True Cramer-Rao bounds for carrier and symbol synchronization

This contribution considers the Cramer-Rao bound (CRB) related to estimating the synchronization parameters (carrier phase, carrier frequency and time delay) of a noisy linearly modulated signal with random data symbols. We explore various scenarios, involving the estimation of a subset of the parameters while the other parameters are either considered as nuisance parameters or a priori known to the receiver. In addition, some results related to the CRB for coded transmission will be presented

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

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

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]

A new bound and algorithm for star 16-QAM carrier phase estimation

Copyright © 2003 IEEEThe true Cramer-Rao lower bound (CRLB) is derived and evaluated for the estimation of carrier phase of Star 16-quadrature amplitude modulation (QAM) and can be simply applied to carrier frequency estimation. Different geometries are investigated by varying the ring ratio (RR). For signal-to-noise ratios (SNRs) between 6-15 dB, the CRLB with RR=3 is lower than that of Square 16-QAM. A modified phase estimator is presented, which closely follows the new CRLB. Investigation of symbol error performance in short packet length reveals Star 16-QAM to be superior to Square 16-QAM for SNR<13 dB, which is a reasonable operating range for a coded system. Although Square 16-QAM and Star RR=1.8 are optimum for a perfect receiver, when the effect of phase estimation is considered, we find Star RR=3 to be better for SNR below 10 dB.Feng Rice, Mark Rice, and Bill Cowle

Advanced methods in automatic modulation classification for emerging technologies

Modulation classification (MC) is of large importance in both military and commercial communication applications. It is a challenging problem, especially in non-cooperative wireless environments, where channel fading and no prior knowledge on the incoming signal are major factors that deteriorate the reception performance. Although the average likelihood ratio test method can provide an optimal solution to the MC problem with unknown parameters, it suffers from high computational complexity and in some cases mathematical intractability. Instead, in this research, an array-based quasi-hybrid likelihood ratio test (qHLRT) algorithm is proposed, which depicts two major advantages. First, it is simple yet accurate enough parameter estimation with reduced complexity. Second the incorporation of antenna arrays offers an effective ability to combat fading. Furthermore, a practical array-based qHLRT classifier scheme is implemented, which applies maximal ratio combining (MRC) to increase the accuracy of both carrier frequency offset (CFO) estimation and likelihood function calculation in channel fading. In fact, double CFO estimations are executed in this classifier. With the first the unknown CFO, phase offsets and amplitudes are estimated as prerequisite for MRC operation. Then, MRC is performed using these estimates, followed by a second CFO estimator. Since the input of the second CFO estimator is the output of the MRC, fading effects on the incoming signals are removed significantly and signal-to-noise ratio (SNR) is augmented. As a result, a more accurate CFO estimate is obtained. Consequently, the overall classification performance is improved, especially in low SNR environment. Recently, many state-of-the-arts communication technologies, such as orthogonal frequency division multiplexing (OFDM) modulations, have been emerging. The need for distinguishing OFDM signal from single carrier has become obvious. Besides, some vital parameters of OFDM signals should be extracted for further processing. In comparison to the research on MC for single carrier single antenna transmission, much less attention has been paid to the MC for emerging modulation methods. A comprehensive classification system is proposed for recognizing the OFDM signal and extracting its parameters. An automatic OFDM modulation classifier is proposed, which is based on the goodness-of-fittest. Since OFDM signal is Gaussian, Cramer-von Mises technique, working on the empirical distribution function, has been applied to test the presence of the normality. Numerical results show that such approach can successfully identify OFDM signals from single carrier modulations over a wide SNR range. Moreover, the proposed scheme can provide the acceptable performance when frequency-selective fading is present. Correlation test is then applied to estimate OFDM cyclic prefix duration. A two-phase searching scheme, which is based on Fast Fourier Transform (FFT) as well as Gaussianity test, is devised to detect the number of subcarriers. In the first phase, a coarse search is carried out iteratively. The exact number of subcarriers is determined by the fine tune in the second phase. Both analytical work and numerical results are presented to verify the efficiency of the proposed scheme

Asymptotic Performance of the Pth Power-Law Phase Estimator

An expression for the true variance of the Pth powerlaw phase estimator, as the number of samples approaches infinity, is given. This expression is an extension to the linear approximation of Moeneclaey and de Jonghe [1] which is known to be inadequate in some practical systems. Our new expression covers general 2π/P-rotationally symmetric constellations that include those of PAM, QAM, PSK, Star M-QAM, MR-DPSK, and others. This expression also generalizes the known expressions for QAM and PSK. Additionally, our expression reduces to the Cramer-Rao bound given by Steendam and Moeneclaey [9], as SNR goes to zero. Monte Carlo simulations provide experimental verification of the theoretical expression for various constellations

New expression for the functional transformation of the vector Cramér-Rao lower bound

Assume that it is desired to estimate α = f(θ), where f(·) is an r-dimensional function. This paper derives the general expression for the functional transformation of the vector Cramér-Rao lower bound (CRLB). The derived bound is a tight lower bound on the estimation of uncoupled parameters, i.e., parameters that can be estimated separately. Unlike previous results in the literature, this new expression is not dependent on the inverse of the Fisher's information matrix (FIM) of the untransformed parameters, θ. Thus, it can be applied to scenarios where the FIM for θ is ill-conditioned or singular. Finally, as an application, the derived transformation is applied to determine the exact CRLB for estimation of channel parameters in amplify-and-forward relaying networks.This research was supported under Australian Research Council’s Discovery Projects funding scheme (project number DP110102548)