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Climent Nadeu, UPC-Barcelona (Spain)
Jaume Padrell, UPC-Barcelona (Spain)
Ignasi Esquerra, UPC-Barcelona (Spain)
The multiwindow approach is a meaningful framework for nonparametric spectral estimation. It also encompasses several conventional methods as WOSA and frequency-averaged periodogram. Recently, some authors claimed that the Slepian windows of the Thomsons method and other related optimal sets of windows show a better performance in terms of resolution, variance and leakage. In this paper, that claim is discussed by means of some simulation examples and by applying the various methods to speech recognition. In conclusion, frequency averaging of the periodogram is a computationally simple method that has a great flexibility for band specification and comparatively shows good performance. Actually, it is the spectral analysis technique most extensively employed for speech recognition.
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Vojin E. Zivojnovic, RWTH Aachen (Germany)
Dominikus Noll, University of Paul Sabatier (France)
Minimizing the Fisher information measure over the set of power spectrum densities fitting a finite number of autocorrelation lag constraints is treated. Due to an explicit control of the derivative values of the densities, the Fisher information measure produces a useful smoothing effect. The Fisher information based estimate exhibits improved characteristics compared to the maximum entropy approach proposed by Burg. We show that the resulting power spectrum estimate is positive, and along with the autocorrelation constraints, satisfies a generalized Riccati differential equation. In general, the true estimate of the power spectrum may be obtained only by numerically integrating the corresponding boundary value problem. For real time applications, we therefore propose a fast and numerically stable approximate solution in explicit trigonometric form. Although suboptimal, the proposed approach has proven to be advantageous especially for flat spectra. The presented theory is verified on simulated examples.
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Guy Poulalion, CEA/CESTA (France)
Sylvain Morvan, CEA/CESTA (France)
Yannick Berthoumieu, ESI, ENSERB (France)
Mohamed Najim, ESI, ENSERB (France)
Classical high resolution (HR) spectral analysis methods for the estimation of complex sinusoids parameters require the a priori knowledge of the number of sinusoids. In most situations, the correctness of this knowledge is crucial. In this paper, we present a new HR subspace method. Its novelty stems from the fact that the analysis of complex sinusoids is considered as a joint "detection-estimation" issue. Simulation results and an application on real radar signals are presented to illustrate the efficiency of this method.
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Tseng Ch. Chien-Cheng, Dept. of Electrical Engineering National Taiwan University (Taiwan)
Pei Soo-Chang, Dept. of Electrical Engineering National Taiwan University (Taiwan)
This paper is concerned with the definition of the discrete fractional Fourier transform (DFRFT). First, an eigendecomposition of the discrete Fourier transform (DFT) matrix is derived by sampling the Hermite Gauss functions which are eigenfunctions of the continuous Fourier transform and by performing a novel error removal procedure. Then, the result of the eigendecomposition of the DFT matrix is used to define a new DFRFT. Finally, a numerical example is illustrated to demonstrate the proposed DFRFT is a better approximation to the continuous fractional Fourier transform than the conventional defined DFRFT.
ic973965.pdf TOPFrederic Barbaresco, THOMSON-CSF AIRSYS (France)
A new recursive eigendecomposition algorithm of Complex Hermitian Toeplitz matrices is studied. Based on Trench's inversion of Toeplitz matrices from their autoregressive analysis, we have developed a fast recursive iterative algorithm that takes into account the rank-one modification of successive order Toeplitz matrices. To speed up the computational time and to increase numerical stability of ill-conditioned eigendecomposition in case of very short data records analysis, we have extended this method by introducing an ago-antagonistic regularized reflection coefficient via Levinson equation. We provide a geometrical interpretation of this new recursive eigendecomposition.
ic973969.pdf ic973969.pdf TOPMaria Hansson, Lund University (Sweden)
Peak matched multiple windows are found as the Karhunen-Loève basis functions of a predefined peaked spectrum. With a penalty function, an optimization procedure can be constrained with resulting control of sidelobes. Weighting factors, included in the averaging of the periodograms, can be designed to fulfill certain constraints. Desirable properties are low variance and small bias. This paper presents a discussion of minimization of variance at the peak compared with the optimization that also include the neighbourhood of the peak.
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André Ferrari, I3S UNSA/CNRS (France)
Jean-Yves Tourneret, I3S UNSA/CNRS (France)
Gérard Alengrin, I3S UNSA/CNRS (France)
This communication tackles the problem of a Gaussian band-limited continuous signal with unknown characteristics sampled with jitter. Under this weak assumption, we demonstrate a relation linking the power spectral density of the continuous signal to the second and fourth order statistics of the measured samples. A fundamental point is that this relation does not require the knowledge of the jitter characteristics. This result can be exploited for the derivation of spectral estimation algorithms when the jitter is unknown or jitter detection tests when the sampled signal is unknown. A simulation of spectral estimation confirms the validity of the result.
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Dunmin Zheng, University of Florida (U.S.A.)
Jian Li, University of Florida (U.S.A.)
Petre Stoica, University of Uppsala (Sweden)
This paper describes how the computationally efficient one-dimensional MODE (1D-MODE) algorithm can be used to estimate the frequencies of two-dimensional complex sinusoids. We show that the 1D-MODE algorithm is computationally more efficient than the asymptotically statistically efficient 2D-MODE algorithm, especially when the numbers of spatial measurements are large. We find that the 1D-MODE algorithm is asymptotically statistically efficient for high signal- to-noise ratio. We also show that although 1D-MODE is no longer statistically efficient when the number of temporal snapshots is large, the performance of 1D-MODE can still be very close to that of the 2D-MODE under mild conditions. Numerical examples comparing the performances of the 1D-MODE and 2D-MODE algorithms are also presented.
ic973981.pdf ic973981.pdf TOPGerman Feyh, Cirrus Logic (U.S.A.)
The estimation of the frequencies of sinusoids in noise is a very common problem. This paper addresses the estimation of sinusoids in a low SNR environment. This sinusoidal frequency estimation problem can be used to find the carrier frequencies and baud rates of communication waveforms after some appropriate nonlinearity. If the underlying signal model is sinusoids in white Gaussian noise and we use the forward/backward prediction framework, then the forward/backward prediction equations force a Toeplitz/Hankel structure on the data matrix. If there are $M$ distinct sinusoids in the data and no noise, then the data matrix has rank $M$. Cadzow and Wilkes enhance a noisy data matrix by enforcing both the structure and the rank of the data matrix, before solving for the coefficient vector of the prediction problem. Besides the Toeplitz/Hankel structure, I also enforce the estimated singular values of the data matrix. Using more information extracted from the original data matrix extends the threshold to lower SNR values.
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Olivier Besson, ENSICA (France)
Petre Stoica, Uppsala University (Sweden)
Sinusoidal signals with random time-varying amplitude show up in many signal processing applications. Amplitude modulation results in degeneracy of the signal subspace, i.e. the signal subspace corresponding to one amplitude modulated sinusoid is no longer spanned by one vector. In this paper, we propose modifications of two subspace-based techniques, namely ESPRIT and MODE for estimating the center frequency of a sinusoidal signal with random time-varying ARMA amplitude. Numerical simulations illustrate the good performance of the methods. Finally, a robustified scheme of the proposed methods is described and succesfully applied to real radar data.
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Shu Hung Leung, City University of Hong Kong (Hong Kong)
Tin Ho Lee, City University of Hong Kong (Hong Kong)
Wing Hong Lau, City University of Hong Kong (Hong Kong)
This paper presents a general total least squares (GTLS) solution for linear prediction to estimate closely spaced sinusoids. It is found that the TLS prediction error is not a good criterion to provide a robust solution. In this paper, a frequency weighted prediction error approach is introduced. Experimental results show that the GTLS solution based on the frequency weighted prediction error can give a robust performance even in very low SNR.
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Mounir Ghogho, ENSEEIHT (France)
Bernard Garel, ENSEEIHT (France)
The problem of detection and estimation of harmonics in multiplicative and additive noise is addressed. The problem may be solved using i) the cyclic mean if the harmonic amplitude is not zero mean or ii) the cyclic variance if the harmonic amplitude is zero mean. Solution ii) may be used when the amplitude of the harmonic is not zero mean while solution i) fails in the case of zero mean harmonic amplitude. This paper answers the following question: given a multiplicative and additive noisy environment, which solution is optimal? The paper determines thresholds on the coherent to non coherent sine powers ratio which delimitate the regions of optimality of the two solutions. Comparison with higher-order cyclic statistics is presented. Gaussian as well as non Gaussian noise sources are studied.
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Magnus Jansson, KTH, Stockholm (Sweden)
Bo Göransson, KTH, Stockholm (Sweden)
Björn Ottersten, KTH, Stockholm (Sweden)
In a previous paper we have presented a novel method for spatial and temporal frequency estimation assuming that the sources are uncorrelated. The current contribution analyzes this method in the case of spatial frequency estimation. In particular an optimal weighting matrix is derived and it is shown that the asymptotic variance of the frequency estimates coincides with the relevant Cramer-Rao lower bound. This means that the estimator is in large samples an efficient subspace-based spatial frequency estimator. The proposed method thus utilizes the a priori knowledge about the signal correlation as opposed to previously known subspace estimators. Moreover, when a uniform linear array is employed, it is possible to implement the estimator in a non-iterative fashion.
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Frédéric Galtier, ENSICA (France)
Olivier Besson, ENSICA (France)
This paper addresses the problem of estimating particle's velocity in the vicinity of an aircraft by means of a laser velocimeter. A model for the signal generated by a particle of air passing through a probe volume consisting of equidistant bright and dark fringes is given. From this model, a frequency estimator based on the phase of the correlation sequence of the signal is proposed. A theoretical analysis of the frequency estimator is presented. In particular, a formula for the variance of the estimate is derived under the assumption of small estimation errors. Numerical examples confirm the validity of the analysis. Finally, the effectiveness of the proposed algorithm is demonstrated on real data.
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