The girth profile is introduced and search algorithms for regular and irregular quasi-cyclic LDPC block codes with both good girth profile and good degree distribution are presented. New QC LDPC block codes of various code rates are obtained and their bit error rate performance is compared with that of the corresponding LDPC block codes defined in the IEEE 802.16 WiMAX standard of the same block length and code rate.

In 1995, Bestpublished a formula for the exact bit error probability for Viterbi decoding of the rate$R=1/2$, memory$m=1$(two-state) convolutional encoder with generator matrix$G(D)=(1 \quad 1+D)$when used to communicate over the binary symmetric channel. Their formula was later extended to the rate$R=1/2$, memory$m=2$(four-state) convolutional encoder with generator matrix$G(D)=(1+D^{2} \quad 1+D+D^{2})$by LentmaierIn this paper, a different approach to derive the exact bit error probability is described. A general recurrent matrix equation, connecting the average information weight at the current and previous states of a trellis section of the Viterbi decoder, is derived and solved. The general solution of this matrix equation yields a closed-form expression for the exact bit error probability. As special cases, the expressions obtained by Bestfor the two-state encoder and by Lentmaierfor a four-state encoder are used. The closed-form expression derived in this paper is evaluated for various realizations of encoders, including rate$R=1/2$and$R=2/3$encoders, of as many as 16 states. Moreover, it is shown that it is straightforward to extend the approach to communication over the quantized additive white Gaussian noise channel.

The relation between parity-check matrices of quasi-cyclic (QC) low-density parity-check (LDPC) codes and biadjacency matrices of bipartite graphs supports searching for powerful LDPC block codes. Using the principle of tailbiting, compact representations of bipartite graphs based on convolutional codes can be found. Bounds on the girth and the minimum distance of LDPC block codes constructed in such a way are discussed. Algorithms for searching iteratively for LDPC block codes with large girth and for determining their minimum distance are presented. Constructions based on all-one matrices, Steiner Triple Systems, and QC block codes are introduced. Finally, new QC regular LDPC block codes with girth up to 24 are given.

A new method using Hamming codes to construct base matrices of (J,K)-regular LDPC convolutional codes with large free distance is presented. By proper labeling the corresponding base matrices and tailbiting these parent convolutional codes to given lengths, a large set of quasi-cyclic (QC) (J,K)-regular LDPC block codes with large minimum distance is obtained. The corresponding Tanner graphs have girth up to 14. This new construction is compared with two previously known constructions of QC (J,K)-regular LDPC block codes with large minimum distance exceeding (J + 1)!. Applying all three constructions, new QC (J,K)-regular block LDPC codes with J = 3 or 4, shorter codeword lengths and/or better distance properties than those of previously known codes are presented.

The relation between the parity-check matrices of quasi-cyclic (QC) low-density parity-check (LDPC) codes and the biadjacency matrices of bipartite graphs supports searching for powerful LDPC block codes. Algorithms for searching iteratively for LDPC block codes with large girth are presented and constructions based on Steiner Triple Systems and short QC block codes are introduced, leading to new QC regular LDPC block codes with girth up to 24.

Next generation wireless systems have to be able to efficiently deal with fast fading environments in order to achieve high spectral efficiency. Using multiple-input multiple-output (MIMO) systems and exploiting receive diversity, the spectral efficiency can be greatly increased. Commonly, the channel is estimated via training symbols, before data detection is carried out based on the obtained channel estimate. While this significantly simplifies the process of data detection, it leads in general to suboptimal results. A better approach is to carry out joint channel estimation and data detection; we turn our attention to joint maximum-likelihood (ML) detection which is the optimal strategy. In this paper, the BEAST - Bidirectional Efficient Algorithm for Searching code Trees - is proposed as an alternative algorithm for joint ML channel estimation and data detection and its complexity is compared with recently published algorithms in the literature.

The relation between low-density parity-check (LDPC) codes and hypergraphs supports searching for powerful LDPC codes based on hypergraphs. On the other hand, coding theory methods can be used in searching for hypergraphs with large girth. Moreover, compact representations of hypergraphs based on convolutional codes can be found. Algorithms for iteratively constructing LDPC codes with large girth and for determining their minimum distance are introduced. New quasi-cyclic (QC) LDPC codes are presented, some having both optimal girth and optimal minimum distance.

A rate R=5/20 hypergraph-based woven convolutional code with overall constraint length 67 and constituent convolutional codes is presented. It is based on a 3-partite, 3-uniform, 4-regular hypergraph and contains rate Rc=3/4 constituent convolutional codes with overall constraint length 5. Although the code construction is based on low-complexity codes, the free distance of this construction, computed with the BEAST algorithm, is dfree=120, which is remarkably large.

Rate R = (c-1)/c convolutional codes of constraint length ν can be represented by conventional syndrome trellises with a state complexity of s = ν or by binary syndrome trellises with a state complexity of s = ν or s = ν + 1, which corresponds to at most 2sstates at each trellis level. It is shown that if the parity-check polynomials fulfill certain conditions, there exist binary syndrome trellises with optimum state complexity s = ν. The BEAST is modified to handle parity-check matrices and used to generate code tables for optimum free distance rate R = (c - 1)=c, c = 3; 4; 5, convolutional codes for conventional syndrome trellises and binary syndrome trellises with optimum state complexity. These results show that the loss in distance properties due to the optimum state complexity restriction for binary trellises is typically negligible.

A common technique to deploy linear prediction to non-stationary signals is time segmentation and local analysis. Variations of a process within such a segment cause inaccuracies. In this paper, we model the temporal changes of linear prediction coefficients (LPCs) as a Fourier series. We obtain a compact description of the vocal tract model limited by the predictor order and the maximum Doppler frequency. Filter stability is guaranteed by all-pass filtering, deploying the human ear's insensitivity to absolute phase. The periodicity constraint induced by the Fourier series is counteracted by oversampling in the Doppler domain. With this approach, the number of coefficients required for the vocal tract modeling is significantly reduced compared to a LPC system with block-wise adaptation while exceeding its prediction gain. As a by-product it is found that the Doppler frequency of the vocal tract is in the order of 10 Hz. A generalization of the algorithm to an auto-regressive moving average model with time-correlated filter coefficients is straight forward.