Author Bernhard Hiptmair Submission Institute of Signal Processing Thesis Supervisor Univ.-Prof. Dr. Mario Huemer Referee DI (FH) Christian Analysis of Error Correction Hofbauer Codes in Unique Word January 2016 OFDM Bachelor’s Thesis to confer the academic degree of Bachelor of Science in the Bachelor’s Program Information Electronics JOHANNES KEPLER UNIVERSITAT¨ LINZ Altenbergerstraße 69 4040 Linz, Osterreich¨ www.jku.at DVR 0093696 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 the performance 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 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 Pr¨afixesverbessert die Leistung des Verfahrens. Der Grund daf¨urist, dass durch die Verwendung des UW eine gewisse Redundanz im Frequenzbereich hinzugef¨ugtwird, 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. F¨urdie 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 f¨urUW-OFDM und CP-OFDM eingebunden. F¨urverschiedene Ubertragungskanal¨ Modelle wurden die Bitfehlerraten (BER) ermittelt und verglichen. CONTENTS 3 Contents 1 Introduction 4 2 Orthogonal Frequency Division Multiplexing 5 2.1 Principles of OFDM . .5 2.2 Cyclic Prefix OFDM . .6 2.3 Unique Word OFDM . .6 3 Error Correction Coding 8 3.1 Convolutional Code . .8 3.2 Block Codes . .9 3.2.1 Low Density Parity Check Code . 10 3.2.2 Reed Solomon Code . 11 4 Simulation Results 14 4.1 Additive White Gaussian Noise Channel . 16 4.2 Frequency Selective Indoor Environment - Channel A . 17 4.3 Frequency Selective Indoor Environment - Channel B . 19 4.4 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 increases continuously. 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 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 exten- sion, 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 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 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. 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 modu- lation, 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, us- ing 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. 2 ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING 6 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 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 2. 2.3 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 TDF T + TGI to TDF T . [2] TGI TDF T TGI TDF T CP1 Data CP1 CP2 Data CP2 CP3 ... TGI TDF T TDF 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 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 donates the N-point DFT. 0 T T T To generate a UW-OFDM symbol in time domain, given as x = [xd xu ] with the unique word forming the tail, two steps are performed. 2 ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING 7 T T T The first step is to generate a zero UW such as x = [xd 0 ] in time domain −1 and relation x = FN ˜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 h iT the data in frequency domain, can be written as ˜x = BP d˜T ˜rT . Thus the redundant subcarriers depend on the data vector, where the expression can be h iT rewritten as ˜x = BP IT TT d˜ = BGd˜. G can be interpreted as a generation matrix for the non zero part of the OFDM symbol. T h T Ti The second step contains the addition of the unique word ˜xu = FN 0 xu . −1 ˜ The complete OFDM symbol is then given by x = FN (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 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 convo- lutional 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.