The impact of the interleaver, embedded in the encoder for a parallel concatenated code, called the turbo code, is studied. The known turbo codes consist of long random interleavers, whose purpose is to reduce the value of the error coefficients. It is shown that an increased minimum Hamming distance can be obtained by using a structured interleaver. For low bit-error rates (BERs), we show that the performance of turbo codes with a structured interleaver is better than that obtained with a random interleaver. Another important advantage of the structured interleaver is the short length required, which yields a short decoding delay and reduced decoding complexity (in terms of memory). We also consider the use of turbo codes as component codes in multilevel codes. Powerful coding structures that consist of two component codes are suggested. Computer simulations are performed in order to evaluate the reduction in coding gain due to suboptimal iterative decoding. From the results of these simulations we deduce that the degradation in the performance (due to suboptimal decoding) is very small.

The encoding and decoding schemes presented are aimed at enabling the transfer of data through a channel in which two types of interference are added to the transmitted signal and the sum is quantized. One of these interferences is known (or can be estimated), whereas the second is an additive white Gaussian noise (AWGN). Since the input of the quantizer is not accessible, the known interference can not be removed from the received signal. We show that the error rate for an uncoded transmission through this channel is unacceptably large, even for low noise levels and linear quantization. It is also shown that the problem becomes even more severe when a nonlinear quantization is present. Therefore, coding is essential and a huge coding gain is achievable in this application. An upper-bound on the error rate, contributed by the component codes of a multilevel code, has been developed for multistage decoding. Results of computer simulations of a practical case with optimal and suboptimal decoding algorithms, both developed in this paper, are presented.

In this correspondence we are concerned with the error rates contributed by component codes in a multilevel code, decoded by a multistage decoder. These error rates are required for estimating the performance of the structure, but more importantly, they can serve as a design tool for achieving high coding gain. The latter objective can be reached by approaching a balance among these error rates. For evaluating the error rate contributed by a component code, it is essential to calculate the corresponding spectrum of Euclidean distances. A method for computing this spectrum is developed in this correspondence. The method is general and applicable to any signal constellation, and convolutional as well as block component codes. The error coefficient, available in the literature for some multilevel codes with an infinite size signal constellation, is shown to be significantly different from that of a practical finite constellation. Note that the complete spectrum is essential when codewords (of a component code) with Hamming distance beyond the minimum distance affect the error rate (see, e.g., Turbo codes).

A tight upper bound on the decoding error probability is derived for block-coded modulation structures where an M-ary phase shift keying (M-PSK) signal constellation is employed. This bound, called a tangential sphere bound, is tight for very low (as well as for high) signal-to-noise ratios (SNRs). Berlekamp's tangential union bound, previously derived for binary codes, can be derived for an M-PSK block coded modulation structure as well. However, it is proven that our tangential sphere bound is tighter than Berlekamp's (1980) tangential bound. For particular schemes, it is shown that for low SNRs our bound is considerably tighter than the tangential bound. As one of the examples, a multistage decoder is considered.

Two techniques for upper bounding the average probability of decoding error in coded modulation structures are presented. The first bound, which is applicable to the additive white Gaussian noise (AWGN) channel, is tighter than the well-known union bound and the minimum distance bound, especially for low signal to noise ratio. It is shown that for the Leech lattice this upper bound is very close to a sphere lower bound. For the second upper bound, which is applicable to any memoryless channel (not necessarily AWGN), a method of random coset coding is presented. For the AWGN channel, a tighter upper bound is obtained by employing the method of random coset coding for calculating the average spectrum of distances of the code, which is required for the computation of the first upper bound.>

Encoding and decoding schemes for concatenated multilevel block codes are presented. By one of these structures, a real coding gain of 5.6-7.4 dB for the bit error range of 10/sup -6/ to 10/sup -9/ is achieved for transmission through the additive white Gaussian noise channel. Also, a rather large asymptotic coding gain is obtained. The new coding schemes have very low decoding complexity and increased coding gain in comparison with the conventional block and trellis coded modulation structures. A few design rules for concatenated (single and) multilevel block codes with large coding gain are also provided.>

The main results are presented of an analysis of the convergence and the steady-state behavior of the DLMS (delayed least mean square) algorithm, with the aim of providing useful insight which may be helpful in the design of such filters. The problem is defined, and some basic definitions are presented. Conditions for convergence, convergence rate, and limits are discussed. Remarks are presented. The implications of the results for the design of DLMS adaptive filters are addressed.>