Cramer-Rao Lower Bound for Non-Data-Aided SNR Estimation of Linear Modulation Schemes

Powerful parameter estimators exhibit a jitter variance which is fairly close to the Cramer-Rao lower bound (CRLB) as the theoretical limit. In contrast to symbol timing and carrier frequency/phase, not very much information is available from the open literature with respect to the signal-to-noise ratio (SNR), i. e., the CRLB has been reported only for the data- aided case and some simple M-PSK examples for non-data-aided estimation of the SNR. Motivated by this background, an efficient algorithm is presented which applies to any M-ary one/two- dimensional modulation scheme with axis/halfplane symmetry and a channel distorted by additive white Gaussian noise. Finally, the performance of different SNR estimators is compared to the derived bound.     

基于改进DFT相位差的正弦波频率估计

针对基于离散傅里叶变换( DFT)相位差的正弦波频率估计方法对频偏敏感的问题,提出了一种改进DFT相位差频率估计方法。首先推导了DFT相位差法频率估计的均方误差,然后提出了基于Rife插值的改进DFT相位差频率估计方法,较好地解决了正弦波频率估计对频偏敏感的问题。仿真实验结果表明,改进方法在各种频偏下均能取得较高的估计精度,估计性能接近克拉美罗限( CRLB)。     

Blind estimation of OFDM carrier frequency offset via oversampling

Blind deterministic estimation of the orthogonal frequency division multiplexing (OFDM) frequency offset via oversampling is proposed in this paper. This method utilizes the intrinsic phase shift of neighboring sample points incurred by the frequency offset that is common among all subcarriers. The proposed method is data efficient - it requires only a single OFDM symbol to achieve reliable estimation, hence making it more suitable to systems with stringent delay requirement and mobility-induced channel variation. The proposed scheme is devised to perfectly retrieve frequency offset in the absence of noise. Quite remarkably, we show that in the presence of channel noise, this intuitive scheme is indeed the maximum likelihood estimate of the carrier frequency offset. The possible presence of virtual carriers are also accommodated in the system model, and some interesting observations are obtained. The Cramer-Rao lower bound is derived for the oversampling-based signal model, and we show through numerical simulation that the proposed algorithm is efficient. Practical issues such as identifiability, the front-end filter bandwidth, and the possible presence of correlated noises are also carefully addressed.     

超奈奎斯特信号载波频偏估计的梯度下降算法

超奈奎斯特（Faster-than-Nyquist, FTN）传输技术是一种基于时域压缩的信号调制技术，具有更高的传输速率及频带利用率，但是却引入了无限长的码间串扰（inter-symbol interferences , ISIs），这对载波频偏估计提出了新的挑战。对于FTN信号的频偏估计，传统算法在门限和精度等方面都有所下降。本文将梯度下降算法应用到载波频偏估计中，通过迭代搜索的方式，获得对频偏的精确估计。估计过程中直接通过数据部分进行估计，不需要借助额外的导频序列。仿真结果表明，梯度下降法与传统算法在性能上相比有了较好的改善，虽然具有较多的迭代次数和运算量，但是却能够较好地适应FTN信号的特性。梯度下降法不仅在门限范围内更靠近克拉美罗界（Cramer-Rao bound，CRB），而且在整个频段上具有更稳定的性能。      

Blind channel and carrier frequency offset estimation usingperiodic modulation precoders

Previous results have shown that blind channel estimators, which are resilient to the location of channel zeros, color of additive stationary noise, and channel order overestimation errors, can be developed for communication systems equipped with transmitter-induced cyclostationarity precoders. The present paper extends these blind estimation approaches to the more general problem of estimating the unknown intersymbol interference (ISI) and carrier frequency offset/Doppler effects using such precoders. An all-digital open-loop carrier frequency offset estimator is developed, and its asymptotic (large sample) performance is analyzed and compared to the Cramer-Rao bound (CRB). A subspace-based channel identification approach is proposed for estimating, in closed-form, the unknown channel, regardless of the channel spectral nulls. It is shown that compensating for the carrier frequency offset introduces no penalty in the asymptotic performance of the subspace channel estimator. Simulations are presented to corroborate the performance of the proposed algorithms     

Two-Dimensional Polynomial Phase Signals: Parameter Estimation and Bounds

This paper considers the problem of parametric modeling and estimation of nonhomogeneous two-dimensional (2-D) signals. In particular, we focus our study on the class of constant modulus polynomial-phase 2-D nonhomogeneous signals. We present two different phase models and develop computationally efficient estimation algorithms for the parameters of these models. Both algorithms are based on phase differencing operators. The basic properties of the operators are analyzed and used to develop the estimation algorithms. The Cramer-Rao lower bound on the accuracy of jointly estimating the model parameters is derived, for both models. To get further insight on the problem we also derive the asymptotic Cramer-Rao bounds. The performance of the algorithms in the presence of additive white Gaussian noise is illustrated by numerical examples, and compared with the corresponding exact and asymptotic Cramer-Rao bounds. The algorithms are shown to be robust in the presence of noise, and their performance close to the CRB, even at moderate signal to noise ratios.     

Estimators of the Nakagami-m parameter and performance analysis

The Nakagami-m parameter is known to capture the envelope distribution of various fading channel conditions in wireless communications. The value of m is indicative of the severity of fading, and is a measure of channel quality, making its estimation necessary in many applications. In this paper, we summarize the existing estimators for the Nakagami-m parameter and propose a new class of estimators, whose performance is analyzed by deriving the asymptotic variance and comparing with the Cramer-Rao bound (CRB). Moreover, we develop a novel estimator robust to the presence of additive white Gaussian noise (AWGN) or any other additive noise with a symmetrical distribution. We also discuss practical issues not previously addressed in the literature including adaptation and computational complexity of these estimators. We conclude that our novel integer-moment based estimators are the best choice from a computational complexity and performance point of view. Simulation results corroborate our analysis.     

Frequency estimation in the presence of Doppler spread: performanceanalysis

We are concerned with the estimation of the frequency of a complex sinusoid that has been corrupted by complex-valued multiplicative and additive noise. This problem is important in many applications including array processing in the case of spatially distributed sources and synchronization in the context of time-selective channels. The multiplicative noise smears the spectral line due to the sinusoid. This smearing, which is often called Doppler spreading, may significantly degrade the estimation accuracy. The goal of this paper is to analytically assess this degradation. The finite-sample Cramer-Rao bounds (CRBs) are derived, and closed-form expressions are given for the large-sample CRB. The latter gives insights into the effective coherent and noncoherent SNRs for frequency estimation. We then analyze the accuracy of frequency estimators that are based on the angles of the sample covariances. Simulations results are presented to illustrate the theoretical results     

Doppler frequency estimation and the Cramer-Rao bound

Addresses the problem of Doppler frequency estimation in the presence of speckle and receiver noise. An ultimate accuracy bound for Doppler frequency estimation is derived from the Cramer-Rao inequality. It is shown that estimates based on the correlation of the signal power spectra with an arbitrary weighting function are approximately Gaussian-distributed. Their variance is derived in terms of the weighting function. It is shown that a special case of a correlation-based estimator is a maximum-likelihood estimator that reaches the Cramer-Rao bound. These general results are applied to the problem of Doppler centroid estimation from SAR (synthetic aperture radar) data     

A computationally efficient method of timing and phase estimation in TDMA systems using a preamble sequence

Optimum and near optimum methods of burst parameter estimation based on the discrete Fourier transform are presented. The methods are applicable to single sample per symbol demodulators when the burst preamble is chosen to be a discrete frequency component at one quarter of the symbol rate. The proposed methods do not require the squaring of the signal, and hence avoid the loss associated with squaring the noise components. The optimum method is shown to reach the Cramer-Rao bound for symbol timing and carrier phase estimation for the chosen preamble, whereas the more computationally efficient method shows a loss of 1.0dB and yields information regarding the timing offset only. The performances of the two methods in the presence of carrier frequency offset and additive white Gaussian noise are analysed, and implementations are given together with a comparison of the respective computational complexities.     

Optimal blind carrier recovery for MPSK burst transmissions

The paper introduces and analyzes the asymptotic (large sample) performance of a family of blind feedforward nonlinear least-squares (NLS) estimators for joint estimation of carrier phase, frequency offset, and Doppler rate for burst-mode phase-shift keying transmissions. An optimal or "matched" nonlinear estimator that exhibits the smallest asymptotic variance within the family of envisaged blind NLS estimators is developed. The asymptotic variance of these estimators is established in closed-form expression and shown to approach the Cramer-Rao lower bound of an unmodulated carrier at medium and high signal-to-noise ratios (SNR). Monomial nonlinear estimators that do not depend on the SNR are also introduced and shown to perform similarly to the SNR-dependent matched nonlinear estimator. Computer simulations are presented to corroborate the theoretical performance analysis.     

基于循环移位前导序列的OFDM载波同步方法

提出一种基于循环移位前导序列的正交频分复用（Orthogonal Frequency Division Multiplexing，OFDM）系统载波同步方法，该方法根据离散傅里叶变换（Discrete Fourier Transform，DFT）的时移性质，构造循环移位前导序列，并设计了两个频偏估计器。由估计器1完成粗同步，而精同步采用两个估计器的最优线性组合完成。在多径信道下优化估计器的设计，消除了均方误差中出现的地板效应。新算法与传统算法相比，估计范围大，运算精度高，且估计范围和精度可灵活调节。同时分析并得出了本估计方法的Cramer—Rao下限。仿真结果表明，新算法降低了均方误差和误码率，有效提高了系统性能。     

An Efficient Maximum Likelihood Carrier Frequency Offset Estimator for OFDM Using Null Subcarriers and Cyclic Prefix

Accurate estimation of carrier frequency offset (CFO) is an important requirement in orthogonal frequency-division multiplexing (OFDM) based wireless communication systems. In this paper, we propose a hybrid procedure to accomplish this task efficiently. One of the key importance of the approach is the judicious combination of two independent estimators so as to reduce the bandwidth overhead and computational complexity over many conventional methods. It employs the cyclic prefix and a few null subcarriers, respectively, for the fractional and integer frequency offset estimations. We also propose a novel null subcarrier allocation scheme based on Fibonacci series. The range of frequency offset that can be estimated by the proposed technique is equal to the full OFDM bandwidth. Furthermore, performance of the proposed CFO estimator is mathematically analyzed by deriving an expression for the bit error probability of the receiver under Rayleigh fading channel and the Cramer-Rao lower bound for the mean square estimation error. For moderate SNRs, our approach is shown to greatly outperform some recently published methods in terms of BER performance, bandwidth overhead and receiver complexity.     

基于周期图和自相关函数的频偏估计方法

提出了一种频差估计的新算法,该算法结合观测序列的周期图和自相关函数进行频率估值。仿真结果表明：对于数据辅助的MPSK信号,信噪比较低时,其估值方差仍接近克拉美劳界,且该算法计算复杂度低,易于实现。     

Frequency estimation in slowly fading multipath channels

This paper concerns the estimation of a frequency offset of a known (pilot) signal propagated through a slowly fading multipath channel, such that channel parameters are considered to he constant over the observation interval. We derive a maximum-likelihood (ML) frequency estimation algorithm for additive Gaussian noise and path amplitudes having complex Gaussian distribution when covariance matrices of the fading and noise are known; we consider in detail the algorithm for the white noise and Rayleigh fading, in particular, for independent fading of path amplitudes and pilot signals with diagonal autocorrelation matrices. For the latter scenario, we also derive an ML frequency estimator when the power delay profile is unknown, but the noise variance and bounds for the path amplitude variances are specified; in particular, this algorithm can be used when path delays and amplitude variances are unknown. Finally, we consider frequency estimators which do not use a priori information about the noise variance; these algorithms are also operable without timing synchronization. All the frequency estimators exploit the multipath diversity by combining periodograms of multipath signal components and searching for the maximum of the combined statistic. For implementation of the algorithms, we use a fast Fourier transform-based coarse search and fine dichotomous search. We perform simulations to compare the algorithms. The simulation results demonstrate high accuracy performance of the proposed frequency estimators in wide signal-to-noise ratio and frequency acquisition range.     

Rapid carrier acquisition from baud-rate samples

Maximum-likelihood estimation of phase and frequency offset involves the maximization of a nonlinear likelihood function. We develop an estimator for carrier acquisition by linearizing the likelihood-function, and show that its performance is close to the Cramer-Rao bound (CRB). The estimator is then extended to blind frequency acquisition and combined with testing to eliminate unlikely hypotheses. With proper modeling of the likelihood resulting from the blind search, simulations show that the algorithm quickly rejects all but the correct frequency hypothesis     

一种基于分散导频块的频偏估计新算法

针对短突发信号解调的需要,提出了一种基于分散导频块的频偏估计新算法。区别于以往基于连续导频的频偏估计算法,该算法除利用了导频本身已知信息外,还充分利用了导频的位置信息。在相同的帧效率下,相比于基于连续导频的频偏估计算法,该算法可以获得更好的估计性能。理论分析和仿真结果都表明,该算法在其无偏估计范围内估计性能达到了相应帧结构下的克拉美罗界。     

Performance of robust symbol‐timing and carrier‐frequency estimation for OFDM systems

In recent years, many maximum likelihood (ML) blind estimators have been proposed to estimate timing and frequency offsets for orthogonal frequency division multiplexing (OFDM) systems. However, the previously proposed ML blind estimators utilizing cyclic prefix do not fully characterize the random observation vector over the entire range of the timing offset and will significantly degrade the estimation performance. In this paper, we present a global ML blind estimator to compensate the estimation error. Moreover, we extend the global ML blind estimator by accumulating the ML function of the estimation parameters to achieve a better accuracy without increasing the hardware or computational complexity. The simulation results show that the proposed algorithm can significantly improve the estimation performance in both additional white Gaussian noise and ITU‐R M.1225 multipath channels. Copyright © 2008 John Wiley & Sons, Ltd.     

Analytic and Asymptotic Analysis of Bayesian CramÉr–Rao Bound for Dynamical Phase Offset Estimation

In this paper, we present a closed-form expression of a Bayesian Cramer-Rao lower bound for the estimation of a dynamical phase offset in a non-data-aided BPSK transmitting context. This kind of bound is derived considering two different scenarios: a first expression is obtained in an offline context, and then a second expression in an online context logically follows. The SNR-asymptotic expressions of this bound drive us to introduce a new asymptotic bound, namely the asymptotic Bayesian Cramer-Rao Bound. This bound is close to the classical Bayesian bound but is easier to evaluate.     

Optimal blind nonlinear least-squares carrier phase and frequency offset estimation for general QAM modulations

This paper introduces a family of blind feedforward nonlinear least-squares (NLS) estimators for joint estimation of the carrier phase and frequency offset of general quadrature amplitude modulated (QAM) transmissions. As an extension of the Viterbi and Viterbi (1983) estimator, a constellation-dependent optimal matched nonlinear estimator is derived such that its asymptotic (large sample) variance is minimized. A class of conventional monomial estimators is also proposed. The asymptotic performance of these estimators is established in closed-form expression and compared with the Cramer-Rao lower bound. A practical implementation of the optimal matched estimator, which is a computationally efficient approximation of the latter and exhibits negligible performance loss, is also derived. Finally, computer simulations are presented to corroborate the theoretical performance analysis and indicate that the proposed optimal matched nonlinear estimator improves significantly the performance of the classic fourth-power estimator.