7. Line Spectra Signal is assumed to consists of sinusoidals as

Transcription

1 7. Line Spectra Signal is assumed to consists of sinusoidals as n y(t) = α p e j(ω pt+ϕ p ) + e(t), p=1 (33a) where α p is amplitude, ω p [ π, π] is angular frequency and ϕ p initial phase. By using β p = α p e jϕ p a simple model is obtained. That is n y(t) = β p e jωpt + e(t). (33b) p=1 The spectrum estimation problem is now reduced to estimation of frequences ω p in the presence of nuisance parameters β p. 92

2 Itisassumedthate(t) is white Gaussian noise with variance σ 2. It can be shown that the true autocorrelation and PSD are n r(k) = α 2 pe jω pk (34a) p=1 φ(ω) =2π n α 2 pδ(ω ω p )+σ 2, p=1 (34b) where δ is the Dirac impulse (delta function), that has property π π F(ω)δ(ω ω p) dω = F(ω p ) for any function that is continuous at ω p. The PSD forms a line spectrum, that is the reason for name of the methods. Also a name discrete spectrum is used. After estimating the unknowns, the spectrum can be drawn. 93

3 The frequency estimation methods introduced here are superresolution methods, i.e., they can separate sinusoidals that are spaced closer than 1/N, that is the limit for the classical periodogram. They are consistent and have low variance (near Cramer-Rao bound), if the assumed model is correct. Assumptions of noise can be relaxed, especially, if nonlinear least squares (NLS) is used. 94

4 Other estimators to be discussed are multiple signal classification (MUSIC) methods, Min-Norm and estimation of signal parameters via rotational invariance (ESPRIT). These are so called subspace methods, that use eigendecomposition of the sample correlation matrix. An improved correlation estimate is also discussed. This is known as forward-backward approach. It is empirically shown to improve frequency estimation accuracy. 95

5 NLS Using notations β =[β 1...β n ] T, y =[y(1)...y(n)] T, e = [e(1)...e(n)] T, ω =[ω 1...ω n ] T and B(ω) =B = ejω 1... e jω n.. e jω1n... e jω nn the signal model (33) can be expressed as Y = Bβ + e. (35) The model is linear with respect to amplitudes β but nonlinear with respect to frequencies ω. NLS solves β and ω by minimizing f(β, ω) = y Bβ 2. (36) 96

6 It was early shown that ^β = ( B H B ) 1 B H y. By substituting this into (36) we get ˆω = arg max ω yh B ( B H B ) 1 B H y. (37) The noise power is estimated as mean of (36). Asymptotically, as N is large, the covariance matrix of estimation errors ˆω ω is given as Cov( ˆω) = 6σ2 1/α (38) N 3 0 1/α 2 n If noise is Gaussian, this is also the Cramer-Rao lower bound. In the Gaussian noise case NLS provides (asymptotically) the most accurate frequency estimates. 97

7 Solution of (36) requires multidimensional search. E.g., Newton or Gauss methods or their variants can be used for search. Wrong initial values may lead the algorithms to convergence to local maximum, not global, or even diverge. There exist several methods to find initial values. One method is obtained by assuming n = 1 (discussed later). Another is the alternating projection (AP) algorithm. NLS solution computes n-dimensional functional f(^β, ω) and finds maximum from that surface. 98

8 As an example, a two dimensional case is computed. N = 32 such that resolution in angular frequency is The frequencies are π/3 and π/ such that separation is below the classical resolution. Amplitudes are 1 and

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10 NLS is a superresolution method. Spectral leakage is not a problem. NLS works even if signal amplitudes are different such that in the periodogran the other signal is buried under the sidelobes of another. In terms of spread spectrum communications, NLS is a near-far robust method. 101

11 NLS when n = 1 If n = 1 we get from (37) 1 ˆω = ˆω 1 = arg max ω N BH y 2, (39) where B C N and 1/N term becomes from fact N = B H B. This can be ignored. This is the periodogram without windowing. The periodogram is NLS estimator if only one signal is present. From the properties of the periodogram it follows that (39) can be used in multiple signal case if: Signal separation is larger than 2π/N and Signal powers are rather equal such that leakage does not form problems. 102

12 In such case NLS estimates are n largest peaks of the periodogram that are not necessary quaranteed to have covariance properties of the NLS. The difference is of order 1/N. The periodgram can be used as an initial value solver for NLS (if leakage is not a problem). 103

13 Eigenanalysis of Correlation Matrix Note that (for n = 1) y(t) =β k e jω kt y(t 1) =e jω kβ k e jω kt which, in a matrix form is.. y(t m + 1) =e j(m 1) β k e jω kt ỹ(t) =a(ω k )x k (t)+ẽ(t), where a(ω) = [1e jω...e j(m 1)ω ] T C m and x k (t) = β k e jωkt. In generel case ỹ(t) =A(ω) x(t)+ẽ(t), (40) where A(ω) =[a(ω 1 )...a(ω n )] T C m n and x(t) =[x 1 (t)...x n (t)] T. 104

14 Since ω k are different, a(ω k ) are linearly independent and rank{a} = n. R has correlation matrix R = E{ỹ(t)ỹ H (t)} = APA H + σ 2 I, where P = α α 2 n and I is a suitable (m m in this case) identity matrix. (41a) (41b) P = lim N N t=1 x(t) xh (t) is the covariance of sinusoidals. Off-diagonal elements consists of average of terms like e jω kt e jω lt = e j(ω k ω t )t = e jω ht, and that is zero (average of sinusoidal). 105

15 Since APA H C m m) has rank n (n <m) it is nonsingular, i.e., its determinant is zero, i.e, APA H = 0. This is equal to R σ 2 I = 0, which means that σ 2 is an eigenvalue of R. This means that Ru i = σ 2 u i, where u i is eigenvector associated with σ 2. Since R = APA H + σ 2 I APA H u i = 0. (42) This means that eigenvector associated with σ 2 lies in the null space of A H. 106

16 Since the dimension of the range space of A rank{apa H } = n, it follows that the dimension of the null space is m n. It follows that eigenvalue σ 2 has multiplicity m n. The remainder n eigenvalues satisfy APA H u i =(λ i σ 2 )u i. (43) Since R is positive semidefinite, its eigenvalues are nonnegative. Thus σ 2 is the minimum eigenvalue. The eigenvalues can be ordrered as λ 1 λ 2...λ n >σ 2 =...σ 2. The n largest eigenvectors form range space of A (if P is full rank). This is also called as signal subspace. The m n smallest eigenvectors form null space of A H.Thisis also called as noise subspace. Analysis hold also for other linearly independed vectors a(ω) if P is full rank. In a general case, P is not diagonal. 107

17 MUSIC (42) means that sinusoidals a(ω) (with true values) are orthogonal to noise subspace. This can be stated as m i=m n a(ω)h u i = 0. Let G =[u m n...n m ] C m m n include eigenvectors that span noise subspace. Then, the true frequencies are solutions to a(ω) H GG H a(ω) = G H a(ω) 2 = 0. (44) In practise, only estimate ˆR = 1/N N t=m ỹ(t)ỹh (t) of R is available. It follows that only estimate Ĝ of G is available and it is better to find frequences as minimizers of f music (ω) = ĜH a(ω) 2. (45) 108

18 Sometimes 1/f music (ω) is called the MUSIC (pseudo)spectrum, since the highest peaks are associated with the sinusoidals. In the MUSIC method a) find sample correlation matrix, b) compute its eigenvectors, c) form noise subspace estimate Ĝ from the eigenvectors associated with m n smallest eigenvalues. The d) frequency estimates are minima of (45). Often the faster way to compute the eigenvectors is to compute the SVD of Y =[ỹ(m)...ỹ(n)]. The decomposition is SVD{Y} = UΣV H and U =[SG]. Let S =[u 1...u n ] contain the rest eigenvectors of R. Since SS H + GG H = I it follows that frequencies can also obtained as maximizers of f music (ω) = ŜH a(ω) 2. This form is usefull if n<m n since the computational com- 109

19 plexity is reduced. The Pisarenko method uses only the eigenvector that is associated with the smallest eigenvalue of ˆR, i.e, is a special case of MUSIC. (No order determination problems but decreased performance.) The root MUSIC computes frequencies as angular positions of n roots of equation a T (z 1 )ĜĜH a(z) =0 (46) that are nearest (inside) the unit circle. a(z) is obtained from a(ω) by replacing e jω by z so that a(z) = [1z 1... z (m 1) ] T. 110

20 Parameter m should be as large as possible (for resolution), but not too close to N in order to compute ^R sufficiently (for variance). Sometimes is is said that ˆR requires (2 4)m vectors. Since number of vectors ỹ(t) is N m, this suggest that m<n/(3-5). Matlab includes pmusic, rootmusic. Amplitudes can be estimated as in NLS. Noise power can be estimated as an average of noise subspace eigenvalues (or some of them). 111

21 As an example, let N = 128, m = N/4 = 32. Now, 1/N angular resolution limit is 0.05, and angular frequencies are π/20 = 0.15 π/3 = 1.05 and π/ Amplitudes are 1, 10 and 10, correspondingly. Angular resolution 1/m is Next pages present eigenvalues and the MUSIC spectrum, of which the highest peaks corresponds to frequency estimates. 112

22 normalized eigenvalue eigenvalue number 113

23 35 separation MUSIC spectrum angular frequency 114

24 Three large eigenvalues can be separated. Can not separate frequencies below 1/N limit. In the next pages separation is 0.15 and

25 35 separation MUSIC spectrum angular frequency 116

26 35 separation MUSIC spectrum angular frequency 117

27 The two close frequencies are separable under 1/m limit (if signal-to-noise ratio is high enough or N large such that SNR becomes high after averaging). Computation of root-music is then addressed. Find Ĝ. Compute GG = ĜĜH (m m). Find the 2m 1 coefficients of (46). Each is obtained as as sum of diagonal elements of GG. The first coefficient is the lowest diagonal (i.e., element (m 1, 1)). The second coefficient is the diagonal above that(i.e., elements (m 1, 1) (m, 2)), etc. (sum(diag(gg,j) j=-(m-1):m-1 in Matlab). Find n roots ξ k that are nearest unit circle but inside it (roots command in Matlab). Angular frequency estimates are ˆω k = ξ k. 118

28 For the previous example we got following results: correct value estimated value Separation Separation Separation correct value estimated value correct value estimated value As observed, the method can separate very close frequencies (at least in this case), that spectral-music was not able to separate

29 In the next page the roots inside the unit cirle are shown (polar(angle(r),abs(r), * ) in Matlab). 120

30

31 Min-Norm Is 1 noise eigenvector sufficient? This would give computational savings, if compared with MUSIC. It is not, if the Pisarenko method is used. Min-Norm tries to do this with similar statistical accurary than MUSIC. Select a vector [ 1 q = ĝ] from the noise subspace, which has minimum Euclidian norm. The Min-Norm frequency estimates are n maxima of the pseudospectrum 1 f min norm (ω) = a H (ω)q 2. (47) 122

32 Also arootversion exist. That solves frequencies as angular positions of the n roots of the polynomial a T (z 1 )q that are located nearest the unit cirle (not necessary inside). To compute root-min-norm, find roots of q. 123

33 The course books gives two ways to determine ĝ. The first one is usefull if n<m n. [ ] [ ] Let Ŝ and Ĝ partitioned as Ŝ = α H S and Ĝ =, i.e, α H and β H are the first rows of Ŝ and Ĝ. The solutions are and ĝ = Sα/(1 α 2 ), β H Ḡ (48a) ĝ = Ḡβ/ β 2 (48b) Note that relation α 2 1 must be valid. It is valid, at least, for large N. 124

34 As an example we repeated examples used to demonstrate MU- SIC with close frequency separations 0.02, 0.15 and

35 3 x 104 separation Min Norm spectrum angular frequency 126

36 800 separation Min Norm spectrum angular frequency 127

37 500 separation Min Norm spectrum angular frequency 128

38 As can be seen, the Min-Norm is not neseccary so reliable. It may lead to wrong conclusions (now that happened when very near signals are more powerful than the separated one). As in the MUSIC case, the pseudospectrum does not give correct amplitudes. The peaks in Min-Norm seems to be narrower than in MUSIC. Then, the root-min-norm is examined. 129

39 For the previous example we got following results: correct value estimated value Separation Separation Separation correct value estimated value correct value estimated value As observed, the method can separate very close frequencies (at least in this case)

40 The roots are shown in the next page for the last case. 131

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42 Rationale behind root-methods This is not discussed in the course book, but added here for educational purposes. Equation (42) means that sinusoidal a(ω) is orthogonal to vectors that lie in the noise subspace. By replazing e jω by z in a(ω) we get u H a(z) =0, where u lies in the noise subspace. Let elements of u be u i. Then we have polynomial y(z) = m 1 i=0 u i z i, which have roots z k = e jω k,k= 1,..., n. These roots lie in the unit circle (remember: if a polynomial x(z) has n roots z i, then it can be presented as x(z) = n i=1 (z z i)). In the root-min-norm q lies in the noise subspace and we select roots of polynomial y(z) = m 1 i=0 q i z i that are closest to unit circle. 133

43 In the root-music we use all noise subspace level eigenvectors. lth elementh of vector G H a(z) is polynomial y l (z) = m 1 i=0 G i+1,l z i, where G l,i denotes (l, i)th elementh of matrix G. kth elementh of vector a(z 1 )G is polynomial x k (z) = m 1 i=0 G i+1,kz i, a(z 1 )GG H a(z) is therefore polynomial r(z) = m m 1 i=0 G j+1,kg i+1,k zj i. k=1 m 1 j=0 r(z) = m 1 i= (m 1) a iz i has n roots z k = e jω k on the unit circle. r(z) can also be factored as r(z) =p(z)p (1/z ), which means that roots are reciprocal. In the root-music we select n roots of r(z) that are inside the unit circle and closest to it. 134

44 How are the coefficient a i selected? Consider first a (m 1), which is obtained if j = 0 and i = m 1. In this case a (m 1) = m k=1 G 1,kG m,k, which means that first row of G is multiplied by the last column of G H, i.e, (1, m)th elementh of GG H, or sum of elements of mth diagonal of GG H. Consider then a (m 2), which is obtained as sum of terms where (j = 0, i = m 2) and (j = 1, i = m 1). This leads to a m (m 2) k=1 (G 1,kG m 1,k + G 2,kG m,k ),whichissumof elements of m 1th diagonal of GG H. This leads to given algorithm. 135

45 ESPRIT Note that m 1 1 vectors 1 a 1 (ω) = e jω. a 2 (ω) = e j(m 2)ω e jω e j2ω. e j(m 1)ω have connection a 2 (ω) =a 1 (ω)e jω. This can be generalized to n signal case by division such that A 1 =[I m 1 0 m 1 ]A A 2 =[0 m 1 I m 1 ]A A 2 = A 1 D (49a) 136

46 where D = e jω e jω n (49b) Relation (49) is known as rotation (D is a unitary matrix). Similarly, define S 1 =[I m 1 0 m 1 ]S S 2 =[0 m 1 I m 1 ]S. Multiply equation (43) from left by [I m 1 0 m 1 ] and [0 m 1 I m 1 ]. Let Λ be diagonal matrix with elements λ i σ 2. Then, the result is A 1 PA H S = S 1 Λ A 2 PA H S = S 2 Λ. 137

47 From the first row we get A 1 = S 1 C 1, where C = PA H SΛ 1. The second row gives S 2 = A 2 C = A 1 DC = S 1 C 1 DC = S 1 Φ, where Φ = C 1 DC. Φ and D have equal eigenvalues ν k = e jω k. In practice, only estimate of S is available, and Φ is obtained as a minimum norm solution. The ESPRIT method a) determines ˆΦ, b) its eigenvalues ˆν k and c) solves frequencies as ˆω = ˆν k. Alternatives for ˆΦ are LS and Total LS (TLS). LS solution is ) 1Ŝ H ˆΦ = (ŜH 1 Ŝ 1 Ŝ 2. (50) 1 138

48 For the previous example we got following results: correct value estimated value Separation Separation Separation correct value estimated value correct value estimated value As observed, the method can separate very close frequencies (at least in this case)

49 As a conclusions: root versions of MUSIC and Min-Norm provide direct frequency estimates. With spectral versions, maxima (or minima) search is required. Amplitudes can be computed like in NLS, after the frequencies have been estimated. Noise power can be estimated as mean estimation error in NLS, or as an average of noise subspace eigenvalues. 140

50 Forward-Backward Method As a last subject, we consider improved correlation matrix estimator that is empirically shown to lead more accurate frequency estimates. The regular correlation matrix estimate ˆR = 1 N y(t). [y (t)... y (t m + 1)] N t=m y(t m + 1) can be seen as a result of a LS problem where newest value y(t) is estimated by m previous values. This is forward prediction. 141

51 As an alternatively, the m + 1 latest value can be estimated by m newer samples. This is known as backward prediction. This leads to sample correlation matrix ˆR b = 1 N y (t m + 1). [y(t m + 1)... y (t)], N t=m y (t) that can be expressed as R b = JˆR T J, where J = is so called exchange matrix. (51) 142

52 The Forward-Backward (FB) correlation matrix estimate is R = 1 2 (ˆR + JˆR T J). (52) This can be used instead of ^R in hope of more accurate results. 143