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     

Cramer–Rao bound and minimum variance unbiased estimator for joint sampling clock offset and channel taps in MC‐CDMA systems

Exact closed‐form expressions of the Cramer–Rao bound (CRB) for joint sampling clock offset and channel taps are obtained in multi‐carrier code division multiple access systems. CRB is undoubtedly the most well known variance's bound to determine. It provides a benchmark against which we can compare the performance of any unbiased estimator. Furthermore, minimum variance unbiased (MVU) estimator for these parameters is proposed. Moreover, maximum likelihood (ML) and least‐squares estimators for joint sampling clock offset and channel taps are presented. Best linear unbiased estimator is also introduced just for channel taps. The performances of the estimators are compared through simulation results with the proposed CRB. Our results show the better performances of MVU and ML estimators with more computational complexity compared with the others. Copyright © 2008 John Wiley & Sons, Ltd.     

Blind Frequency Synchronization in OFDM via Diagonality Criterion

In this paper, we address the problem of blind carrier frequency offset (CFO) estimation in orthogonal frequency-division multiplexing (OFDM) systems in the case of frequency-selective channels. CFO destroys the orthogonality between the carriers leading to nondiagonal signal covariance matrices in frequency domain. The proposed blind method enforces a diagonal structure by minimizing the power of nondiagonal elements. Hence, the orthogonality property inherent to OFDM transmission with cyclic prefix is restored. The method is blind since it does not require a priori knowledge of the transmitted data or the channel, and does not need any virtual subcarriers. A closed-form solution is derived, which leads to accurate and computationally efficient CFO estimation in multipath fading environments. Consistency of the estimator is proved and the convergence rate as a function of the sample size is analyzed as well. To assess the large sample performance, we derive the CramÉr–Rao bound (CRB) for the blind CFO estimation problem. The CRB is derived assuming a general Gaussian model for the OFDM signal, which may be applied to both circular and noncircular modulations. Finally, simulation results on CFO estimation are reported using a realistic channel model.     

Carrier phase and frequency estimation for pilot-symbol assisted transmission: bounds and algorithms

In this paper we consider the Cramer-Rao lower bound (CRB) for the joint estimation of the carrier phase and the frequency offset from a noisy linearly modulated burst signal containing random data symbols (DSs) as well as known pilot symbols (PSs). We point out that the CRB depends on the location of the PSs in the burst, the number of PSs, the number of DSs, the signal-to-noise ratio (SNR), and the data modulation scheme. Distributing the PSs symmetrically about the center of the burst and estimating the carrier phase in the center of the burst interval decouples the frequency and phase estimation, making the CRB for phase estimation independent of the specific location of the PSs. At low and moderate SNR, the CRBs for both phase and frequency estimation decrease as the fraction of the PSs in the burst increases. In addition, the CRB for frequency estimation decreases as the PSs are separated with more DSs. Numerical evaluation of the CRB indicates that the carrier phase and frequency of a "hybrid" burst (i.e., containing PSs and DSs) can be estimated more accurately when exploiting both the presence of the DSs and the a priori knowledge about the PSs, instead of using only the knowledge about the PSs (and ignoring the DSs), or considering all the received symbols (PSs and DSs) as unknown (and ignoring the knowledge about the PSs). Comparison of the CRB with the performance of existing carrier synchronizers shows that the iterative soft-decision-directed (sDD) estimator with data-aided (DA) initialization performs very closely to the CRB and provides a large improvement over the classical non-data-aided (NDA) estimator at lower SNR.     

Joint Maximum-Likelihood CFO and Channel Estimation for OFDMA Uplink Using Importance Sampling

In this paper, the optimal carrier frequency offset (CFO) and channel estimator for an orthogonal frequency-division multiple-access (OFDMA) uplink based on a joint maximum-likelihood (ML) criterion is derived. Direct implementation of the resultant estimation scheme is challenging due to the need for a multidimensional exhaustive search in multi-CFO estimation. To solve this problem, an optimization theorem is exploited, and a computationally efficient method using an importance sampling technique is proposed. Without the need to provide an initial estimate, the proposed estimator guarantees the generation of the global optimal solution. Simulation results clearly verify the effectiveness of the proposed estimation scheme.      

A hybrid joint ML technique for CFO,timing offset,and channel estimations in an OFDMA uplink

In this paper, we propose a novel hybrid joint maximum‐likelihood estimator for carrier frequency offset (CFO), timing offset, and channel response of all users in the uplink of an orthogonal frequency division multiple access (OFDMA) system. The proposed estimation method significantly reduces complexity of this multiparameter, multidimensional optimization problem, using the concept of separation of the different user signals by means of newly defined projection operators. This projection technique is combined with the alternating projection method available in the literature to arrive at a new hybrid algorithm that offers significant performance advantages in terms of computational complexity and estimator performance. The joint estimation of the CFOs, timing offsets, and channel coefficients for all active users together at the base station of the OFDMA uplink is a rarely addressed task. The proposed method also offers the flexibility of application to any subcarrier assignment scheme used in OFDMA systems. Extensive simulation studies corroborate the advantages of the new hybrid method for all three estimation requirements in the multiuser OFDMA uplink. Copyright © 2012 John Wiley & Sons, Ltd.     

Joint linear timing and carrier phase estimation of DS-CDMAmultiuser communications

Previous work on multiuser synchronization assumed that either timing recovery or carrier phase recovery schemes were available. Therefore, a generalized joint linear timing and carrier-phase estimation scheme to effectively estimate both the received timings and carrier phases at the same time is of paramount interest. Based on the idea to linearly combine the sufficient statistics of timings and carrier phases, we introduce the generalized joint linear timing and carrier phase estimation scheme with commensurate complexity of the generalized linear timing estimator. To avoid the difficulty of the design by minimizing the mean-square error (MSE), a simple and novel design criterion for this generalized joint linear estimation is proposed. With a discrete timing assumption, we not only mathematically prove that the MSE of the proposed estimator converges to zero as noise vanishes for an arbitrary number of users, but also conduct simulations to verify it. On the other hand, for the general case with a continuous timing assumption, arguments and simulations are provided to confirm the effectiveness of our proposed estimator for an arbitrary number of users. We further show that such an efficient linear-complexity joint estimation scheme is effective in the near-far environment. Consequently, the nonpolynomial (NP) hard multiuser synchronization over additive white Gaussian noise (AWGN) channel is successfully solved     

The true Crame/spl acute/r-Rao lower bound for data-aided carrier-phase-independent time-delay estimation from linearly Modulated waveforms

In this paper, we present the derivation and analysis of the true Crame/spl acute/r-Rao lower bound (CRB) for the variance of unbiased, data-aided (DA) symbol-timing estimates, obtained from a block of K samples of a linearly modulated information signal, transmitted through an additive white Gaussian noise channel with random carrier phase. We consider a carrier-phase-independent time-delay estimation scenario wherein the carrier phase is viewed as an unwanted or nuisance parameter. The new bounds require only a moderate computational effort and are tighter than the CRB for the variance of unbiased time-delay estimates obtained under the assumption that the carrier phase is known. These bounds are particularly useful to assess the ultimate accuracy that can be achieved by pilot-assisted symbol synchronizers. Conversely, they may be used to evaluate data sequence suitability for the purpose of time-delay estimation. Comparison of the actual variance of a DA feedforward timing estimator with the new bounds show that these are attainable by practical synchronizers.     

Blind code-timing estimation for CDMA systems with bandlimited chip waveforms in multipath fading channels

In this paper, we present a blind code-timing estimator for asynchronous code-division multiple-access (CDMA) systems that use bandlimited chip waveforms. The proposed estimator first converts the received signal to the frequency domain, followed by a frequency deconvolution to remove the convolving chip waveform, and then calculates the code-timing estimate from the output of a narrowband filter with a sweeping center frequency, which is designed to suppress the overall interference in the frequency domain. The proposed estimator is near-far resistant, and can deal with time- and frequency-selective channel fading. It uses only the spreading code of the desired user, and can be adaptively implemented for both code acquisition and tracking. We also derive an unconditional Crame/spl acute/r-Rao bound (CRB) that is not conditioned on the fading coefficients or the information symbols. It is a more suitable lower bound than a conditional CRB for blind code-timing estimators which do not assume knowledge of the channel or information symbols. We present numerical examples to evaluate and compare the proposed and several other code-timing estimators for bandlimited CDMA systems.     

On the performance limits of data-aided synchronization

This paper addresses data-aided (DA) synchronization, in which the reference parameter acquisition is aided by a training sequence known to the receiver. The Cramer-Rao lower bound (CRB) for the DA timing and/or carrier phase recovery is presented. For DA parameter estimation, the CRB typically varies with the training sequence. This indicates that different training sequences offer fundamentally different performance. In the literature, the widely cited closed-form CRB for timing and carrier phase recovery was derived under the assumption that the training sequence is independent and identically distributed (i.i.d.) and sufficiently long. We derive a closed-form CRB for timing and carrier phase recovery with respect to an arbitrary training sequence and pulse shaping function for the over and under sampling cases. It turns out that the CRB is a weighted summation of the aperiodic correlation of the training sequence and the weighting factor is determined by the pulse shaping filter. Therefore, this paper reveals the fundamental link between a training sequence and its corresponding performance limit.     

Cramér-Rao Bound for SAGE-based Residual Frequency Offset Estimation Algorithm in OFDM Systems

The Cramér-Rao bound (CRB) is a powerful tool in estimation theory as it gives a performance lower bound for parameter estimation problems. In this paper, a much tighter CRB for Lee’s residual frequency offset (RFO) estimation method (IEEE Transactions on Communications 54:765, 2006) is first given. The tighter low bound is obtained by considering the ICI that affects the performance of space-alternating generalized expectation-maximization (SAGE) based RFO estimator. It can be concluded that the performance of SAGE based RFO estimation method decreases as the normalized RFO increases and increases with the increasing of signal-to-noise (SNR). Simulation results show that the proposed CRB of SAGE based RFO estimator is extremely tight. It approximates closely the MSE performance obtained by Monte Carlo simulation.     

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     

Timing Synchronization in Decode-and-Forward Cooperative Communication Systems

Cooperative communication systems have attracted much attention recently due to their desirable performance gain while using single antenna terminals. This paper addresses the joint timing and channel estimation problem, and furthermore the resynchronization of multiple timing offsets in a cooperative relay system. The estimations of timing and channel are conducted in two phases and the associated CramÉr-Rao bounds (CRB) are derived for both phases. It is demonstrated that the conventional CRB is not valid for timing parameters under fading conditions, and a new bound called Weighted Bayesian CRB is proposed. With the timing and channel estimates, a general framework of the resynchronization filter design is developed in order to compensate the multiple timing offsets at the destination. The proposed methods are applied to different scenarios with varying degrees of timing misalignment and are numerically shown to provide excellent performances that approach the perfectly synchronized case.      

Decision-aided-based multiple blind PN code estimation algorithm of asynchronous CDMA signal

Aiming at the problem of blind estimation of multi-user pseudo-code of multi-path asynchronous code division multiple access (CDMA) signal,using the finite element (FA) characteristic of transmitted symbol,a decision aid (DA) method based on maximum likelihood was proposed.A two-step iterative estimation structure for PN code,transmitted bit and channel was designed to realize the estimation of multiple user code in asynchronous CDMA system with superior performance.Moreover,the Cramer-Rao bound (CRB) of PN code estimation in asynchronous CDMA signal was derived to evaluate the performance of proposed method.Simulation indicates significant improvement in performance which is close to the theory bound of the proposed method compared to commonly-used iterative least square with projection (ILSP) method.The method can also achieve superior estimation when the user number increases.     

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.     

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.     

Timing Recovery With Frequency Offset and Random Walk: Cramér–Rao Bound and a Phase- Locked Loop Postprocessor

We consider the problem of timing recovery for bandlimited, baud-rate sampled systems with intersymbol interference and a timing offset that can be modeled as a combination of a frequency offset and a random walk. We first derive the Crameacuter-Rao bound (CRB), which is a lower bound on the estimation error variance for any timing estimator. Conventional timing recovery is based on a phase-locked loop (PLL). We compare the conventional timing-recovery method with the CRB for realistic timing parameters for the magnetic recording channel, and observe a 7 dB signal-to-noise ratio gap between the two. Next, we propose a PLL postprocessor based on the maximum a posteriori estimation principle that performs to within 1.5 dB of the CRB. This postprocessor performs time-invariant filtering and time-varying scaling of the PLL timing estimates. The refined timing estimates from the postprocessor are then used to get refined samples by interpolating the samples taken at the PLL's timing estimates. Finally, we present suboptimal implementations that allow a performance-complexity tradeoff     

The true Cramer-Rao bound for carrier frequency estimation from a PSK signal

This paper considers the Cramer-Rao bound (CRB) related to estimating the carrier frequency of a noisy phase-shift keying signal. The following scenarios are discussed: 1) carrier frequency estimation irrespective of the carrier phase, based on either known or random data and 2) joint carrier phase and frequency estimation, based on either known or random data. Ideal symbol timing is assumed. We compare the results obtained from a (commonly used) simplified observation model against those resulting from the correct model. Because of the presence of nuisance parameters (random data and/or random carrier phase), the analytical computation of the corresponding CRBs is often not feasible. Here we present results that are based upon a combined analytical/numerical approach. Our results show that the choice of the observation model has essentially no effect on the CRBs at moderate and high signal-to-noise ratios. We also show that of the two scenarios considered, joint frequency and phase estimation yields the smaller CRB; the penalty resulting from frequency estimation, irrespective of the carrier phase, decreases with increasing observation interval.     

Blind channel estimation for multicarrier systems with narrowband interference suppression

We present herein a novel blind channel estimator for multicarrier (MC) systems in the presence of unmodeled narrowband interference. A generalized multichannel minimum variance principle is invoked to design an equalizing filterbank that preserves desired signal components and suppresses the overall interference. While a channel estimate may be obtained by directly maximizing the filterbank output power through multidimensional nonlinear searches, such an approach is computationally prohibitive and suffers local convergence. To overcome this difficulty, we derive an asymptotically (in SNR) tight lower bound of the filterbank output power and use it for channel estimation, which reduces the problem to a quadratic minimization. Numerical examples show that the proposed scheme compares favorably with a subspace blind channel estimator in the presence of unknown narrowband interference.     

Estimation of synchronization impairments in multi-relay DF cooperative networks

Cooperative communication systems can effectively increase channel capacity and combat fading. Effective cooperation requires synchronization impairments such as multiple timing offsets and multiple carrier frequency offsets to be accurately estimated and mitigated. This paper seeks to address the joint estimation of synchronization impairments in multi-relay decode-and-forward （DF） cooperative networks. Firstly, a simple yet effective estimation method based on the devised training signals is presented for achieving synchronization. Then, an iterative algorithm is further derived in order to improve the performance associated with the estimation of synchronization impairments. Our proposed algorithm converts the difficult multiple parameter estimation problem into more tractable sub-problems of estimating many individual impairments pairs for the independent relays. Simulations indicate that, the proposed estimator can asymptotically achieve the mean square error （MSE） for the perfectly timing or frequency synchronized case.