Power Spectrum and Correlation (779817), страница 5

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Consider a signal y(m) composed of P complex-valued sinusoids andadditive white noise:Py (m) = ∑ Ak e − j ( 2πFk m+φk ) + n(m)(9.104)k =1The ESPIRIT algorithm exploits the deterministic relation betweensinusoidal component of the signal vector y(m)=[y(m), . .

., y(m+N–1]T andthat of the time-shifted vector y(m+1)=[y(m+1), . . ., y(m+N)]T. The signalcomponent of the noisy vector y(m) may be expressed asx (m) = S a(9.105)where S is the complex sinusoidal matrix and a is the vector containing theamplitude and phase of the sinusoids as in Equations (9.91) and (9.92). Aj 2πFi mcomplex sinusoid ecan be time-shifted by one sample throughj 2πFi. Hence the time-shifted sinusoidalmultiplication by a phase term esignal vector x(m+1) may be obtained from x(m) by phase-shifting eachcomplex sinusoidal component of x(m) asx (m + 1) = SΦ a(9.106)where Φ is a P × P phase matrix defined asΦ = diag[e j 2πF1 , e j 2πF2 ,, e j 2πFP ](9.107)The diagonal elements of Φ are the relative phases between the adjacentsamples of the sinusoids.

The matrix Φ is a unitary matrix and is known asa rotation matrix since it relates the time-shifted vectors x(m) and x(m+1).The autocorrelation matrix of the noisy signal vector y(m) can be written asR y ( m) y ( m) = SPS H + σ 2n I(9.108)High-Resolution Spectral Estimation293where the matrix P is diagonal, and its diagonal elements are the powers ofthe complex sinusoids P = diag[ A12 , , AP2 ] = aa H . The cross-covariancematrix of the vectors y(m) and y(m+1) isR y ( m) y ( m+1) = SPΦ H S H + Rn( m ) n( m+1)(9.109)where the autocovariance matrices Ry(m)y(m+1) and Rn(m)n(m+1) are defined asryy ( 2)ryy (3) ryy (1)ryy (1)ryy ( 2) ryy (0)ryy ( 0)ryy (1)R y ( m) y ( m+1) =  ryy (1) r ( N − 2) r ( N − 3) r ( N − 4)yyyy yyand0 0 0 0 20 0 0σ nRn( m) n( m+1) =  0 σ n2 0 0    00 σ n2 0  ryy ( N )  ryy ( N − 1)  ryy ( N − 2) ryy (1) (9.110)(9.111)The correlation matrix of the signal vector x(m) can be estimated asR x ( m) x ( m) = R y ( m) y ( m) − Rn( m) n( m) = SPS H(9.112)and the cross-correlation matrix of the signal vector x(m) with its timeshifted version x(m+1) is obtained asR x ( m) x ( m+1) = R y ( m) y ( m+1) − Rn( m) n( m+1) = SPΦ H S H(9.113)− j 2πFiSubtraction of a fraction λi = eof Equation (9.113) from Equation(9.112) yieldsR x ( m) x ( m) − λ i R x ( m) x ( m+1) = SP (I − λ i Φ H ) S H(9.114)294Power Spectrum and CorrelationFrom Equations (9.107) and (9.114), the frequencies of the sinusoids can beestimated as the roots of Equation (9.114).9.7 SummaryPower spectrum estimation is perhaps the most widely used method ofsignal analysis.

The main objective of any transformation is to express asignal in a form that lends itself to more convenient analysis andmanipulation. The power spectrum is related to the correlation functionthrough the Fourier transform. The power spectrum reveals the repetitiveand correlated patterns of a signal, which are important in detection,estimation, data forecasting and decision-making systems. We began thischapter with Section 9.1 on basic definitions of the Fourier series/transform,energy spectrum and power spectrum. In Section 9.2, we considered nonparametric DFT-based methods of spectral analysis. These methods do notoffer the high resolution of parametric and eigen-based methods. However,they are attractive in that they are computationally less expensive thanmodel-based methods and are relatively robust. In Section 9.3, weconsidered the maximum-entropy and the model-based spectral estimationmethods.

These methods can extrapolate the correlation values beyond therange for which data is available, and hence can offer higher resolution andless side-lobes. In Section 9.4, we considered the eigen-based spectralestimation of noisy signals. These methods decompose the eigen variablesof the noisy signal into a signal subspace and a noise subspace. Theorthogonality of the signal and noise subspaces is used to estimate the signaland noise parameters. In the next chapter, we use DFT-based spectralestimation for restoration of signals observed in noise.BibliographyBARTLETT M.S. (1950) Periodogram Analysis and Continuous Spectra.Biometrica. 37, pp. 1–16.BLACKMAN R.B.

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