Bayesian Cramer-Rao Lower Bound Research Articles

Phase Noise in MIMO Systems: Bayesian Cramér–Rao Bounds and Soft-Input Estimation

This paper addresses the problem of estimating time varying phase noise caused by imperfect oscillators in multiple-input multiple-output (MIMO) systems. The estimation problem is parameterized in detail and based on an equivalent signal model its dimensionality is reduced to minimize the overhead associated with phase noise estimation. New exact and closed-form expressions for the Bayesian Cramer-Rao lower bounds (BCRLBs) and soft-input maximum a posteriori (MAP) estimators for online, i.e., filtering, and offline, i.e., smoothing, estimation of phase noise over the length of a frame are derived. Simulations demonstrate that the proposed MAP estimators' mean-square error (MSE) performances are very close to the derived BCRLBs at moderate-to-high signal-to-noise ratios. To reduce the overhead and complexity associated with tracking the phase noise processes over the length of a frame, a novel soft-input extended Kalman filter (EKF) and extended Kalman smoother (EKS) that use soft statistics of the transmitted symbols given the current observations are proposed. Numerical results indicate that by employing the proposed phase tracking approach, the bit-error rate performance of a MIMO system affected by phase noise can be significantly improved. In addition, simulation results indicate that the proposed phase noise estimation scheme allows for application of higher order modulations and larger numbers of antennas in MIMO systems that employ imperfect oscillators.

Superimposed training design based on Bayesian optimisation for channel estimation in two-way relay networks

In this study, the superimposed training strategy is introduced into orthogonal frequency division multiplexing-modulated amplify-and-forward two-way relay network (TWRN) to perform two-hop transmission-compatible individual channel estimation. Through the superposition of an additional training vector at the relay under power allocation, the separated source–relay channel information can be directly obtained at the destination and then used to estimate the channels. The closed-form Bayesian Cramer-Rao lower bound (CRLB) is derived for the estimation of block-fading frequency-selective channels with random channel parameters, and orthogonal training vectors from the two source nodes are required to keep the Bayesian CRLB simple because of the self-interference in the TWRN. A set of optimal training vectors designed from the Bayesian CRLB are applied in an iterative linear minimum mean-square-error channel estimation algorithm, and the mean-square-error performance is provided to verify the Bayesian CRLB results.

CRLB Based Optimal Noise Enhanced Parameter Estimation Using Quantized Observations

In this letter, optimal additive noise is characterized for parameter estimation based on quantized observations. First, optimal probability distribution of noise that should be added to observations is formulated in terms of a Cramer-Rao lower bound (CRLB) minimization problem. Then, it is proven that optimal additive ¿noise¿ can be represented by a constant signal level, which means that randomization of additive signal levels is not needed for CRLB minimization. In addition, the results are extended to the cases in which there exists prior information about the unknown parameter and the aim is to minimize the Bayesian CRLB (BCRLB). Finally, a numerical example is presented to explain the theoretical results.

Optimization of Spatiotemporal Apertures in Channel Sounding

In this paper, we investigate the impact of the spatiotemporal aperture of a channel sounding system equipped with antenna arrays at the transmitter and receiver on the accuracy of joint estimation of Doppler frequency and bidirection. The contribution of this paper is threefold. First, we state a spatiotemporal model that can describe parallel as well as switched sounding systems. The proposed model is applicable for arbitrary layouts of the spatial arrays. To simplify the derivations, we investigate the special case of linear spatial arrays. However, the results obtained for linear arrays can be generalized to arbitrary arrays. Secondly, we give the necessary and sufficient conditions for a spatiotemporal array to yield the minimum Cramer-Rao lower bound (CRLB) in the single-path case and Bayesian CRLB in the multipath case. The obtained conditions amount to an orthogonality condition on the spatiotemporal array. Thirdly, we define the Doppler-bidirection ambiguity function and derive the necessary and sufficient conditions for a linear spatiotemporal array to be ambiguous. Based on the ambiguity function, we define the normalized side-lobe level, which we propose to use as a figure of merit in the design of spatiotemporal arrays.

MAP/ML Estimation of the Frequency and Phase of a Single Sinusoid in Noise

The problem of estimating the frequency and carrier phase of a single sinusoid observed in additive, white, Gaussian noise is addressed. Much of the work in the literature considers maximum likelihood (ML) estimation. However, the ML estimator given by the location of the peak of a periodogram in the frequency domain shown in D.C. Rife and R. R. Boorstyn, Single-tone parameter estimation from discrete-time observations, IEEE Transactions on Information Theory, vol. IT-20, pp. 591-598, Sep. 1974, has a very high computational complexity. This paper derives the explicit structure of the ML estimator for data processing in the time domain, assuming only reasonably high signal-to-noise ratio (SNR). The result of this approximate ML estimator shows that both the phase and the magnitude of the noisy signal samples are utilized in the and the phase data alone as assumed in S. A. Tretter, Estimating the frequency of a noisy sinusoid by linear regression, IEEE Transactions on Information Theory, vol. IT-31, pp. 832-835, Nov. 1985 and S. Kay, A fast and accurate single frequency estimator, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 39, pp. 1203-1205, May 1991, is not a sufficient statistic. The sample-by-sample iterative processing nature of the estimator enables us to propose a novel, recursive phase-unwrapping algorithm that allows the estimator to be implemented efficiently. To facilitate the performance analysis, a new, linearized observation model for the instantaneous signal phase that is more accurate than that of S. A. Tretter, Estimating the frequency of a noisy sinusoid by linear regression, IEEE Transactions on Information Theory, vol. IT-31, pp. 832-835, Nov. 1985 and of S. Kay, A fast and accurate single frequency estimator, vol. 39, pp. 1203-1205, May 1991, is proposed. This new model explains physically why the phase data are weighted by the magnitude information in the ML estimator. Moreover, by incorporating a priori knowledge via the a priori probability density function of the unknown frequency and the carrier phase, the explicit structure of the approximate maximum a posteriori probability (MAP) estimator is derived, and the Bayesian Cramer-Rao lower bound (BCRLB) on the mean-square error (mse) is obtained. Our analysis shows that the mse performance of the MAP estimator can approach the BCRLB very closely