Analysis of Error Correction Codes in Unique Word OFDM

Transcription

1 Author Bernhard Hiptmair Submission Institute of Signal Processing Thesis Supervisor Univ.-Prof. Dr. Mario Huemer Analysis of Error Correction Codes in Unique Word OFDM Referee DI (FH) Christian Hofbauer January 2016 Bachelor s Thesis to confer the academic degree of Bachelor of Science in the Bachelor s Program Information Electronics JOHANNES KEPLER UNIVERSITÄT LINZ Altenbergerstraße Linz, Österreich DVR

2 2 Abstract Unique word orthogonal frequency division multiplexing (UW-OFDM) is an improved version of the well known modulation method cyclic prefix orthogonal frequency division multiplexing (CP-OFDM). The implementation of a known sequence, the so called unique word (UW) instead of the cyclic prefix improves the performance of the transmission. The reason is that due to the UW, a certain redundancy is introduced in frequency domain, which can be beneficially exploited. In this thesis the impact of different error correction codes in a UW-OFDM simulation chain is analysed and compared against a CP-OFDM system. For the analysis, three error correction codes were used, a convolution code, a low density parity check (LDPC) code and a Reed Solomon (RS) Code. To evaluate the impact, these codes have been implemented in existing Matlab frameworks for UW-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 eine verbesserte Version des bekannten Modulationsverfahren Cyclic Prefix Orthogonal Frequency Division Multiplexing (CP-OFDM). Die Verwendung einer bekannten Sequenz, genannt Unique Word (UW) anstatt des cyklischen Präfixes verbessert die Leistung des Verfahrens. Der Grund dafür ist, dass durch die Verwendung des UW eine gewisse Redundanz im Frequenzbereich hinzugefügt wird, die vorteilhaft genutzt werden kann. In dieser Arbeit wird die Auswirkung von verschiedenen Fehlerkorrekturverfahren in einer UW-OFDM Simulation untersucht und mit CP- OFDM verglichen. Für 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 in bestehende Matlab Simulationen für UW-OFDM und CP-OFDM eingebunden. Für verschiedene Übertragungskanal Modelle wurden die Bitfehlerraten (BER) ermittelt und verglichen.

3 CONTENTS 3 Contents 1 Introduction 4 2 Orthogonal Frequency Division Multiplexing Principles of OFDM Cyclic Prefix OFDM Unique Word OFDM Error Correction Coding Convolutional Code Block Codes Low Density Parity Check Code Reed Solomon Code Simulation Results Additive White Gaussian Noise Channel Frequency Selective Indoor Environment - Channel A Frequency Selective Indoor Environment - Channel B Frequency Selective Indoor Environment - Multiple Channels Conclusion 22

4 1 INTRODUCTION 4 1 Introduction Data communication is getting more and more important nowadays, because everyone 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 increases continuously. One significant part of data communication are wireless local area networks (WLAN), with the physical layer defined by the IEEE standard [1]. This standard includes the usage of Orthogonal Frequency Division Multiplexing (OFDM), where a cyclic prefix is added to every transmitted data block. Due to the cyclic extension, the linear convolution with the channel impulse response becomes a circular convolution and also inter symbol interferences (ISI) are eliminated. But the cyclic extension with random data can not be used to increase the transmission performance. To improve this technology, this cyclic prefix is replaced by a deterministic sequence called unique word (UW). Due to this known extension, several parameters can be estimated and also synchronisation behaviour can be improved at the receiver [2]. To improve the quality of transmission, error correction coding are normally used in a digital communication system. Although convolutional codes are well known and already quite powerful, even more powerful block codes and non binary codes like low density parity check codes (LDPC) and Reed-Solomon (RS) codes became more attractive, because of the rising processing power of chips and the latest research. The goal of this thesis is to analyse the impact of different error correction codes to the improvement of the unique word implementation in Orthogonal Frequency Division Multiplexing systems.

5 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 split up in several sub-carriers (SC). To avoid inter carrier interference, the SCs are separated, but this is not very efficient. By using orthogonal carriers, the SCs are allowed to overlap by 50 % without any crosstalk between the SCs, leading to a high spectral efficiency. This concept is known as orthogonal frequency division multiplexing (OFDM). By introducing a guard interval (GI), the orthogonality can be maintained even over dispersive channels. OFDM is an effective parallel data transmission scheme, which is robust against narrowband interferences, but sensitive to frequency offset and phase offset. Figure 1 shows a simple OFDM transmission scheme. The scheme can be split up in three groups, where only the first part is relevant for this thesis. The first part covers symbol mapping and forward error correction coding, which is further explained in section 3. The second part is the actual OFDM part including modulation, where the data given in frequency domain is transformed into time domain using inverse discrete Fourier transform (IDFT). That part also includes the guard interval (GI) insertion at the transmitter side, GI removal and demodulation, using discrete Fourier transform (DFT) at the receiver side. The third part is the RF-modulation/demodulation and transmission over the communication channel [3]. Figure 1: Simple OFDM transmission block diagram.

6 2 ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING 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 than the channel impulse response, the symbols can not interfere with each other. If the guard interval in empty, the inter carrier interference (ICI) can still arise and the 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, as shown in Figure Unique Word OFDM In UW-OFDM, known sequences (unique words) are inserted instead of a cyclic prefix. 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 to that the symbol duration in UW-OFDM reduces from T DF T + T GI to T DF T. [2] T GI T DF T T GI T DF T CP1 Data CP1 CP2 Data CP2 CP3... T GI T DF T T DF T UW Data UW Data UW... Figure 2: Transmit data structure using CPs (above) or UWs (below). Unique Word Implementation Notation Lower-case bold face variables indicate vectors, whereas upper-case bold face variables indicate matrices. To distinguish between time and frequency domain variables, a tilde is used to express frequency domain vectors and matrices. F N donates the N-point DFT. To generate a UW-OFDM symbol in time domain, given as x = [xd T the unique word forming the tail, two steps are performed. x T u ] T with

7 2 ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING 7 The first step is to generate a zero UW such as x = [xd T 0 T ] T in time domain and relation x = F 1 N 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 which consists of zero-rows at the position of the zero subcarriers. For the generation of the redundant subcarriers, a permutation matrix P is introduced, therefore the data in frequency domain, can be written as x = BP [ dt r T] T. Thus the redundant subcarriers depend on the data vector, where the expression can be rewritten as x = BP [ I T T T]T d = BG d. G can be interpreted as a generation matrix for the non zero part of the OFDM symbol. [ ] T. The second step contains the addition of the unique word x u = F N 0 T xu T The complete OFDM symbol is then given by x = F 1 N (BG d + x u ). Figure 3: Time- and frequency-domain view of an OFDM symbol in UW-OFDM.

8 3 ERROR CORRECTION CODING 8 3 Error Correction Coding During transmission of data over a communication channel, errors will naturally occur. By appending redundancy, these errors can be detected and/or corrected upon the received data. The addition of redundancy decreases the data rate, but increases 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 coding rate of convolutional codes is given by R = k/n, where n represents the number of input bits and k the number of output bits. The common encoders are either systematic, or non-systematic or recursive convolutional encoders. Figure 4(a) shows a systematic encoder, where the coded output consists of the input data and a modulo 2 sum of states. Figure 4(b) displays a non-systematic encoder, where the input data is no longer visible in the codeword and is replaced by a modulo 2 sum of states. The encoder in figure 4(c) though is systematic, but the state of the encoder is a result of the previous state and the input data, which is called a recursive encoder. A convolutional code can also be seen as a finite state machine or described as polynomial, but the most common representation is the trellis diagram. Each point represents a state and each state transition is represented by a line, which is labelled by the output of the encoder. Figure 5 shows an exemplary trellis diagram of a convolutional encoder with an initial state (000). The most common algorithm for decoding convolutional codes, is the Viterbi algorithm. This algorithm is based on the search of the codeword which has the shortest Hamming distance to the received word. Puncturing Puncturing is used to obtain different coding rates from one code. It involves using 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 the encoder and is therefore easy to implement. At the decoder, the bits which are not transmitted, have to be filled by neutral elements. For example by using binary detection (+1 for logical 0 and -1 for logical 1 ), the null value is taken as neutral element. For the decoder this neutral element contains no information

9 3 ERROR CORRECTION CODING 9 (a) (b) (c) Figure 4: Example of a (a) systematic, (b) non-systematic and (c) recursive convolutional 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 n symbols. The ratio k/n is called the coding rate. The dataword as well as the codeword often take their values out of a Galoise field GF q with q elements. A linear block code, where the mapping is a linear function, has the feature that the

10 3 ERROR CORRECTION CODING 10 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 GF 2 and 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, leading to c T = d T G. The permutation of the rows or columns of the generation matrix produces the same set of codewords, which means that the generation matrix is not unique. As a consequence the generator matrix can be brought in a form G = [I k P], where I k is the k k identity matrix. The code is then systematic and the dataword is directly visible in the codeword. Now we define a dual linear block code H T, which satisfies that any codeword of the dual code is orthogonal to any codeword of the original code. This is given, when their scalar product result to null. This condition leads to 0 = c T H T = d T GH T, hence GH T = 0. That feature is used in the receiver to check if a received data is a valid codeword. The matrix H is called the parity check matrix [5] Low Density Parity Check Code Low density parity check (LDPC) codes are powerful codes, which are capable of closely approaching the channel capacity. Their performance is comparable to turbo codes. Due to their iterative decoding algorithms, they are easy to implement. The coding rate is selectable by specifying the shape of the parity check matrix. LDPC codes have very sparse parity check matrices and are linear block

11 3 ERROR CORRECTION CODING 11 codes. In this thesis, I focus on binary LDPC codes. A LDPC code is regular, if the column and the row weight, which is equivalent with the number of non zero elements 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. Figure 6 shows an exemplary schematic of a Tanner graph. A Tanner graph has two different types of nodes, variable nodes c i and parity nodes e i. The n variable nodes represent the bits of the codeword (columns of H), the m parity nodes represent the parity constraints (rows of H). Two nodes are connected, if the parity check matrix H contains a 1 on the corresponding position. Bipartite graphs contain 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 decoding performance. Figure 6: Exemplary schematic diagram of a Tanner graph. For decoding an LDPC code, the belief propagation algorithm is used. In every iteration, the a priori information is sent from the variable nodes to the parity nodes. Further, the parity nodes compute and return the extrinsic information. This steps are performed till all parity equations are satisfied or the maximum number of iterations is reached. For soft decoding, log likelihood ratios are used as a priori information. The performance of the belief propagation algorithm increases with the length of the smallest cycle of the graph and gets optimal for a cycle free graph [5] Reed Solomon Code Reed-Solomon (RS) codes are block codes with non-binary symbols. The coefficients of the datawords and codewords take there values in a Galois field GF q with q = 2 m, where each dataword is encoded by m binary symbols. The addition of

12 3 ERROR CORRECTION CODING 12 two codewords form another codeword and also the cyclic rotation of a codeword leads 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, which equals their constriction distance. The length of the codeword of such codes is n=q-1, the length of the dataword is given by k = n-2t with t, the number of errors, which can be corrected. The generator polynomial is defined by his d-1 roots α j, α j+1, α j+2,..., α j+d 2 and can be written as product of minimal polynomials g(x) = d 2 i=0 (x + α j+i ). The parameter j is either set to 0 or 1, like for Bose Chaudhuri Hocquenghem (BCH) codes. For encoding, the k data symbols are represented as polynomial D(x) = m 1 i=0 c i x i, multiplied by x n k and divided by the generator polynomyal. D(x) x n k g(x) = q(x) + r(x) g(x) The codeword C(x) is than found by C(x) = M(x) x n k +r(x), with the remainder r(x) of the division. Figure 7 shows a schematic diagram of a RS encoder with generation polynomial g(x) = x 4 + α 3 x 3 + α 6 x 2 + α 3 x + α 10. Figure 7: Exemplary schematic diagram of a RS encoder. For decoding RS codes, syndroms S i are calculated. Thus the reminder was added in the encoding process, a valid codeword is divisible by the generation polynomial and so by any factor of it, without a reminder. If no errors occur, the fallowing equation leads to no syndrom, otherwise the error pattern can be determined out of the syndrom. The following equation shows the mathematical background, R(x) are the received symbols, (x + α i ) a minimal polynomial of the generator polynomial.

13 3 ERROR CORRECTION CODING 13 R(x) (x + α i ) = Q(x) + S i(x) x + α i The syndroms do not depend on the data, but on the error pattern, which relieves the error correction. For more than t = (n-k)/2 errors, the code is exceeded and the errors can not be corrected [5].

14 4 SIMULATION RESULTS 14 4 Simulation Results As mentioned is section 2, we consider two approaches, the cyclic prefix OFDM and the unique word OFDM. In CP-OFDM the symbols were cyclic extended to avoid inter carrier interferences. As reference for the classical CP-OFDM system, the IEEE a WLAN standard is used. In UW-OFDM the cyclic extension is replaced by a known sequence, the unique word, which can be used to improve transmission behaviour. In table 1 the most important parameters of the used frameworks are confronted. The difference at the number of data subcarriers is due to the difference in additional subcarriers. In CP-OFDM 4 pilot subcarriers are used for synchronisation, in UW-OFDM synchronisation is done with the redundant subcarriers, which also defines the unique word. The second difference is the symbol duration, hence the guard interval is part of the DFT/IDFT interval in UW-OFDM. As a result of the unique word, there is also a gain in performance. We compare three error correcting codes, presented in section 3, about the preservation of this gain. We take the convolutional code, with generator polynomial (133, 171) oct, as reverence and compare a LDPC and a RS code to it. For the LDPC code a codeword length of 648 bits is used, being aware that LDPC codes become more powerful the longer they get. The codeword length of the used RS code is 255 bits. For all simulations binary phase-shift keying (BPSK) is applied for modulation and code rates 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 the bit error rate (BER) over the bit-energy to noise ratio (E b /N 0 ) is plotted. Table 1: Main physical parameters of the IEEE a and UW-OFDM framework. Data IEEE a UW-OFDM Number of total subcarriers Number of zero subcarriers Additional subcarriers 4 (pilot) 16 (redundant) Number of data subcarriers Guard interval duration 800 ns 800 ns FFT/IFFT duration 3.2 µs 3.2 µs OFDM symbol duration 4 µs 3.2 µs sample frequency 20 MHz 20 MHz Modulation BPSK BPSK Code rate 1/2, 3/4 1/2, 3/4

15 4 SIMULATION RESULTS 15 At the transmission of data over a channel multiple effects occur. For example there is always a superposition of Additive White Gaussian Noise (AWGN). This effect can mathematically be written as simple addition of a noise vector n with the same length as the sent data vector s. The received signal r is than given by r = s + n. The noise vector for AWGN is assumed to be a zero-mean complex Gaussian random variable with variance σ 2 n. Another major effect is the multipath propagation, which is modelled as a Finite Impulse Response (FIR) filter with the Channel Impulse Response (CIR) h = [h 0 h 1 h 2... h Nk 1] T. The length N k has to be smaller or equal than the length of the guard interval to eliminate inter symbol interferences. Each entry of the CIR has uniformly distributed phase, Rayleigh distributed magnitude and the power is decreasing exponentially. Figure 8 shows two channel snapshots featuring an delay spread of 100ns. Channel A which involves two deep fading holes within the system bandwidth, whereas channel B does not involve such fading holes. Figure 8: Frequency-domain representation of two multipath channel snapshots (channel A, channel B).

16 4 SIMULATION RESULTS Additive White Gaussian Noise Channel The first simulation setup is an AWGN channel, where just noise is added to the sent data, no further multipath disorders are considered. For all simulations BPSK modulation is used. Comparison of the Error Correction Codes Figure 9 shows the BER behaviour of the different error correction codes for the CP-OFDM and the UW-OFDM system. For both cases, three different codes are used, a convolutional, a LDPC and a RS code. The profit of the higher coding rate R=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 E b /N 0 level, which is the reason why they outperform the convolutional code. A disadvantage of the LDPC codes is the bad behaviour at low levels of E b /N 0, where they do not work properly. That behaviour leads to a worse BER for the higher coding rate, till a E b /N 0 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 RS code drops 3 db after the LDPC code. (a) (b) Figure 9: AWGN - BER comparison between convolutional, LDPC and RS code for (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

17 4 SIMULATION RESULTS 17 R=3/4 is presented in sub-figure 10(b). In both cases, the introduction of the UW enhances the BER behaviour of approximately 1 db for a BER value of In AWGN environment all simulated error correction codes show the same amount of gain, due to the unique word. (a) (b) Figure 10: AWGN - BER comparison between UW-OFDM and CP-OFDM at different coding rates, (a) coding rate 1/2, (b) coding rate 3/ Frequency Selective Indoor Environment - Channel A In this simulation, we determine a frequency selective environment. Channel A is a multipath channel involving two deep fading holes in the frequency response and a delay spread of 100ns. A snapshot of the channel is shown in figure 8. Comparison of the Error Correction Codes Figure 11 shows the BER behaviour of the different error correction codes for the CP-OFDM and the UW-OFDM system. In both cases, again three different codes are used, a convolutional, a LDPC and a RS code. Due to the difference, the profit of the coding rates is significant, although there is a slight difference for the CPand the UW-OFDM system, for the convolutional code. The characteristic drop of the LDPC code and the drop of the RS code 3 db afterwards, is clearly evident. Therefore the LDPC code outperforms the convolutional code.

18 4 SIMULATION RESULTS 18 (a) (b) Figure 11: Channel A - BER comparison between convolutional, LDPC and RS code for (a) CP-OFDM, (b) UW-OFDM system. Comparison of the UW-OFDM and the CP-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, coding rate R=3/4 is presented in sub-figure 12(b). In both cases, the introduction of the UW enhances the BER behaviour of approximately 1 db for a BER value of 10 4, convolutional code and for the LDPC code. At the coding rate of R=3/4 the gain of the convolutional code increases with rising E b /N 0 level and also the gain for the RS code slightly differs. (a) (b) Figure 12: Channel A - BER comparison between UW-OFDM and CP-OFDM at different coding rates, (a) coding rate 1/2, (b) coding rate 3/4.

19 4 SIMULATION RESULTS Frequency Selective Indoor Environment - Channel B In this simulation we determine another frequency selective environment. Channel B is a multipath channel involving no deep fading holes in the frequency response and a delay spread of 100ns. A snapshot of the channel is shown in figure 8. Comparison of the Error Correction Codes Figure 13 shows the BER behaviour of the different error correction codes for the CP-OFDM and the UW-OFDM system. In both cases, again three different codes are used, a convolutional, a LDPC and a RS code. Due to the difference, the profit of the coding rates significant, although there is a major difference for CP- and UW-OFDM for the RS code. The characteristic drop of 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 13: Channel B - BER comparison between convolutional, LDPC and RS code for (a) CP-OFDM, (b) UW-OFDM system. Comparison of the UW-OFDM and the CP-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 rate R=3/4 is presented in sub-figure 14(b).

20 4 SIMULATION RESULTS 20 (a) (b) Figure 14: Channel B - BER comparison between UW-OFDM and CP-OFDM at different coding rates, (a) coding rate 1/2, (b) coding rate 3/ Frequency Selective Indoor Environment - Multiple Channels In this simulation averaging over frequency selective channels is done. All channels feature a delay spread of 100ns and different frequency responses. To calculate the average, 1000 bytes are sent over each channel. Comparison of the Error Correction Codes The BER behaviour of the different error correction codes for the CP-OFDM and the UW-OFDM system is displayed in figure 15. In both cases, again three different 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 of the LDPC and the RS code, the drastic drop is not that obvious. Therefore the convolutional code outperforms the LDPC code. Comparison of the UW-OFDM and the CP-OFDM system 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, coding rate R=3/4 is presented in sub-figure 16(b). The gain of the UW implementation remains constant for the convolutional code and the LDPC code, but increases significant for the RS code.

21 4 SIMULATION RESULTS 21 (a) (b) Figure 15: Multiple channels - BER comparison between convolutional, LDPC and 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.

22 5 CONCLUSION 22 5 Conclusion The goal of this thesis is to determine the influence of different error correction codes in orthogonal frequency division multiplexing systems. Therefore, two systems are used, a classical cyclic prefix OFDM system defined by the IEEE a WLAN standard as reference, and the novel unique word OFDM system, using a known sequence instead of the cyclic prefix, which can be used do estimate several transmission parameters. A convolutional code is used as reference to analyse the effect of implementing a low density parity check code or a Reed Solomon code. To determine the behaviour of these codes, the bit error rates are simulated and visualised. First the variation of 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 Gaussian noise, which always appears by transmitting data. The second model represents a frequency selective indoor environment. The first disturbance is a simple addition of a noise vector to the data vector, the second interference is modelled as a finite channel impulse response. Due to the waterfall characteristic of the LDPC code, the convolutional code gets outperformed by the LDPC code in AWGN environment. The RS code features the same characteristic, but because of the hard decision demapping the RS code is difficult to compare. The frequency selective environment shows similar behaviour. For a channel with deep spectral notches (channel A), the advance of the LDPC decreases. In channel environment without spectral notches (channel B) the advantage, the rapid drop of the LDPC code, is more evident. The averaging over frequency selective channels result in a unexpected way. The beneficial behaviour of the LDPC and the RS code, is not obtained any more. Therefore the convolution code slightly outperforms the LDPC code. Due to the implementation of the unique word instead of the cyclic prefix, there is a certain benefit in performance. Now we are interested in the dependants of this benefit based on the error correction code. For the convolutional code and the LDPC code, that benefit is obtained in AWGN as well as in frequency selective environments. The RS code shows some slight variations, but there is always a benefit recognizable.

23 5 CONCLUSION 23 The output of the analysis therefore is, that the benefit of the novel UW-OFDM is just slightly pending on the kind of error correction coding. So the error correction code can be chosen to fit the present channel environment best, without effecting the unique word concept.

24 LIST OF FIGURES 24 List of Figures 1 Simple OFDM transmission block diagram Transmit data structure using CPs (above) or UWs (below) Time- and frequency-domain view of an OFDM symbol in UW- OFDM Example of a (a) systematic, (b) non-systematic and (c) recursive convolutional encoder Trellis diagram of a convolutional encoder Exemplary schematic diagram of a Tanner graph Exemplary schematic diagram of a RS encoder Frequency-domain representation of two multipath channel snapshots (channel A, channel B) AWGN - BER comparison between convolutional, LDPC and RS code for (a) CP-OFDM, (b) UW-OFDM system AWGN - BER comparison between UW-OFDM and CP-OFDM at different coding rates, (a) coding rate 1/2, (b) coding rate 3/ Channel A - BER comparison between convolutional, LDPC and RS code for (a) CP-OFDM, (b) UW-OFDM system Channel A - BER comparison between UW-OFDM and CP-OFDM at different coding rates, (a) coding rate 1/2, (b) coding rate 3/ Channel B - BER comparison between convolutional, LDPC and RS code for (a) CP-OFDM, (b) UW-OFDM system Channel B - BER comparison between UW-OFDM and CP-OFDM at different coding rates, (a) coding rate 1/2, (b) coding rate 3/ Multiple channels - BER comparison between convolutional, LDPC and RS code for (a) CP-OFDM, (b) UW-OFDM system Multiple channels - BER comparison between UW-OFDM and CP- OFDM at different coding rates, (a) coding rate 1/2, (b) coding rate 3/ List of Tables 1 Main physical parameters of the IEEE a and UW-OFDM framework

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