The true Cramer-Rao bound for estimating the carrier phase of a convolutionally encoded PSK signal

This contribution considers the true Cramer-Rao bound (CRB) related to estimating the carrier phase of a noisy linearly modulated signal in the presence of encoded data symbols. Timing delay and frequency offset are assumed to be known. A generall expression and computational method is derived to evaluate the CRB in the presence of codes for which a trellis diagram can be drawn (block codes, trellis codes, convolutional codes,...). Results are obtained for several minimum free distance non-recursive convolutional (NRC) codes, and are compared with the CRB obtained with random (uncoded) data [1] and with the modified Cramer-Rao bound (MCRB) from [2]. We find that for small signal-to-noise ratio (SNR) the CRB is considerably smaller for coded transmission than for uncoded transmission. We show that the SNR at which the CRB is close to the MCRB decreases as the coding gain increases, and corresponds to a bit error rate (BER) of about 0.001. We also compare the new CRBs with the simulated performance of (i) the (code-independent) Viterbi & Viterbi phase estimator [3] and (ii) the recently developed turbo synchronizer [4,5]

CRB derivation and new Code-Aided timing recovery technique for QAM modulated signals

International audience— * In this paper, we propose a maximum likelihood based Code-Aided (CA) timing recovery algorithm for square-QAM modulated signals. We also theoretically derive the analytical expression of the CA Cramer-Rao Bound for time delay estimation. Our simulations show that the proposed CA approach realizes a performance equivalent to the Data-Aided (DA) approach over a large interval of signal to noise ratio (SNR) values

Performance Study of a Near Maximum Likelihood Code-Aided Timing Recovery Technique

International audienceIn this paper, we propose a new code-aided (CA) timing recovery algorithm for various linear constant modulus constellations based on the Maximum Likelihood (ML) estimator. The first contribution is the derivation of a soft estimator expression of the transmitted symbol instead of its true or hard estimated value which is fed into the timing error detector (TED) equation. The proposed expression includes the Log-Likelihood Ratios (LLRs) obtained from a turbo decoder. Our results show that the proposed CA approach achieves almost as good results as the data-aided (DA) approach over a large interval of SNR values while achieving a higher spectral efficiency. We also derive the corresponding CA Cramer Rao Bounds (CRB) for various modulation orders. Contrarily to former work, we develop here the CRB analytical expression for different M-PSK modulation orders and validate them through comparison to empirical CRB obtained by Monte Carlo iterations. The proposed CA estimator realizes an important gain over the non data-aided approach (NDA) and achieves a smaller gap when compared to its relative CA CRB, especially at moderate SNR values where modern systems are constrained to work

Synchronization in digital communication systems: performance bounds and practical algorithms

Communication channels often transfer signals from different transmitters. To avoid interference the available frequency spectrum is divided into non-overlapping frequency bands (bandpass channels) and each transmitter is assigned to a different bandpass channel. The transmission of a signal over a bandpass channel requires a shift of its frequency-content to a frequency range that is compatible with the designated frequency band (modulation). At the receiver, the modulated signal is demodulated (frequency shifted back to the original frequency band) in order to recover the original signal. The modulation/demodulation process requires the presence of a locally generated sinusoidal signal at both the transmitter and the receiver. To enable a reliable information transfer, it is imperative that these two sinusoids are accurately synchronized. Recently, several powerful channel codes have been developed which enable reliable communication at a very low signal-to-noise ratio (SNR). A by-product of these developments is that synchronization must now be performed at a SNR that is lower than ever before. Of course, this imposes high requirements on the synchronizer design. This doctoral thesis investigates to what extent (performance bounds) and in what way (practical algorithms) the structure that the channel code enforces upon the transmitted signal can be exploited to improve the synchronization accuracy at low SNR

On the Cramer-Rao bound for carrier frequency estimation in the presence of phase noise

We consider the carrier frequency offset estimation in a digital burst-mode satellite transmission affected by phase noise. The corresponding Cramer-Rao lower bound is analyzed for linear modulations under a Wiener phase noise model and in the hypothesis of knowledge of the transmitted data. Even if we resort to a Monte Carlo average, from a computational point of view the evaluation of the Cramer-Rao bound is very hard. We introduce a simple but very accurate approximation that allows to carry out this task in a very easy way. As it will be shown, the presence of the phase noise produces a remarkable performance degradation of the frequency estimation accuracy. In addition, we provide asymptotic expressions of the Cramer-Rao bound, from which the effect of the phase noise and the dependence on the system parameters of the frequency offset estimation accuracy clearly result. Finally, as a by-product of our derivations and approximations, we derive a couple of estimators specifically tailored for the phase noise channel that will be compared with the classical Rife and Boorstyn algorithm, gaining in this way some important hints on the estimators to be used in this scenario

OFDM Waveform Optimisation for Joint Communications and Sensing

Radar systems are radios to sense objects in their surrounding environment. These operate at a defined set of frequency ranges. Communication systems are used to transfer information between two points. In the present day, proliferation of mobile devices and the advancement of technology have led to communication systems being ubiquitous. This has made these systems to operate at the frequency bands already used by the radar systems. Thus, the communication signal interferes a radar receiver and vice versa, degrading performance of both systems. Different methods have been proposed to combat this phenomenon. One of the novel topics in this is the RF convergence, where a given bandwidth is used jointly by both systems. A differentiation criterion must be adopted between the two systems so that a receiver is able to separately extract radar and communication signals. The hardware convergence due to the emergence of software-defined radios also motivated a single system be used for both radar and communication. A joint waveform is adopted for both radar and communication systems, as the transmit signal. As orthogonal frequency-division multiplexing (OFDM) waveform is the most prominent in mobile communications, it is selected as the joint waveform. Considering practical cellular communication systems adopting OFDM, there often exist unused subcarriers within OFDM symbols. These can be filled up with arbitrary data to improve the performance of the radar system. This is the approach used, where the filling up is performed through an optimisation algorithm. The filled subcarriers are termed as radar subcarriers while the rest as communication subcarriers, throughout the thesis. The optimisation problem minimises the Cramer--Rao lower bounds of the delay and Doppler estimates made by the radar system subject to a set of constraints. It also outputs the indices of the radar and communication subcarriers within an OFDM symbol, which minimise the lower bounds. The first constraint allocates power between radar and communication subcarriers depending on their subcarrier ratio in an OFDM symbol. The second constraint ensures the peak-to-average power ratio (PAPR) of the joint waveform has an acceptable level of PAPR. The results show that the optimised waveform provides significant improvement in the Cramer--Rao lower bounds compared with the unoptimised waveform. In compensation for this, the power allocated to the communication subcarriers needs to be reduced. Thus, improving the performances of the radar and communication systems are a trade-off. It is also observed that for the minimum lower bounds, radar subcarriers need to be placed at the two edges of an OFDM symbol. Optimisation is also seen to improve the estimation performance of a maximum likelihood estimator, concluding that optimising the subcarriers to minimise a theoretical bound enables to achieve improvement for practical systems

Blind LDPC encoder identification

Nowadays, adaptive modulation and coding (AMC) techniques can facilitate flexible strategies subject to dynamic channel quality. The AMC transceivers select the most suitable coding and modulation mechanisms subject to the acquired channel information. Meanwhile, a control channel or a preamble is usually required to synchronously coordinate such changes between transmitters and receivers. On the other hand, low-density parity-check (LDPC) codes become more and more popular in recent years due to their promising capacity-approaching property. The broad range of variations in code rates and codeword lengths for LDPC codes makes them ideal candidates for future AMC transceivers. The blind encoder identification problem emerges when the underlying control channel is absent or the preamble is not allowed in AMC systems. It would be quite intriguing for one to build a blind encoder identification technique without spectrum-efficiency sacrifice. Therefore, in this thesis, we investigate blind LDPC encoder identification for AMC systems. Specifically, we would like to tackle the blind identification of binary LDPC codes (encoders) for binary phase-shift keying (BPSK) signals and nonbinary LDPC codes for quadrature-amplitude modulation (QAM) signals. We propose a novel blind identification system which consists of three major components, namely expectation-maximization (EM) estimator for unknown parameters (signal amplitude, noise variance, and phase offset), log-likelihood ratio (LLR) estimator for syndrome a posteriori probabilities, and maximum average-LLR detector. Monte Carlo simulation results demonstrate that our proposed blind LDPC encoder identification scheme is very promising over different signal-to-noise ratio conditions

A performance lower bound for quadratic timing recovery accounting for the symbol transition density

The symbol transition density in a digitally modulated signal affects the performance of practical synchronization schemes designed for timing recovery. This paper focuses on the derivation of simple performance limits for the estimation of the time delay of a noisy linearly modulated signal in the presence of various degrees of symbol correlation produced by the various transition densities in the symbol streams. The paper develops high- and low-signal-to-noise ratio (SNR) approximations of the so-called (Gaussian) unconditional Cramér–Rao bound (UCRB), as well as general expressions that are applicable in all ranges of SNR. The derived bounds are valid only for the class of quadratic, non-data-aided (NDA) timing recovery schemes. To illustrate the validity of the derived bounds, they are compared with the actual performance achieved by some well-known quadratic NDA timing recovery schemes. The impact of the symbol transition density on the classical threshold effect present in NDA timing recovery schemes is also analyzed. Previous work on performance bounds for timing recovery from various authors is generalized and unified in this contribution.Peer Reviewe