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]

Near Optimum Low Complexity Smoothing Loops for Dynamical Phase Estimation—Application to BPSK Modulated Signals

International audience—This correspondence provides and analyzes a low complexity, near optimum, fixed-interval smoothing algorithm that approaches the performance of an optimal smoother for the price of two low complexity sequential estimators, i.e., two phase-locked loops (PLLs). Based on a linear approximation of the problem, a theoretical performance evaluation is given. The theoretical results are compared to some simulation results and to the Bayesian and hybrid Cramér–Rao bounds. They illustrate the good performance of the proposed smoothing PLL (S-PLL) algorithm. Index Terms—Dynamical phase estimation, phase-locked loop (PLL), smoothing algorithm

Signal to noise ratio estimation using the Expectation Maximization Algorithm

Signal to Noise Ratio (SNR) estimation when the transmitted symbols are unknown is a common problem in many communication systems, especially those which require an accurate SNR estimation. For instance, modern wireless communication systems usually require accurate estimate of SNR without knowledge of the transmitted symbols. In addition, SNR estimation is required in order to perform efficient signal detection, power control, and adaptive modulation In this study, Non data Aided (NDA) SNR estimation for Binary Phase Shift Keying (PBSK) and Quadrature Phase Shift Keying (QPSK) using the Expectation Maximization (EM) algorithm is developed. The assumption here is that the received data samples are drawn from a mixture of Gaussians distribution and they are independent and identically distributed (i.i.d.). The quality of the proposed estimator is examined via the Cramer-Rao Lower Bound (CRLB) of NDA SNR estimator. It is also assumed that the channel gain is constant during each symbol interval, and the noise is Additive White Gaussian (AWGN). Maximum Likelihood estimator is being used if we have access to the complete data, in this case the problem would be much easier since we get the exact closed form solution, but when the observed data are incomplete or partially available, the EM algorithm will be used. This approach is an iterative method to get an approximated result which is either an approximated global maximum or local maximum. However, in the NDA SNR estimation, we only have a global maximum since our assumption is that the distribution is a mixture of Gaussians. This is being investigated for the cases of Single Input Single Output (SISO) and Single Input Multiple Output (SIMO). The main concern about the receive diversity is the cost, size, and power, that is why we resort to the transmit diversity such as Multiple Input single Output(MISO) with space time block codes (STBC). The base station usually serves hundreds to thousands of remote units which is the sole reason of using transmit diversity at the base station instead of at every remote unit covered by the base station. It is more economical in this case to add equipment to the base station instead of the remote units. Alamouti used a simple transmit diversity technique and assumed in his paper that the receiver has perfect knowledge of the channel transition matrix. However, this assumption may seem highly unrealistic. One of our contributions is to estimate the channel information, as well as the noise variance which would be used in estimating the SNR and deriving the CRLB for both DA and NDA case. The performance of our estimator would be empirically assessed using Monte-Carlo simulations, with CRLB as a performance metric

Bornes de Cramer Rao de DOA pour signaux BPSK et QPSK en présence de bruit non uniforme

Cet article est consacré au calcul de la borne de Cramer-Rao (CRB) stochastique des directions d'arrivée (DOA) de signaux de modulation BPSK et QPSK en présence de bruit additif gaussien circulaire de matrice de covariance spatiale diagonale arbitraire. Une expression analytique de cette borne est donnée dans le cas d'une seule source grâce au découplage entre les paramètres puissance et les paramètres de phase de la matrice d'information de Fisher (FIM). Cette borne est comparée à diverses bornes dérivées selon diverses connaissances a priori. Ces résultats et comparaisons sont ensuite étendus au cas de deux sources indépendantes où des expressions analytiques de CRB sont obtenues pour des grandes valeurs de rapport signal sur bruit (SNR). Nous avons mis en évidence de grandes différences de comportement par rapport à la borne standard obtenue dans le cas de signaux gaussiens circulaires, démontrant ainsi l'intérêt des approches EM dans le cas de connaissances a priori sur la distribution discrète des sources

Power delay profile and noise variance estimation for OFDM

In this letter, we present cyclic-prefix (CP) based noise-variance and power-delay-profile estimators for Orthogonal Frequency Division Multiplexing (OFDM) systems. Signal correlation due to the use of the CP is exploited without requiring additional pilot symbols. A heuristic estimator and a class of approximate maximum likelihood (ML) estimators are proposed. The proposed algorithms can be applied to both unitary and non-unitary constellations. These algorithms can be readily used for applications such as minimum mean-square error (MMSE) channel estimation

Non-Data-Aided Parameter Estimation in an Additive White Gaussian Noise Channel

Non-data-aided (NDA) parameter estimation is considered for binary-phase-shift-keying transmission in an additive white Gaussian noise channel. Cramer-Rao lower bounds (CRLBs) for signal amplitude, noise variance, channel reliability constant and bit-error rate are derived and it is shown how these parameters relate to the signal-to-noise ratio (SNR). An alternative derivation of the iterative maximum likelihood (ML) SNR estimator is presented together with a novel, low complexity NDA SNR estimator. The performance of the proposed estimator is compared to previously suggested estimators and the CRLB. The results show that the proposed estimator performs close to the iterative ML estimator at significantly lower computational complexity