Tor Aulin

Serially concatenated continuous phase modulation with iterative decoding

Serially concatenated and interleaved continuous phase modulation (CPM) with iterative decoding is investigated. An a posteriori probability (APP) algorithm for CPM is developed based on the classic APP algorithm for channel codes. The system is analyzed through upper bounds on the average bit error probability. For coded and interleaved minimum shift keying, the weight spectrum is computed, resulting in a transfer function bound. This is cumbersome for a general CPM system; instead, only the most significant error events contributing to the weight spectrum are identified. Simulations show that, firstly, these events give a satisfactory view of system performance when equal outer codes are used, and secondly, that remarkably good performance can be obtained for some simple systems. Finally, power spectral densities and bandwidths are computed, allowing for a bandwidth/performanee comparison of different combinations.

A Class of Reduced-Complexity Viterbi Detectors for Partial Response Continuous Phase Modulation

Partial response continuous phase modulation (CPM) gives constant envelope digital modulation schemes with excellent power spectra. Both narrow main lobe and low spectral tails can be achieved. When these signals are detected in an optimum coherent maximum likelihood sequence detector (Viterbi detector), power efficient schemes can also be designed, sometimes at the expense of receiver complexity. This paper describes a general class of simple Viterbi detectors with reduced complexity compared to the optimum case. The key idea is that the approximate receiver is based on a less complex CPM scheme than the transmitted scheme. The asymptotically optimum reduced-complexity receiver is found for a variety of transmitted schemes and various complexity reduction factors, for a specific class of receivers and modulation indexes. A new distance measure is introduced for the performance analysis. Smooth schemes based on raised cosine pulses are analyzed and simulated for the case of simplified reception. A graceful performance degradation occurs with the reduction of complexity.

Breadth-first maximum likelihood sequence detection: basics

The problem of performing breadth-first maximum likelihood sequence detection (MLSD) under given structural and complexity constraints is solved and results in a family of optimal detectors. Given a trellis with S states, these are partitioned into C classes where B paths into each class are selected recursively in each symbol interval. The derived result is to retain only those paths which are closest to the received signal in the Euclidean (Hamming) distance sense. Each member in the SA(B, C) family of sequence detectors (SA denotes search algorithm) performs complexity constrained MLSD for the additive white Gaussian noise (AWGN) (BSC) channel. The unconstrained solution is the Viterbi algorithm (VA). Analysis tools are developed for each member of the SA(B, C) class and the asymptotic (SNR) probability of losing the correct path is associated with a new Euclidean distance measure for the AWGN case, the vector Euclidean distance (VED). The traditional Euclidean distance is a scalar special case of this, termed the scalar Euclidean distance (SED). The generality of this VED is pointed out. Some general complexity reductions exemplify those associated with the VA approach.

On the minimum Euclidean distance for a class of signal space codes

The minimum Euclidean distance for a class of constant envelope phase modulation codes is studied. Bandwidth and power efficient signals with continuous phase are considered. The information carrying phase varies piecewise linearly and the slopes are cyclically changed for successive symbol time intervals, yielding the so-called multi- h signals. It has previously been shown that this class of signals contains bandwidth and power efficient signals when coherent maximum likelihood sequence detection is used. Bounds on the achievable Euclidean distance for signals in the above class are given. Upper bounds are calculated as well as minimum distance results for specific multilevel multi- h signals. It is concluded that quaternary and octal muiti- h schemes considerably outperform the binary schemes. Furthermore in the important small modulation index region, 2-h codes gain the maximum 3 dB. Larger gains are not available by increasing the number of h values.

Iterative detectors for trellis-code multiple-access

Trellis-code multiple-access (TCMA) is a narrow-band multiple-access scheme based on trellis-coded modulation. There is no bandwidth expansion, so K users occupy the same bandwidth as one single user. The load of the system, in number of bits per channel use, is therefore much higher than the load in, for example, conventional code-division multiple-access systems. Interleavers are introduced as a new feature to separate the users. This implies that the maximum-likelihood sequence detector (MLSD) is now too complex to implement. Iterative detectors are therefore suggested as an alternative to the joint MLSD. The conventional interference cancellation (IC), detector has lower complexity than the MLSD, but its performance is shown to be far from acceptable. Even after a novel improvement of the IC detector, the performance is unsatisfactory. Instead of using IC, another iterative detector is suggested. This detector updates the branch metric for every iteration, and avoids the standard Gaussian approximation. Simulations show that the performance of this detector can be close to single-user performance, even when the interleaver and the phase offset are the only user-specific features in the TCMA system.

Minimum Euclidean distance for combinations of short rate 1/2 convolutional codes and CPFSK modulation

Continuous phase frequency shift keying (CPFSK) is a constant amplitude modulation method with good spectral sidelobe properties. Good error probability properties can be obtained with coherent maximum-likelihood detection. In this paper we study the Euclidean distance properties of signals formed by a conventional rate 1/2 convolutional encoder followed by a binary or 4 -level CPFSK modulator. The minimum Euclidean distance is calculated for these signal sets as a function of the modulation index and the observation interval length. The optimum detector is discussed for rational modulation index values. The best obtainable codes are found for the case of short rate 1/2 codes with binary or 4 -level CPFSK modulation. Lists of the best codes are given. Among the results are that the noncatastrophic rate 1/2 convolutional codes with optimum free Hamming distance do not in general give the best Euclidean distance with CPFSK.

The application of semidefinite programming for detection in CDMA

In this paper, a detection strategy based on a semidefinite relaxation of the optimal maximum-likelihood problem is investigated. Cutting planes are also used to strengthen the approximation. The semidefinite program arising from the relaxation can be solved efficiently by interior point methods with polynomial complexity in the number of users. The simulated bit error rate performance shows that this approach provides a good approximation to the ML detector.

Serially concatenated continuous phase modulation with convolutional codes over rings

In this paper, we investigate serially concatenated continuous phase modulation (SCCPM) with convolutional codes (CC) over rings. The transmitted signals are disturbed by additive white Gaussian noise. The properties for systems with both infinite and finite block lengths are investigated. For an infinite-length system, we check the convergence threshold using the extrinsic information transfer chart. For a finite-length system, we use union-bound techniques to estimate the error floors. In the union-bound analysis, we consider both the order and the position of nonzero permuted symbols. A simple method for determining a CPM error event through the sum of the input symbol sequence is shown. Thus, we can determine if the output symbol sequence of an error event in the ring CC can form an error event in CPM. Two properties concerning the interleaver gain (IG) are investigated. A recursive search algorithm for the maximal IG is shown. Compared with previous SCCPM with a binary CC, the proposed system shows an improvement concerning the convergence threshold or error floors

Serially Concatenated Continuous-Phase Modulation with Iterative Decoding

Serially concatenated continuous phase modulation (SCCPM), or coded and interleaved CPM, is examined. Like turbo codes, SCCPM allows for iterative decoding based on a posteriori probability algorithms. The system is analyzed through an upper bound on the average bit error probability with maximum likelihood sequence detection, showing that SCCPM is capable of providing interleaver gain - and thus turbo-like performance - if a proper mapping of the CPM symbols is used. Simulations indicate that firstly, the analysis gives a satisfactory view of system performance when equal outer codes are used, and secondly, that remarkably good performance can be obtained with some simple systems. For unequal outer codes, the convergence of the iterative decoder will also have a significant effect on performance. Finally, a bandwidth/performance comparison of some selected systems shows that SCCPM can achieve large gains compared to the best known coded CPM systems without interleaving. Keywords: concatenated codes; serial concatenation; continuous phase modulation; iterative decoding; APP algorithm

Orthogonalization by principal components applied to CPM

The orthonormal basis for the space spanned by a given signal set can be chosen in many different ways. However, when the basis is truncated to fewer dimensions, the quality of the approximated signals differs, depending on the choice of the original basis. We study two energy-related quality measures and show that the optimal lower-dimensional approximation is given by the principal components (PC) method, which is also a simple and efficient alternative to Gram-Schmidt techniques. In addition, we derive and bound the average decrease in squared Euclidean distances over one symbol interval caused by the PC method. This measure is relevant in serially concatenated continuous phase modulation, where a manifold of signal pairs contributes to the bit-error rate for low-to-medium signal-to-noise ratios. By a numerical evaluation, we find that for this measure, the decrease is lower than that of a previous method by J. Huber and W. Liu (see IEEE J. Select. Areas Commun., vol.7, p.1437-49, 1989). Finally, we compare the minimum squared Euclidean distance for error events, which is relevant for uncoded CPM systems. Here, the loss with the PC method is generally larger than with Huber and Lius method, although examples of the opposite exist.