Practical Spectral Estimation

Transcription

1 Digital Signal Processing/F.G. Meyer Lecture 4 Copyright 2015 François G. Meyer. All Rights Reserved. Practical Spectral Estimation 1 Introduction The goal of spectral estimation is to estimate how the total power of a signal is distributed over the range of frequencies. The estimation is usually performed from a single recording of the signal. 1.1 Energy spectral density of deterministic signals Consider a signal with finite energy then we can consider the Fourier transform n= X(e jω ) = x[n] 2 <, (1) n= x[n]e jωn. (2) Because of Parseval theorem X(e jω ) 2 dω = x[n] 2 < (3) n= we can define to be the energy spectral density. X(e jω ) 2 (4) We can also consider the autocovariance ρ[k] that measures how similar x[n] and x[n k] are, ρ[k] = x[n]x [n k]. (5) n= We can show that k= ρ[k]e jωk = X(e jω ) 2 (6) 1

2 1.2 Power spectral density of random signals Signals measured in practice cannot be determined with infinite precision. A better model for such signals takes into account the stochastic nature of the measurement, and considers that the measured signal is one particular realization of a class of possible random signals with certain noise characteristics. The question becomes: can we estimate from one realization some general properties about the class of signals under consideration? In the following we assume that the signal is a stationary stochastic process with mean zero, Definition 1. The autocovariance sequence r[k] is defined by Ex[n] = 0, for all n. (7) r[k] = Ex[n]x [n k] (8) We have the following properties and r [k] = r[ k] (9) r[0] r[k], for all k. (10) In general, stochastic signals need not have finite energy. However they usually have finite power. Definition 2. We define the power spectral density as follows, Φ(ω) = lim E 1 N 2 x[n]e jωn. (11) N N As in the deterministic case, we have Theorem 1. The power spectral density Φ(ω) is the Fourier transform of the autocovariance, Φ(ω) = r[k]e jkω. (12) k= Because of the definition of the power spectral density, we have Φ(ω) 0 for all ω. (13) Finally, we can show that if the signal is filtered with the impulse response h[n] then y[n] = x h[n] (14) and Φ y (ω) = H(e jω ) 2 Φ x (ω), (15) where H(e jω ) is the frequency response of h[n]. n=1 2

3 1.3 The spectral estimation problem The goal of spectral estimation is to construct an estimate of Φ(ω) from a finite, and possibly small, number of samples of a realization of the stochastic process x. Note that even if we had an infinite number of samples for x[n], we would still not be able to compute Φ y (ω). Indeed, the definition of Φ(ω) requires that we compute the expectation over all realizations, something impractical. We describe in these notes three different approaches to the estimation of Φ y (ω): 1. the nonparametric methods. These methods assume no specific parametric model for Φ y (ω) of x[n]. Such methods can be applied to any existing signal. 2. parametric methods that assume that Φ y (ω) can be approximated with a rational function of the form P k= P β ke jkω Q k= Q α ke jkω (16) 3. parametric methods that assume that the signal of interest is a linear combination of sinusoidal functions, K x[n] = µ k e jω kn+ϕ k (17) 2 Periodogram methods k=1 These methods are based on the definition of the power spectral density, where only a finite number of samples is used, and the expectation is computed over only one realization. Definition 3. The periodogram of the signal {x[n]} N n=1 is defined by Φ P (ω) = 1 N 2 x[n]e jωn N n=1 (18) We can also define a finite sample correlogram. Definition 4. The correlogram of the signal {x[n]} N n=1 is defined by Φ c (ω) = N 1 k= (N 1) where ˆr[k] is an estimate of the the autocovariance r[k], defined by ˆr[k]e jkω, (19) ˆr[k] = 1 N and for negative indices, we define N x[n]x [n k] k = 0,..., N 1 (20) n=k+1 ˆr[ k] = ˆr [k] (21) 3

4 The above estimator of the autocovariance matrix is known as the standard biased estimate. An unbiased version is available and is defined by ˆr[k] = 1 N k N n=k+1 x[n]x [n k] k = 0,..., N 1. (22) This estimator is more accurate for small sample sizes. However, as soon as N is large, one uses the biased estimate. Indeed, for large k, N k is small, but one would expect that ˆr(k) to be large. We have the equivalent of theorem 2 Theorem 2. The Fourier transform, Φ c, of the standard biased autocovariance ˆr[k] is the periodogram Φ P. 2.1 Properties of the periodogram We can estimate the mean squared error defined by { E ΦP (ω) Φ(ω) 2} { = E ΦP (ω) E Φ P (ω) 2} + E Φ P (ω) Φ(ω) 2 (23) The first term is the variance and the second term is the bias squared Bias analysis of the periodogram If we define the hat (Bartlett) function by { 1 k k = N, N + 1,..., N N w B [n] = 0 otherwise (24) then we have E Φ P (ω) = k= Because of the product-convolution theorem, we have E Φ P (ω) = 1 2π Unfortunately, the Fourier transform of the hat window (Fejer kernel), π π w B [k]r[k]e jωk (25) Φ(θ)W B (ω θ)dθ (26) W B (ω) = 1 N [ ] 2 sin(ωn/2) (27) sin(ω/2) is very different from a Dirac impulse, and therefore the bias is significant. Nevertheless, lim N W B (ω) = δ(ω), and therefore lim Φ P (ω) = Φ(ω). (28) N The periodogram is therefore an unbiased estimate of the power spectral density. 4

5 2.1.2 Variance analysis of the periodogram It is much more difficult to estimate the small sample variance of the periodogram. In the case where the signal is generated by filtering some Gaussian white noise through a filter, we can show that lim E ΦP (ω) Φ(ω) 2 = Φ 2 (ω). (29) N The periodogram is an inconsistent estimator of the power spectral density since the variance does not converge to 0 as the number of samples goes to infinity. In fact, it can be shown that the values of Φ P (ω) become uncorrelated for large N. Therefore the periodogram exhibits and erratic behavior MATLAB implementation [Pxx,w] = periodogram(x) returns the power spectral density (PSD) estimate Pxx of the sequence x using a periodogram. The power spectral density is calculated in units of power per radians per sample. The corresponding vector of frequencies w is computed in radians per sample, and has the same length as Pxx. 2.2 Improved periodogram methods All these methods aim at reducing the variance of the periodogram Blackman-Tukey method The basic idea is to decrease the importance of ˆr[k] for large lags k. Indeed, for large k N, ˆr[k] is very inaccurate. The Blackman-Tukey estimator is given by Φ BT (ω) = N 1 k= (N 1) In the frequency domain this relationship becomes Φ BT (ω) = 1 2π π π w[k]ˆr[k]e jωk. (30) Φ P (θ)w (ω θ)dθ. (31) The Blackman-Tukey spectral estimator at a given ω is a weighted average of different values of the periodogram around this ω. It is possible to show that the variance of the estimator convergences to zero, at the price of increasing the bias. The window w[n] should be chosen in order to minimize the following two criteria 1. the equivalent time width 2. the equivalent bandwidth n = 1 w[0] N 1 n= (N 1) w[k] (32) ω = 1 π W (e jω dω (33) W [0] π 5

6 We can show that Theorem 3. The equivalent time-bandwidth product is constant MATLAB implementation n ω = 1. (34) [Pxx,w] = periodogram(x,window) returns the PSD estimate Pxx computed using the modified periodogram method. The vector window specifies the coefficients of the window used in computing a the Blackman-Tukey method. Both input arguments must be vectors of the same length. When you don t supply the second argument window, or set it to the empty vector [], a rectangular window (rectwin) is used by default. In this case the standard periodogram is calculated. 2.3 Bartlett and Welch method Yet another method to decrease the variance is use the data several times to compute several high variance periodograms, and then average these estimators. The idea behind the Bartlett method is divide the data into non overlapping window over which an estimate of the power spectral density Φ j (ω) is computed. The final estimate is the average over all time segments, Φ B (ω) = 1 L L Φ l (ω) (35) l=1 This approach has a slightly higher variance than the Blackman-Tukey method. The Welch approach is a refinement of the Bartlett method where the data segments are now allowed to overlap. The theoretical analysis is more involved. It has been shown in practice that the variance was comparable to the variance of the Blackman-Tukey estimator MATLAB implementation [Pxx,w] = pwelch(x,window,noverlap) divides x into segments according to window, and uses the integer noverlap to specify the number of signal samples (elements of x) that are common to two adjacent segments. The function estimates the power spectral density Pxx of the input signal vector x using Welch s averaged modified periodogram method of spectral estimation. noverlap must be less than the length of the window you specify. If you specify noverlap as the empty vector [], then pwelch determines the segments of x so that there is 50% overlap (default). The power spectral density is calculated in units of power per radians per sample. The corresponding vector of frequencies w is computed in radians per sample, and has the same length as Pxx. A real-valued input vector x produces a full power one-sided (in frequency) PSD (by default), while a complex-valued x produces a two-sided PSD. 6

7 3 Parametric methods for rational spectra 3.1 Introduction We consider the problem of approximating the power spectral density with a rational function of the form P k= P β ke jkω Q k= Q α. (36) jkω ke The geometric intuition is that: high peaks in Φ(ω) can be captured with poles at the corresponding frequencies, low values of Φ(ω) can be captured with zeroes at the corresponding frequencies. We could use a trigonometric polynomial, but it would require a larger order to approximate the peaks of Φ(ω). Because Φ(ω) 0 we can write where Φ(ω) = σ 2 B(ejω ) A(e jω ) (37) A(e jω ) = 1 + a 1 e jω + + a Q e jqω, and B(e jω ) = 1 + b 1 e jω + + b P e jp ω. (38) In the z-domain this becomes where Φ(z) = σ 2 B(z)B (1/z ) A(z)A (1/z ) (39) A(z) = 1 + a 1 z a Q z Q, and B(z) = 1 + b 1 z b P z P. (40) Equation (37) can be interpreted as follows: the power spectral density of the signal x[n] obtained by filtering a white noise e[n] through the filter H(z) = A(z)/B(z), In the time domain this last equation becomes x[n] + X(z) = B(z) E(z). (41) A(z) Q a i x[n i] = P b i e[n i]. (42) Let h[n] be the impulse response of the filter H(z). One can compute the autocovariance of x[n], and we get Q P r[n] + a i r[n i] = σ 2 b i h i n (43) If we choose h[n] to be causal, we have r[n] + Q a i r[n i] = 0 for n > P. (44) 7

8 3.2 All-pole signals We consider the case where B(z) = 1. We have from (43), r[0] + Q a i r[ i] = σ 2 (45) If we combine this equation with (44) we get the following system of linear equation in a 1,..., a Q : r[0] r[ 1]... r[ Q] 1 σ 2 r[1] r[0]. a r[ 1]. = 0. r[q]... r[0] a Q 0 The Yule-Walker equations can be solved for a 1,..., a Q provided one estimates r[k] with ˆr[k]. The final power spectral density is MATLAB implementation Φ(ω) = σ 2 (46) Q k=1 a ke jkω 2 (47) Pxx = pyulear(x,q) implements the Yule-Walker algorithm and returns Pxx, an estimate of the power spectral density (PSD) of the vector x. The entries of x represent samples of a discrete-time signal. a is the integer specifying the order of the all-pole system. The power spectral density is calculated in units of power per radians per sample. Real-valued inputs produce full power one-sided (in frequency) PSDs (by default), while complex-valued inputs produce two-sided PSDs. 3.3 Pole-zero model There are no well established algorithms that are optimal from a theoretical and practical perspective. There exist practical estimators that work well. The Modified Yule-Walker method is a two-stage procedure that starts with equation (44) to estimate the coefficients of the denominator. The coefficient of the numerator are then obtained. 4 Parametric methods for line spectra We assume that the signal takes the form x[n] = K µ k e jω kn+ϕ k. (48) k=1 We are only able to measure y[n] = x[n] + e[n] (49) where e[n] is a white Gaussian noise, with variance σ 2, that corrupts the measurement. We need to estimate the frequencies ω k, the amplitudes µ k, and the initial phases ϕ k. The initial phases are often 8

9 considered to be nuisance parameters since they only depend on the temporal alignment of the different sinusoidal functions. We assume that the initial phases are independent random variables distributed on ( π, π]. We can show that the autocovariance is given by r[l] = Ey[n]y [n l] = K µ 2 k e jωkl + σ 2 δ l,0 (50) k=1 and therefore Φ(ω) == sπ K µ 2 kδ(ω ω k ) + σ 2. (51) Several techniques are available to estimate the frequencies ω k and the amplitudes µ k Nonlinear least-squares method The unknown parameters are the solution of min We note that the squared error depends on the ω k in a nonlinear way Pisarenko and MUSIC methods k=1 N K y[n] 2 µ k e jω kn+ϕ k. (52) n=1 k=1 These methods rely on the fact that the autocovariance completely encodes the frequencies ω 1,..., ω K. Let 1 e jω a(ω) = (53). be a L 1 vector. Consider the L K matrix e jlω A = [ a(ω 1 ) a(ω K ). ] (54) Finally define µ P =.... (55) We have 0 µ 2 K y[1] R = E. [ y[1] y[n] ] y[n] = APA + σ 2 I (56) The vectors a(ω 1 ),..., a(ω K ) are the eigenvectors of R σ 2 I. The Pisarenko and MUSIC methods rely on this property to determine the frequencies ω 1,..., ω K. 9

10 4.0.3 MATLAB implementation [S,w] = pmusic(x,k) implements the MUSIC (Multiple Signal Classification) algorithm and returns S, the pseudospectrum estimate of the input signal x, and a vector w of normalized frequencies (in rad/sample) at which the pseudospectrum is evaluated. The pseudospectrum is calculated using estimates of the eigenvectors of a correlation matrix associated with the input data x, where x is a row or column vector representing one observation of the signal. The parameter K is the number of complex exponentials in the model. Note that if the model is specified in terms of sines and cosines, than K should be twice the number of sines/cosines. 10