Non-data-aided Maximum Likelihood Research Articles

Maximum Likelihood SNR Estimation of Hyper Cubic Signals Over Gaussian Channel

In this letter, we address the problem of closed-form data aided (DA) and non-data aided (NDA) maximum likelihood (ML) signal-to-noise ratio (SNR) estimation for hyper-cubic modulated signals over Gaussian channel. The requirement of high SNR is exploited to derive closed-form expression of unbiased NDA ML SNR estimator. Exact expressions for the DA and NDA Cramer-Rao lower bound (CRLB) on the variance of these estimators are also determined analytically. The Monte Carlo simulation method is used to compute the normalized mean squared error (NMSE) of the estimators, and the results show that the NMSEs approach the respective normalized CRLBs.

Data-Aided and Non-Data-Aided Maximum Likelihood SNR Estimators for CPM

Data-aided (DA) and non-data-aided (NDA) maximum likelihood (ML) estimators for the SNR of CPM in the presence of phase and frequency offset are derived and analyzed. Cramer-Rao bounds for both are obtained and compared to the simulated performance of these estimators for a full response example, CPSFK and a partial response example, GMSK. Analysis and simulations show that the performance of the DA ML estimator suffers from an unremovable bias caused by uncompensated frequency offset due to frequency offset estimation errors. As a consequence, the estimator error variance of the DA ML estimator is not able to achieve its lower bound. In contrast, the estimator error variance of the NDA ML estimator is capable of achieving its lower bound (although the lower bound for NDA ML estimator is higher than the lower bound for the DA ML estimator). This is because the NDA ML estimator is not burdened with the requirement of relying on estimates of nuisance parameters. The NDA estimator only achieves its lower bound when the observation length or true SNR are sufficiently large.

Maximum Likelihood SNR Estimation of Linearly-Modulated Signals Over Time-Varying Flat-Fading SIMO Channels

In this paper, we tackle for the first time the problem of maximum likelihood (ML) estimation of the signal-to-noise ratio (SNR) parameter over time-varying single-input multiple-output (SIMO) channels. Both the data-aided (DA) and the non-data-aided (NDA) schemes are investigated. Unlike classical techniques where the channel is assumed to be slowly time-varying and, therefore, considered as constant over the entire observation period, we address the more challenging problem of instantaneous (i.e., short-term or local) SNR estimation over fast time-varying channels. The channel variations are tracked locally using a polynomial-in-time expansion. First, we derive in closed-form expressions the DA ML estimator and its bias. The latter is subsequently subtracted in order to obtain a new unbiased DA estimator whose variance and the corresponding Cram\'er-Rao lower bound (CRLB) are also derived in closed form. Due to the extreme nonlinearity of the log-likelihood function (LLF) in the NDA case, we resort to the expectation-maximization (EM) technique to iteratively obtain the exact NDA ML SNR estimates within very few iterations. Most remarkably, the new EM-based NDA estimator is applicable to any linearly-modulated signal and provides sufficiently accurate soft estimates (i.e., soft detection) for each of the unknown transmitted symbols. Therefore, hard detection can be easily embedded in the iteration loop in order to improve its performance at low to moderate SNR levels. We show by extensive computer simulations that the new estimators are able to accurately estimate the instantaneous per-antenna SNRs as they coincide with the DA CRLB over a wide range of practical SNRs.

SNR Estimation Over SIMO Channels From Linearly Modulated Signals

In this paper, we address the problem of data-aided (DA) and nondata-aided (NDA) per-antenna signal-to-noise ratio (SNR) estimation over wireless single-input multiple-output (SIMO) channels from linearly modulated signals. Under constant channels and additive white Gaussian noise (AWGN), we first derive the DA maximum-likelihood (ML) SNR estimator in closed-form expression. The performance of the DA ML estimator is analytically carried out by deriving the closed-form expression of its bias and variance. Besides, in order to compare its performance with the fundamental limit, we derive the DA Cramér-Rao lower bound (CRLB) in closed-form expression. In the NDA case, the expectation-maximization (EM) algorithm is derived to iteratively maximize the log-likelihood function. The performance of the NDA ML estimator is empirically assessed using Monte Carlo simulations. Moreover, we introduce an efficient algorithm, which applies to any one/two-dimensional <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -ary constellation, to numerically compute the NDA CRLBs. In this paper, the noise components are assumed to be spatially uncorrelated over all the antenna elements and temporally white. In both cases, we show that our new inphase and quadrature I/Q-based estimators offer substantial performance improvements over the single-input single-output (SISO) ML SNR estimator due to the optimal usage of the statistical dependence between the antenna branches, and that it reaches the corresponding CRLB over a wide SNR range. We also show that the use of the I/Q-based ML estimators can lead to remarkable performance improvements over the moment-based estimators for the same antenna-array size. Moreover, it is shown that SIMO configurations can contribute to decreasing the required number of iterations of the EM algorithm.

Open‐loop analysis and self‐noise performance of an NDA synchroniser for carrier‐independent symbol timing recovery at baud rate

AbstractFeedback loops for symbol timing recovery are an attractive solution for continuous transmission of data, where rapid acquisition is not so important as in burst‐mode systems. In this context, carrier‐independent non‐data‐aided (NDA) timing error detectors, operated at only one sample per symbol, can be a most interesting option. Assuming M‐ary PSK signals and focusing on a linearised loop model for analytical work, such an algorithm has been discussed already in the technical literature. On the other hand, the linear approach does not apply for larger deviations from the stable equilibrium point as they are encountered during initial acquisition. In this case, the detector characteristic (S‐curve) must be known. Since not available from the open literature, it is derived in this paper not only for PSK but for linear modulation schemes in general. The slope in the stable equilibrium point is given in closed form such that the linearised tracker can be specified immediately. Furthermore, it turned out that the complexity of the detector, but also the computation of both S‐curve and slope, is most simple for a special value of the additional design parameter characterizing the algorithm. Based on this, the self‐noise variance is derived in closed form. For M‐PSK schemes, it is proved that they experience no jitter floor due to pattern (data) noise. On the other hand, for non‐constant modulus signals, like QAM or APSK constellations, it is shown that the performance is proportional to the loop bandwidth. Finally, a detailed comparison with NDA maximum‐likelihood and Gardner algorithm, as standard method for carrier‐blind NDA tracking, is presented. Copyright © 2009 John Wiley &amp; Sons, Ltd.

Symbol timing estimation in MIMO correlated flat‐fading channels

AbstractIn this paper, the data aided (DA) and non‐data aided (NDA) maximum likelihood (ML) symbol timing estimators and their corresponding conditional Cramer–Rao bound (CCRB) and modified Cramer–Rao bound (MCRB) in multiple‐input‐multiple‐output (MIMO) correlated flat‐fading channels are derived. It is shown that the approximated ML algorithm in References [4,13] is just a special case of the DA ML estimator; while the extended squaring algorithm in Reference [14] is just a special case of the NDA ML estimator. For the DA case, the optimal orthogonal training sequences are also derived. It is found that the optimal orthogonal sequences resemble the Walsh sequences, but present different envelopes. Simulation results under different operating conditions (e.g. number of antennas and correlation between antennas) are given to assess and compare the performances of the DA and NDA ML estimators with respect to their corresponding CCRBs and MCRBs. It is found that (i) the mean square error (MSE) of the DA ML estimator is close to the CCRB and MCRB, (ii) the MSE of the NDA ML estimator is close to the CCRB but not to the MCRB, (iii) the MSEs of both DA and NDA ML estimators are approximately independent of the number of transmit antennas and are inversely proportional to the number of receive antennas, (iv) correlation between antennas has little effect on the MSEs of DA and NDA ML estimators and (v) DA ML estimator performs better than NDA ML estimator at the cost of lower transmission efficiency and higher implementation complexity. Copyright © 2004 John Wiley &amp; Sons, Ltd.

Non—data—aided ML carrier frequency and phase synchronization in OFDM systems

AbstractIn this paper non—data—aided(NDA), maximum likelihood(ML) algorithms are derived for the carrier frequency and phase offset, separately, for OFDM systems employing M—PSK modulation scheme. NDA ML estimation algorithm for frequency offset estimation exploits the redundant information contained in the cyclic prefix proceeding the OFDM symbols, thus reducing the need for pilots. Its mean—squared performance is obtained analytically and compared with simulation results. It is observed that the resulting algorithm generates very accurate estimation even when the offset is high. It is also shown that the frequency estimator may be used in a tracking mode. The ML algorithm derived for the carrier phase estimation is also a non—data—aided(NDA) and maximizes the low SNR limit of the likelihood function averaged over M—PSK signal constellation. It is shown that for sufficiently small SNR the ML phase estimator obtained reduces to the familiar M/th order power synchronizer which belongs to the class of NDA feedforward carrier synchronizers introduced earlier in the literature. Its mean—squared performance is obtained analytically and compared with simulation results. We observe that the resulting algorithm generates very accurate estimation even when the phase offset is high, that the self noise is absent and the performance of the algorithm is basically the same as die Cramer—Rao bound for moderate to high SNR. Finally we note mat me error variance derived for the mean—squared performance of this NDA ML synchronizer is an extension of the approximate variance formula appeared in Reference 20,equation(14) for M—PSK.