Efficient frequency estimation of a single real tone based on principal singular value decomposition

doi:10.1016/j.dsp.2012.05.010

Digital Signal Processing

Volume 22, Issue 6, December 2012, Pages 1005-1009

https://doi.org/10.1016/j.dsp.2012.05.010 Get rights and content

Abstract

The problem of single-tone frequency estimation for a discrete-time real sinusoid in white Gaussian noise is addressed. We first show that the frequency information is embedded in the principal singular vectors of a matrix which stores the observed data with no repeated entry. The technique of weighted least squares is then utilized for finding the frequency from the singular vectors. It is proved that the variance of the frequency estimate approaches Cramér–Rao lower bound when the data observation length tends to infinity. The computational efficiency and estimation accuracy are demonstrated via computer simulations.

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Fast procedures for accurate parameter estimation of sine-waves affected by noise and harmonic distortion
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Moreover the required processing effort is quite high [3,4]. Other state-of-the-art time-domain algorithms, such as estimators based on linear prediction [5,6], singular value decomposition [7], or autocorrelation [8] are affected by the same drawbacks. Time-domain methods widely adopted in Analog-to-Digital (ADC) testing are the three-parameter sine-fit (3PSF) and the four-parameter sine-fit (4PSF) algorithms [1,2,9–12], which are named according to the number of parameters to be estimated.
In this paper two procedures are proposed for the estimation of the frequency, amplitude, and phase of sine-waves affected by either wideband noise or by both noise and harmonic distortion, respectively. The procedures are very simple and suitable for real-time applications. According to them, firstly the signal parameters are estimated by means of the new Corrected Interpolated Discrete Fourier transform (IpDFTc) algorithm, which compensates the contribution of the spectral interference from the fundamental image component and harmonics on the parameter estimates returned by the classical IpDFT algorithm. Then, a linear sine-fit algorithm is applied to optimize the noise robustness of the estimators, which is worsened by signal windowing employed to reduce the effect of spectral leakage on the IpDFTc estimates. In the paper, the Hann window is applied since it exhibit both the highest noise robustness among the Maximum Sidelobe Decay (MSD) windows and a good reduction of long-range spectral leakage. It has been shown that both proposed procedures almost attain the Cramér-Rao Lower Bounds (CRLBs) for unbiased estimators when at least about 1.5 sine-wave cycles are observed so that the effect of interfering tones on the IpDFTc estimates can be effectively compensated by windowing. The performances of both procedures are analysed through computer simulations and experimental results.
A sliding-window DFT based algorithm for parameter estimation of multi-frequency signal
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Estimating the parameters of a multi-frequency signal, including frequency, amplitude, phase angle, etc., has attracted considerable research interest due to its demand and importance in applications, such as channel sounding, power quality analysis, conformance test and wireless communication systems. A number of estimators have been proposed: discrete Fourier transform (DFT) [1–21], wavelet transform (WT) [22,23] based methods, Hilbert-Huang transform (HHT) [24], singular value decomposition (SVD) [25], QR decomposition [26], estimation of signal parameters via rotational invariance technique (ESPRIT) [27,28], multiple signal classification (MUSIC) [29–31] and the Prony method [32–35]. According to the linear prediction (LP) property, the smart discrete Fourier transform (SDFT) is presented to estimate the frequency of a real-valued sinusoidal signal [11–15].
A sliding-window DFT (SWDFT) based two-step algorithm is proposed for parameter estimation of multi-frequency signal with high accuracy. In the first step, the SWDFT sequence in frequency-domain varied with time instant is obtained by performing the SWDFT on the input signal, where the coarse frequency is estimated. It has been proved that the SWDFT of a multi-frequency signal possesses the same linear relationship as the original time-domain signal and the signal-to-noise ratio (SNR) of each frequency component is enhanced by the SWDFT. In the second step, the parameters are estimated with high accuracy by a linear prediction from the SWDFT sequence, where the complex-valued least squares (CLS) framework is used. Computer simulation and practical measurement results show that the proposed algorithm can estimate parameters of a multi-frequency signal distorted by noise with high accuracy.
Effect of windowing and noise on the amplitude and phase estimators returned by the Taylor-based Weighted Least Squares
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They can be classified either as time-domain or frequency-domain based algorithms. Time-domain based methods, such as the Pisarenko method [1,2], the maximum likelihood methods [3,4], the singular value decomposition method [5], and the sine-fit algorithms [6–8], exhibit high frequency selectivity, but require a high processing effort. Moreover, they are sensitive to the model adopted to describe the acquired signal.
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The parameters of a sine-wave are often estimated by means of the classical Interpolated Discrete Fourier Transform (IpDFTc) method based on cosine windows since this method provides high accuracy while requiring low computational effort. When the number of acquired sine-wave cycles is small, estimation accuracy is heavily affected by the interference due to spectral leakage from both the sine-wave image component and other disturbance tones such as signal harmonics. To reduce both these detrimental effects new cosine windows are proposed in this paper. Their spectra exhibit a deep null close to the image component interfering frequency and the highest allowed sidelobe decay rate, so assuring minimization of long range spectral leakage. These windows are called Maximum Image interference Rejection windows with Rapid Sidelobe Decay rate (RSD) or simply, MIR-RSD windows. The analytical expressions of the sine-wave parameter estimators provided by the IpDFTc method based on the H-term MIR-RSD window ( $H \geq 2$ ) are also derived. To assess the effectiveness of the MIR-RSD windows in rejecting the spectral interference when analyzing noisy or harmonically distorted sine-waves, the accuracy of the proposed frequency estimator is compared with those provided by other IpDFT frequency estimators through simulation and experimental results.
Frequency estimation of a sinusoidal signal via a three-point interpolated DFT method with high image component interference rejection capability
2014, Digital Signal Processing: A Review Journal
Citation Excerpt :
To estimate this parameter different time-domain and frequency-domain methods have been developed [1–21]. When the Signal-to-Noise Ratio (SNR) is low, the parametric (model-based) methods [1,3–7], the Maximum Likelihood (ML) approach [2,8], or the so-called robust Discrete Fourier Transform (DFT) based methods [9–11] are expected to provide better performances. Conversely, when sinusoidal signals with high SNRs are concerned, as often occurs in fields like instrumentation and measurements, the standard DFT-based methods are normally preferred.
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H.C. So was born in Hong Kong. He obtained the B.Eng. degree from City University of Hong Kong and the Ph.D. degree from The Chinese University of Hong Kong, both in electronic engineering, in 1990 and 1995, respectively. From 1990 to 1991, he was an Electronic Engineer at the Research and Development Division of Everex Systems Engineering Ltd., Hong Kong. During 1995–1996, he worked as a Post-Doctoral Fellow at The Chinese University of Hong Kong. From 1996 to 1999, he was a Research Assistant Professor at the Department of Electronic Engineering, City University of Hong Kong, where he is currently an Associate Professor. His research interests include statistical signal processing, fast and adaptive algorithms, signal detection, parameter estimation, and source localization. He has been on the editorial boards of IEEE Transactions on Signal Processing, Signal Processing, Digital Signal Processing and ISRN Applied Mathematics as well as a member in Signal Processing Theory and Methods Technical Committee of the IEEE Signal Processing Society.

Frankie K.W. Chan received the B.Eng. degree in computer engineering and the Ph.D. degree from the City University of Hong Kong in 2002 and 2008, respectively. He is currently a Research Fellow in the same university. His research interests include parameter estimation, optimization and distributed processing, with particular attention to frequency estimation and node localization in wireless sensor network.

Weize Sun received the B.S. degree in Electronic Information Science and Technology from SUN YAT-SEN University, China, in 2005. He is currently a Research Student in City University of Hong Kong. His research interests include statistical signal processing, parameter estimation, tensor algebra, with particular attention to frequency estimation.

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Efficient frequency estimation of a single real tone based on principal singular value decomposition

Abstract

Section snippets

Signal Process.

Signal Process.

Signal Process.

Signal Process.

Signal Process.

Signal Process.

Fundamentals of Statistical Signal Processing: Estimation Theory