the BER behaviour of the different error correction codes for theCP-OFDM and the UW-OFDM system. In both cases, again three different codesare used, a convolutional, a LDPC and a RS code. Due to the difference, the profitof the coding rates significant, although there is a major difference for CP- andUW-OFDM for the RS code. The characteristic drop of the LDPC code and thedrop of the RS code 3 dB afterwards, is clearly evident. Therefore the LDPC codeoutperforms the convolutional code.

(a) (b)

Figure 13: Channel B - BER comparison between convolutional, LDPC and RScode for (a) CP-OFDM, (b) UW-OFDM system.


Figure 14 shows a direct comparison of the CP-OFDM and the UW-OFDM system.In sub-figure 14(a) the comparison for coding rate R=1/2 is presented, coding rateR=3/4 is presented in sub-figure 14(b).


(a) (b)

Figure 14: Channel B - BER comparison between UW-OFDM and CP-OFDM atdifferent coding rates, (a) coding rate 1/2, (b) coding rate 3/4.

4.4 Frequency Selective Indoor Environment - MultipleChannels

In this simulation averaging over 10 000 frequency selective channels is done. Allchannels feature a delay spread of 100ns and different frequency responses. Tocalculate the average, 1000 bytes are sent over each channel.


The BER behaviour of the different error correction codes for the CP-OFDMand the UW-OFDM system is displayed in figure 15. In both cases, again threedifferent codes are used, a convolutional, a LDPC and a RS code. For the CP-OFDM system, as well as the UW-OFDM system, the major characteristic ofthe LDPC and the RS code, the drastic drop is not that obvious. Therefore theconvolutional code outperforms the LDPC code.


Figure 16 shows a direct comparison of the CP-OFDM and the UW-OFDM system.In sub-figure 16(a) the comparison for coding rate R=1/2 is presented, codingrate R=3/4 is presented in sub-figure 16(b). The gain of the UW implementationremains constant for the convolutional code and the LDPC code, but increasessignificant for the RS code.


(a) (b)

Figure 15: Multiple channels - BER comparison between convolutional, LDPCand RS code for (a) CP-OFDM, (b) UW-OFDM system.

(a) (b)

Figure 16: Multiple channels - BER comparison between UW-OFDM and CP-OFDM at different coding rates, (a) coding rate 1/2, (b) coding rate 3/4.

5 CONCLUSION 22

5 Conclusion

The goal of this thesis is to determine the influence of different error correctioncodes in orthogonal frequency division multiplexing systems. Therefore, two sys-tems are used, a classical cyclic prefix OFDM system defined by the IEEE 802.11aWLAN standard as reference, and the novel unique word OFDM system, using aknown sequence instead of the cyclic prefix, which can be used do estimate severaltransmission parameters.

A convolutional code is used as reference to analyse the effect of implementing alow density parity check code or a Reed Solomon code. To determine the behaviourof these codes, the bit error rates are simulated and visualised. First the variationof the codes are compared, then the influence on the whole systems are analysed.

Two different channel models are used. The first one is additive white Gaussiannoise, which always appears by transmitting data. The second model represents afrequency selective indoor environment. The first disturbance is a simple additionof a noise vector to the data vector, the second interference is modelled as a finitechannel impulse response.

Due to the waterfall characteristic of the LDPC code, the convolutional code getsoutperformed by the LDPC code in AWGN environment. The RS code featuresthe same characteristic, but because of the hard decision demapping the RS codeis difficult to compare. The frequency selective environment shows similar be-haviour. For a channel with deep spectral notches (channel A), the advance of theLDPC decreases. In channel environment without spectral notches (channel B)the advantage, the rapid drop of the LDPC code, is more evident. The averagingover frequency selective channels result in a unexpected way. The beneficial be-haviour of the LDPC and the RS code, is not obtained any more. Therefore theconvolution code slightly outperforms the LDPC code.

Due to the implementation of the unique word instead of the cyclic prefix, thereis a certain benefit in performance. Now we are interested in the dependants ofthis benefit based on the error correction code. For the convolutional code and theLDPC code, that benefit is obtained in AWGN as well as in frequency selectiveenvironments. The RS code shows some slight variations, but there is always abenefit recognizable.

5 CONCLUSION 23

The output of the analysis therefore is, that the benefit of the novel UW-OFDM isjust slightly pending on the kind of error correction coding. So the error correctioncode can be chosen to fit the present channel environment best, without effectingthe unique word concept.

LIST OF FIGURES 24

List of Figures1 Simple OFDM transmission block diagram. . . . . . . . . . . . . . . 52 Transmit data structure using CPs (above) or UWs (below). . . . . 63 Time- and frequency-domain view of an OFDM symbol in UW-

OFDM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Example of a (a) systematic, (b) non-systematic and (c) recursive

convolutional encoder. . . . . . . . . . . . . . . . . . . . . . . . . . 95 Trellis diagram of a convolutional encoder. . . . . . . . . . . . . . . 106 Exemplary schematic diagram of a Tanner graph. . . . . . . . . . . 117 Exemplary schematic diagram of a RS encoder. . . . . . . . . . . . 128 Frequency-domain representation of two multipath channel snap-

shots (channel A, channel B). . . . . . . . . . . . . . . . . . . . . . 159 AWGN - BER comparison between convolutional, LDPC and RS

code for (a) CP-OFDM, (b) UW-OFDM system. . . . . . . . . . . . 1610 AWGN - BER comparison between UW-OFDM and CP-OFDM at

different coding rates, (a) coding rate 1/2, (b) coding rate 3/4. . . . 1711 Channel A - BER comparison between convolutional, LDPC and

RS code for (a) CP-OFDM, (b) UW-OFDM system. . . . . . . . . 1812 Channel A - BER comparison between UW-OFDM and CP-OFDM

at different coding rates, (a) coding rate 1/2, (b) coding rate 3/4. . 1813 Channel B - BER comparison between convolutional, LDPC and

RS code for (a) CP-OFDM, (b) UW-OFDM system. . . . . . . . . 1914 Channel B - BER comparison between UW-OFDM and CP-OFDM

at different coding rates, (a) coding rate 1/2, (b) coding rate 3/4. . 2015 Multiple channels - BER comparison between convolutional, LDPC

and RS code for (a) CP-OFDM, (b) UW-OFDM system. . . . . . . 2116 Multiple channels - BER comparison between UW-OFDM and CP-

OFDM at different coding rates, (a) coding rate 1/2, (b) codingrate 3/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

List of Tables1 Main physical parameters of the IEEE 802.11a and UW-OFDM

framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

REFERENCES 25

References

[1] Supplement to IEEE standard for Information technology– telecommunicationsand information exchange between systems– local and metropolitan area net-works – specific requirements: Part 11 : wireless LAN medium access control(MAC) and physical layer (PHY) specifications : High-speed physical layer inthe 5 GHz band. New York, N.Y., USA: Institute of Electrical and ElectronicsEngineers, 1999.

[2] M. Huemer, C. Hofbauer, and J. B. Huber, “The potential of unique wordin ofdm,” Proceedings of the International OFDM-Workshop, Hamburg, Ger-many, pp. 140–144, 2010.

[3] R. Prasad, OFDM for wireless communications systems, ser. Artech Houseuniversal personal communications series. Boston: Artech House, 2004.

[4] R. van Nee and R. Prasad, OFDM for wireless multimedia communications,ser. Artech House universal personal communications library. Boston andLondon: Artech House, 2000.

[5] C. Berrou, Codes and turbo codes, ser. IRIS international series. Paris andNew York: Springer-Verlag Paris, 2010.

[6] M. Huemer, C. Hofbauer, and J. B. Huber, “Non-systematic complex number rscoded ofdm by unique word prefix,” IEEE Transactions on Signal Processing,vol. 60, no. 1, pp. 285–299, 2012.

[7] M. Huemer, A. Onic, and C. Hofbauer, “Classical and bayesian linear dataestimators for unique word ofdm,” IEEE Transactions on Signal Processing,vol. 59, no. 12, pp. 6073–6085, 2011.

[8] M. Huemer, C. Hofbauer, A. Onic, and J. B. Huber, “Design and analysis ofuw-ofdm signals,” International Journal of Electronics and Communications(AEU), vol. 68, no. 10, pp. 958–968, 2014.

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Analysis of Error Correction Codes in Unique Word OFDM · sion, the linear convolution with the channel impulse response becomes a circular convolution and also inter symbol interferences