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Performance of ESPRIT for Estimating Mixtures of
Complex Exponentials Modulated by Polynomials:
Supporting Document
Roland Badeau, Gaël Richard, Bertrand David
To cite this version:
Roland Badeau, Gaël Richard, Bertrand David. Performance of ESPRIT for Estimating Mixtures of
Complex Exponentials Modulated by Polynomials: Supporting Document. 2007. �hal-00180015v2�
Performance of ESPRIT for Estimating Mixtures of
Complex Exponentials Modulated by Polynomials:
Supporting Document
Roland Badeau, Gaël Richard, and Bertrand David
Abstract—High Resolution (HR) methods are known to
pro-vide accurate frequency estimates for discrete spectra [1]. The Polynomial Amplitude Complex Exponentials (PACE) model was presented as the most general model tractable by HR methods. A subspace-based estimation scheme was recently proposed in [2], derived from the ESPRIT algorithm [3]. In [4], we focused on the performance of this estimator. We first presented some asymptotic expansions of the estimated parameters, obtained at the first order under the assumption of a high signal-to-noise ratio (SNR). Then the performance of the generalized ESPRIT algorithm for estimating the parameters of this model was analyzed in terms of bias and variance, and compared to the Cramér-Rao bounds. In this supporting document, we present the proofs of the theoretical results introduced in [4]. This document, written as a sequel of [4], is not intended to be read separately. It is organized as follows: section I is devoted to the perturbation analysis, then the performance of the estimators is analyzed in section II.
Performance d’ESPRIT pour l’estimation de mélanges d’exponentielles complexes modulées par des polynômes:
document de support
Résumé—Les méthodes à Haute Résolution (HR) sont connues
pour fournir une estimation fréquentielle précise des spectres discrets [1]. Le modèle d’Exponentielles Complexes à modulation d’Amplitude Polynomiale (PACE) a été présenté comme le modèle le plus général que l’on puisse traiter par des méthodes HR. Une procédure d’estimation de type sous-espace a été récemment pro-posée dans [2], développée à partir de l’algorithme ESPRIT [3]. Dans [4], nous avons porté notre attention sur les performances de cet estimateur. Nous avons d’abord présenté des développe-ments asymptotiques des paramètres estimés, obtenus au premier ordre sous l’hypothèse d’un rapport signal-à-bruit (RSB) élevé. Les performances de l’algorithme ESPRIT généralisé visant à estimer les paramètres de ce modèle ont ensuite été analysées en terme de biais et de variance, et comparées aux bornes de Cramér-Rao. Dans ce document de support, nous présentons les preuves des résultats théoriques introduits dans [4]. Ce document, conçu pour être lu en complément de [4], est structuré de la façon suivante : la section I porte sur l’analyse des perturbations, puis les performances des estimateurs sont analysées dans la section II.
Index Terms—ESPRIT, high resolution (HR), multiple
eigen-values, performance analysis, perturbation theory, polynomial modulation.
This document was initially posted to the Hyper Article on Line (HAL) server of the French Centre pour la Communication Scientifique Directe (CCSD) on October 17, 2007.
The authors are with the Department of Signal and Image Processing (TSI), Télécom Paris (ENST) - CNRS LTCI, 75634 Paris Cedex 13, France (e-mail: roland.badeau@enst.fr).
I. PERTURBATION ANALYSIS A. Perturbation of the spectral matrix
Proof of proposition 10: First, note that the function
ε 7→ detW(ε)H ↓W(ε)↓
is continuous. As W↓ is full-rank, it is non-zero at ε = 0. Thus it remains non-zero in a neighborhood of 0. Consequently, W↓(ε) is full-rank in this neighborhood. Moreover, according to proposition 9, the function ε 7→ W (ε) is C∞. Thus the function Φ(ε) =
W(ε)H ↓W(ε)↓
−1 W(ε)H
↓W(ε)↑ is also C∞. Thus equa-tions (34) to (36) can be derived by calculating the first order expansion of the equality
W(ε)H ↓W(ε)↓
Φ(ε) = W (ε)H↓W(ε)↑.
B. Perturbation of the Jordan matrix
Proof of corollary 11: Left-multiplying equation (34) by
G−1and right-multiplying it by G yields (37) and (38), where ∆J⊥= G−1∆Φ⊥G. (55) Left-multiplying equation (36) by G−1, right-multiplying it by G, and substituting equations (55), (7), and (8) yields
∆J⊥= −Vn ↓ † ∆W⊥ ↓G J + J V n ↑ † ∆W⊥ ↑G. (56) In other respects, equation (4) yields S† = Vl†TD−1Vn†. Thus substituting equations (29) and (7) into (56) yields
∆J⊥ = −Vn↓ † In↓− Vn↓V n† ∆S Vl†TD−1J +J Vn↓ † In↑− Vn↑V n† ∆S Vl†TD−1. From this last equation, lemma 15 below proves equation (39).
Lemma 15. The Pascal-Vandermonde matrix Vn satisfies the following identities: h Vn↓†, 0(n×1) i − Vn†= −v′ ↓e′↓ H (57) h 0(n×1), Vn ↑ † i − Vn†= −v′↑e ′ ↑ H . (58) where • v′↓= Z −1v ↓ 1−vH ↓Z−1v↓ • v′↑= Z −1v ↑ 1−vH ↑Z−1v↑ • e↓= [0 . . . 0, 1]T − VnZ−1v↓
2 TÉLÉCOM PARIS (ENST) - CNRS LTCI, TECHNICAL REPORT
• e↑= [1, 0 . . . 0]T− VnZ−1v↑ • Z= VnHVn.
Proof: Let us show equation (57). Since the
ma-trix Vn ↓ is full-rank, V n ↓ † = Vn↓HVn↓−1Vn↓H. Be-sides, Vn ↓ H
Vn↓ = Z − v↓vH↓. Applying the matrix inver-sion lemma [5, pp. 18-19] to this last equality and right-multiplying it by Vn ↓ H yields Vn↓†= Z−1Vn↓H+ v′↓v H ↓Z −1Vn ↓ H . (59) Besides, Vn†=h Z−1Vn↓H, 1 − vH ↓Z −1v ↓ v′ ↓ i . (60) Substituting equations (60) and (59) into the left member of equation (57) finally leads to the right member of equa-tion (57). Equaequa-tion (58) can be derived in a similar way.
C. Perturbation of the poles
Proof of proposition 12: Remember that the matrix J
in equation (37) is block-diagonal, with blocks of dimensions Mk× Mk. The perturbation theory shows that for all ε in the neighborhood of 0, the matrix J(ε) in equation (37) can also be block-diagonalized, with blocks of the same dimensions, and that the functions which associate each of these blocks to εare C∞. More precisely, the function which associates the Mk× Mk block related to the pole zk to ε admits the first order expansion: Jk(ε) = Jk+ ε ∆J⊥k + A ′ kJk− JkA′k + O(ε2). However the sum of the eigenvalues z(m)k (ε) is equal to the trace of this block. Thus the function ε 7→ zk(ε) is C∞ and admits the first order expansion
zk(m)(ε) = zk+ Mεktrace ∆J⊥ k + A ′ kJk− JkA′k + O(ε2). (61) Since matrix products can be commuted inside the trace op-erator, trace A′
kJk− JkA′k
= 0. Therefore equations (40) and (41) directly follow from equation (61). Moreover, sub-stituting equation (39) into equation (41) shows that
Mk∆zk= trace v′ ↓k× e′ ↓ H ∆S Vl†TD−1J − traceJ v′ ↑k ×e′ ↑ H ∆S Vl†TD−1 = tracee′ ↓ H ∆S Vl†TD−1J× v′ ↓k − tracee′ ↑ H ∆S Vl†TD−1×J v′ ↑k = e′ ↓ H ∆S Vl†TD−1J v′ ↓k− e′↑ H ∆S Vl†TD−1J v′ ↑k, from which equations (42) and (43) are derived1.
Proof of proposition 3: The complex logarithm is a C∞ -diffeomorphism of C into R×] − π, π[, thus the functions ε7→ δk(ε) and ε 7→ fk(ε) are C∞. A first order expansion
1It can be noticed a scaling factor 1
α(Mk −1)k is artificially introduced in
equation (42), and counterbalanced in equation (43). This trick makes the vectors f↓kand f↑knot depend on the complex amplitudes, in the particular
case of single poles.
yields, by using equation (40), ln(zk(ε)) = ln(zk) + ε ∆zk
zk + O ε2. Finally, equation (12) can be derived by substituting equation (44) into this first order expansion.
D. Perturbation of the amplitudes and phases
Proof of lemma 13: The coefficients of the matrix VN(ε) are powers of the estimated poles zk(ε). Since these poles are C∞ functions of the variable ε, the function ε 7→ VN(ε) is also C∞. In other respects, the column of VN
(ε) related to the pole zk(ε) at index m < Mk ism!1 d
m v
dzm(zk(ε)), where v(z) = [1, z, . . . , zN −1]T. Consequently, its first order expansion is
1 m! dmv dzm(zk) + ε 1 m! dm+1v dzm+1(zk) dzk dε (0) + O(ε 2) = v(m)k + ε(m + 1)∆zkv (m+1) k + O(ε2). where v(m) k is the column of V N
related to the pole zk at index m ≤ Mk. Thus equations (46) to (49) can be derived columnwise.
Proof of proposition 14: Since the poles zk are distinct and since the functions ε 7→ zk(ε) are continuous, they have distinct values in a neighborhood of 0. Therefore, the Pascal Vandermonde matrix VN(ε) remains full-rank in this neighborhood. Moreover, according to lemma 13, the function ε7→ VN(ε) is C∞, thus the function ε 7→ VN
(ε)†is also C∞. Therefore the function ε 7→ α(ε) = VN(ε)†s(ε) is C∞ in a neighborhood of 0. Moreover, the first order expansion of the equality VN(ε) α(ε) = s(ε) yields
VN∆α + ∆VNα= ∆s.
Substituting equation (47) into this last equality yields ∆α = VN †∆s − VN∆Z α. (62) Besides, substituting equations (44), (48) and (49) into equa-tion (62), a simple rewriting shows that
∆Z α =hBT0, . . . , BTK−1i T
∆s (63)
(where the matrices Bkare defined in equation (54)). Finally, equations (52) and (53) are derived by substituting equa-tion (63) into equaequa-tion (62).
Proof of proposition 4: It is supposed that |α(m)k (0)| = a(m)k 6= 0. Then since the function ε 7→ α(m)k (ε) is C∞, the function ε 7→ a(m)
k (ε) = |α (m)
k (ε)| is also C
∞ in the neigh-borhood of zero. Moreover, substituting the first row of equa-tion (13) into the equaequa-tion a(m)k (ε) = a(m)k
r α(m)k (ε) α(m)k α(m)k (ε)∗ α(m)k ∗ , yields its first order expansion.
In other respects, the complex logarithm is a C∞ -diffeomorphism of C into R×] − π, π[, thus the function ε7→ φ(m)k (ε) is C∞. A first order expansion yields, by using equation (51), lnα(m)k (ε)= lnα(m)k + ε∆α (m) k α(m)k + O ε 2
from which the first order expansion of the function ε 7→ φ(m)k (ε) can be derived.
II. PERFORMANCE OF THE ESTIMATORS A. First order performance
The results presented in section IV-A in [4] are proved below.
Proof of proposition 5: Since the signal ∆s(t) is
cen-tered, equation (44) yields E[∆zk] = 0, thus the estimator bzkis unbiased at the first order. Moreover, var (bzk) ∼ σ2E[|∆zk|2] (where we have defined σ = ε), from which expression (15) can be derived by using equation (44).
In other respects, since the signal ∆s(t) is centered, sub-stituting equation (12) into equation (11) shows that the estimators bδkand bfkare unbiased at the first order. Moreover,
var(bδk) ∼ σ2 M2 k E ℜ ukH∆s zkα(M k−1) k !2 . Substituting the identity (ℜ(z))2= 1
2(|z|2+ ℜ(z2)) into this last equation shows that var(bδk) is asymptotic to
σ2 2M2 k uk HE[∆s∆sH]u k zkα(M k−1) k 2 + ℜ uk HE[∆s∆sT]u k∗ zkα(M k−1) k 2 . However, since ∆s is a circular complex random vector, E[∆s∆sT] = 0N ×N. Therefore var(bδk) ∼ σ2 2M2 k e −2δk a(Mk−1) k 2ukHΓuk+ 0 , from which equation (16) is derived. Finally, equation (17) can be derived in a similar way, by using the identity (ℑ(z))2 = 1
2(|z|2− ℜ(z2)).
B. Asymptotic performance
The results presented in section IV-B in [4] are proved below. In order to keep the developments concise, it will be supposed that all poles are single, although the results remain valid if only the pole of interest is single. Before proving corollaries 7 and 8, we present a lemma used in both proofs.
Lemma 16. For all k∈ {0 . . . K − 1}, the coefficients of the
vector uk defined in equation (45) admit the second order
expansion2 uk(τ ) = 1(τ ≥n−1) zkt−l+τ nl − 1(τ ≤l−1) zt−l+τk nl + O 1 N3 . (64)
Proof:Let Vn be the n × r Vandermonde matrix
intro-duced in definition 5. The inverse of the matrix Z = VnHVn involved in corollary 11 admits the asymptotic expansion3
2The function 1
(.)is one if its argument is true and zero otherwise. 3More generally, in presence of multiple poles, the matrix Z−1 admits
the expansion Z−1= 1 nZ −1 0 + O 1 n2
, where the matrix Z−1 0 is block-diagonal. Z−1 = 1 nIr+ O 1 n2
. Consequently, the vectors introduced in this corollary verify
e↑ = [1, 0 . . . 0]T + O 1 n (65) e↓ = [0 . . . 0, 1]T + O 1 n (66) v′ ↑ = 1 nv↑+ O 1 n2 (67) v′ ↓ = 1 nv↓+ O 1 n2 . (68)
Then, substituting equations (67) and (68) into equa-tion (43), and noticing that the matrix Hk introduced in proposition 1 satisfies Hk= zkt−l+1α(0)k , yields
f↓k = zk−t+l−n+1 nl v l(z∗ k) + O 1 N3 (69) f↑k = z −t+l k nl v l(z∗ k) + O 1 N3 (70) where vl(z) = [1, z . . . zl−1]T. Finally, substituting equa-tions (65), (66), (69) and (70) into equation (45) yields expression (64).
Now let us prove corollaries 7 and 8.
Proof of corollary 7: Lemma 16 shows that
uH kuk = n22l+ O N14 if n ≥ l = nl22 + O N14 if n ≤ l
Equations (21) and (22) are obtained by substituting this result into equations (16) and (17), where Γ = IN (white noise hypothesis). The minimum variance under the constraint n + l= N + 1 is reached for n = 2l = 23(N + 1) or for l = 2n =
2
3(N + 1).
Proof of corollary 8:
First, let us simplify the expression of the matrix B defined in equation (53). Since all poles are supposed to be single, the matrices Bk introduced in equation (54) have dimension 2 × N and are equal to [0, 1]TuH
k. Besides, V
N is the N × K Vandermonde matrix, and VN is the corresponding N × 2K Pascal-Vandermonde matrix. Since the pseudo-inverse of VN satisfies VN †
= 1 NV
N H + O( 1
N2), it can be verified that
VN †VN B0 .. . BK−1 = N2JHUH+ O 1 N2 ,
where J is the Jordan matrix4introduced in section II-B in [4] (here the K × K matrix J is diagonal since all poles are supposed to be single), and U is the N × K matrix whose columns are the vectors uk. By substitution in equation (53), one obtains BH = 1 NV N H − N 2J H UH+ O 1 N2 .
4 TÉLÉCOM PARIS (ENST) - CNRS LTCI, TECHNICAL REPORT Therefore BHBis equal to 1 N2V N H VN+N 2 4 J H UHU J−ℜVN HU J+O 1 N2 . (71) Besides, VN HVN
= N Ir + O(1). Moreover, lemma 16 yields UHU = 2 max(n, l)2min(n, l)Ir+ O 1 N4 . Lastly, lemma 16 also shows that the coefficient of indices (k1, k2) in the matrix VN HU is VN HU (k1,k2) = vN(z k1) Hu k2 = z t−l k2 nl n+l−2P τ=max(n−1,l) zk2 zk1 τ − min(l−1,n−2)P τ=0 zk2 zk1 τ ! + O 1 N3 (where vN(z) = [1, z . . . zN −1]T). This last expression is equal to O 1 N3 if zk1 = zk2, or O 1 N2 if not. Therefore ℜVN HU J= O N12
. Finally, equation (71) yields BHB= 1 N + N2 2 max(n, l)2min(n, l) Ir+ O 1 N2 . Equations (23) and (24) are obtained by substituting this result into equations (19) and (20), where Γ = IN (white noise hypothesis). It can be noticed that the first order terms of the diagonal coefficients of BHB are equal, and that the minimum of their common value under the constraint n+ l = N + 1 is reached for n = 2l = 23(N + 1) or for l= 2n = 2
3(N + 1).
REFERENCES
[1] A.-J. Van der Veen, E. F. Deprettere, and A. L. Swindlehurst, “Subspace based signal analysis using singular value decomposition,” Proc. of IEEE, vol. 81, no. 9, pp. 1277–1308, Sept. 1993.
[2] R. Badeau, B. David, and G. Richard, “High resolution spectral analysis of mixtures of complex exponentials modulated by polynomials,” IEEE
Trans. Signal Processing, vol. 54, no. 4, pp. 1341–1350, Apr. 2006.
[3] R. Roy, A. Paulraj, and T. Kailath, “ESPRIT–A subspace rotation ap-proach to estimation of parameters of cisoids in noise,” IEEE Trans.
Acoust., Speech, Signal Processing, vol. 34, no. 5, pp. 1340–1342, Oct. 1986.
[4] R. Badeau, B. David, and G. Richard, “Performance of ESPRIT for Esti-mating Mixtures of Complex Exponentials Modulated by Polynomials,”
IEEE Trans. Signal Processing, vol. 56, no. 2, pp. 492–504, Feb. 2008. [5] R. A. Horn and C. R. Johnson, Matrix analysis. Cambridge University
Press, 1985.
Roland Badeau was born in Marseilles, France
on August 28, 1976. He received the State Engineer-ing Degree from the École Polytechnique, Palaiseau in 1999, the State Engineering Degree from the École Nationale Supérieure des Télécommunica-tions (ENST), Paris in 2001, the M.Sc. degree in applied mathematics from the École Normale Supérieure (ENS), Cachan in 2001, and the Ph.D degree from the ENST, Paris in 2005, in the field of signal processing.
In 2001, he joined the Department of Signal and Image Processing, GET-Télécom Paris (ENST), as an Assistant Professor, where he became Associate Professor in 2005. His research interests include high resolution methods, adaptive subspace algorithms, audio signal processing and music information retrieval.
Gaël Richard received the State Engineering
de-gree from the École Nationale Supérieure des Télé-communications (ENST), Paris, France, in 1990, the Ph.D degree from LIMSI-CNRS, University of Paris-XI, in 1994 in speech synthesis and the
Habilitation à Diriger des Recherchesdegree from the University of Paris XI in September 2001.
After his Ph.D, he spent two years at the CAIP Center, Rutgers University, Piscataway, NJ, in the speech processing group of Prof. J. Flanagan, where he explored innovative approaches for speech pro-duction. Between 1997 and 2001, he successively worked for Matra Nortel Communications, Bois d’Arcy, France, and for Philips Consumer Comunications, Montrouge, France. In particular, he was the project manager of several large-scale European projects in the field of audio and multimodal signal processing. In September 2001, he joined the Department of Signal and Image Processing, GET-Télécom Paris (ENST), where he is now full Professor in audio signal processing and Head of the Audio, Acoustics and Waves research group. He is co-author of over 70 papers and inventor in a number of patents, he is also one of the expert of the European commission in the field of audio signal processing and man/machine interfaces. Pr. Richard is a member of the EURASIP, senior member of the IEEE and Associate Editor of the IEEE Transactions on Audio, Speech and Language Processing.
Bertrand David was born on March 12, 1967
in Paris, France. He received the M.Sc. degree from the University of Paris-Sud, in 1991, and the Agrégation, a competitive French examination for the recruitment of teachers, in the field of applied physics, from the École Normale Supérieure (ENS), Cachan. He received the Ph.D. degree from the University of Paris 6 in 1999, in the fields of musical acoustics and signal processing of musical signals. He formerly taught in a graduate school in elec-trical engineering, computer science and communi-cation. He also carried out industrial projects aiming at embarking a low complexity sound synthesizer. Since September 2001, he has worked as an Associate Professor with the Signal and Image Processing Department, GET-Télécom Paris (ENST). His research interests include parametric methods for the analysis/synthesis of musical and mechanical signals, music information retrieval, and musical acoustics.
