Power Spectrum and Correlation (779817), страница 3
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.,K–1where D is the overlap. For each segment of length 2N, the correlationfunction in the range of 0 ≥ m ≥ N is given byrˆxx (m) =1N −1∑ x i( k ) xi (k + m) ,N k=0m = 0, 1, . . ., N–1(9.46)Power Spectrum and Correlation278In Equation (9.46), the estimate of each correlation value is obtained as theaveraged sum of N products.9.5 Model-Based Power Spectrum EstimationIn non-parametric power spectrum estimation, the autocorrelation functionis assumed to be zero for lags | m |≥ N , beyond which no estimates areavailable.
In parametric or model-based methods, a model of the signalprocess is used to extrapolate the autocorrelation function beyond the range| m |≤ N for which data is available. Model-based spectral estimators have abetter resolution than the periodograms, mainly because they do not assumethat the correlation sequence is zero-valued for the range of lags for whichno measurements are available.In linear model-based spectral estimation, it is assumed that the signalx(m) can be modelled as the output of a linear time-invariant system excitedwith a random, flat-spectrum, excitation. The assumption that the input hasa flat spectrum implies that the power spectrum of the model output isshaped entirely by the frequency response of the model.
The input–outputrelation of a generalised discrete linear time-invariant model is given byPQk =1k =0x(m) = ∑ a k x(m − k ) + ∑ bk e(m − k )(9.47)where x(m) is the model output, e(m) is the input, and the ak and bk are theparameters of the model. Equation (9.47) is known as an auto-regressivemoving-average (ARMA) model. The system function H(z) of the discretelinear time-invariant model of Equation (9.47) is given byQH ( z) =B( z )=A( z )∑ bk z −kk =0P1− ∑ a k z(9.48)−kk =1where 1/A(z) and B(z) are the autoregressive and moving-average parts ofH(z) respectively. The power spectrum of the signal x(m) is given as theproduct of the power spectrum of the input signal and the squaredmagnitude frequency response of the model:Model-Based Power Spectrum EstimationPXX ( f ) = PEE ( f ) H ( f ) 2279(9.49)where H(f) is the frequency response of the model and PEE(f) is the inputpower spectrum.
Assuming that the input is a white noise process with unitvariance, i.e. PEE(f)=1, Equation (9.49) becomesPXX ( f ) = H ( f ) 2(9.50)Thus the power spectrum of the model output is the squared magnitude ofthe frequency response of the model. An important aspect of model-basedspectral estimation is the choice of the model. The model may be an autoregressive (all-pole), a moving-average (all-zero) or an ARMA (pole–zero)model.9.5.1 Maximum–Entropy Spectral EstimationThe power spectrum of a stationary signal is defined as the Fouriertransform of the autocorrelation sequence:PXX ( f ) =∞∑ r xx (m)e− j 2πfm(9.51)n= − ∞Equation (9.51) requires the autocorrelation rxx(m) for the lag m in the range± ∞ . In practice, an estimate of the autocorrelation rxx(m) is available onlyfor the values of m in a finite range of say ±P.
In general, there are aninfinite number of different correlation sequences that have the same valuesin the range | m | ≤ P | as the measured values. The particular estimate usedin the non-parametric methods assumes the correlation values are zero forthe lags beyond ±P, for which no estimates are available.
This arbitraryassumption results in spectral leakage and loss of frequency resolution. Themaximum-entropy estimate is based on the principle that the estimate of theautocorrelation sequence must correspond to the most random signal whosecorrelation values in the range | m | ≤ P coincide with the measured values.The maximum-entropy principle is appealing because it assumes no morestructure in the correlation sequence than that indicated by the measureddata. The randomness or entropy of a signal is defined asPower Spectrum and Correlation2801/ 2H [PXX ( f )]=∫ ln PXX ( f ) df(9.52)−1 / 2To obtain the maximum-entropy correlation estimate, we differentiateEquation (9.53) with respect to the unknown values of the correlationcoefficients, and set the derivative to zero:∂ H [ PXX ( f )] 1 / 2 ∂ ln PXX ( f )df = 0= ∫∂ rxx (m)∂r(m)xx−1 / 2for |m| > P(9.53)Now, from Equation (9.17), the derivative of the power spectrum withrespect to the autocorrelation values is given by∂ PXX ( f )= e − j 2πfm∂ rxx (m)(9.54)From Equation (9.51), for the derivative of the logarithm of the powerspectrum, we have∂ ln PXX ( f )−1( f ) e − j 2πfm= PXX(9.55)∂ rxx (m)Substitution of Equation (9.55) in Equation (9.53) gives1/ 2−1∫ PXX ( f ) e− j 2πfmdf = 0for |m| > P(9.56)−1 / 2−1 ( f )Assuming that PXXis integrable, it may be associated with anautocorrelation sequence c(m) as∞−1PXX( f )=∑ c(m) e − j 2πfm(9.57)m = −∞where1/ 2c( m) =−1∫ PXX ( f ) e−1 / 2j 2πfmdf(9.58)Model-Based Power Spectrum Estimation281From Equations (9.56) and (9.58), we have c(m)=0 for |m| > P.
Hence, fromEquation (9.57), the inverse of the maximum-entropy power spectrum maybe obtained from the Fourier transform of a finite-length autocorrelationsequence as−1 ( f ) =PXXP∑ c(m) e − j2 πfm(9.59)m= − Pand the maximum-entropy power spectrum is given byMEPˆXX( f )=1P∑ c(m)e− j 2πfm(9.60)m= − PSince the denominator polynomial in Equation (9.60) is symmetric, itfollows that for every zero of this polynomial situated at a radius r, there is azero at radius 1/r. Hence this symmetric polynomial can be factorised andexpressed asP1∑ c(m) z −m = σ 2 A( z ) A( z −1 )(9.61)m= − Pwhere 1/σ2 is a gain term, and A(z) is a polynomial of order P defined asA( z ) =1+ a1 z −1 + + a p z − P(9.62)From Equations (9.60) and (9.61), the maximum-entropy power spectrummay be expressed asσ2MEPˆXX(f)=(9.63)A( z ) A( z −1 )Equation (9.63) shows that the maximum-entropy power spectrum estimateis the power spectrum of an autoregressive (AR) model.
Equation (9.63)was obtained by maximising the entropy of the power spectrum with respectto the unknown autocorrelation values. The known values of theautocorrelation function can be used to obtain the coefficients of the ARmodel of Equation (9.63), as discussed in the next section.Power Spectrum and Correlation2829.5.2 Autoregressive Power Spectrum EstimationIn the preceding section, it was shown that the maximum-entropy spectrumis equivalent to the spectrum of an autoregressive model of the signal.
Anautoregressive, or linear prediction model, described in detail in Chapter 8,is defined asPx ( m ) = ∑ a k x ( m − k ) + e( m )(9.64)k =1where e(m) is a random signal of variance σ e2 . The power spectrum of anautoregressive process is given byσ e2ARPXX(f)=P(9.65)21− ∑ a k e − j 2πfkk =1An AR model extrapolates the correlation sequence beyond the range forwhich estimates are available. The relation between the autocorrelationvalues and the AR model parameters is obtained by multiplying both sidesof Equation (9.64) by x(m-j) and taking the expectation:E [ x( m) x (m − j )] =P∑ akE [ x (m − k ) x (m − j )] + E [e( m)x ( m − j )](9.66)k =1Now for the optimal model coefficients the random input e(m) is orthogonalto the past samples, and Equation (9.66) becomesr xx ( j) =P∑ ak rxx ( j − k) ,j=1, 2, .
. .(9.67)k =1Given P+1 correlation values, Equation (9.67) can be solved to obtain theAR coefficients ak. Equation (9.67) can also be used to extrapolate thecorrelation sequence. The methods of solving the AR model coefficients arediscussed in Chapter 8.Model-Based Power Spectrum Estimation2839.5.3 Moving-Average Power Spectrum EstimationA moving-average model is also known as an all-zero or a finite impulseresponse (FIR) filter. A signal x(m), modelled as a moving-average process,is described asQx(m) = ∑ bk e(m − k )(9.68)k =0where e(m) is a zero-mean random input and Q is the model order.
Thecross-correlation of the input and output of a moving average process isgiven byrxe (m) = E [x( j )e( j − m)]Q= E ∑ bk e( j − k ) e( j − m) = σ e2 bmk =0(9.69)and the autocorrelation function of a moving average process is 2 Q −|m|bk bk + m , | m | ≤ Qσrxx (m) = e k∑=0| m | >Q0,(9.70)From Equation (9.70), the power spectrum obtained from the Fouriertransform of the autocorrelation sequence is the same as the power spectrumof a moving average model of the signal. Hence the power spectrum of amoving-average process may be obtained directly from the Fouriertransform of the autocorrelation function asQMA =PXX∑ rxx (m) e − j2πfm(9.71)m = −QNote that the moving-average spectral estimation is identical to theBlackman–Tukey method of estimating periodograms from theautocorrelation sequence.Power Spectrum and Correlation2849.5.4 Autoregressive Moving-Average Power SpectrumEstimationThe ARMA, or pole–zero, model is described by Equation (9.47).
Therelationship between the ARMA parameters and the autocorrelationsequence can be obtained by multiplying both sides of Equation (9.47) byx(m–j) and taking the expectation:QPr xx ( j) = − ∑ ak r xx ( j − k ) + ∑ bk r xe ( j − k)k =1(9.72)k=0The moving-average part of Equation (9.72) influences the autocorrelationvalues only up to the lag of Q. Hence, for the autoregressive part ofEquation (9.72), we havePr xx (m) = − ∑ ak rxx (m − k ) for m > Q(9.73)k =1Hence Equation (9.73) can be used to obtain the coefficients ak, which maythen be substituted in Equation (9.72) for solving the coefficients bk. Oncethe coefficients of an ARMA model are identified, the spectral estimate isgiven byQARMAPXX(f) = σ e2∑ bk e2− j 2πfkk =0P1+ ∑ a k e2(9.74)− j 2πfkk =1where σ e2 is the variance of the input of the ARMA model.
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