Power Spectrum and Correlation (Vaseghi - Advanced Digital Signal Processing and Noise Reduction), страница 2

Hence thefrequency resolution of the DFT spectrum ∆f, i.e. the space betweensuccessive frequency samples, is given byû1 =F11== sû% NTs N(9.14)Note that the frequency resolution ∆f and the time resolution ∆T areinversely proportional in that they cannot both be simultanously increased;in fact, ∆T∆f=1. This is known as the uncertainty principle.9.3.3 Energy-Spectral Density and Power-Spectral DensityEnergy, or power, spectrum analysis is concerned with the distribution ofthe signal energy or power in the frequency domain. For a deterministicdiscrete-time signal, the energy-spectral density is defined as2X(f ) =∞∑ x(m)e2− j 2πfm(9.15)m =−∞The energy spectrum of x(m) may be expressed as the Fourier transform ofthe autocorrelation function of x(m):X ( f ) 2 = X ( f )X *( f )=∞∑ rxx (m)e − j 2πfm(9.16)m= −∞where the variable r xx (m) is the autocorrelation function of x(m).

TheFourier transform exists only for finite-energy signals. An importantFourier Transform: Representation of Aperiodic Signals271theoretical class of signals is that of stationary stochastic signals, which, as aconsequence of the stationarity condition, are infinitely long and haveinfinite energy, and therefore do not possess a Fourier transform. Forstochastic signals, the quantity of interest is the power-spectral density,defined as the Fourier transform of the autocorrelation function:PXX ( f ) =∞∑ rxx (m)e − j 2πfm(9.17)m = −∞where the autocorrelation function rxx(m) is defined asr xx (m) = E [ x(m)x(m + k)](9.18)In practice, the autocorrelation function is estimated from a signal record oflength N samples asrˆxx (m) =1 N −| m| −1∑ x(k ) x(k + m) , k =0, .

. ., N–1N −| m | k = 0(9.19)In Equation (9.19), as the correlation lag m approaches the record length N,the estimate of rˆxx (m) is obtained from the average of fewer samples andhas a higher variance. A triangular window may be used to “down-weight”the correlation estimates for larger values of lag m. The triangular windowhas the form1 − | m | ,| m | ≤ N −1w(m) = (9.20)N0,otherwiseMultiplication of Equation (9.19) by the window of Equation (9.20) yieldsrˆxx (m) =1NN −| m | −1∑ x ( k ) x( k + m)(9.21)k =0The expectation of the windowed correlation estimate rˆxx (m) is given byPower Spectrum and Correlation2721E [rˆxx (m)] =NN −|m|−1∑ E [x(k ) x(k + m)]k =0(9.22)m rxx (m)= 1 −NIn Jenkins and Watts, it is shown that the variance of rˆxx (m) is given byVar[rˆxx (m)]≈1N∞∑ [rxx2 (k ) + rxx (k − m)rxx (k + m)](9.23)k = −∞From Equations (9.22) and (9.23), rˆxx (m) is an asymptotically unbiased andconsistent estimate.9.4 Non-Parametric Power Spectrum EstimationThe classic method for estimation of the power spectral density of an Nsample record is the periodogram introduced by Sir Arthur Schuster in 1899.The periodogram is defined as1PˆXX ( f ) =NN −1∑ x ( m )e2− j 2πfmm =0(9.24)1X(f )2=NThe power-spectral density function, or power spectrum for short, defined inEquation (9.24), is the basis of non-parametric methods of spectralestimation.

Owing to the finite length and the random nature of mostsignals, the spectra obtained from different records of a signal varyrandomly about an average spectrum. A number of methods have beendeveloped to reduce the variance of the periodogram.9.4.1 The Mean and Variance of PeriodogramsThe mean of the periodogram is obtained by taking the expectation ofEquation (9.24):Non-Parametric Power Spectrum Estimation[]1E X(f ) 2NN −11  N −1= E  ∑ x(m)e − j 2πfm ∑ x(n)e j 2πfn N m=0n =0E [ PˆXX ( f )] ==273(9.25)N −1m1 − rxx (m)e − j 2πfmN m = − ( N −1) ∑As the number of signal samples N increases, we havelim E [ PˆXX ( f )] =N →∞∞∑ rxx (m)e − j 2πfm = PXX ( f )(9.26)m = −∞For a Gaussian random sequence, the variance of the periodogram can beobtained as  sin 2πfN  2 2ˆVar[ PXX ( f )] = PXX ( f ) 1 +    N sin 2πf  (9.27)As the length of a signal record N increases, the expectation of theperiodogram converges to the power spectrum PXX ( f ) and the variance of2 (f)Pˆ XX ( f ) converges to PXX.

Hence the periodogram is an unbiased butnot a consistent estimate. The periodograms can be calculated from a DFTof the signal x(m), or from a DFT of the autocorrelation estimates rˆxx (m) . Inaddition, the signal from which the periodogram, or the autocorrelationsamples, are obtained can be segmented into overlapping blocks to result ina larger number of periodograms, which can then be averaged. Thesemethods and their effects on the variance of periodograms are considered inthe following.9.4.2 Averaging Periodograms (Bartlett Method)In this method, several periodograms, from different segments of a signal,are averaged in order to reduce the variance of the periodogram. The Bartlettperiodogram is obtained as the average of K periodograms asPower Spectrum and Correlation2741 K ˆ (i )BˆPXX ( f ) = ∑ PXX ( f )K i =1(9.28)(i )where PˆXX ( f ) is the periodogram of the ith segment of the signal. Theexpectation of the Bartlett periodogram Pˆ B ( f ) is given byXX(i )B( f )] = E[ PˆXX( f )]E[ PˆXX=N −1m− j 2πfm1 − rxx (m)eNm = − ( N −1) 1=N∑1/ 2(9.29)2 sin π ( f − ν ) N ∫ PXX (ν )  sin π ( f − ν )  dν−1 / 22where (sin πfN sin πf ) N is the frequency response of the triangularwindow 1–|m|/N.

From Equation (9.29), the Bartlett periodogram isB (f)asymptotically unbiased. The variance of PˆXXis 1/K of the variance ofthe periodogram, and is given by  sin 2πfN  2 1 2BˆVar PXX ( f ) = PXX ( f ) 1+  K  N sin 2πf  [](9.30)9.4.3 Welch Method: Averaging Periodograms from Overlappedand Windowed SegmentsIn this method, a signal x(m), of length M samples, is divided into Koverlapping segments of length N, and each segment is windowed prior tocomputing the periodogram. The ith segment is defined asxi (m) = x(m + iD) ,m=0, .

. .,N–1, i=0, . . .,K–1(9.31)where D is the overlap. For half-overlap D=N/2, while D=N corresponds tono overlap. For the ith windowed segment, the periodogram is given byNon-Parametric Power Spectrum Estimation1(i )PˆXX( f )]=NUN −1275∑ w(m) xi (m) e2− j 2πfm(9.32)m =0where w(m) is the window function and U is the power in the windowfunction, given by1 N −1 2U = ∑ w ( m)(9.33)N m =0The spectrum of a finite-length signal typically exhibits side-lobes due todiscontinuities at the endpoints. The window function w(m) alleviates thediscontinuities and reduces the spread of the spectral energy into the sidelobes of the spectrum.

The Welch power spectrum is the average of Kperiodograms obtained from overlapped and windowed segments of asignal:1 K −1 (i )WPˆXX( f ) = ∑ PˆXX(f)K i =0(9.34)W (f)Using Equations (9.32) and (9.34), the expectation of PˆXXcan beobtained as(i )W( f )] = E [ PˆXX( f )]E[ PXX=1 N −1 N −1∑ ∑ w(n) w(m)E[ xi (m) xi (n)]e − j 2πf (n−m)NU n=0 m=01 N −1 N −1w(n) w(m)rxx (n − m)e − j 2πf ( n−m)=∑∑NU n=0 m=01/ 2=∫ PXX (ν )W (ν − f )dν−1 / 2(9.35)where1W( f )=NUN −1∑ w(m)e2− j 2πfmm=0and the variance of the Welch estimate is given by(9.36)Power Spectrum and Correlation2761WVar[ PˆXX( f )] =K2(i )( j)W( f ) PˆXX( f )] − (E [PˆXX( f ) ])∑ ∑ E [PˆXXK −1 K −12(9.37)i =0 j = 0Welch has shown that for the case when there is no overlap, D=N,WVar[ PXX(i )2Var[ PXX( f )] PXX(f)( f )] =≈K1K1(9.38)and for half-overlap, D=N/2 ,92WVar[ PˆXX( f )] =PXX( f )]8K 2(9.39)9.4.4 Blackman–Tukey MethodIn this method, an estimate of a signal power spectrum is obtained from theFourier transform of the windowed estimate of the autocorrelation functionasBTPˆXX( f )=N −1∑ w(m) rˆxx (m) e − j 2πfm(9.40)m= − ( N −1)For a signal of N samples, the number of samples available for estimation ofthe autocorrelation value at the lag m, rˆxx (m) , decrease as m approaches N.Therefore, for large m, the variance of the autocorrelation estimateincreases, and the estimate becomes less reliable.

The window w(m) has theeffect of down-weighting the high variance coefficients at and around theend–points. The mean of the Blackman–Tukey power spectrum estimate isBT( f )] =E[ PˆXXN −1∑ E [rˆxx (m)]w(m) e − j 2πfm(9.41)m = − ( N −1)Now E[rˆxx (m)] =rxx (m) wB (m) , where wB (m) is the Bartlett, or triangular,window. Equation (9.41) may be written asNon-Parametric Power Spectrum Estimation277N −1BT( f )] =E [ PˆXX∑ rxx (m)wc (m) e − j 2πfm(9.42)m =− ( N −1)where wc (m)= wB (m) w(m) . The right-hand side of Equation (9.42) can bewritten in terms of the Fourier transform of the autocorrelation and thewindow functions as1/ 2BT ( f )]=E[ PˆXX∫ PXX (ν)Wc ( f − ν)dν(9.43)−1 / 2where Wc(f) is the Fourier transform of wc(m). The variance of theBlackman–Tukey estimate is given byU 2BTVar[ PˆXX( f )]≈ PXX(f)N(9.44)where U is the energy of the window wc(m).9.4.5 Power Spectrum Estimation from Autocorrelation ofOverlapped SegmentsIn the Blackman–Tukey method, in calculating a correlation sequence oflength N from a signal record of length N, progressively fewer samples areadmitted in estimation of rˆxx (m) as the lag m approaches the signal lengthN.

Hence the variance of rˆxx (m) increases with the lag m. This problem canbe solved by using a signal of length 2N samples for calculation of Ncorrelation values. In a generalisation of this method, the signal record x(m),of length M samples, is divided into a number K of overlapping segments oflength 2N. The ith segment is defined asxi (m) = x( m + iD),(9.45)m = 0, 1, . . ., 2N–1i = 0, 1, . .