There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Contention resolution diversity slotted ALOHA (CRDSA) is a simple but effective improvement of slotted ALOHA. CRDSA relies on MAC bursts repetition and on interference cancellation (IC), achieving a peak throughput T ≅ 0.55, whereas for slotted ALOHA T ≅ 0.37. In this paper we show that the IC process of CRDSA can be conveniently described by a bipartite graph, establishing a bridge between the IC process and the iterative erasure decoding of graph-based codes. Exploiting this analogy, we show how a high throughput can be achieved by selecting variable burst repetition rates according to given probability distributions, leading to irregular graphs. A framework for the probability distribution optimization is provided. Based on that, we propose a novel scheme, named irregular repetition slotted ALOHA, that can achieve a throughput T ≅ 0.97 for large frames and near to T ≅ 0.8 in practical implementations, resulting in a gain of ~ 45% w.r.t. CRDSA. An analysis of the normalized efficiency is introduced, allowing performance comparisons under the constraint of equal average transmission power. Simulation results, including an IC mechanism described in the paper, substantiate the validity of the analysis and confirm the high efficiency of the proposed approach down to a signal-to-noise ratio as a low as Eb/N0=2 dB.

Graph-Based Analysis and Optimization of Contention Resolution Diversity Slotted ALOHA

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Detection, Estimation, and Modulation Theory

IEEE International Solid-State Circuits Conference

Fundamentals of statistical signal processing: Estimation theory: by Steven M. KAY; Prentice Hall signal processing series; Prentice Hall; Englewood Cliffs, NJ, USA; 1993; xii + 595 pp.; $65; ISBN: 0-13-345711-7

This paper is devoted to turbo synchronization, that is to say the use of soft information to estimate parameters like carrier phase, frequency offset or timing within a turbo receiver. It is shown how maximum-likelihood estimation of those synchronization parameters can be implemented by means of the iterative expectation-maximization (EM) algorithm [A.P. Dempster, et al., 1977]. Then we show that the EM algorithm iterations can be combined with those of a turbo receiver. This leads to a general theoretical framework for turbo synchronization. The soft decision-directed ad-hoc algorithm proposed in V. Lottici and M. Luise, [2002] for carrier phase recovery turns out to be a particular instance of this implementation. The proposed mathematical framework is illustrated by simulations reported for the particular case of carrier phase estimation combined with iterative demodulation and decoding [S. ten Brink, et al., 1998].

/pdf/turbo-synchronization-an-em-algorithm-interpretation-j1umvryznj.pdf

Turbo synchronization: an EM algorithm interpretation

This contribution considers turbo synchronization, that is to say, the use of soft data information to estimate parameters like carrier phase, frequency, or timing offsets of a modulated signal within an iterative data demodulator. In turbo synchronization, the receiver exploits the soft decisions computed at each turbo decoding iteration to provide a reliable estimate of some signal parameters. The aim of our paper is to show that such "turbo-estimation" approach can be regarded as a special case of the expectation-maximization (EM) algorithm. This leads to a general theoretical framework for turbo synchronization that allows to derive parameter estimation procedures for carrier phase and frequency offset, as well as for timing offset and signal amplitude. The proposed mathematical framework is illustrated by simulation results reported for the particular case of carrier phase and frequency offsets estimation of a turbo-coded 16-QAM signal.

/pdf/a-theoretical-framework-for-soft-information-based-1xm0a2y6ae.pdf

A theoretical framework for soft-information-based synchronization in iterative (Turbo) receivers

The introduction of turbo and low-density parity-check (LDPC) codes with iterative decoding that almost attain Shannon capacity challenges the synchronization subsystems of a data modem. Fast and accurate signal synchronization has to be performed at a much lower value of signal-to-noise ratio (SNR) than in previous less efficiently coded systems. The solution to this issue is developing specific synchronization techniques that take advantage of the presence of the channel code and of the iterative nature of decoding: the so-called turbo-synchronization algorithms. The aim of this paper within this special issue devoted to the turbo principle is twofold: on the one hand, it shows how the many turbo-synchronization algorithms that have already appeared in the literature can be cast into a simple and rigorous theoretical framework. On the other hand, it shows the application of such techniques in a few simple cases, and evaluates improvement that can be obtained from them, especially in the low-SNR regime.

Code-Aided Turbo Synchronization

This paper derives the Cramer-Rao bound (CRB) related to the estimation of the time delay of a linearly modulated bandpass signal with unknown carrier phase and frequency. We consider the following two scenarios: joint estimation of the time delay, the carrier phase, and the carrier frequency; and joint estimation of the time delay and the carrier frequency irrespective of the carrier phase. The transmit pulse is a bandlimited square-root Nyquist pulse. For each scenario, the transmitted symbols constitute either an a priori known training sequence or an unknown random data sequence. In spite of the presence of random data symbols and/or a random carrier phase, we obtain a relatively simple expression of the CRB, from which the effect of the constellation and the transmit pulse are easily derived. We show that the penalty resulting from estimating the time delay irrespective of the carrier phase decreases with increasing observation interval. However, the penalty, caused by not knowing the data symbols a priori, cannot be reduced by increasing the observation interval. Comparison of the true CRB to existing symbol synchronizer performance reveals that decision-directed timing recovery is close to optimum for moderate-to-large signal-to-noise ratios.

/pdf/true-cramer-rao-bound-for-timing-recovery-from-a-bandlimited-4yelf8ubki.pdf

True Cramer-Rao bound for timing recovery from a bandlimited linearly Modulated waveform with unknown carrier phase and frequency

In this letter, we express the Cramer-Rao bound (CRB) for carrier phase estimation from a noisy linearly modulated signal with encoded data symbols, in terms of the marginal a posteriori probabilities (APPs) of the coded symbols. For a wide range of classical codes (block codes, convolutional codes, and trellis-coded modulation), these marginal APPs can be computed efficiently by means of the Bahl-Cocke-Jelinke-Raviv (BCJR) algorithm, whereas for codes that involve interleaving (turbo codes and bit interleaved coded modulation), iterated application of the BCJR algorithm is required. Our numerical results show that when the BER of the coded system is less than about 10/sup -3/, the resulting CRB is essentially the same as when transmitting a training sequence.

/pdf/the-cramer-rao-bound-for-phase-estimation-from-coded-2rs9fehy3l.pdf

Nele Noels

Papers

Turbo synchronization: an EM algorithm interpretation

A theoretical framework for soft-information-based synchronization in iterative (Turbo) receivers

Code-Aided Turbo Synchronization

True Cramer-Rao bound for timing recovery from a bandlimited linearly Modulated waveform with unknown carrier phase and frequency

The Cramer-Rao bound for phase estimation from coded linearly modulated signals