Power Spectrum and Correlation (779817), страница 4
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In general, thepoles model the resonances of the signal spectrum, whereas the zeros modelthe anti-resonances of the spectrum.9.6 High-Resolution Spectral Estimation Based on SubspaceEigen-AnalysisThe eigen-based methods considered in this section are primarily used forestimation of the parameters of sinusoidal signals observed in an additivewhite noise. Eigen-analysis is used for partitioning the eigenvectors and theHigh-Resolution Spectral Estimation285eigenvalues of the autocorrelation matrix of a noisy signal into twosubspaces:(a) the signal subspace composed of the principle eigenvectorsassociated with the largest eigenvalues;(b) the noise subspace represented by the smallest eigenvalues.The decomposition of a noisy signal into a signal subspace and a noisesubspace forms the basis of the eigen-analysis methods considered in thissection.9.6.1 Pisarenko Harmonic DecompositionA real-valued sine wave can be modelled by a second-order autoregressive(AR) model, with its poles on the unit circle at the angular frequency of thesinusoid as shown in Figure 9.5.
The AR model for a sinusoid of frequencyFi at a sampling rate of Fs is given byx(m) = 2 cos(2πFi / Fs ) x(m − 1)− x(m − 2)+ Aδ (m − t 0 )(9.75)where Aδ(m–t0) is the initial impulse for a sine wave of amplitude A. Ingeneral, a signal composed of P real sinusoids can be modelled by an ARmodel of order 2P as2Px ( m ) = ∑ a k x ( m − k ) + Aδ ( m − t 0 )(9.76)k =1PoleX( f )ω0−ω0F0fFigure 9.5 A second order all pole model of a sinusoidal signal.Power Spectrum and Correlation286The transfer function of the AR model is given byH ( z) =A2P1− ∑ a k z=−kk =1AP∏ (1 − e− j 2πFk −1z )(1 − e+ j 2πFk −1(9.77)z )k =1± j 2πFkwhere the angular positions of the poles on the unit circle, e,correspond to the angular frequencies of the sinusoids. For P real sinusoidsobserved in an additive white noise, we can writey(m) = x(m) + n(m)2P=∑ ak x(m − k) + n(m)(9.78)k =1Substituting [y(m–k)–n(m–k)] for x(m–k) in Equation (9.73) yields2P2Pk =1k =1y( m) − ∑ ak y (m − k ) = n (m)− ∑ ak n( m − k )(9.79)From Equation (9.79), the noisy sinusoidal signal y(m) can be modelled byan ARMA process in which the AR and the MA sections are identical, andthe input is the noise process.
Equation (9.79) can also be expressed in avector notation asy Ta = nTa(9.80)where yT=[y(m), . . ., y(m–2P)], aT=[1, a1, . . ., a2P] and nT=[n(m), . . .,n(m–2P)]. To obtain the parameter vector a, we multiply both sides ofEquation (9.80) by the vector y and take the expectation:orE [ yy T ] a =E[ yn T ]a(9.81)Ryy a = Ryn a(9.82)where E [ yy T ]= R yy , and E [ yn T ]= R yn can be written asHigh-Resolution Spectral Estimation287R yn = E[( x + n)n T ]= E[nn T ] = Rnn = σ n2 I(9.83)where σ n2 is the noise variance. Using Equation (9.83), Equation (9.82)becomesR yy a =σ n2 a(9.84)Equation (9.84) is in the form of an eigenequation. If the dimension of thematrix Ryy is greater than 2P × 2P then the largest 2P eigenvalues areassociated with the eigenvectors of the noisy sinusoids and the minimumeigenvalue corresponds to the noise variance σ n2 .
The parameter vector a isobtained as the eigenvector of Ryy, with its first element unity and associatedwith the minimum eigenvalue. From the AR parameter vector a, we canobtain the frequencies of the sinusoids by first calculating the roots of thepolynomial1 + a1 z −1 + a 2 z −2 + + a 2 z −2 P + 2 + a1 z −2 P +1 + z −2 P = 0(9.85)Note that for sinusoids, the AR parameters form a symmetric polynomial;that is ak=a2P–k. The frequencies Fk of the sinusoids can be obtained fromthe roots zk of Equation (9.85) using the relationz k = e j 2πFk(9.86)The powers of the sinusoids are calculated as follows.
For P sinusoidsobserved in additive white noise, the autocorrelation function is given byPryy (k ) = ∑ Pi cos 2kπFi + σ n2 δ (k )(9.87)i =12where Pi = Ai / 2 is the power of the sinusoid Ai sin(2πFi), and white noiseaffects only the correlation at lag zero ryy(0). Hence Equation (9.87) for thecorrelation lags k=1, .
. ., P can be written asPower Spectrum and Correlation288cos 2πF2 cos 2πF1cos 4πF2 cos 4πF1 cos 2 PπF1 cos 2 PπF2 cos 2πFP P1 ryy (1) cos 4πFP P2 ryy (2) = cos 2 PπFP PP ryy ( P) (9.88)Given an estimate of the frequencies Fi from Equations (9.85) and (86), andan estimate of the autocorrelation function rˆyy (k) , Equation (9.88) can besolved to obtain the powers of the sinusoids Pi.
The noise variance can thenbe obtained from Equation (9.87) asσ n2 = r yy (0) −P∑ Pi(9.89)i=19.6.2 Multiple Signal Classification (MUSIC) Spectral EstimationThe MUSIC algorithm is an eigen-based subspace decomposition methodfor estimation of the frequencies of complex sinusoids observed in additivewhite noise. Consider a signal y(m) modelled asPy (m) = ∑ Ak e − j( 2πFk m+φk ) + n(m)(9.90)k =1An N-sample vector y=[y(m), . .
., y(m+N–1)] of the noisy signal can bewritten asy = x + n= Sa + n(9.91)where the signal vector x=Sa is defined as x( m) e j2πF1m j2πF ( m+1) x( m + 1) = e 1 F m N x( m + N − 1) e j2π 1 ( + −1)e j2πF2me j2πF2 ( m+1)ej2πF2 ( m + N −1) A1e j2πφ1 e j 2πFP ( m+1) A2 e j2πφ 2 e j2πFP ( m+ N −1) AP e j2πφ P (9.92)e j2πFP mHigh-Resolution Spectral Estimation289The matrix S and the vector a are defined on the right-hand side of Equation(9.92). The autocorrelation matrix of the noisy signal y can be written as thesum of the autocorrelation matrices of the signal x and the noise asR yy = R xx + Rnn= SPS H + σ 2n I(9.93)where Rxx=SPSH and Rnn=σn2I are the autocorrelation matrices of thesignal and noise processes, the exponent H denotes the Hermitian transpose,and the diagonal matrix P defines the power of the sinusoids asP = aa H = diag[ P1 , P2 ,, PP ](9.94)− j 2πFiwhere Pi = Ai2 is the power of the complex sinusoid e.
Thecorrelation matrix of the signal can also be expressed in the formPR xx = ∑ Pk s k s kH(9.95)k =1j2 πFkH, ,e j2 π( N −1) Fk ] . Now consider an eigen-decompositionwhere s k =[1, eof the N × N correlation matrix RxxNR xx = ∑ λ k v k v kHk =1P=∑(9.96)λ k v k v kHk =1where λk and vk are the eigenvalues and eigenvectors of the matrix Rxxrespectively. We have also used the fact that the autocorrelation matrix Rxxof P complex sinusoids has only P non-zero eigenvalues, λP+1=λP+2, ...,λN=0.
Since the sum of the cross-products of the eigenvectors forms anidentity matrix we can also express the diagonal autocorrelation matrix ofthe noise in terms of the eigenvectors of Rxx asPower Spectrum and Correlation290Rnn =σ n2 I =σ n2N∑ v k v kH(9.97)k =1The correlation matrix of the noisy signal may be expressed in terms of itseigenvectors and the associated eigenvalues of the noisy signal asPNk =1Pk =1R yy = ∑ λ k v k v kH + σ 2n ∑ v k v kH(=∑ λk +k =1σ n2)v k v kH+ σ n2(9.98)N∑v k v kHk = P +1From Equation (9.98), the eigenvectors and the eigenvalues of thecorrelation matrix of the noisy signal can be partitioned into two disjointsubsets (see Figure 9.6).
The set of eigenvectors {v1, . . ., vP}, associatedwith the P largest eigenvalues span the signal subspace and are called theprincipal eigenvectors. The signal vectors si can be expressed as linearcombinations of the principal eigenvectors. The second subset ofeigenvectors {vP+1, . . ., vN} span the noise subspace and have σ n2 as theireigenvalues. Since the signal and noise eigenvectors are orthogonal, itfollows that the signal subspace and the noise subspace are orthogonal.Hence the sinusoidal signal vectors si which are in the signal subspace, areorthogonal to the noise subspace, and we haveEigenvaluesλ 1+ σ n22λ 2+ σ nλ 3+ σ n2λP + σ nλ P+1 = λ P+2 = λ P+3 =...2...Principal eigenvaluesNoise eigenvaluesλN= σ n2indexFigure 9.6 Decomposition of the eigenvalues of a noisy signal into the principaleigenvalues and the noise eigenvalues.High-Resolution Spectral Estimations iH ( f)v k =291N −1∑ vk (m)e − j 2πF m = 0i = 1,,Pik = P + 1,, N(9.99)m =0Equation (9.99) implies that the frequencies of the P sinusoids can beobtained by solving for the zeros of the following polynomial function ofthe frequency variable f:N∑ s H ( f )v k(9.100)k = P +1In the MUSIC algorithm, the power spectrum estimate is defined asNPXX ( f )=∑ s H ( f )v k2(9.101)k = P +1where s(f) = [1, ej2πf, .
. ., ej2π(N-1)f] is the complex sinusoidal vector, and{vP+1, . . . ,vN} are the eigenvectors in the noise subspace. From Equations(9.102) and (9.96) we have thatPXX ( f i ) = 0 ,i = 1, . . ., P(9.102)Since PXX(f) has its zeros at the frequencies of the sinusoids, it follows thatthe reciprocal of PXX(f) has its poles at these frequencies. The MUSICspectrum is defined asMUSICPXX( f )=1N∑sk = P +1H( f )v k2=1s H ( f )V ( f )V H ( f ) s ( f )(9.103)where V=[vP+1, . . . ,vN] is the matrix of eigenvectors of the noise subspace.PMUSIC(f) is sharply peaked at the frequencies of the sinusoidal componentsof the signal, and hence the frequencies of its peaks are taken as the MUSICestimates.Power Spectrum and Correlation2929.6.3 Estimation of Signal Parameters via Rotational InvarianceTechniques (ESPRIT)The ESPIRIT algorithm is an eigen-decomposition approach for estimatingthe frequencies of a number of complex sinusoids observed in additive whitenoise.
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