Design and applications of Filterbank structures implementing Reed-Solomon codes

In nowadays communication systems, error correction provides robust data transmission through imperfect (noisy) channels. Error correcting codes are a crucial component in most storage and communication systems ? wired or wireless ?, e.g. GSM, UMTS, xDSL, CD/DVD. At least as important as the data integrity issue is the recent realization that error correcting codes fundamentally change the trade-offs in system design. High-integrity, low redundancy coding can be applied to increase data rate, or battery life time or by reducing hardware costs, making it possible to enter mass market. When it comes to the design of error correcting codes and their properties, there are two main theories that play an important role in this work. Classical coding theory aims at finding the best code given an available block length. This thesis focuses on the ubiquitous Reed-Solomon codes, one of the major achievements of classical coding theory. Nowadays, their applications span from storage devices (including magnetic tape, CD, DVD, barcodes, etc) over highspeed modems (such as ADSL, xDSL, etc wireless and mobile communications (including cellular telephones, microwave links, etc) to satellite and deep space communications. However, RS codes have some drawbacks either; A well know problem, addressed in this thesis, is the lack of an efficient soft decoder. With the introduction of Turbo codes, soft decoding attracted a lot of attention, leading to another theory, commonly referred to as modern coding theory. This theory results from the ambition to approach the ultimate limits of communication (Shannon limit) as close as possible. Codes belonging to this class are concatenated codes, typically with a very long block length. This block length is a crucial issue regarding their performance, but prevents application of these codes in consumer electronics, where severe implementation constraints rule. Reducing this block length is not a good option, since the overall code performance of a concatenated code with a small block length is seriously affected by its particular structure. In this thesis, a novel solution to this problem is presented. Instead of concatenating several small codes, we start from RS codes (with a maximum distance) which are subsequently broken into a number of component codes. Then, algorithms of modern coding theory (Gallager?s algorithm) can be applied to obtain a soft RS decoder. We refer to this technique as coordinated concatenation, since the component codes are carefully designed and concatenated such that the overall code is an RS code (with optimal distance properties given a finite block length). Hence, the thesis can best be situated in between classical and modern code theory, since it combines the optimal distance properties of classical codes with the performance of soft decoding methods originating from modern code theory. The development of a soft RS decoder is the most important contribution from a practical point of view. The RS code decomposition is a crucial aspect in the soft RS decoder. Therefore, the critically subsampled filter bank structure behind RS codes is unveiled, which is the main contribution from a theoretical point of view. The critical subsampling is crucial, and allows us to separate the subband codes (that add redundancy) from the synthesis bank code (that combines the redundancy). The existence of a critically subsampled filter bank is remarkable noting that it can never perform linear filtering operations. In fact, the cyclic character of the RS code is called upon, which matches with the periodically time varying behavior of the filter bank resulting from the critical subsampling. Furthermore, an extension towards the family of BCH codes is presented. A second (similar) problem arises when an error correcting code is concatenated with an advanced modulator, e.g. an OFDM modulator which is frequently used to tackle the frequency selective character of a communication channel. Although OFDM with bit-loading is known to achieve capacity (i.e. in concatenation with an infinite length, rate 0 theoretical code), the situation is quite different if a finite length code is applied. In he latter case, the distance properties of the overall code must be carefully assessed and optimized. In this thesis, a new approach regarding this problem is presented, which is once more based on RS codes and their filter bank representation. It is explained how an OFDM scheme can seamlessly be merged with an RS code, leading to a new scheme referred to as RS-OFDM. More specifically, an RS-OFDM system (re)uses the DFT hidden in the synthesis bank of the filter bank representation of an RS code in the OFDMmodulator. The overall optimal scheme so obtained is referred to as RS-OFDM, and shows an RS code that is matched to the OFDM-modulator. Apart from a performance gain, the RS-OFDM scheme shows a reduced PAPR.