Parameter Estimation of a Noisy Real Sinusoid Using Quartic Polynomial Approach

As Gaussian minimum shift keying (GMSK) modulation scheme is adopted for wireless body area network (WBAN) applications, simple and noise-robust synchronisation algorithms for GMSK are desired to meet the limited resources of power, storage and size in WBAN. However, the current synchronisation algorithms for GMSK are either too complicated or have unsatisfactory performance for low signal-to-noise ratio (SNR), especially for coded GMSK systems. The authors will propose the synchronisation algorithms for general GMSK systems with three unknown synchronisation parameters, namely, timing, frequency and phase offsets estimation. By incorporating noise/interference-reduction methods, the authors propose low-complexity synchronisation algorithms for GMSK that can achieve satisfactory performance in low SNR region. Furthermore, their new feedforward timing and frequency recovery algorithms can provide much better performance than those in the literature by using reliability-measuring mechanism for timing estimation and two-phase frequency estimation. The data-aided Cramer–Rao bounds (CRBs) for the joint estimation of GMSK synchronisation parameters are also derived. The numerical results show that the performance of their proposed synchronisation algorithms is close to the CRBs. The overall bit error rate performance for both uncoded and Bose-Chaudhuri-Hocquenhem BCH-coded GMSK systems are examined. The performance loss owing to synchronisation errors can be reduced to about 1 dB with their proposed algorithms.

Accurate frequency estimation of multiple complex and real sinusoids based on iterative interpolation

A new pilot assisted (PA) carrier frequency offset (CFO) estimation algorithm for OFDM communication system is presented in this paper. This algorithm extends the principle of the discrete Fourier transform (DFT) based single carrier frequency estimation algorithm in S. Reisenfeld and E. Aboutanios (2003). The proposed algorithm comprises of two steps, first a coarse CFO estimator such as the one presented in M. Li and W. Zhang (2003) is employed to estimate the integer component of the CFO. Assuming frequency corresponding to the DFT bin number which is the integer component of the CFO has been estimated correctly, the residue CFO will be bounded between -0.5T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</sub> to 0.5T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</sub> , where T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</sub> is the OFDM symbol period. In this paper, a new fractional frequency estimation algorithm is presented. Unlike the fractional frequency estimation algorithm presented in M. Li and W. Zhang (2003) and P.H. Moose (1994), this estimator can estimate the frequency offset on a symbol by symbol bases which is ideal for environments where the channel characteristics change rapidly. It will be shown that under additive white Gaussian noise (AWGN), the proposed frequency estimation algorithm has a variance performance that approaches the Cramer Rao lower bound (CRLB) at high signal to noise ratio (SNR) and has an acquisition range of -0.5T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</sub> to 0.5T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</sub> Hz

This paper discusses the problem of Non Data Aided (NDA) Signal to Noise Ratio (SNR) estimation of Binary Phase Shift keying (BPSK) modulated signals using the Expectation Maximization (EM) Algorithm. In addition, the Cramer-Rao Lower Bounds (CRLB) for the estimation of Data Aided (DA) and Non Data Aided (NDA) Signal to Noise Ratio (SNR) estimation is derived. Multiple Input Single Output (MISO) channels with Space Time Block Codes (STBC) is used. The EM algorithm is a method that finds the Maximum Likelihood (ML) solution iteratively when there are unobserved (hidden or missing) data. Extension of the proposed approach to other types of linearly modulated signals in estimating SNR is straight forward. The performance of the estimator is assessed using the NDA CRLBs. Alamouti coding technique is used in this paper with two transmit antennas and one receive antenna. The authors’ assumption is that the received signal is corrupted by additive white Gaussian noise (AWGN) with unknown variance, and scaled by fixed unknown complex channel gain. Monte Carlo simulations are used to show that the proposed estimator offers a substantial improvement over the conventional Single Input Single Output (SISO) NDA SNR estimator due to the use of the statistical dependences in space and time. Moreover, the proposed NDA SNR estimator works close to the NDA SNR estimator over Single Input Multiple Output (SIMO) channels.

For the Sinusoid Signals with additive Gauss white noise, a frequency estimation algorithm based on discrete Fourier transform (DFT) interpolation algorithm is proposed in this paper. Based on the classical interpolation algorithm, the algorithm of this paper takes full use of the Peak Spectral Frequency and its neighbor spectral lines to estimate the frequency of the signal. The analysis and simulation results indicate that the performance of the algorithm approaching to the Cramer-Rao lower bound (CRLB) when the signal to noise ratio (SNR) is above −5 dB. Compared with the classical Rife algorithm, the performance of the proposed algorithm is better and more stable under the same conditions. Besides the algorithm is easy to implement and apply with low computational complexity.

Improved Frequency Estimation by Interpolated DFT Method

In this paper we discuss the performance of a correlator based frequency estimator (CE). The estimator is statistically analysed for single-tone frequency estimation for an additive white Gaussian noise (AWGN) channel. The CE is a maximum likelihood estimator (ML) S. M. Kay (1993) which theoretically achieves the Cramer-Rao lower bound (CRB) S. M. Kay (1993) if properly implemented. Frequency estimates on the received single-tone signal are made by correlating the received signal with a locally generated signal by varying the frequency, and maximizing the energy at the output of the correlator. The statistical distribution of the frequency estimates, made by the CE is studied; we consider four different cases as discussed in the paper for the distribution, whilst providing simulation results to verify the analysis.

This paper presents a data-aided frequency offset estimator for a single complex sinusoidal signal corrupted by complex additive white Gaussian noise (AWGN). Kay [4] developed a maximum likelihood (ML) estimator based on differential phase measurements which achieves Cramer-Rao lower bound (CRLB) at high signal-to-noise ratios (SNRs); however, it exhibits large outliers at SNRs lower than 10-15 dB. In this paper, we propose an idea to divide the observation vector into adjacent segments. The samples within the same segment are then added together to produce a new sample. The closed-form of the proposed estimator is derived by applying the ML estimation over the new set of samples in the same way as of Kay's estimator. Simulation results show that with the proposed estimator, the threshold reduces 3-4 dB as the segment length is doubled. The estimation variance remains close to the CRLB even at SNR values as low as 0 dB. The proposed estimator is also compared with another family of frequency estimators that exploits the sample correlations.

The problem of two-dimensional (2D) frequency estimation of a complex sinusoid embedded in a additive white Gaussian noise is addressed. A new frequency estimator based on a least squares plane fitting of the estimated autocorrelation phase of the signal is derived. This algorithm requires a 2D phase unwrapping step which can be easily done. This algorithm is shown to be unbiased and attains the Cramer Rao bounds for high signal-to-noise ratio (SNR > 0 dB). Accuracy and robustness of this new 2D frequency estimator are statistically assessed by Monte Carlo simulations. The results obtained show that a good local frequency estimation can be achieved with a very simple algorithm, and a very small number of points are used for the autocorrelation estimation.

In this paper, we present a new computationally efficient and accurate so-called explicit modulated Lagrange adaptive frequency estimation (EMLAFE) algorithm that is based on a variable delay element of Etter and Hush and the modulated Lagrange adaptive frequency estimator (MLAFE) of Dooley and Nandi. Adopting a similar idea as in So et al., on explicit time delay estimation, we introduce explicit frequency estimation in our EMLAFE algorithm in order to speed up the frequency estimation. Our development of the EMLAFE algorithm is based on the fact that the cost function of the modulated Lagrange interpolation delay filter (MLIDF), which is used as a FIR delay system in the frequency estimator, is sinusoidal in form with respect to the product of signal frequency and time delay under a high signal to noise ratio (SNR) condition. This leads to a simple formula relating the estimated delay to the unknown frequency, and thus to an efficient frequency estimator. The superior performance of the new algorithm is demonstrated in the context of tracking and estimating a chirp or constant frequency signal in noise. We also derive the Cramer-Rao bound and the steady state variance, for the frequency estimate.

It is well known that the singular value decomposition (SVD), as the best rank-revealing factorization, furnishes the best rank-k approximation to a dense matrix with expensive computational cost of (9(M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> N + N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> M + min(M, N)3). Moreover, it is hard to implement in parallel, which challenges the memory storage in wireless communications data-driven system. In this letter, a fast rank-revealing technique, namely, bilateral random projections (BRP) with (9(MN) operations, is exploited for two-dimensional (2-D) frequency estimation of a complex sinusoid in noisy environment. Based on the resulting data matrix, whereafter, two-stage QR factorization frequency estimation method with the weighted least squares (WLS) as solver, is proposed to reduce the computational complexity of those SVD-based frequency estimators. Simulation results demonstrate the efficiency of the proposed algorithm in comparison with several frequency estimation approaches and the Cramér-Rao lower bound (CRLB) as benchmark.

The problem of distributed parameter estimation from binary quantized observations is studied when the unquantized observations are corrupted by combined multiplicative and additive Gaussian noise. These results are applicable to sensor networks where the sensors observe a parameter in combined additive and nonadditive noise and to a case where dispersed receivers are employed with analog communication over fading channels where the receivers employ binary quantization before noise-free digital communications to a fusion center. We first discuss the case in which all the quantizers use an identical threshold. The parameter identifiability condition is given, and, surprisingly, it is shown that unless the common threshold is chosen properly and the parameter lies in an open interval, the parameter will not generally be identifiable, in contrast to the additive noise case. The best achievable mean square error (MSE) performance is characterized by deriving the corresponding Cramer-Rao Lower Bound (CRLB). A closed-form expression describing the corresponding maximum likelihood (ML) estimator is presented. The stability of the performance of the ML estimators is improved when a nonidentical threshold strategy is utilized to estimate the unknown parameter. The thresholds are designed by maximizing the minimum asymptotic relative efficiency (ARE) between quantized and unquantized ML estimators. Although the ML estimation problem is nonconvex, it is shown that one can relax the optimization to make it convex. The solution to the relaxed problem is used as an initial solution in a gradient algorithm to solve the original problem. Next, the case where both the variances of the additive noise and multiplicative noise are unknown is studied. The corresponding CRLB is obtained, and the ML estimation problem is transformed to a convex optimization, which can be solved efficiently. Finally, numerical simulations are performed to substantiate the theoretical analysis.

A novel algorithm is proposed for frequency estimation of a single complex exponential signal or a real sinusoid signal in additive white Gaussian noise (AWGN). The proposed algorithm consists of a coarse search and a fine search, where the coarse search can be accomplished by maximum likelihood (ML) estimation, while the fine search performs a frequency interpolation via time-domain zero padding before applying discrete Fourier transform (DFT) with a novel residue frequency estimator proposed. Simulation results demonstrate that improved performance can be achieved by our proposed estimator based on zero-padding data sequence.

The Discrete Fourier Transform (DFT), which can be calculated efficiently by the Fast Fourier Transform (FFT), is one of the most commonly used tools for frequency estimation of a multi-frequency exponential signal. However, the performance of the FFT is inherently affected by spectral leakage when non-coherent sampling occurs. The Chirp-Z Transform (CZT), which can refine the spectrum to the narrowband range of the source signal accurately, can be generalized to a special form of DFT. This paper proposes a CZT based algorithm which can estimate frequencies of the multi-frequency exponential signal accurately. In our method, the CZT bin, as a nonlinear equation, is firstly used to describe the correspondence among the frequencies, amplitudes, phases and the observed spectrum. Then, the above nonlinear equation is rewritten as a linear equation with unknown frequencies. We further extend the proposed method to the Discrete-time Fourier Transform (DTFT), zero-padding FFT and DFT, which are the special cases of the CZT. We also give the derivation procedure of theoretical variance of our method. In our method, the spectrum leakage is fully considered as useful information to establish the linear equation set. Therefore, it can estimate the frequencies of multi-frequency exponential signal with high accuracy. Simulation results verify the performance of the proposed method.

&lt;p&gt;Nuclear Magnetic Resonance spectroscopy (NMR) is a powerful technique for rapid and efficient quantitation of compounds in chemical samples. NMR causes the nuclei in the molecules to resonate and various chemical arrangements appear as peaks in the Fourier spectrum of a free induction decay (FID). The spectral parameters elicited from the peaks serve as a fingerprint of the chemical components contained in the molecule. These fingerprints can be employed to understand the chemical structure. Signal acquired from a NMR spectrometer is ideally modelled as a superposition of multiple damped complex exponentials (cisoids) in Additive White Gaussian Noise (AWGN). The number as well as the spectral parameters of the cisoids need to be estimated for characterisation of the underlying chemicals. The estimation, however, suffers from numerous difficulties in practice. These include: unknown number of cisoids, large signal length, large dynamic range, large peak density, and numerous distortions caused by experimental artefacts. This thesis aims at the development of estimators that, in view of the above-mentioned practical features, are capable of rapid, high-resolution and apriori-information-free quantitation of NMR signals. Moreover, for the analytic evaluation of the performance of such estimators, the thesis aims to derive interpretable analytic results for the fundamental estimation theory tool for assessing the performance of an unbiased estimator: the Cramer Rao Lower Bound (CRLB). By such results, we mean those that analytically allow the determination, in terms of the CRLB, of the impact of the free model parameters on the estimator performance. For the CRLB, we report analytic expressions on the variance of unbiased parameter estimates of damping factors, frequencies and complex amplitudes of an arbitrary number of damped cisoids embedded in AWGN. In addition to the CRLB, analytic expressions for the determinant and the condition number of the associated Fisher Information Matrix (FIM) are also reported. Further results, in similar order, are reported for two special cases of the damped cisosid model: the Magnetic Resonance Relaxometry model and the amplitude-only model (employed in quantitative NMR - qNMR). Some auxiliary results for the above-mentioned models are also presented, i.e., on the multiplicity of the eigenvalues and the factorisation of the characteristic polynomial associated with their respective FIMs. These results have not been previously reported. The reported theoretical results successfully account for various physical and chemical phenomena observed in experimental NMR data, and quantify their impact on the accuracy of an unbiased estimator as a function of both model and experimental parameters, e.g., influence of prior knowledge, peak multiplicity, multiplet symmetry, solvent peak, carbon satellites, etc. For rapid, high-resolution and apriori-information-free quantitation of NMR signals, a sub-band Steiglitz-McBride algorithm is reported. The developed algorithm directly converts the time-domain FID data into a table of estimated amplitudes, phases, frequencies and damping factors, without requiring any previous knowledge or pre-processing. A 2D sub-band Steiglitz-McBride algorithm, for the quantitation of 2D NMR data in a similar manner, is also reported. The performance of the developed algorithms is validated by their application to experimental data, which manifests that they outperform the state-of-the-art in terms of speed, resolution and apriori-information-free operation.&lt;/p&gt;