相位编码信号码元宽度估计的修正克拉美—罗限

推导了相位编码信号码元宽度估计的修正克拉美-罗限(MCRB).采用Parseval定理将单位冲激函数δ(t)平方的积分转换到频域计算,得到脉冲成形函数是矩形脉冲时码元宽度估计的修正克拉美-罗限.当脉冲成形函数是升余弦脉冲时则进行了数值计算.计算表明,当滚降系数为0.5时,升余弦脉冲和矩形脉冲对应的码元宽度估计性能相差1dB左右.     

On the true Cramer-Rao lower bound for data-aided carrier-phase-independent frequency offset and symbol timing estimation

In this letter we present new and simple closed form expressions for the true Cramer-Rao lower bound (CRB) for data-aided (DA) joint and individual carrier frequency offset and symbol timing estimation from a linearly modulated waveform transmitted over an AWGN channel. The bounds are derived under a carrier-phase-independent (CPI) estimation strategy wherein the carrier phase is viewed as a nuisance parameter and assumed to have a worst-case noninformative uniform distribution over [-ππ]. The computation of these CRBs requires only a single numerical integration. In addition, computationally simpler yet highly accurate asymptotic lower bounds are presented. As particularizations, new bounds for individual CPI frequency estimation with known symbol timing from M-PSK and continuous wave (CW) signals are also reported.     

一种低信噪比下PSK信号载波频率估计方法

提出了一种利用自适应陷波器对单频信号分量衰减的特性估计PSK信号的载波频率的方法.信号通过陷波器滤波,输出能量最小对应的频点即为信号的载波估计频率.仿真结果表明,在低信噪比环境下该方法的估计误差小于1％.     

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     

The Cramer-Rao bound for phase estimation from coded linearly modulated signals

In this letter, we express the Cramer-Rao bound (CRB) for carrier phase estimation from a noisy linearly modulated signal with encoded data symbols, in terms of the marginal a posteriori probabilities (APPs) of the coded symbols. For a wide range of classical codes (block codes, convolutional codes, and trellis-coded modulation), these marginal APPs can be computed efficiently by means of the Bahl-Cocke-Jelinke-Raviv (BCJR) algorithm, whereas for codes that involve interleaving (turbo codes and bit interleaved coded modulation), iterated application of the BCJR algorithm is required. Our numerical results show that when the BER of the coded system is less than about 10/sup -3/, the resulting CRB is essentially the same as when transmitting a training sequence.     

True Cramer-Rao bound for timing recovery from a bandlimited linearly Modulated waveform with unknown carrier phase and frequency

This paper derives the Cramer-Rao bound (CRB) related to the estimation of the time delay of a linearly modulated bandpass signal with unknown carrier phase and frequency. We consider the following two scenarios: joint estimation of the time delay, the carrier phase, and the carrier frequency; and joint estimation of the time delay and the carrier frequency irrespective of the carrier phase. The transmit pulse is a bandlimited square-root Nyquist pulse. For each scenario, the transmitted symbols constitute either an a priori known training sequence or an unknown random data sequence. In spite of the presence of random data symbols and/or a random carrier phase, we obtain a relatively simple expression of the CRB, from which the effect of the constellation and the transmit pulse are easily derived. We show that the penalty resulting from estimating the time delay irrespective of the carrier phase decreases with increasing observation interval. However, the penalty, caused by not knowing the data symbols a priori, cannot be reduced by increasing the observation interval. Comparison of the true CRB to existing symbol synchronizer performance reveals that decision-directed timing recovery is close to optimum for moderate-to-large signal-to-noise ratios.     

The modified Cramer-Rao bound in vector parameter estimation

In this paper we extend the scalar modified Cramer-Rao bound (MCRB) to the estimation of a vector of nonrandom parameters in the presence of nuisance parameters. The resulting bound is denoted with the acronym MCRVB, where “V” stands for “vector”. As with the scalar bound, the MCRVB is generally looser than the conventional CRVB, but the two bounds are shown to coincide in some situations of practical interest. The MCRVB is applied to the joint estimation of carrier frequency, phase, and symbol epoch of a linearly modulated waveform corrupted by correlated impulsive noise (encompassing white Gaussian noise as a particular case), wherein data symbols and noise power are regarded as nuisance parameters. In this situation, calculation of the conventional CRVB is infeasible, while application of the MCRVB leads to simple useful expressions with moderate analytical effort. When specialized to the case of white Gaussian noise, the MCRVB yields results already available in the literature in fragmentary form and simplified contexts     

Minimax lower bound for time-varying frequency estimation ofharmonic signal

Estimation of the instantaneous frequency and its derivatives is considered for a harmonic complex-valued signal with the time-varying phase and time-invariant amplitude. The asymptotic minimax lower bound is derived for the mean squared error of estimation, provided that the phase is an arbitrary m-times piecewise differentiable function of time. It is shown that this lower bound is different only in a constant factor from the upper bound for the mean squared errors of the local polynomial periodogram with the optimal window size. The time-varying phases “worst” for estimation of the instantaneous frequency and its derivatives are obtained as a solution of the minimax problem     

一种卫星信号载波频率精确估计算法

针对卫星载波频率高精度估计问题,阐述了现有的频率估计实时处理方法,结合类Rife频率修正算法,并考虑卫星信号特点,给出了适用于数字方法实现的卫星信号载波频率高精度估计算法流程。对其进行了仿真实现,其结果表明,该算法能有效提高卫星信号载频估计精度。     

The Cramer-Rao bound on frequency estimates of signals closelyspaced in frequency

This paper examines the Cramer-Rao (CR) lower bound on the variance of frequency estimates for the problem of n signals closely spaced in frequency. The main results presented are simple analytic expressions for the CR bound in terms of the maximum frequency separation, δω, SNR, and the number of data vectors, N, that are valid for small δω. The results are applicable to the conditional (deterministic) signal model. The results show that the CR bound on frequency estimates is proportional to (δω)^-2(n-1)/N×SNR. Therefore, the bound increases rapidly as the signal separation is reduced. Examples indicate that the expressions closely approximate the exact CR bounds whenever the signal separation is smaller than one resolution cell. Based upon the results, it is argued that the threshold SNR at which an unbiased estimator can resolve n closely spaced signals is at least proportional to (δω)^-2n/N. The results are quite general and apply to many different types of temporal and spatial sampling grids     

Low-SNR limit of the Cramer-Rao bound for estimating the carrierphase and frequency of a PAM, PSK, or QAM waveform

We consider the Cramer-Rao bound (CRB) for the estimation of the carrier phase and frequency of a noisy linearly modulated signal with random data symbols. The observation vector consists of the matched filter output samples taken at the symbol rate, assuming known symbol timing. Because of the presence of the random data, the evaluation of this CRB is quite tedious. Instead, here we derive a simple closed-form expression for the limit of the CRB at low-signal-to-noise ratio (SNR), which holds for arbitrary PAM, PSK, and QAM constellations     

一种基于频率校正的BPSK信号高精度载频估计算法

为了实现BPSK信号载频的高精度估计,提出一种基于频率校正的高精度BPSK信号载频估计算法。算法通过非线性变换将BPSK信号的载频估计问题转化为对点频信号的估计,再用DFT对点频信号进行粗估计,针对DFT估计频率分辨率差的问题,采用相位差校正法对DFT估计频率进行校正,从而实现BPSK信号载频的高精度估计。最后,通过性能分析和计算机仿真验证了算法的有效性,其结果表明:该算法在低信噪比情况下能获得较好的估计精度,在高信噪比情况下,估计精度大幅提高。     

A compact Cramer-Rao bound expression for parametric estimation ofsuperimposed signals

The problem of parameter estimation of superimposed signals in white Gaussian noise is considered. Closed-form expressions of the Cramer-Rao bound for real or complex signals with vector parameters are derived, extending recent results by P. Stoica and A. Nehorai (1989)     

On the generalized Cramer-Rao bound for the estimation of thelocation

It is shown that the generalized Gaussian distribution maximizes the generalized Cramer-Rao (CR) bound for the pth absolute central moment of any classical location parameter unbiased estimator. The underlying maximization is taken over the class of distributions with fixed and finite pth-order moment and exhibits particular utility in minimax designs as well as in worst-case performance analysis. The relationship between the generalized Gaussian density and the generalized CR bound is further examined for the model of a mixture of generalized Gaussian distributions as well as for scenarios where multiple independent generalized Gaussian observations are involved     

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     

Data-aided frequency estimation for PSK signaling infrequency-selective fading

A new data-aided frequency estimator is introduced for phase-shift keying signals transmitted over frequency-selective fading channels. This estimator is developed based on a maximum likelihood criterion. It assumes the use of a special class of pilots, called near-i.i.d. (independent identically distributed) sequences, with impulsive fourth-order moments. With the help of such pilots, the proposed method can estimate frequency offsets without the need for channel information. The pilots of GSM and IS-136 mobile communication systems have been observed as being near-i.i.d., and statistical analysis indicates that the proposed estimate is almost unbiased if the pilot is near-i.i.d. The advantage of the proposed estimator over conventional methods is demonstrated via computer simulation     

The Cramer-Rao lower bound for bilinear systems

Estimation of the unknown parameters that characterize a bilinear system is of primary importance in many applications. The Cramer-Rao lower bound (CRLB) provides a lower bound on the covariance matrix of any unbiased estimator of unknown parameters. It is widely applied to investigate the limit of the accuracy with which parameters can be estimated from noisy data. Here it is shown that the CRLB for a data set generated by a bilinear system with additive Gaussian measurement noise can be expressed explicitly in terms of the outputs of its derivative system which is also bilinear. A connection between the nonsingularity of the Fisher information matrix and the local identifiability of the unknown parameters is exploited to derive local identifiability conditions of bilinear systems using the concept of the derivative system. It is shown that for bilinear systems with piecewise constant inputs, the CRLB for uniformly sampled data can be efficiently computed through solving a Lyapunov equation. In addition, a novel method is proposed to derive the asymptotic CRLB when the number of acquired data samples approaches infinity. These theoretical results are illustrated through the simulation of surface plasmon resonance experiments for the determination of the kinetic parameters of protein-protein interactions.     

Low-SNR limit of the Cramer-Rao bound for estimating the time delayof a PSK, QAM, or PAM waveform

In this letter we consider the Cramer-Rao bound (CRB) for the estimation of the time delay of a noisy linearly modulated signal with random data symbols and random carrier phase. Because of the presence of the nuisance parameters (i.e., data symbols and carrier phase), a closed-form expression of this CRB is very hard to obtain for arbitrary PSK, QAM or PAM constellations and a band-limited transmit pulse. Instead, we derive a simple expression for the limit of the CRB at low signal-to-noise ratio (SNR), which is a relevant benchmark for timing recovery algorithms operating at small E_s/N₀     

On the cramer-rao bound for carrier frequency estimation in the presence of phase noise

We consider the carrier frequency offset estimation in a digital burst-mode satellite transmission affected by phase noise. The corresponding Cramer-Rao lower bound is analyzed for linear modulations under a Wiener phase noise model and in the hypothesis of knowledge of the transmitted data. Even if we resort to a Monte Carlo average, from a computational point of view the evaluation of the Cramer-Rao bound is very hard. We introduce a simple but very accurate approximation that allows to carry out this task in a very easy way. As it will be shown, the presence of the phase noise produces a remarkable performance degradation of. the frequency estimation accuracy. In addition, we provide asymptotic expressions of the Cramer-Rao bound, from which the effect of the phase noise and the dependence on the system parameters of the frequency offset estimation accuracy clearly result. Finally, as a by-product of our derivations and approximations, we derive a couple of estimators specifically tailored for the phase noise channel that will be compared with the classical Rife and Boorstyn algorithm, gaining in this way some important hints on the estimators to be used in this scenario     

OFDM系统中盲、半盲以及非盲信道估计方法的Cramér-Rao界

从OFDM信号系统的矩阵表示出发，推导了OFDM系统中盲、半盲以及非盲信道估计方法的CRB，其中，半盲信道估计的CRB是一般的表达式，经过外推可以得到盲信道估计和非盲信道估计的CRB。在推导盲信道估计的CRB时，证明了线性约束和二次约束两种情况下的CRB是一致的。针对OFDM系统的两种不同方案CP-OFDM和ZP-OFDM，分别进行了信道估计的性能仿真，验证了所推导CRB的正确性，并且利用CRB对一些信道估计方法进行了性能评价。