LOCAL LEGENDRE POLYNOMIAL MODELING FOR BOTH AM AND FM OF A SINGLE COMPONENT

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1 LOCAL LEGENDRE POLYNOMIAL MODELING FOR BOTH AM AND FM OF A SINGLE COMPONENT Abstract François Léonard 1, Michelle Vieira, Nadine Martin, Meryem Jabloun 1 Institut de recherche d Hydro-Québec, 18 boul. Lionel-Boulet, Varennes (QC) Canada, J3X 1S1, leonard.francois@ireq.ca Laboratoire des Images et des Signaux, Site Campus, 961 rue de la Houille Blanche, BP 46, 384 Saint Martin d'hères, France,michelle.vieira@lis.inpg.fr, nadine.martin@lis.inpg.fr, meryem.jabloun@lis.inpg.fr Second-order Legendre polynomials are used locally for modeling amplitude modulation (AM) and frequency modulation (FM) over a short time segment. Since both modulations are modeled by three parameters, a total of seven parameters, including phase, are used to model a single sinusoidal component. This model allows quadratic AM and FM signal matching and yields accurate instantaneous amplitude (IA) and instantaneous frequency (IF) estimates. Monte Carlo simulations show estimation results close to the Cramer-Rao bound. On a measured multi-component signal, the estimation of one component can be followed by its subtraction from the signal, leaving residual noise under the removed component trajectory. Keywords: Legendre polynomial modeling, time-frequency, parametric estimation, simulated annealing. I. INTRODUCTION Most signals encountered in engineering applications are inherently non-stationary. These signals may have a single component with time-varying frequency and/or amplitude or they may be a sum of such signal, whose time location and duration are unknown. These signals [1] cover a wide range, including speech [-5], music [6,7], seismic [8,9], radar [1,11], sonar [1], medical [13,14] and mechanical [15-17]. The introduction of instantaneous frequency several decades ago [18,19] was followed by numerous well-established non-parametric and parametric methods for the analysis of these signals. Some early nonparametric methods assumed stationary signals within short time intervals such as the Short Time Fourier Transform (STFT) []. Current approaches to analyzing multi-component AM- FM signals use mainly nonparametric time-frequency distributions (TFD), Cohen's class distributions, for example [1]. Some of these techniques are based on the direct definition of the discrete IF and make no implicit assumptions about the signal, for this reason; they therefore exhibit a high variance. In order to improve the tracking ability of TFDs when the IF is not a linear function of time, nonparametric methods such as the matched spectrogram [], polynomial classes of TFDs [3] and local polynomial time-frequency transform [4,5] have been developed. However, the two wellknown drawbacks of these approaches are the existence of a time-frequency resolution limitation and cross-terms in the TF plane. Unlike nonparametric methods, parametric approaches are not resolution-limited [6]. In the last fifteen years, many parametric approaches have been deployed to track multi-component AM/FM [17,7-9]. Recently, Francos and Porat [3] introduced an algorithm using a second-order polynomial phase model for describing each frequency component. In the latter algorithm, a leastsquares fit using the Minimum Cross Entropy TFD (MCE-TFD) is followed by an improved estimation of the phase parameter from the dechirped signal. The proposed model by Francos and Porat appears well adapted to linear chirps. However, many measured signals of physical processes are far from being a simple linear chirp. There is therefore a clear need for a model designed to fit quadratic chirp, since a continuous FM signal can be modeled by successive quadratic FM segments with no discontinuity of the IF. In light of this, in our work, we propose a three-step process: 1. The continuous signal is segmented into short time signals for further local estimation;. In the local estimation, each segment is modeled by a second-order Legendre polynomial approximation of the instantaneous amplitude and the instantaneous frequency functions; 3. Finally, the contiguous modeled segments are merged, yielding the estimated continuous component. 1

2 For the second step, a first-order modeling of the IF and IA cannot be used since the merge of linear chirps yields unacceptable discontinuities at segment joints over the derivative of IF and IA functions. Note that the three parameters for the FM trajectories plus the phase parameter is equivalent to a third-order polynomial phase model. This paper focuses on the second-order Legendre polynomial modeling of both IF and IA discrete functions. Steps 1 and 3 will be presented in detail in a future paper. II. PROBLEM FORMULATION AND SIGNAL MODELING II-A Signal description Consider the general case of a discrete-time, real-value signal yn ( ) consisting of the sum of superimposed deterministic AM-FM components and additive noise en ( ) such that: K yn ( ) = sk ( n) + en ( ), n N -1 (1) k = 1 where each of the components is of general form: sk( n) = Ak( n) cos ( Φ k( n) ), nstartk < n < nendk () with: K is an unknown number of components of the observed signal. Ak ( n) and Φ k ( ) are respectively the sampled values of the unknown time-varying instantaneous amplitude and phase of the k th component. The unknown time instants, nstart and nend define the time location and duration ( nend - k nstart k +1) of the k th component. This birth and death concept was first introduced by McAulay and Quatiery []. en ( ) corresponds to the difference between the observed signal and the sum of the estimated AM-FM components. In this work, en ( ) is assumed to be a real white Gaussian noise process with zero mean and variance σ. Moreover, we assume that instantaneous amplitude and instantaneous frequency (IF) are continuous differentiable functions such that: The IA is narrowband relative to the sampling frequency. The IF is positive and does not exceed the half Nyquist rate such that IF<.5. All components are well separated in the time-frequency plane [31] showing no sideband overlapping. If the different components are well separated in the time-frequency plane, adequate numerical processing can be used to discriminate each component, as will be demonstrated in section IV. For a single component, the measurement consists of N noisy observations y( n) = s( n) + e( n), with s ( n) = A( n) cos( Φ ( n) ) for n N 1. The goal is to estimate the latter discrete-time amplitude A( n ) and phase Φ ( n). For this purpose, we propose a local modeling of the segmented signal with a parametric polynomial function for both AM and FM. II-B Local Legendre polynomial approximation Locally, consider the m th segment of a single AM-FM component sm( l) = s( l+ nm), with l [, L 1], starting at the sample n m and its corresponding measurement ym( l) = y( l+ nm). For the sake of simplicity, we will focus on the estimation of one segment and drop the m indices: sm () l becomes s(l), and y m (l) becomes y(l). We will describe the segment-merging process in a later paper. k k The signal segment s(l) is estimated by the numerical generated signal segment l L/ g () l = A() l cos θ+ π F() i π F() i i= i= where: A() l is the instantaneous amplitude, F() i is the instantaneous frequency, (3)

3 θ is the phase of the AM-FM component referenced to the center of the observation window at l = L/. From the Cramer-Rao bound (CRB) for a continuous tone, var{} θ var{ πf} ( T( nr n )) + var{} θ is f the variance of the estimated phase when the frequency f is unknown [3], where: var πf is the variance of the estimated frequency, { } T is the sampling period with f = 1 T, R n is the reference time sample for the phase, n is the center of the observation window, var{ } s θ is the variance of the estimated phase when the frequency f is known. f The phase variance, expressed as a function of the reference n R, appears minimum for nr = n. In fact, referencing the phase to the window center minimizes the contribution of the frequency estimation error. Significant reductions in the variance of estimation can be achieved by incorporating a priori knowledge into the estimation procedure. In counterpart, this incorporated knowledge reduces the solution domain. One means of doing this is to assume that the IF and IA models may be expressed as finite-order polynomials. Such modeling is allowed by the Weierstraβ approximation theorem, which formulates that any continuous function on a closed and bounded interval can be uniformly approximated by polynomials to any degree of accuracy. The choice of the polynomial order is fixed by the a priori information expected in the signal segment: if the signal's IF (or IA) is known to be slowly changing in the segment, low orders can be chosen, whereas high orders can be appropriate if the model is known to be changing rapidly. When we can choose the segment length, a compromise appears: increasing the segment length enhances the frequency resolution, giving more rejection between components, whereas decreasing the segment length allows a fair estimation by a lower-order model. Higher-order modeling shows problems with the iteration convergence, affecting the stability of the results and imposing an unreasonable computation time. Moreover, a higher-order model facilitates noise-fitting, introducing additional noise in the estimation. In our experience, the second order is a good compromise between frequency resolution and computational complexity. In a second-order classical modeling, the IA and IF are given by: A(l) = α j l j and F(l) = υ j l j (4) j= j= where α j and υ j are respectively the real polynomial amplitude coefficients and real polynomial frequency coefficients of the AM-FM component. In this particular modeling, the corresponding space representations for α j and υ j are not orthogonal. IA and IF must be remodeled in order to decouple the parameters as much as possible. This is why we use one of the better known sets of orthonormal basis functions for a prescribed interval: the normalized Legendre polynomials. Figure 1: Decomposition of FM (AM is modeled similarly).. The new formulation of the IA and IF using discrete-time Legendre polynomials is Al () = ap() l and Fl () = fp() l (5a) j= for the interval l [, L 1] { a, a1, a, f, f1, f } and ( ) 1. j j j= j j with the parameters set P l = (5b) L 1 P1 () l = l L 1 (5c) 1 1 L 1 P () l = 1 l 1. ( L 1)( L+ 1) (5d) This formulation decomposes both the IA and the IF modulations in the sum of three curves centered in the middle of the window, like the example presented in Fig. 1. 3

4 The local estimation problem addressed in this paper can now be posed as follows: for L noisy observations y() l consisting of the sum of the a single AM-FM component s( l ) and the additive white Gaussian noise el ( ), the goal is to estimate the parameter set {, a, a, a, f, f, f } θ. 1 1 III. PARAMETER ESTIMATION III-A Least-squares estimator Guided by the constraints on IF and IA, the nonlinear least-squares (LS) estimator finds the θ, a, a, a, f, f, f which minimizes set { } LS 1 1 L / [ yl () gl ()] l = (6) l= L/ where g() l is defined by Eq. 3 with A() l and F() i given by Eq. 5. This LS estimator is originally expressed in the R 7 domain for the seven elements of the parameter set. There is no closed-form solution. Consequently, we are compelled to solve the least squares numerically. Iterative search is a commonly used procedure here. III-B Reduction of the dimensionality of the solution domain In an iterative process, the computational complexity is directly related to the dimension of the solution domain. Reducing the search range requires strong confidence in the initial conditions or a prior knowledge of the solution range. Instead of targeting a sub-set of the solution domain, it seems preferable to reduce the dimensionality of the domain. In order to eliminate the search for the parameter a, we replace the LS minimization (Eq. 6) with the maximization of the correlation coefficient γ y,h = l y(l) h(l) ( l y (l)) l h (l) ( ) (7) with a ' arbitrary fixed at 1. for the function hl (). This maximization over six parameters yields the parameter set { θ, a1, a, f, f1, f}. The right amplitude values are obtained by the renormalization yl () hl () l ai = a i, i=,1,. (8) h () l ( ) l A correlation followed by a renormalization gives the same results as the LS minimization. By doing an optimization over a R 6 domain instead R 7, the correlation algorithm is more robust and reduces the processing time by a factor of or more. a) f parameter b) f parameter f parameter f parameter Figure : Typical correlation patterns (1 samples, θ=., a=1., a1=.5, a=.4, f=.156 fs, f1=. fs, f=.87 fs) III-C Iterative process and simulated annealing (SA) The mean frequency f and the phase value θ estimated by a FFT fix the initial conditions of the iterative process. These initial conditions { θ, a1, a, f, f1, f} = { θ FFT,,, ffft,,} define the start launch coordinates of the iteration process in the R 6 solution domain. The correlation domain shows many local maximums. Figure shows the correlation pattern s l for a typical quadratic FM signal. Figure a illustrates a D plane cutting the between y( l ) and ( ) 4

5 solution domain with fixed θ and f 1 values. Figure b illustrates a plane defined by fixed θ and f values. These cuts are perpendicular and pass through the true summit. Numerous secondary summits are present, many of which resemble a long hyper-tunnel waving through the solution domain. Moreover, this complex pattern varies a lot with the other parameter values. Here, a huge number of iterations are required to systematically sample the solution domain and find the true summit. The chosen iteration strategy is based on a global stochastic optimization technique called simulated annealing (SA). At each iteration, a set of six random values is generated, like a thermal agitation, and added to the previous parameter set { θ, a1, a, f, f1, f}. This new generated set is then tested and used to replace the previous set if the correlation increases. In SA, the temperature is gradually reduced until it reaches a stable solution. The goals in SA setting are: To choose the best distribution law for parameter agitation; To find the appropriate cooling function (dispersion reduction of the latter distribution). ( 1-K ) K The three distribution laws, namely uniform, Gaussian and the K,6, each in x family for { } turn offering greater efficiency for parameter agitation, were tested. The cooling function has a search range starting at ± 5% of the full parameter ranges and follows a ( ii ) diminishing law with the i th iteration progression where the parameter I is the minimal number of iterations. When the generated sinusoid jumps to a new parameter set, the i count is re-evaluated then i = max ( I J sqrt( I), i), where J corresponds to the counted number of jumps. The maximum number of iterations is.5 I( sqrt( I) 1). These functions fix the rate of convergence and the probability of a false summit occurring. III-D Fine-search algorithm At the end of the SA, the true summit is reached with confidence. The iteration gives a result somewhere under the summit lobe, not far from the summit itself. Since the path from this point to the summit is monotonic, deterministic step algorithms can be used effectively to reach the summit. At the end of the iteration process, we added the following fine-search algorithm. This algorithm has two states: a walking state and a search range state. In the walking state, a step is tried in each parameter dimension. When no step occurs, the search range is reduced. The number of iterations is not deterministic since the walking process may have numerous steps when the ridge is narrow. MSE Fine Search No Fine Search SNR (db) Figure 3: Monte Carlo simulation results, MSE with and without fine-search algorithm, 3 samples, realizations of random modulated signals, I=. The fine search dramatically reduces the number of iterations required for high SNR. The results illustrated in Fig. 3 show a resolution limited by a fixed number of iterations and the improvement provided by the fine-search algorithm. For the 4 db SNR case, statistically over realizations, 14,76 iterations were done in the simulated annealing plus,85 iterations for the fine-search algorithm. Without fine search, 15,3 iterations are required to reach the same standard deviation (STD). IV. NUMERICAL AND REAL MEASUREMENT EXAMPLES IV-A Modulation effect on estimation accuracy Figure 4 shows the influence of the AM-FM modulations on the accuracy of the segment estimation. This accuracy is given by ( s() l g() l ), the mean square error (MSE) between the signal and its estimate. For the maximum modulation results, the frequency is randomly chosen near 5

6 .5 f s and the modulation is forced to 95% by fixing the values a 1 + a and f1 + f close to the negative modulation limits. Since the Monte Carlo simulation results appear to be a function of the sampling distribution of the signal domain, we must sample the widest domain allowed by the respect of non-negative modulation conditions uniformly. Consequently, for the Random Modulation case (Fig. 4), the parameter values are randomly selected with a uniform density probability distribution. Figure 5 illustrates some signal examples for random parameter selection before the addition of white noise. The no modulation case corresponds to a continuous tone and amplitude sinusoid. In these π, π. The Cramer-Rao bound simulations, the phase parameter is also randomly selected over ] ] CRB L ( L L 1 ) ( ) ( ) ( ) = σ + +, illustrated in Fig. 4, corresponds to an un-modulated sinusoid []. In conclusion, the resulting MSEs are located 3.6 db over the Cramer Rao bound and show low sensitivity to modulation. MSE Maximum modulation Random modulation No modulation CRB 1/ SNR (db) Figure 4: Monte Carlo simulation results for 3 samples, realizations of AM-FM random modulated signals, I= with fine search Time (samples number) Figure 5: Typical realizations of AM-FM random modulated signals. IV-B MSE increases with model order Figure 6 illustrates the effect of introducing a new parameter in the model, showing the results for the 3-parameter set{, a, f } θ, a, a, f, f and the 7-parameter set {, a, a, a, f, f, f } θ. 1 1 θ, the 5-parameter set { } 1 1 MSE parameters 5 parameters 3 parameters CRB 1/ Time (Sample number) MSE {theta,a,a1,f,f1,f} {theta,a,a1,f,f1} {theta,a,f,f1} {theta,a,a1,a,f,f1} {theta,a,a1,f} CRB 1/ Number of parameters Figure 6: Monte Carlo analysis results, CRB and estimated standard deviation for different model orders for a constant amplitude and tone signal, SNR=1 db, 4 realizations. Figure 7: Monte Carlo analysis results, mean estimated standard deviation versus the number of model parameters for a constant amplitude and tone signal, SNR=1 db, 4 realizations. 6

7 When a parameter is introduced, the generated sinusoid is better able to fit the signal as well as the noise. Therefore, for a constant amplitude and tone sinusoid, the standard deviation increases with the model order. Roughly, we observe the relation σ εtotal ( number of parameters) (9) N for the variance of s( n) g( n). The latter equation is illustrated by the dashed curve plotted in Fig. 7. Indeed, models able to fit quadratic chirp show a higher variance than a lower-order model which fits linear chirp or constant frequency tone. IV-C Required number of iterations The number of iterations is a function of the required precision of the estimate and the density of false summits. With the fine search algorithm disabled, for a given SNR, adding iterations to the SA process increases the probability of finding the right summit and reduces the average distance to the top summit. Figure 8 illustrates the MSE for different SNRs as a in function of the iteration parameter I. MSE (relative) db db 3 db I parameter Figure 8: Monte Carlo analysis results, estimated standard deviation for different SNRs as a function of the I parameter, 3 samples, realizations. Wrong solution count I=1 I= I=4 I= Number of time samples L Figure 9: Number of false summits reached for 1 realizations as a function of the number of samples, db SNR. With the fine search enabled, the precision is jeopardized mainly by the occurrence of false summits in the output of the simulated annealing algorithm. Indeed, the probability of stopping on a false summit increases with the density of false summits. Therefore, the probability density of false summit increases with the number of time samples. For a fixed I parameter and the case when the right summit is selected, the MSE diminishes with an increasing number of samples L, following the behavior of the CRB defined in section IV-A. However, in the general case, the MSE decreases with L until it reaches a limit where the probability of false summits occurring shows a sudden increase like those illustrated in Fig. 9. IV-D Identification of a component in a simulated multi-component signal For a multi-component signal, identification starts with the largest amplitude component or a userselected component. A component is then estimated and subtracted successively from the signal. We call extraction the process of estimating followed by a subtraction. Figure 1 shows the extraction of a linear chirp from a multi-component signal. Here, the signal generated ( ) yn ( ) =.6963 n n e j(.1n.3 n ) π ( ) j( π.37n. n ) + 1 cos(1 π n/18) e + e( n), (1) for [ ],17 n, nearly matches example 5 proposed by Francos and Porat in [3] with an average of 15 db SNR. For the first component extracted, the accuracy of the estimation is enhanced compared to results obtained by these authors [3, Fig. 8] as illustrated in Fig

8 Frequency (f/fs) Frequency (f/fs) Z axis linear scale Time (samples number) Figure 1: Extraction of a chirp into from a multicomponent signal. Top: noisy signal. Bottom: noisy signal with extracted chirp. Instantaneous Amplitude Generated IA Estimated IA Time (sample number) Generated IF Estimated IF Figure 11: IF and IA of the extracted component illustrated on the left. Instantaneous Frequency IV-E Identification of a component in a measured multi-component signal Our vibro-mechanical measurement database does not contain fast modulated signals corresponding to the signal description in section II-A. On the other hand, slow modulated signals show distinct components but do not constitute good examples to challenge the algorithm. Finally, to demonstrate component extraction on real measurements, we chose a canary chirp exhibiting a fast amplitude and frequency modulation with very low levels of distortion and noise. Originally sampled at 44.1 ks/s, this sound track was decimated to 14.7 ks/s. For the selected segment on the sound track and the selected component for estimation, the IF and IA modulations are respectively 7% and 53% for a 15- cycle signal length: a drop from 16 Hz to 8 Hz in less than.1 s. Figure 1 illustrates the time trace of the signal, the estimated component and the difference. The IA and IF of the estimated component are seen in Fig. 13, which shows clearly that the IA and IF modulations are significantly quadratic. 15 Signal Estimated component Signal-Est. component Signal Sample number Figure 1: Estimation of one sinusoidal component over a -sample segment of a canary chirp containing many components. IA IA IF Sample number IF (Hz) Figure 13: IF and IA of the estimated component illustrated in Fig. 1. A better appreciation of the estimation accuracy is given by the comparison of the signal spectrogram before and after extraction of the estimated component. Note the vertical flumes on each extremity of the extracted zone. These flumes correspond to the discontinuities appearing at the transition between the signal and the residual signal after extraction of the estimated component. 8

9 Time (s) Signal-Component Time (s) Figure 14: Spectrograms of original canary chirp and of the chirp with the estimated component removed. Under the extracted component in Fig. 14 (right), the residual spectral power has similar amplitude to that of the surrounding noise. At this location, the second harmonic component is more easily distinguished after extraction of the first harmonic. Here, extraction of the second harmonic will be facilitated by the removal of the first harmonic component. In this example, the algorithm extracts only one component from one segment. It is also possible to estimate the same component over many segments, merge estimations and extract the component over these segments. The reader can find other examples and some applications for measured data in [33] for an old version of the algorithm. 1.5 V. CONCLUSION The Legendre polynomial provides an efficient third order phase / second order amplitude local model for an AM-FM modulated component. Replacement of the least-squares minimization by a correlation maximization reduces the dimensionality of the solution domain and increases computational performance. The solution domain contains numerous summits with narrow valleys and crests. Simulated annealing locates the right summit with confidence when the appropriate number of iterations is fixed. This number increases with N, the number of time samples in the analyzed segment. The proposed fine-search algorithm climbs efficiently to the summit, reducing the computational expense. Monte Carlo simulations show that the MSEs are 3.5 db higher than the CRB for to 45 db SNR and have a low sensitivity to extreme modulation. Moreover, these simulations demonstrate that the MSE increases linearly with the number of model parameters, starting at the CRB level for an unmodulated sinusoid modeling (three model parameters). Instead of the usual algorithms, the one proposed is not limited to linear chirps but also be used for quadratic AM-FM modulated multicomponent signals. VI. REFERENCES [1] Proc. IEEE, Special Issue on Time-Frequency Analysis, Vol. 84, September [] R.J. McAulay and T.F. Quatieri, Speech Analysis/Synthesis Based on a Sinusoidal Representation, IEEE Transactions on Acoustics, Speech, and Signal processing, Vol. 34, No. 4, Aug [3] T.F. Quatieri and R.J. McAulay, Peak-to-RMS Reduction of Speech Based on a Sinusoidal Model, IEEE Transactions on Signal Processing, Vol. 39, No., Feb [4] P. Maragos, J. Kaiser and T. Quatieri, Energy Separation in Signal Modulation with Application to Speech Analysis, IEEE Transactions on Signal Processing, Vol. 41, No. 1, Oct [5] S. Lu and P. Doerschuk, Nonlinear Modeling and Processing of Speech Based on Sums of AM-FM Formant Models, IEEE Transactions on Signal Processing, Vol. 44, No. 4, April [6] R. Gribonval, E. Bacry, S. Mallat, Ph. Depalle and X. Rodet, Analysis of Sound Signals with High Resolution Matching Pursuit, Proceedings of Third Int. Symposium on Time-Frequency and Time-Scale Analysis, pp , June [7] X. Rodet, Musical Sound Signal Analysis/Synthesis: Sinusoidal+Residual and Elementary Waveform Models, IEEE Time-Frequency and Time-Scale Workshop 97, August [8] D. Maiwald, J.W. Dalle Molle, J.F. Böhme, Model Identification and Validation of Nonstationary Seismic Signal, IEEE Signal Processing Workshop on High-Order Statistics, pp , June

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