Bayesian and Hybrid Cramér–Rao Bounds for the Carrier Recovery Under Dynamic Phase Uncertain Channels

Abstract

In this paper, we study Bayesian and hybrid Cramé r-Rao bounds (BCRB and HCRB) for the code-aided (CA), the data-aided (DA) and the non-data-aided (NDA) dynamical phase estimation of QAM modulated signals. We address the bounds derivation for both the off-line scenario, for which the whole observation frame is used, and the on-line which only takes into account the current and the previous observations. For the CA scenario we show that the computation of the Bayesian information matrix (BIM) and of the hybrid information matrix (HIM) is NP hard. We then resort to the belief-propagation (BP) algorithm or to the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm to obtain some approximate values. Moreover, in order to avoid the calculus of the inverse of the BIM and of the HIM, we present some closed form expressions for the various CRBs, which greatly reduces the computation complexity. Finally, some simulations allow us to compare the possible improvements enabled by the off-line and the CA scenarios.

On a hybrid preamble/soft-output demapper approach for time synchronization for IEEE 802.15.6 narrowband WBAN

“…Numerical results are presented in Table IV and V. The saturation of the MSE at the right side of Fig. 5 and 6 is due to the self-noise of the updating error (11). Similar results are obtained for other roll-off factors.…”

“…In this case, no pilot signal is needed. Also, the timing synchronizer and the channel decoder can improve each other progressively by exchanging information [10], [11]. However, this technique cannot be held in a WBAN context due to its implementation complexity.…”

Near MAP Dynamical Delay Estimator and Bayesian CRB for Coded QAM Signals

“…To find a compromise between the spectral efficiency and the synchronizer reliability, CA techniques have been proposed. These techniques take advantage of the decoder soft output to reduce the estimator error in the timing recovery [7]- [15] as well as in the carrier frequency and the phase synchronization process [16], [17]. In [7], [8], the authors deal with the time synchronization problem for a constant time delay and they suggest to use a CA TED derived from the Maximum Likelihood (ML) estimator.…”

“…However, the performance evaluation has been made only at low SNR regime, based on the assumption that inter-symbol interference (ISI) could be approximated by an additive Gaussian noise as in [9]. To evaluate one estimator relevance, its MSE is traditionaly compared to lower bounds such as the Cramer-Rao Bounds (CRB) [19] as the one derived for the unknown random phase offset problem in [16], [20]. Closed form expressions of the CRB have been derived in [21] for CA carrier frequency and phase offset estimation for turbo-coded Square-QAM modulated signals.…”

A constrained hybrid Cramér-Rao bound for parameter estimation

“…One can cite, for example, the Gaussian generalized linear model [9], array shape calibration [1], time-delay estimation in radar signal [4], phase estimation in binary phase-shift keying transmission in a nondata-aided context [10], phase estimation of QAM modulated signals [11], cisoid frequency estimation [12], joint estimation of the pair dynamic carrier phase/Doppler shift and the time-delay in a digital receiver [13], parameters estimation in long-code DS/CDMA systems [14], bearing estimation for deformed towed arrays in the fluid mechanics context [15]. It is therefore the aim of this paper to provide an extension of the deterministic CCRB [16] to the hybrid parameter context yielding the Constrained HCRB (CHCRB).…”