Performance of correlation-based frequency estimation methods in the presence of multiplicative noise

This paper provides a comprehensive performance evaluation of a correlation-based frequency estimator under different fading channel conditions. The efficiency of the estimator in terms of the Cramer-Rao lower bound (CRB) is investigated for different fading conditions, i.e., slow fading, fast fading, Rayleigh fading, and Ricean fading. The asymptotic CRB, which provides an insight to the frequency estimation in fading channels, has been evaluated via a frequency-domain approach. The optimal correlation lag value suitable for frequency estimation under different fading conditions is determined, and closed-form expressions for the variance of the estimates are provided. For proper implementation of the estimator, the Doppler spread of the signal is also estimated. It is shown that using the Doppler spread information, frequency estimation accuracy at low signal-to-noise ratios can be improved using multiple lags of correlations.     

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     

Frame-frequency estimation in ultrawideband systems

This paper investigates the estimation of the frame frequency in ultrawideband (UWB) communication systems. An estimation method is proposed that exploits the transmission of a periodic pulse sequence at the frame frequency. The samples of the received waveform are used to compute a cost function that depends on a trial value of the incoming pulse frequency. The location of the maximum provides an estimate of the transmitted frequency. The performance of the estimator is assessed theoretically and is compared to the Cramer-Rao lower bound. It is shown that in certain conditions the estimator achieves the bound at high SNR values. Simulations validate the theory and show the degradations in the estimation performance caused by multiple-access interference. They also give an idea of the estimation accuracy needed in a correlation receiver.     

A digital DS spread-spectrum receiver with joint channel andDoppler shift estimation

A digital spread-spectrum receiver design is presented for communication over multipath channels with severe Doppler shifts. The characteristics of the underwater channel relevant to spread-spectrum system design are discussed, and a channel model for short-range communications (less than 10 km) is defined. The receiver considered uses a digital coherent RAKE combiner, coupled with an extended Kalman filter (EKF)-based estimator for channel parameters and pseudonoise code delay. Receiver performance is evaluated by computing average bit-error rate (BER) versus iterations of the EKF joint estimator, using both fixed and time-varying channels. It is shown that the BER obtained using the EKF joint estimator closely tracks the optimum BER obtained when the channel, delay, and Doppler parameters are known exactly. Finally, the Cramer-Rao lower bound for time-invariant joint channel, delay, and Doppler estimation is derived, and compared with the ensemble averaged mean-squared error of the EKF estimator     

Estimation of the Parameters of Sinusoidal Signals in Non-Gaussian Noise

Accurate estimation of the amplitude and frequency parameters of sinusoidal signals from noisy observations is an important problem in many signal processing applications. In this paper, the problem is investigated under the assumption of non-Gaussian noise in general and Laplace noise in particular. It is proven mathematically that the maximum likelihood estimator derived under the condition of Laplace white noise is able to attain an asymptotic Cramer-Rao lower bound which is one half of that achieved by periodogram maximization and nonlinear least squares. It is also proven that when applied to non-Laplace situations, the Laplace maximum likelihood estimator, which may also be referred to as the nonlinear least-absolute-deviations estimator, can achieve an even higher statistical efficiency especially when the noise distribution has heavy tails. A computational procedure is proposed to overcome the difficulty of local extrema in the likelihood function. Simulation results are provided to validate the analytical findings.     

The estimation of time delay and Doppler stretch of widebandsignals

The problem of estimating the time delay and the Doppler stretch for wideband signals from a moving target is considered. The Cramer-Rao bound and the maximum likelihood (ML) method of estimation are derived. Due to the uncertainty of the reflection coefficient, the ML method may not be practicable. An alternative method involving the location of the peak of the wideband ambiguity function of the signal is suggested. The performance of the method is analysed, and, under high signal-to-noise ratios (SNRs), the method is asymptotically unbiased, and the variances of the estimates are closed to the Cramer-Rao bound for a large variety of signals. Optimum signals for the joint estimation of the time delay and the Doppler stretch under practical constraints are designed and, through computer simulations, their performance are shown to be superior to the commonly used signals     

Frequency-offset estimation for HIPERLAN

Frequency-offset correction is considered for a HIPERLAN (HIgh-PErformance Radio LAN) system over the indoor radio channel. Since the multipath channel response is not known a priori, a viable frequency-offset estimator should not depend on such knowledge. Such an estimator, using a single sample per symbol, is derived for HIPERLAN. The estimator is shown to approach the Cramer-Rao bound for frequency-offset estimation over a multipath channel. A HIPERLAN system simulation example shows that the performance with an offset of 150 kHz is within 0.5 dB of that of a system with zero frequency offset     

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.     

On Parameter Estimates of the Lossy Wave Equation

The Cramer-Rao bound (CRB) and maximum-likelihood estimates in Gaussian noise for the parameters of a sampled signal that satisfies the lossy wave equation (such as the voltage sampled along a transmission line) are derived. Results for both equally and unequally spaced samples are given. It is shown that for certain values of the wave number (frequency) the estimates are sensitive to the amount of the ldquoreflected waverdquo in the signal, while for other values of the wave number the estimates are insensitive. It is also shown that for certain special cases it is possible to improve on standard frequency estimation (up to 3.6 dB) by reflecting some of the (noise-free) signal energy into a backward-propagating wave and applying the maximum-likelihood estimator (MLE) to the new measured signal.     

On parameter estimation of MIMO flat-fading channels with frequency offsets

We address the frequency offsets and channel gains estimation problem for a multi-input multi-output (MIMO) flat-fading channel using a training sequence. The general case where the frequency offsets are possibly different for each transmit antenna is considered. The Cramer-Rao bound (CRB) for the problem at hand is derived. Additionally, we present a simple, closed-form expression for the large-sample CRB and show that it depends in a simple way on the channel parameters. Next, the parameters estimation issue is investigated. First, the maximum likelihood estimator (MLE), which entails solving an n-dimensional maximization problem where n is the number of transmit antennas, is derived. Then, we show that the likelihood function can be written as the product of n one-dimensional (1-D) functions if a suitable choice of the training sequence is made. Based on this fact, we suggest two computationally simpler methods. Numerical examples that illustrate the performance of the estimators and compare it with the CRB are provided.     

Computationally Efficient Single-Frequency Estimation Scheme in High-Rate WPAN Systems

Sub-optimal algorithms that avoid the complexity of the maximum likelihood scheme for estimating a frequency offset have been developed based on samples of the estimated auto-correlation function. However, their computation burden is still heavy for near-optimal performance. To overcome this problem, this paper proposes a reduced-complexity algorithm of single-frequency estimation for a high-rate wireless personal area network application. Accuracy and robustness of our frequency estimator are statistically assessed by Monte Carlo simulations. The results indicate that the proposed complexity effective algorithm closely conforms to the Cramer-Rao bound.     

Parameter estimation for random amplitude chirp signals

We consider the problem of estimating the parameters of a chirp signal observed in multiplicative noise, i.e., whose amplitude is randomly time-varying. Two methods for solving this problem are presented. First, an unstructured nonlinear least-squares approach (NLS) is proposed. It is shown that by minimizing the NLS criterion with respect to all samples of the time-varying amplitude, the problem reduces to a two-dimensional (2-D) maximization problem. A theoretical analysis of the NLS estimator is presented, and an expression for its asymptotic variance is derived. It is shown that the NLS estimator has a variance that is very close to the Cramer-Rao bound. The second approach combines the principles behind the high-order ambiguity function (HBF) and the NLS approach. It provides a computationally simpler but suboptimum estimator. A statistical analysis of the HAF-based estimator is also carried out, and closed-form expressions are derived for the asymptotic variance of the HAF estimators based on the data and on the squared data. Numerical examples attest to the validity of the theoretical analyzes and establish a comparison between the two proposed methods     

Aperiodic complementary sets of sequences-based MIMO frequency selective channel estimation

Accurate estimation of MIMO frequency selective fading channels is important for reliable communication. In this letter, a new channel estimator which relies on aperiodic complementary sets of sequences is proposed. Theoretical analysis and Monte-Carlo simulation have shown that it achieves the minimum possible Cramer-Rao lower bound. A low-complexity hardware implementation of the estimator is also provided.     

Sensor array processing based on subspace fitting

Algorithms for estimating unknown signal parameters from the measured output of a sensor array are considered in connection with the subspace fitting problem. The methods considered are the deterministic maximum likelihood method (ML), ESPRIT, and a recently proposed multidimensional signal subspace method. These methods are formulated in a subspace-fitting-based framework, which provides insight into their algebraic and asymptotic relations. It is shown that by introducing a specific weighting matrix, the multidimensional signal subspace method can achieve the same asymptotic properties as the ML method. The asymptotic distribution of the estimation error is derived for a general subspace weighting, and the weighting that provides minimum variance estimates is identified. The resulting optimal technique is termed the weighted subspace fitting (WSF) method. Numerical examples indicate that the asymptotic variance of the WSF estimates coincides with the Cramer-Rao bound. The performance improvement compared to the other techniques is found to be most prominent for highly correlated signals     

Cramer-Rao bound for Gaussian random processes and applications toradar processing of atmospheric signals

Calculations of the exact Cramer-Rao bound (CRB) for unbiased estimates of the mean frequency, signal power, and spectral width of Doppler radar/lidar signals (a Gaussian random process) are presented. Approximate CRBs are derived using the discrete Fourier transform (DFT). These approximate results are equal to the exact CRB when the DFT coefficients are mutually uncorrelated. Previous high SNR limits for CRBs are shown to be inaccurate because the discrete summations cannot be approximated with integration. The performance of an approximate maximum likelihood estimator for mean frequency approaches the exact CRB for moderate SNR and moderate spectral width     

Phase Estimators with Predetection Feedback for Optical Communication

Causal, minimum mean-square error (MMSE) estimators of a Gauss-Markov process observed through a conditional Poisson process whose rate parameter is a linear function of the estimation error are presented. Although the conditional estimation performance is data dependent, precomputable upper bounds on the average estimation performance are obtained. Approximate expressions are also presented for the Cramer-Rao lower bound, and it is shown that the estimator performance achieves that bound with equality when the estimator is operating considerably "above threshold." The estimator structure is applied to the problem of phase-tracking receivers for optical communication. Two receiver structures that use predetector phase feedback are considered: one uses local reference fields to allow the detector to observe phase error (homodyne/heterodyne), while the other is a novel direct detection receiver that depends explicitly on the closed-loop nature of the phase estimator. It is concluded that a large local oscillator amplitude is desirable to improve the phase-tracking performance in the homodyne/heterodyne case, and that as few as 4/8 detected signal photons per phase coherence time are required to keep the estimator above threshold. The direct detection scheme achieves the same performance as the homodyne system only in the limit of no dark current-background noise counts, and in general may require considerably more signal photons to keep the estimator "locked."     

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     

SNR estimation in time-varying fading channels

Signal-to-noise ratio (SNR) estimation is considered for phase-shift keying communication systems in time-varying fading channels. Both data-aided (DA) estimation and nondata-aided (NDA) estimation are addressed. The time-varying fading channel is modeled as a polynomial-in-time. Inherent estimation accuracy limitations are examined via the Cramer-Rao lower bound, where it is shown that the effect of the channel's time variation on SNR estimation is negligible. A novel maximum-likelihood (ML) SNR estimator is derived for the time-varying channel model. In DA scenarios, where the estimator has a simple closed-form solution, the exact performance is evaluated both with correct and incorrect (i.e., mismatched) polynomial order. In NDA estimation, the unknown data symbols are modeled as random, and the marginal likelihood is used. The expectation-maximization algorithm is proposed to iteratively maximize this likelihood function. Simulation results show that the resulting estimator offers statistical efficiency over a wider range of scenarios than previously published methods.     

基于特征分析和二次规划的窄带信号超分辨率频率估计

本文研究了窄带信号的超分辨率频率估计.在单次实验数据和信号包络函数形式未知的条件下,通过推广应用于分布源DOA估计的特征分析方法,并利用最优化理论中的二次规划算法,提出了一种窄带信号的超分辨率频率估计方法,本文称之为EQP方法.通过仿真分析,将该方法和其它方法进行了比较,同时还将估计结果的均方误差(MSE)的数值仿真结果和克拉美-罗界(CRB)作了比较,都表明本文提出的方法有效实现了窄带信号的超分辨率频率估计.     

Cramer-Rao lower bounds for QAM phase and frequency estimation

In this paper, we present the true Cramer-Rao lower bounds (CRLBs) for the estimation of phase offset for common quadrature amplitude modulation (QAM), PSK, and PAM signals in AWGN channels. It is shown that the same analysis also applies to the QAM, FSK, and PAM CRLBs for frequency offset estimation. The ratio of the modulated to the unmodulated CRLBs is derived for all QAM, PSK, and PAM signals and calculated for specific cases of interest. This is useful to determine the limiting performance of synchronization circuits for coherent receivers without the need to simulate particular algorithms. The hounds are compared to the existing true CRLBs for an unmodulated carrier wave (CW), BPSK, and QPSK. We investigated new and existing QAM phase estimation algorithms in order to verify the new phase CRLB. This showed that new minimum distance estimator performs close to the QAM bound and provides a large improvement over the power law estimator at moderate to high signal-to-noise ratios