Sparse Semi-Parametric Estimation of Harmonic Chirp Signals

J. SWARD,¨ J. BRYNOLFSSON, A. JAKOBSSON AND M. HANSSON-SANDSTEN

Published in: IEEE Transactions on Signal Processing doi:10.1109/TSP.2015.2404314

Lund 2015

Mathematical Statistics Centre for Mathematical Sciences Lund University 1 Sparse Semi-Parametric Estimation of Harmonic Chirp Signals Johan Sward¨ ∗, Johan Brynolfsson∗, Andreas Jakobsson∗, and Maria Hansson-Sandsten∗

Abstract—In this work, we present a method for estimating may then be refined using image processing techniques to the parameters detailing an unknown number of linear, possibly fit a linear chirp model [13]–[15]. The latter methods seem harmonically related, chirp signals, using an iterative sparse to render good estimates, although they typically require reconstruction framework. The proposed method is initiated by a re-weighted group-sparsity approach, followed by an it- rather large data sets to do so. The reassignment method will erative relaxation-based refining step, to allow for high resolu- yield perfect localization of the IF for each chirp component, tion estimates. Numerical simulations illustrate the achievable given enough noise-free observations. Regrettably, it is quite performance, offering a notable improvement as compared to sensitive to noise corruption [16]. Furthermore, in [17] a other recent approaches. The resulting estimates are found to be LASSO-based framework to estimate linear chirp signals was statistically efficient, achieving the corresponding Cramer-Rao´ lower bound. proposed that showed more robustness to noise as well as allowed for estimating an unknown number of unrelated linear Index Terms—Harmonic chirps, multi-component, Block spar- chirps. Also, some efforts have been made to use a compressed sity, LASSO, Cramer-Rao´ lower bound sensing approach [18], where a dictionary containing a small number of chirps is formed and the final estimates are found I.INTRODUCTION using an FFT-based algorithm. The size of the dictionary was limited to the signal length, thus impairing the estimation abilities. Also, the method did not allow for any modeling ANY forms of everyday signals, ranging from radar of additional signal structure. and biomedical signals to seismic measurements and M Often, the nonparametric methods have the advantage of human speech, may be well modeled as signals with instanta- computational efficiency, but generally also suffer from the neous frequencies (IF) that varies slowly over time [1]. Such poor resolution and high variance as is inherent to the spec- signals are often modeled as linear chirps, i.e., periodic signals trogram (see, e.g. [19]). The parametric methods on the other with an IF that changes linearly with time. Given the preva- hand often have good performance and resolution, but gener- lence of such signals, much effort has gone into formulating ally require a priori knowledge of the number of components efficient estimation algorithms of the start frequency and rate in the signal. Furthermore, it is not uncommon that one of development, and then, in particular, for signals only con- also needs to have good initial estimates to be able to use taining a single (complex-valued) chirp. One noteworthy such such methods; otherwise, the algorithm might suffer from method is the phase unwrapping algorithm presented by Djuric convergence problems. and Kay [2]; further development of this method can be found Many naturally occurring signals show a harmonic structure, in e.g. [3]. Other methods presented for single component i.e., a fundamental frequency with a number of overtones that estimation are, for example, based on Kalman filtering [4], [5], are integer multiples of the fundamental frequency. For such or sample covariance matrix estimates [6]. Similarly, in [7], the signals there are many proposed algorithms (see e.g. [20]– authors utilized the idea of single chirp modeling in detecting [22]). However, in many signals the signal structure suffers non-stationary phenomena in very noisy data. Recent work has from inharmonicities, such that the spectral components are to a larger extent focused on also identifying multi-component not exactly harmonic. Recently, two works have also examined chirp signals, such as the maximum likelihood technique extensions to the case of a single source harmonic chirp, presented in [8], the fractional Fourier transform method [9]– containing a set of harmonically related chirps signals [23], [11], and the multitapered synchrosqueezed transform [12]. [24]. These signals have lately started to attract interest due to Others have used some Fourier based time-frequency estimate, their ability to model non-stationary harmonical signals, such e.g, the Wigner-Ville distribution, the reassigned spectrogram, as many forms of audio signals [23]. In these works, both a or a Gabor dictionary, as a rough initial estimate, which nonlinear least squares (NLS) [23] and a maximum likelihood This work was supported in part by the Swedish Research Council, Knut solution [24] were examined. In this work, we extend upon and Alice Wallenberg’s and Carl Trygger’s foundations. This work has been and generalize the findings in [17], to account for an harmonic presented in part at the Asilomar Conference on Signals, Systems, and structure, where both the number of sources and the number Computers 2014. ∗Dept. of Mathematical Statistics, Lund Univer- of harmonic overtones for each source are unknowns, as well sity, P.O. Box 118, SE-221 00 Lund, Sweden, emails: as allow for the case when some of the harmonics are missing. {js,johanb,aj,sandsten}@maths.lth.se. The algorithm requires very few samples to get an accurate Copyright (c) 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be estimate of the parameters, which allows the method to also obtained from the IEEE by sending a request to [email protected]. model short segments of even highly non-linear chirp signals 2

0 as being piecewise linear over each of the segments, yielding consists of estimating K, Lk, fk , and rk, as well as, in the a quite accurate local signal representation. Furthermore, as process, also the phase, ϕk,` , ∠αk,`, and the magnitude, 0 long as the sampling times are known, the algorithm will ak,` , |αk,`|. Finally, we assume that min {`φk(t)} ≥ 0 and 0 also handle irregularly sampled data. Typically, most existing max {`φk(t)} ≤ Fs, ∀(k, `), where Fs denotes the sampling works rely on available a priori knowledge of the order of the frequency, in order to ensure that all frequencies in the signal models, although such details are in general unavailable, and are observable, i.e., fulfilling the Nyquist-Shannon sampling are notoriously hard to estimate [21]. Recently, some efforts theorem. on alleviating these assumptions have been made for purely harmonic signals [22], wherein a block-sparse framework is III.ALGORITHM utilized to form the estimates. The here presented work extends In order to form an efficient algorithm for estimating the on these efforts, also allowing for inharmonic sources, using unknown parameters in (1), one may rewrite (1) as the ideas introduced in [23]. We demonstrate the performance of the proposed method using both real and simulated data, y = D˜ a˜ + e (4) and compare the results with the corresponding Cramer-Rao´ lower bound (CRLB), which is also presented, as well as where T with competing algorithms. To improve on the computational y = y(t0) . . . y(tN−1) (5) complexity, we present an efficient implementation, utiliz- T a˜ = α . . . α . . . α (6) ing the alternating direction method of multipliers (ADMM) 1,1 1,L1 K,LK ˜ framework (see, e.g. [25]). D = d1,1 ... d1,L1 ... dK,LK (7) In this paper, scalars will be denoted with lower case T d = ei2π`φk(t0) . . . ei2π`φk(tN−1) (8) symbols, e.g. x, whereas vectors will be denoted with bold k,` lower case, x. Matrices will be denoted with bold upper case and where e is formed in the same manner as y. To allow letter, X. Furthermore, (·)T , (·)H , <, and = will be used to for an unknown number of components, we expand the signal denote the transpose, the conjugate transpose, the real part, representation in (4) into one formed using a large dictionary PK and the imaginary part, respectively. containing P k=1 Lk candidate chirps, such that The paper is structured as follows: In the next section, y ≈ Da (9) we introduce the signal model for harmonic chirp signals. Then, in section III, we derive the proposed algorithms and where D is an N ×P dictionary matrix, and a the correspond- present some heuristics for setting the user parameters. In sec- ing amplitudes, which thus mostly contains zeros, but with (at tion IV, we present efficient implementations of the algorithms, PK least) k=1 Lk non-zero elements. It is here assumed that P is whereas in section V, we illustrate the available performance selected sufficiently large so that the corresponding dictionary of the introduced methods using numerical results. Finally, in elements are close to the location of the true components and section VI, we conclude upon our work. also spans the the relevant parameter space, e.g. ranging from 0 to Fs for the starting frequency parameter (see also [26], II.SIGNAL MODEL [27] for a related discussion). Solving (9) using ordinary least Consider squares, if feasible, would yield a non-sparse solution, i.e.,

K Lk most of the indexes of a would be non-zero. Instead, we here X X i2π`φk(t) y(t) = αk,le + e(t), t = t0, . . . , tN−1 (1) impose the harmonic structure upon the solution by forcing it k=1 `=1 to choose between the different candidate chirp groups, while allowing for one or many of the overtones to be missing. To where K and L denote the unknown number of fundamental k impose this structure, we form the minimization chirps and the number of unknown harmonics for the kth component, respectively, whereas N denotes the number of Q 2 X available samples, t the sample times, which may be irregular, minimize ||y − Dx||2 + λ||x||1 + γ ||x[q]||2 (10) x αk the complex valued amplitude, φk(t) the time dependent q=1 frequency function, and e(t) an additive noise term, here where x[q] selects all elements in x corresponding to block q assumed to be white, circularly symmetric, and Gaussian in D, and Q denotes the number of blocks considered, where distribution. Furthermore, the chirp signal is assumed to be each block contains a fundamental chirp and its overtones, i.e., reasonable linear, at least under short time intervals, which for block q, x[q] denotes the elements of x that corresponds allows φk(t) to be modeled as to 0 rk 2 φk(t) = fk t + t (2) d ... d (11) 2 q,1 q,Lq yielding the IF function in the dictionary. The first term in (10) measures the distance between the signal and the model, the second term enforces φ0 (t) = f 0 + r t (3) k k k an overall sparsity between the available chirp candidates and 0 where fk and rk denote the starting frequency and the thus limits the number of chirps that may be part of the frequency rate, i.e., the frequency slope of the chirp, for solution. The third term in (10) acts as a sparsity enhancer chirp component k, respectively. The considered problem for the number of harmonically related chirp groups that are 3 allowed in the solution, thus promoting a solution that has Algorithm 1 The HSMUCHES algorithm fewer number of activated groups. Together, the two last terms 1: Initiate wp = 1, for p = 1,...,P , and vq = 1, in (10) promotes a solution that has few harmonically related for q = 1,...,Q chirp groups, and also allows for chirps within a group to be 2: for b = 1,... do sparse.This optimization problem is convex as it is a sum of 3: Solve (16) convex functions, and the solution may thus be found using 4: Update (14) and (15) standard interior-point methods, such as, e.g., SeDumi [28] 5: end for and SDPT3 [29]. Furthermore, γ and λ are tuning parameters 6: Compute (12) controlling the sparsity of the groups and the sparsity within 7: for j = 1,. . . do the groups, respectively. It is worth noting that if setting 8: for k = 1,..., Kˆ do (j) (j) γ = 0, one solves the problem of finding unrelated chirps 9: z = r + D(·, IKˆ (k)) x(IKˆ (k)) (j) (j) in the signal. Even though P is finely spaced, the quality 10: Using z, update D(·, IKˆ (k)) and x(IKˆ (k)) via of the solution obtained from (10) will depend on the grid NLS structure of D, i.e., if the true components are not contained 11: Subtract the refined estimates from z in the dictionary, the components that are the closest to the 12: end for true chirps will be activated, ensuring that the corresponding 13: end for indices in x will be non-zero. Therefore, the solution attained from (10) will be biased in accordance to the chosen grid structure of D. To avoid this bias, the estimation procedure updated as involves an additional step consisting of a nonlinear least (b) 1 squares (NLS) search to further increase the resolution. In wp = , p = 1,...,P (14) |x(b−1)| + order to do so, let the residual from (10) be formed as p 1 1/2 v(b) = , q = 1,...,Q (15) r = y − Dx (12) q (b−1) 2 ||x [q]||2 + Then, each harmonic chirp component may be iteratively where the superscript b denotes the iteration number, and > 0 updated by first adding one component to the residual formed is a small offset parameter, which prevents the solution from in (12), conducting a NLS search for the parameter estimates, diverging. At each iteration, one thus solves initiated using the estimates found from (10), and then remove Q 2 (b) X (b) the found component using (12). When all components have minimize ||y − Dx||2 + λ||W x||1 + γ vq ||x[q]||2 x been updated in this way, one may continue updating the q=1 residual with the newly refined estimates. The final estimates (16) are found by iterating the entire refinement procedure a few The resulting algorithm is outlined in Algorithm 1, where times. D(·, k) and x(k) denote the kth column and the kth index In the above algorithm, the user has to select a value for of the matrix D and the vector x, respectively. Furthermore, the parameters γ and λ. Of these, the value of γ penalizes the let Kˆ denote the number of non-zero elements in the solution number of harmonic chirps allowed in the solution, meanwhile from (10), and let the corresponding indices in x make up the the value of λ penalizes the overall number of chirps, thus index set IKˆ . Clearly, one must select an appropriate stopping allowing for sparsity within each harmonic chirp component. criteria for the second loop in Algorithm 1. This may be done The values of γ and λ are commonly chosen through cross- in various ways, such as when the parameter estimates does validation [30], or by some data dependent heuristics. In the no longer improve significantly in each iteration, or by setting case of γ = 0, we herein suggest selecting a maximum number of iterations. Empirically, we found that ||y||2 10 iterations where enough for convergence and through out λ = 2 (13) this work, we used this as stopping criteria. It should be noted 2N that the re-weighted approach introduces the tuning parameter which has empirically been shown to provide a reliable choice . In this paper, we have set to be of λ, for the here considered data lengths. When both tuning N parameters are active, the problem of setting good values = (17) ||y||2 becomes more complicated, since the two penalties interact. 2 We have empirically found that, as long as λ is reasonably which is in accordance with the discussion in [31], and which small, one may use (13) as a rule of thumb for also setting γ. has been empirically shown to yield reliable estimates. To further increase the robustness to the choice of γ and λ, It should be noted that if γ = 0, the estimator does and to further enhance the sparsity, we propose a re-weighted not assume any harmonic structure, and therefore constitutes approach, based on the technique introduced in [31]. In this solely a multi-chirp estimator; we term this the Sparse MUlti- approach, one solves the minimization iteratively, where, at component Chirp EStimator (SMUCHES). In the case γ > 0, every iteration, two weight matrices, W and V, with weights the estimator also allows for the possibility of harmonic chirp w1, . . . , wP and v1, . . . , vQ on the diagonals and zeros components; we term this the Harmonic Sparse MUlticompo- elsewhere, are used. The diagonal elements in W and V are nent Chirp EStimator (HSMUCHES). 4

IV. EFFICIENT IMPLEMENTATION 1 True components We proceed to examine efﬁcient implementations of the pro- 0.9 SMUCHES posed estimators using the ADMM framework. The discussion 0.8 here is focused on the HSMUCHES estimator, although the 0.7 implementation also works for the SMUCHES algorithm by 0.6 simply setting γ = 0. In general, an ADMM solves problems 0.5 in the form 0.4 Frequency (Hz) minimize f(x) + g(z) 0.3 x,z 0.2 subject to Ax + Bz = c (18) 0.1

2 0 In our case, A = I, B = −I, c = 0, f(x) = ||y − Dx||2, 0 2 4 6 8 10 12 14 16 18 PQ Time (s) and g(z) = λ||z||1 + γ q=1 ||z[q]||2, where I denotes the identity matrix of size N × P . The augmented Lagrangian for this minimization is formed as Fig. 1. The figure shows the true (solid) and the estimated (dashed) IF. ρ 2 Lρ(x, z, u) = f(x) + g(z) + ||x − z + u||2 (19) 2 algorithm requires a computational complexity of O(KNPˆ ). where u is the scaled dual variable, and ρ is the penalty parameter, penalizing the distance between z and x. The It may be noted that a dictionary similar to (7) was proposed ADMM finds the solution to (16) by iteratively solving (19) in [18]; in this case, the dictionary was restricted to only for each variable separately. The steps in the ADMM are contain N candidate chirps. As a result, the dictionary expe- rienced low correlation between the columns, for which case (k+1) ρ (k) (k) 2 x = argmin f(x) + ||x − z + u ||2 (20) the restricted isometry properties (RIP) will hold, suggesting x 2 that the signal may be recovered with high probability (see, (k+1) ρ (k+1) (k) 2 z = argmin g(z) + ||x − z + u ||2 (21) e.g., [36]). The same result would hold for the dictionary in z 2 (7), if restricted in the same manner. However, to allow for u(k+1) = x(k+1) − z(k+1) + u(k) (22) high resolution estimates, the dictionary should, as discussed, To find the solution to (20), one differentiates (19) with respect be extended to contain many more chirp candidates, indicating to x and put it equal to zero, yielding that the dictionary columns will be highly correlated, thereby no longer satisfying the RIP. Fortunately, as is also shown −1 x(k+1) = DH D + ρI DH y + ρ z(k) − u(k) in the next section, practical evidence indicate that even (23) highly correlated dictionaries enjoy excellent signal recovery properties. To solve (21), one needs to take some further care as g(z) is not differentiable at z = 0. However, it can be shown (see e.g. V. NUMERICAL RESULTS [32]) that the solution to (21) is In order to evaluate the performance of the proposed algo- (k+1) (k+1) (k) z = S S x + u , λ/ρ , γ/ρ (24) rithms, we examine their behavior on both real and simulated data, comparing them both to other alternative techniques, and where S and S are soft thresholds defined as to the CRLB (as derived in Appendix A). All the following xj S(x, κ) = max(|xj| − κ, 0) (25) root mean squared error (RMSE) curves are based on 1000 |xj| Monte Carlo simulations. x[q] Initially, we examine a simulated uniformly sampled signal S(x, κ) = max(||x[q]||2 − κ, 0) (26) ||x[q]||2 of length N = 20, consisting of two non-harmonic chirp for q = 1,...,Q, where S should be interpreted element- components, as depicted in Figure 1, which is corrupted by wise. Observing that f(x) and g(z) are closed, proper, and white circularly symmetric Gaussian noise with a signal to convex functions, and given ρ > 0, then, under some mild noise ratio (SNR) of 10 dB, which is here defined as assumptions, if there is a solution to (16), then the algorithm P SNR = 10log y (27) will converge to this solution [33], [34]. Also, the choice 10 σ2 of ρ will only effect the convergence rate, not whether or 2 not the method will converge. Using this implementation, the where Py denotes the power of the signal, and σ the variance computational complexity for SMUCHES is, for the Lasso of the additive noise. The resulting estimates from the pro- part, O(N 3 + N 2P ). The computations in this part are posed SMUCHES method and for the reassigned spectrogram dominated by (23), which only needs to be calculated once [16] are shown in Figures 1 and 2, respectively. As can be seen throughout the minimization. Furthermore, the computational in Figure 2, the reassigned spectrogram finds the two chirp complexity of the inverse is significantly decreased using the components, but the estimates are blurred, as well exhibiting Woodbury matrix identity [35]. The NLS part of the proposed jumps in the frequencies. On the other hand, as can be seen 5

0.9 True IF 0.8 SMUCHES

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Frequency (Hz) 0.3

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0 0 20 40 60 80 100 120 Time (s)

Fig. 2. The ﬁgure shows the estimated time-frequency distribution of the Fig. 3. The estimated chirp in dashed lines as compared to the true chirp. chirp signals using the reassigned spectrogram.

SMUCHES in Figure 1, the proposed method manages to ﬁnd the chirp 0 Djuric-Kay 10 DCFT components without any such ambiguities. CRLB We continue by showing how the proposed SMUCHES method may be used in tracking a non-linear chirp. In this 10-1 example, we simulated an exponential chirp component de-

ﬁned as RMSE 10-2 rt − 1 φ(t) = f (28) log(r) 0 10-3 where f0 and r are parameters determining the starting frequency and the exponential rate of change, respectively. The 5 10 15 20 25 signal, containing N = 105 samples, was divided in 7 equally SNR (dB) sized sections, such that each segment may be reasonably well Fig. 4. Performance of the proposed SMUCHES method, as compared with modeled as a linear chirp. The signal was corrupted by a white the Djuric-Kay method, the DCFT method, and the CRLB, when estimating circularly symmetric Gaussian noise with SNR = 20 dB. The the starting frequency of a single chirp. proposed algorithm was applied on each signal segment. The resulting chirp estimate is depicted in Figure 3, where it is clearly shown how the proposed method manages to estimate assert a fair comparison, the simulation where the proposed the evolving parameters of the non-linear chirp, showing that method estimated the wrong model order was removed from the local linear approximation is valid. all methods, and is thus not included in the RMSE graphs. Next, we examine the estimation performance of the As is clear from Figures 4 and 5, the SMUCHES method, SMUCHES method as a function of SNR. In this example, the without using any prior knowledge about the number of chirps, simulated signal contains only a single chirp component, with manages to attain the CRLB, as well as outperforming the starting frequency f 0 = 0.6/π, frequency rate r = 0.03/π, Djuric-Kay algorithm, even though the latter has been allowed amplitude α = 1, and a uniformly distributed random phase oracle model order information. Furthermore, it can be seen 1 1 ϕ ∈ U[− 2 , 2 ), which was randomized for each simulation. that the DCFT algorithm is stuck to its initial grid, which The sample length is set to N = 20. suggests why it does not manage to improve beyond a certain Figures 4 and 5 show the RMSE of the SMUCHES esti- limit when the SNR increases. Examining the computational mator, where λ and were selected using (13) and (17), as complexities, it was found that the Djuric-Kay and the DCFT well as the discrete chirp Fourier transform algorithm (DCFT) algorithms (given oracle model orders) are notably faster [9], the algorithm presented by Djuric and Kay in [2], both to compute than the presented SMUCHES implementation, being allowed oracle knowledge of the number of chirps in the requiring on average (computed over 1000 simulations on a signal, and the CRLB. It should be noted that the proposed regular PC, for SNR= 20 dB) 2.3 · 10−4, 5.1 · 10−3, and methods do not assume any model order information, as they 5.0 · 10−1 seconds to execute, respectively. are estimating this as part of the optimization; clearly, this also We proceed by examining the performance on multicompo- implies that the method may estimate the wrong model orders. nent chirp signals. Since the competing methods, which we However, the proposed SMUCHES method only estimated previously compared with, cannot be used on multicomponent the wrong number of components in 1 out of the 1000 data, we only show the results for the proposed method as simulations, and this at the SNR = 5 dB level. For the other compared to the corresponding CRLB. Figure 6 depicts the SNR levels, the order estimations were without any errors. To RMSE of the parameter estimations, as a function of SNR. 6

10-1 SMUCHES SMUCHES 0 Djuric-Kay CRLB 10 DCFT CRLB

10-1 10-2

-2 RMSE 10 RMSE

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10-4 10-4 5 10 15 20 25 5 10 15 20 25 SNR (dB) SNR (dB)

Fig. 5. Performance of the proposed SMUCHES method, as compared Fig. 6. Performance of the proposed SMUCHES method when estimating with the the Djuric-Kay method, the DCFT method, and the CRLB, when the starting frequencies (top curves) and the frequency rates (bottom curves) estimating the frequency rate of a single chirp. of two non-crossing linear chirps, as compared to the CRLB.

18 0 The starting frequency of the chirps were f1 = 0.6/π and 0 16 f2 = 1.2/π, and the slope rates were r1 = 0.03/π and r2 = 0.09/π. The amplitudes were set to unity and the phase 14 1 1 were drawn as ϕ ∈ U[− 2 , 2 ) at each simulation. Once again, 12

λ and were chosen using (13) and (17). As one can note from 10 Figure 6, the proposed method follows the CRLB for SNR levels greater or equal to 10 dB. In this case, the proposed 8 Sample times method estimated the wrong model order 26 times out of the 6 1000 simulations, all for the SNR = 5 dB case, and not at 4 all for higher SNRs. Again, these simulations were removed 2 from the proposed method’s RMSE, and the CRLB was 0 0 5 10 15 20 adjusted correspondingly. Next, we examine the performance Index on irregularly sampled data constituting of 20 observations from a chirp signal with the same chirp components as in the Fig. 7. The distribution of the sample times. previous example. The sampling times where drawn from a rectangular distribution in the range (0, 20] and are depicted 1 1 in Figure 7. The phase was drawn from U[− 2 , 2 ) for each overtones. Here, HSMUCHES estimated the wrong model simulation. Figure 8 shows the resulting RMSE results. As for order 54 times out of 1000 at SNR = 5 dB, 5 times out the earlier examples, for SNR greater than 5 dB, the proposed of 1000 at SNR = 10 dB, and made no mistakes at higher method attains the CRLB. The main difference to the uniform SNRs. sampled case is that the resulting RMSE for SNR = 5 dB As SMUCHES does not take the harmonicity inherent in is worse. Also, the number of times the proposed method the signal in account, there are 18 parameters (model order, estimated the wrong model order increased to 51 times out noise variance, and four parameters for each component) to of 1000, again, only for the SNR = 5 dB case. As before, for estimate using only 20 samples, whereas HSMUCHES only SNR greater than 5 dB, no errors in the model order estimation has to estimate twelve parameters (model order, number of were made. Though slightly more sensitive to non-uniformly overtones, starting frequency, frequency slope, phase, noise sampled data, it can be concluded that the proposed method variance, and four amplitudes). As a result, it can be expected is suitable to use also for non-uniformed sampled data. that SMUCHES will make more order estimation mistakes We proceed to examine the performance on simulated than HSMUCHES, which was also found to be the case. Out harmonic data. The simulated chirp signal consist of one fun- of the 1000 simulations, SMUCHES made 906 model order damental frequency and 3 overtones (K = 1 and L1 = 4), each errors at SNR=5 dB, 261 at SNR=10 dB, 21 at SNR=15 dB, with unit amplitude and uniformly distributed random phase. 8 at SNR = 20 dB, and 3 errors at SNR = 25 dB. The tuning The fundamental starting frequency was set to f 0 = 0.2 ∗ 3/π parameters for SMUCHES were selected using (13) and (17), and the frequency slope to r = −0.004 ∗ 3/π. The resulting and for HSMUCHES γ using (13), with λ = 0, and = 10−4. RMSE are shown in Figures 9 and 10, as a function of SNR, Finally, we show the performance on real data, containing when using N = 20 samples. The RMSEs for both the sounds from bats [37]. Many forms of audio sources, such starting frequency and the frequency slope, are measured as as voiced speech and many forms of music, may be well mean value of the RMSE for each of the four components in modelled as harmonic signals. Thus, it should be expected the signal, i.e., for the fundamental frequency and the two that the sound from a bat may contain a harmonic structure. 7

0 10-1 10 SMUCHES HSMUCHES CRLB SMUCHES −1 10 CRLB

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10-4 5 10 15 20 25 5 10 15 20 25 SNR (dB) SNR (dB)

Fig. 8. Performance of the proposed SMUCHES method when estimating Fig. 10. Performance of the proposed HSMUCHES methods applied to an the starting frequencies (top curves) frequency rates (bottom curves) of two harmonic chirp signal with one fundamental frequency and three overtones, non-crossing linear chirps for irregularly sampled data, as compared to the as compared with the SMUCHES method and the CRLB, when estimating CRLB. the frequency slopes.

HSMUCHES

0 SMUCHES 10 CRLB

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−2 RMSE 10

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−4 10 5 10 15 20 25 SNR (dB)

Fig. 9. Performance of the proposed HSMUCHES methods applied to an Fig. 11. The ﬁgure shows the spectrogram of the bat chirp. harmonic chirp signal with one fundamental frequency and three overtones, as compared with the SMUCHES method and the CRLB, when estimating the starting frequencies. non-linear chirps can be assumed to be reasonably linear. Numerical examples illustrate the preferable performance on The spectrogram of the bat signal is shown in Figure 11, both real and simulated signals. suggesting that the signal contains one fundamental chirp with, at most, two overtones. Figure 12 shows the estimated VII.ACKNOWLEDGEMENT harmonic structure when using HSMUCHES. Comparing the ﬁgures, it is clear that the HSMUCHES algorithm is well The author wishes to thank Curtis Condon, Ken White, and able to capture the changing frequencies in the harmonic Al Feng of the Beckman Institute of the University of Illinois signal, achieving a substantially better resolution than the for the bat data and for permission to use it in this paper. spectrogram. As before, the tuning parameters for SMUCHES were selected using (13) and (17), and for HSMUCHES, γ APPENDIX A was set to two times (13), λ = 0.01, and = 10−4. CRAMER´ -RAO LOWER BOUND The CRLB for a multi-component chirp signal has been VI.CONCLUSION derived in multiple papers, see e.g. [8]. Here, we derive the In this paper, we have proposed two semi-parametric algo- CRLB for the case of both regular frequencies and irregular rithms for estimating the parameters of an unknown number sampling, as well as harmonic overtones. The Fisher infor- of chirp and harmonic chirp components in noisy data, respec- mation matrix (FIM) for any signal observed under complex tively. The methods are shown to work well even for very short valued additive white noise, with variance σ2, can be set up signals, and allow for both uniform and non-uniform sampled block-wise as data. The methods are shown to attain the corresponding N−1 CRLB for both cases. Furthermore, it is shown in the paper 2 X ∂<{y(tn)}) ∂<{y(tn)}) ∂={y(tn)}) ∂={y(tn)}) Jij = + that the methods can be also used to approximate non-linear σ2 ∂θ ∂θ ∂θ ∂θ n=0 i j i j chirps, by dividing the data into small sections, in which the (29) 8

each pairwise Fisher information is found as 70 Lk Lj X X I(f , f 0) = α α 4π2`mt2 cos ∆Ψ (t ) 60 k j k,` j,m n k,`,j,m n `=1 m=1

Lk Lj 50 0 X X 2 3 I(fk , rj ) = αk,`αj,m2π `mtn cos ∆Ψk,`,j,m(tn) `=1 m=1

40 Lk 0 X 2 I(fk , ϕj,m) = αk,`αj,m4π `tn cos ∆Ψk,`,j,m(tn)

Frequency (Hz) 30 `=1 Lk 0 X 20 I(fk , αj,m) = −αk2π`tn sin ∆Ψk,`,j,m(tn) `=1

0.5 1 1.5 2 Lk Lj 0 X X 2 3 Time (s) I(rk, fj ) = αk,`αj,m2π `mtn cos ∆Ψk,`,j,m(tn) `=1 m=1

Fig. 12. The ﬁgure shows the estimated time-frequency content of the bat Lk Lj X X 2 4 signal using the proposed HSMUCHES algorithm. I(rk, rj ) = αk,`αj,mπ `mtn cos ∆Ψk,`,j,m(tn) `=1 m=1

Lk X 2 2 I(rk, ϕj,m) = αk,`αj,m2π `tn cos ∆Ψk,`,j,m(tn) `=1 where θ = [f 0 , r , ϕ , ... , ϕ , α , ... , α ]T , L k k k k,1 k,Lk k,1 k,Lk k Lk X 2 is the number of harmonics, and αk,` is the kth amplitude of I(rk, αj,m) = −αk,lπ`tn sin ∆Ψk,`,j,m(tn) the `th harmonic. Hence, the FIM will have (K × K) blocks, `=1 L such that Lk j 0 X X 2 I(ϕk, fj ) = αk,`αj,m4π mtn cos ∆Ψk,`,j,m(tn) `=1 m=1

Lk Lj   X X 2 2 J 11 J 12 ··· J 1K I(ϕk, rj ) = αk,`αj,m2π mtn cos ∆Ψk,j,`,m(tn) `=1 m=1 J 21 J 22 ··· J 2K   2 J =   (30) I(ϕk,`, ϕj,m) = αk,`αj,m4π cos ∆Ψk,`,j,m(tn)  .. .   ······ . .  Lk X J K1 J K2 ··· J KK I(ϕk, αj,m) = −αk,`2π sin ∆Ψk,`,j,m(tn) `=1

Lj 0 X I(αk,`, fj ) = αj,m2πmt sin ∆Ψk,`,j,m(tn) By denoting the Fisher information between the two parame- m=1 Lj ters u and v as X 2 I(αk,`, rj ) = αj,mπmtn sin ∆Ψk,`,j,m(tn) m=1 I(αk,`, ϕj,m) = αj,m2π sin ∆Ψk,`,j,m(tn) ∂<{y(t )} ∂<{y(t )} ∂={y(t )} ∂={y(t )} I(α , α ) = cos ∆Ψ I(u, v) n n + n n k,` j,m k,`,j,m , ∂u ∂v ∂u ∂v (31) Finally, the CRLB is found as the inverse of the FIM. each block in the FIM may be found as REFERENCES

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Maria Hansson-Sandsten (M’96) was born in Swe- den 1966. She received the MSc degree in Electrical Engineering in 1989 and a PhD degree in Signal Processing in 1996, both from Lund University in Sweden. Currently she is Professor in Mathematical Statistics at the Centre for Mathematical Sciences at Lund University. Her research interests include time- frequency analysis, multitaper spectrum analysis and feature extraction for detection, estimation and clas- siﬁcation of biological and biomedical signals.