Psi_B energy operator is an extension of the cross Teager-Kaiser energy operator which is an non-linear energy tracking operator to deal with complex signals and its usefulness for non-stationary signals analysis has been demonstrated. In this letter two new properties of Psi_B are established. The first property is the link between Psi_B and the dynamic signal which is a generalization of the Instantaneous Frequency (IF). The second property obtained for frequency modulated signals is a simple way to estimate the IF. These properties confirm the interest of Psi_B operator to track the non-stationary of a signal. Results of IF estimation in noisy environment of a non-linear FM signal are presented and comparison to Wigner-Ville distribution and Hilbert transform-based method is provided

BOUDRAA, Abdel-Ouahab

Electronics Letters

English

Name not available

Estimation de la fréquence instantanée  des signaux FM par opérateur d'énergie Psi_B

SAM;  the Arts et Métiers ParisTech open access repository

SAM : Science Arts et Métiers

Science Arts & Métiers (SAM)is an open access repository that collects the work of Arts et Métiers ParisTechresearchers and makes it freely available over the web where possible.This is an author-deposited version published in: http://sam.ensam.euHandle ID: .http://hdl.handle.net/10985/8859To cite this version :Abdel-Ouahab BOUDRAA - Instantaneous frequency estimation of FM signals by Psi_B-energyoperator - Electronics Letters - Vol. 47, n°10, p.623-624 - 2011Any correspondence concerning this service should be sent to the repositoryAdministrator : archiveouverte@ensam.euInstantaneous frequency estimation of FM signalsby ΨB-Energy OperatorAbdel-Ouahab BoudraaEcole Navale, IRENav, BCRM Brest CC 600,29240 BREST Cedex 9, France.boudra@ecole.navale.frAbstractΨB energy operator is an extension of the cross Teager-Kaiser energy opera-tor which is an non-linear energy tracking operator to deal with complex signalsand its usefulness for non-stationary signals analysis has been demonstrated.In this letter two new properties of ΨB are established. The first property is thelink between ΨB and the dynamic signal which is a generalization of the Instan-taneous Frequency (IF). The second property obtained for frequency modulatedsignals is a simple way to estimate the IF. These properties confirm the interestof ΨB operator to track the non-stationary of a signal. Results of IF estimationin noisy environment of a non-linear FM signal are presented and comparisonto Wigner-Ville distribution and Hilbert transform-based method is provided.1 IntroductionΨB-energy operator is an extension of the cross Teager-Kaiser operator which is anon-linear tracking energy operator [1] to deal with complex signals [2]. Based onthe Lie bracket, ΨB is well suited to measure the instantaneous interaction betweentwo complex signals [3]. The output of ΨB is related to the instantaneous CrossCorrelation (CC) of complex signals x(t) and y(t), Rxy(t, τ), as follows [2]:ΨB(x(t), y(t)) = −∂2Rxy(t, τ)∂τ 2∣∣∣∣τ=0−∂2R∗xy(t, τ)∂τ 2∣∣∣∣τ=0(1)Relation (1) shows that ΨB is a cross-energy function of two signals and thus links totransforms using the concept of instantaneous CC, such as Time-Frequency Represen-tations (TFRs) can be found. In fact, relationships between ΨB operator and some12TFRs such as cross ambiguity function or cross-Wigner-Ville Distribution (WVD)show that ΨB is well suited to study non-stationary signals [2]-[4]. ΨB has found ap-plications in both signal and image processing such as time series analysis, gene timeseries expression data clustering, transient detection or time delay estimation [5]-[8].ΨB operator is a symmetric bilinear form defined as follows [2]:ΨB(x, y) = 0.5[ΨC(x, y) + ΨC(y, x)] (2)where the associated quadratic form ΨC(x, x) = ΨB(x, x) is given byΨC(x, y) = 0.5[x˙∗y˙ + x˙y˙∗]− 0.5[xy¨∗ + x∗y¨] (3)In this letter new properties of ΨB are established. We show the link between ΨBand the dynamic signal which describes the rate of the log-magnitude and phase [9].The second relationship is a simple way to estimate the IF of a Frequency Modulated(FM) signal. Thus, it is natural to use ΨB a local operator, based of the signal timederivatives, as a basis for tracking instantaneous features such as IF function. In thefollowing, for x(t) = y(t) the notation ΨB(x(t), y(t)) ≡ ΨB(x(t)) is used.2 Dynamic signal and ΨB operatorThe dynamic signal is similar to the definition of complex cepstrum in the frequencydomain and can be viewed as a generalization of the IF of a signal with varyingmagnitude [9]. Consider an AM-FM signalx(t) = a(t)ejφ(t) (4)a(t) is instantaneous amplitude and φ(t) instantaneous phase. The IF is given byfi(t) =12πφ˙(t) (5)The dynamic signal is defined by [9]β(t) =ddtlog x(t) =a˙(t)a(t)+ jφ˙(t) (6)and the instantaneous bandwidth is given byib(t) =∣∣∣∣ a˙(t)a(t)∣∣∣∣ (7)If one substitutes x(t) (Eq. 4) in equation (2), then one obtainsΨB(x(t)) = −| x(t) |22[(β∗(t)− β(t))2 + (β˙∗(t) + β˙(t))](8)3which shows that ΨB can be written in terms of dynamic signal. Furthermore, equa-tion (8) can also be written asΨB(x(t)) =| x(t) |2[8π2f 2i (t) + ib2(t)− a¨(t)a(t)](9)where there is relationship between ΨB, the IF and the instantaneous bandwidth.Finally, links given by relations (8) and (9) show that ΨB conveys precious localinformation about the instantaneous behavior, over the time, of a signal.For a FM signal (a(t) = A), using equation (9) the IF is given byfi(t) =√ΨB(x(t))2πA√2(10)where A is a constant. Relation (10) is a simple way to estimate the IF of a FMsignal. This IF estimation is obtained without involving integral transform and withno a priori knowledge about the phase φ(t) of the signal. An advantage of ΨB operatoris its localization property. This localization is the consequence of the differentiationbased method.3 ResultsConsider a noisy Polynomial Phase Signal (PPS):y(t) = z(t) + n(t) = Aejφ(t) + n(t)= Aej∑Pk=0 aktk+ n(t), t ∈ [0, T ] (11)where z(t) is a noise-free signal, n(t) is a complex white Gaussian noise, φ(t) issignal phase, P is order of polynomial and ak are corresponding polynomial coef-ficients. T is signal duration. A fourth-order PPS, with parameters A = 2 and(a0 = 1, a1 = 10, a2 = 10, a3 = 2, a4 = −5), is selected and Monte-Carlo simulationsimplemented to show the effectiveness of ΨB (Eq. 10) as an IF estimator. Estima-tion is evaluated for input Signal to Noise Ratio (SNR) ranging from -6dB to 40dBwith an interval of 2dB. Hundred realizations are performed for each SNR and MeanSquare Error (MSE) between the estimate value and desired value chosen as perfor-mance criteria. Since both ΨB (Eq. 10) and unwrapped phase of the analytical signal(Hilbert Transform (HT)) (Eq. 5) involves derivatives, numerical differentiations.For very noisy signals, traditional Euler derivatives are useless. A better option isto use derivatives filters such as Savitzky-Golay differentiation filter [11]. This filteruses a polynomial fit across a moving window that preserves higher order moments ofthe signal. Derivatives are calculated with 4th order SG filter and 33 point window.Figure 1 shows the results of IF tracking using HT, WVD and ΨB-based methods4for SNR=30dB. Compared to HT and VWD, the best extraction is given by ΨB ap-proach. Figure 1(b) shows an agreement of VWD estimate with true IF but withfluctuations of high magnitude due essentially to cross-terms corruption in presenceof noise [10]. HT shows a good match but with ripples (modulations errors) (Fig.1(c)) compared to ΨB (Fig. 1(d)). In figure 2, we show the MSE as function of inputSNR for HT, WVD and ΨB-based method. This result shows the ability of ΨB toresolve a non-linear FM component. As seen in Fig. 2, across of different SNRs,ΨB approach provides a significant performance improvement over HT and VWD ap-proaches. Of particular interest are the results at low SNRs for which WVD approachperforms less better than ΨB and HT. This is expected, as pointed out in [12]. Asthe SNR decreases WVD produces biased representation of signal energy as well spu-rious components (artifacts). Except for higher SNRs (≥ 30dB), HT outperformsWVD approach mainly due to the use of an integral transform which does implicitsmoothing (low pass filtering) [10]. For SNR ≥ 36 dB, ΨB and DWV have similarperformances. It is important to mention that ΨB like most local approaches or thosebased on signal derivatives, is sensitive in very noisy environment. For moderatelynoisy data, classical Euler discretization schemes are robust to noise. For low SNRs,the use of derivatives filters such as SG is necessary for efficient tracking of the IF byΨB or HT. It was found that for different polynomial orders ≥ 3 the obtained resultsare similar but the precision of estimates both by ΨB and HT are dependent on thesize of the SG sliding window used.0.2 0.4 0.6 0.8 1810121416182022Time (sec)Frequency (Hz)  True IF(a)0.2 0.4 0.6 0.8 1810121416182022Time (sec)Frequency (Hz)  WVD(b)0.2 0.4 0.6 0.8 1810121416182022Time (sec)Frequency (Hz)  HT(c)0.2 0.4 0.6 0.8 1810121416182022Time (sec)Frequency (Hz)  PsiBSG(d)Figure 1: IF estimate for fourth-order PPS. (a) True. (b) WVD. (c) HT. (d) ΨBoperator.5−5 0 5 10 15 20 25 30 35 40−100102030405060708090SNR (dB)MSE (dB)  Psi_BWVDHTFigure 2: MSE in polynomial IF estimation for HT, WVD and ΨB operator.4 ConclusionIn this letter two new properties of ΨB energy operator are presented. The link be-tween ΨB and the dynamic signal shows that ΨB conveys precious local informationabout the instantaneous behavior, over the time, of a signal. The second propertyobtained for FM signals is a simple way to estimate the IF with no a priori knowl-edge about the phase of the signal. Preliminary results show that ΨB is effectivefor estimating the IF of a non-linear FM signal compared to HT and WVD basedapproaches. Further, the computational cost of ΨB is much lower. As future work weplan to apply ΨB to a wide range of synthetic and real signals to confirm the obtainedresults and to explore new properties of ΨB.References[1] J.F. Kaiser, ”Some useful properties of Teager’s energy operators,” Proc.ICASSP, vol. 3, pp. 149-152, 1993.[2] J.C. Cexus and A.O. Boudraa, ”Link between cross-Wigner distribution andcross-Teager energy operator,” Elec. Lett., vol. 40, no. 12, pp. 778-780, 2004.[3] A.O. Boudraa, S. Benramdane, J.C. Cexus and T. Chonavel, ”Some useful prop-erties of cross-ΨB-energy operator,” Int. J. Electron. Comm., vol. 63, issue 9, pp.728-735, 2009.6[4] A.O. Boudraa, ”Relationships between ΨB-energy operator and some time-frequency representations,” IEEE Sig. Proc. Lett., vol. 17, no. 6, pp. 527-530,2010.[5] A.O. Boudraa, J.C. Cexus and H. Zaidi, ”Functional segmentation of dynamicnuclear medicine images by cross-ΨB-energy operator,” Comput. Meth. Prog.BioMed., vol. 84, no 2-3, pp 146-152, 2006.[6] A.O. Boudraa, J.C. Cexus and K. Abed-Meraim, ”Cross-ΨB-energy operator-based signal detection,” J. Acoust. Soc. Am., vol. 123, no. 6, pp. 4283-4289,2008.[7] W.F. Zhang, C.C. Liu and H. Yan, ”Clustering of temporal gene expressiondata by regularized spline regression an energy based similarity measure,” Patt.Recong., vol. 43, pp. 3969-3976, 2010.[8] A.O. Boudraa, J.C. Cexus, M. Groussat and P. Brunagel, ”An energy-basedsimilarity measure for time series,” Adv. in Sig. Proc., ID 135892, 8 pages, 2008.[9] M.A. Poletti, ”Instantaneous frequency and conditional moments in the time-frequency plane,” IEEE Trans. Sig. Proc., vol. 39, no. 3, pp. 755-756, 1991.[10] L. Cohen, Time-Frequency Analysis, Prentice-Hall, 1995.[11] A. Savitzky and M.J.E. Golay, ”Smoothing and differentiation of data by sim-plified least squares procedures,” Anal. Chem., vol. 36, no. 8, pp. 1627-1639,1964.[12] P. Rao and F. Taylor, ”Estimation of instantaneous frequency using the discreteWigner distribution,” Elec. Lett., vol. 26, no. 4, pp. 246-248, 1990.

An energy-based similarity measure for time series,” Adv.

Clustering of temporal gene expression data by regularized spline regression an energy based similarity measure,”

Cross-ΨB-energy operatorbased signal detection,”

Estimation of instantaneous frequency using the discrete Wigner distribution,”

Functional segmentation of dynamic nuclear medicine images by cross-ΨB-energy operator,”

Instantaneous frequency and conditional moments in the timefrequency plane,”

Link between cross-Wigner distribution and cross-Teager energy operator,”

Relationships between ΨB-energy operator and some timefrequency representations,”

Smoothing and diﬀerentiation of data by simpliﬁed least squares procedures,”

Some useful properties of cross-ΨB-energy operator,”

Some useful properties of Teager’s energy operators,”

Time-Frequency Analysis,

http://sam.ensam.eu/bitstream/handle/10985/8859/IRENav_IEE_Letters_2011_mai_Boudraa.pdf?sequence=1

Estimation de la fréquence instantanée des signaux FM par opérateur d'énergie Psi_B

Abstract

Similar works

Full text

Available Versions

Name not available

SAM; the Arts et Métiers ParisTech open access repository

SAM : Science Arts et Métiers