MinimumNormSingularCaponSpectralEstimator.doc

精品论文大全minimum norm singular capon spectral estimatorzhigang su1,2 , yingning peng1 and renbiao wu21. department of electronic engineering, tsinghua universitybeijing 100084, china. 2. tianjin key lab for advanced signal processing, civil aviation university of china, tianjin 300300, chinaabstractin this paper, an efficient spectral estimator is presented as a solution of spectral estima- tion with several snapshots available. due to considering the design of the narrowband filter and the formation of the filtering vector simultaneously, the proposed method can give the better spectral estimates. simulation results exhibit the performance of the new spectral estimator.1 introductionspectral estimation is very important for many applications, such as, radar, sonar, communications, and so on. because nonparametric data-adaptive finite-impulse response (fir) filtering-based approaches1 retain the robust nature of the nonparametric methods and at the same time improve the spectral estimates by having narrower spectral peaks and lower sidelobes than those nonparametric data-independent ones, they attract many researchers attention. in order to achieve higher resolution of amplitude spectrum, the filter length is often chosen to be quite large, which results in a few “snapshots” available. sometimes, the sample covariance matrix is even rank-deficient because there are too less snapshots available.the spectral estimator by directly using singular sample covariance matrix is first proposed for synthetic aperture radar (sar) image formation, which is referred to as high-definition imaging (hdi)2. another similar estimator is the rank-deficient robust capon filter bank (rcf) spectral estimator3. both of them are only concerned about how to design the filter but do not consider the effects of noise and interferences in the filtering vector formed by using several snapshots.in this paper, we not only study the design of the narrowband filter but also consider how to form the better filtering vector with high signal-to-interference-plus-noise ratio (sinr). therefore, the developed estimator, which is referred to as the minimum norm singular capon (mins-capon) estimator, can give the accurate spectral estimates.5精品论文大全2 problem formulationwe consider herein the filter bank estimator for estimating the spectrum of one-dimensional (1-d) datan=0sequence yn n 1 . assume the filter length is m , then, the snapshot, yl = yl , yl+1 , yl+m1 t , canbe formed by partitioning the observation data with l = 0, 1, , l 1, where superscript t denotes the transpose of a vector or a matrix. therefore, we have l = n m + 1 snapshots. for angular frequency ( m ), the filtering vector gf () resembles one sinusoid signal, and the capon filter is always be employed as the corresponding narrowband filter. however, if only a few snapshots are available, the filtering vector gf () contains large residual terms of the interference and noise, and the sample covariance matrix, denoted r , is even singular. to obtain the fine spectral estimate, the new filtering vector, which is with high sinr, should be obtained by efficiently weighting the snapshots. at the same time, the trouble introduced by the singular matrix r , must be overcome before the narrowband filter is designed.3 determination of the pure filtering datathe expected weight w should minimize the terms of the interference and noise in the new filtering vector, which is obtained by weighting the snapshots. so,min |yw()|2 s.t. at ()w() = 1. (3)wllet rt = yh y/m . the minimizing problem in eqn. (3) can be rewritten asmin wh ()rt w() s.t.wh ()a = 1. (4)wltherefore, eqn. (4) has an analytical solution given byr1 tw = t al () . (5)at 1 therefore, the new filtering vector isl ()r al()yr1 a ()gm () = t l . (6)at 1 tl ()r al()4 proposed methodby following the idea of capon filter, which passes the desired signal with less distortion and at same time attenuates the terms of other frequencies as much as possible1, the narrowband filter can be designed viamin hh ()r h() s.t. hh ()am () = 1(7)hthough the optimization problem in eqn. (7) has the same form as the capon estimator, the sample covariance matrix r is singular (non-invertible) in case of a few snapshots available (l m ).assume that the singular matrix r is with rank r m . therefore, the eigendecomposition of r can be written asr = s, g 0 # sh #(8)0 0 g hwhere is a r r diagonal matrix whose diagonal elements represent the nonzero eigenvalues of r , unitary matrices s and g are respective m r and m (m r) matrices, which are composed of eigenvectors associated nonzero and zero eigenvalues of r . obviously, the column vectors in s span the observed data space and the ones in g span the orthogonal complement of the observed data space.again, let hs = sh h() and as = sh am () are the projections of h() and am () on the observed data space while hg = g h h() and ag = g h am () denote their respective projections on the cor- responding orthogonal complement space. so, the optimization problem in eqn. (7) can be reexpressed asmin hh ()hss.t. hh as + hh ag = 1(9)hsss g 1 ()asgif hh as = , then hh ag = 1 . consequently, we obtain hs = and hg = (1 ) a.ssg ah 1 ()as|ag |2the euclidean norm of h() indicates the noise gain. hence, the optimal filter should be with theminimum norm. minimizing the norm of h() respect to yields(ah 1 as )2 = s (ah 1 as )2 + |ag |2 (ah 2 as )(10)ssbecauses() = hh ()gm () = (shs + g hg )h gm () = hh sh gm () (11)the new spectral estimator is explicitly(ah 1 as )(ah 1 sgm )() = s s (ah 1 as )2 + |ag |2 (ah 2 as )(12)ss5 numerical exampleone simulation is provided herein to show the new filtering vector, obtained by weighting the snapshots with the non-fourier weight, is with high sinr. assume the data is generated by 12 sinusoids in thegaussian white noise with the standard deviation e = 1mv . the amplitudes of the 12 sinusoids all equal to 10mv and zero phases. one sinusoid with frequency 0.1 is considered as the useful signal and the others are the interferences uniformly distributed in the frequency interval of (0.1 + 1/m ) to (0.1 + 3/m ). as shown in figure 1, the new filtering vector, obtained via about 15 snapshots, is withthe best sinr.1012 modified filtering vectororiginal filtering vectornormalized amplitude810 minscapon8sinr66442201020304050snapshots number000.40.5normalized frequencyfig. 1. comparison of sinrs of two fil-tering vectors.fig. 2.modulus of 1-d spectralestimates(n = 64).another simulation is used to show the estimation performance of the mins-capon estimator. assume two closely sinusoids with amplitude 5mv are at frequencies 0.09 and 0.1. the interferences are 51 sinu- soids, which are uniformly spaced between 0.30 and 0.35 with amplitude 10mv. the spectral estimates, obtained via the rank-deficient rcf and the mins-capon estimator, are shown in fig. 2. obviously, the mins-capon gives two closely spaced spectral lines well separated, while the rank-deficient rcf not.6 conclusionin this paper, we present the mins-capon estimator for estimating the spectrum from several snapshots. since the modified filtering vector is w