Overview of Signal Instantaneous Frequency Estimation Methods

Transcription

1 1 Overview of Signal Instantaneous Estimation Methods Jonatan Lerga, dipl. ing. el. University of Rijeka - Faculty of Engineering, Vukovarska 58, HR-51 Rijeka, Croatia jlerga@riteh.hr Abstract This paper gives an overview of existing instantaneous frequency (IF) estimation methods divided into several major categories. One of the first IF estimation approaches was based on the spectrogram, the resolution of which is highly window size dependant. In order to improve the IF estimation accuracy, the iterative method was proposed, such that each iteration gives more accurate estimation of the IF and better timefrequency representation (TFR) of the signal. For the case of the polynomial frequency modulated (FM) signals, several different methods using the polynomial Wigner-Ville distribution (WVD) and higher order ambiguity functions (HAFs) were proposed. The Wigner-Ville spectrum (WVS) was shown to be optimal for linear FM signals in multiplicative noise, while the polynomial WVD was used for high-order polynomial FM signals corrupted by multiplicative noise. Another way to estimate signal s IF was based on localization of peaks from adaptive lag window TFRs. Various monocomponent IF estimation methods were upgraded and applied to the IF estimation from multicomponent signals. Furthermore, some image processing techniques were applied to the IF localization and estimation from various suitable TFRs. Index Terms Instantaneous frequency estimation, timefrequency signal analysis, time-frequency representations, timefrequency distributions. I. INTRODUCTION Instantaneous frequency (IF) is one of the basic signal parameters which provides important information about the time-varying spectral changes in non-stationary signals. The concept of IF finds its usage in various technical fields and applications such as seismic, radar, sonar, communications and biomedical applications [1]. Different approaches for its estimation were considered, beginning with the one using the spectrogram whose time-frequency resolution is window dependant. In order to improve the spectrogram-based IF estimation method, iterative algorithm provided by the first moment of the spectrogram was proposed [1]. For the polynomial frequency modulated (FM) signals corrupted by additive noise, IF estimation methods based on maxima of the polynomial Wigner-Ville distribution (WVD) and higher-order ambiguity functions (HAFs) were proposed. The polynomial WVD was also used for the multiplicative noise or random amplitude modulation and non-linear FM signals, while the WVD was proposed for the linear FM signals [1]. Another approach to estimate IF was based on adaptive algorithm for window width selection and localization of peaks of the time-frequency representations (TFRs) of the monocomponent signals. This approach was then extended to multicomponent signal IF estimation using different reduced interference distributions (RIDs), such as the modified B- distribution (MBD) [2], [3]. However, one of the most demanding IF estimation problems is finding the components IF of multicomponent nonstationary FM signal. Different techniques for their IF estimation were proposed, including some image processing algorithms applied to the various TFRs in order to extract the information about the signal components and their IF laws. The concept of the IF in communication theory is often connected with the frequency modulation. It can be described as the frequency of sinusoid which locally fits the signal in considered time instant t [1]. Gabor and Ville defined the IF f i (t) of real signal x(t) as the derivation of phase of its analytic signal x a (t) [4]: f i (t) = 1 d 2π dt arg(x a(t)). (1) Introduction of the analytic signal was caused by the fact that the real signal contains positive-frequency and negativefrequency components causing unwanted cross-terms in TFRs. However, X( f) can be obtained from X(f) as X( f) = X (f) (where X(f) is Fourier transform of x(t)) and hence it can be eliminated from the signal frequency representation without losing information [1]. The analytic signal x a (t), such that X a (f) = for f <, can be obtained as: x a (t) = x(t) + jh{x(t)}, (2) where H{x(t)} = F 1 t f {( jsgnf)f t f {x(t)}} is the Hilbert transform (F denotes the Fourier transform). In TFR analysis the IF of the signal at time instant t can be interpreted as the weighted average of all frequencies which occur in the signal at the considered time instant t [5]: fρ(t, f)df f i (t) =, (3) ρ(t, f)df where ρ(t, f) is the TFR. The IF finds its use in various applications based on timefrequency analysis. For example, matched spectrogram using chirp windows which was shown to outperform regular spectrogram for chirp-like signals if the chirp rate of window is matched to the signal IF [6] or it can be used to increase concentration of TFRs along the signal IF for monocomponent signals (such as the pure tones and amplitude-modulated and frequency-modulated signals [7]). Fig. 1 gives example of monocomponent and multicomponent

2 2 x 1 (t) Fourier transform magnitude Fourier transform phase (a) (b) (b) (d) (e) x 2 (t) Fourier transform magnitude Fourier transform phase (f) (b) (b) (i) (j) Fig. 1. Example of monocomponent x 1 (t) (first row) and multicomponent x 2 (t) (second row) signal. (a) x 1 (t); (b) Fourier transform magnitude of x 1 (t); (c) Fourier transform phase of x 1 (t); (d) WVD of x 1 (t); (e) The IF of x 1 (t); (f) x 2 (t); (g) Fourier transform magnitude of x 2 (t); (h) Fourier transform phase of x 2 (t); (i) WVD of x 2 (t); (j) The IF of x 2 (t). signal, magnitude and phase of their Fourier transform, signals WVD and IF. II. ITERATIVE IF ESTIMATION ALGORITHM The iterative IF estimation approach (proposed by M. K. Emresoy and A. E1-Jaroudi [8]) in its first step detects IF from the TFR using (3). The obtained IF is then used to recalculate the TFR. The procedure which consists of the recalculation of the TFR using the IF estimated in the previous step and the IF estimation from the obtained TFR is repeated until convergence is reached. As it was shown in [8], each iteration gives more accurate IF estimation and TFR more matched to the signal IF. Consider the complex signal: z(t) = a(t)e jφ(t), (4) where a(t) is time-varying amplitude, and φ(t) the phase of the signal. The iterative approach comprises of signal demodulation in each iteration l, except the first one which is done using the whole frequency axis [8]. This demodulation can be realized by subtracting the phase estimate (obtained by integrating the IF estimate) from the phase of the signal [8]: z l (t) = a(t)e j(φ(t) ˆφ l (t)). (5) The effect of the demodulation is shifting the spectrum of the nonstationary signal around zero frequency. Thus, after the first iteration the signal under analysis has non-redundant information over positive and negative frequencies [1]. For the real signals, authors have used the analytic signal in the first iteration in order to avoid the first IF estimate being zero. However, the signal obtained after subtracting the phase estimate from the phase of the signal in each iteration following the first one may not be analytic. The obtained TFR in each iteration (authors have used spectrogram) ρ l (t, f) of z was then used to get a new matched spectrogram estimate used in the next iteration [8]: ˆρ l+1 (t, f) = ρ l (t, f ˆf l (t)). (6) The algorithm ends once the convergence is reached. Criterion for the convergence was realized as the threshold applied to the difference between consecutive iterations. The convergence is affected by the selected window width, as well as by the rate of change of the IF [8]. It was shown in [1] that the convergence of the algorithm for the wide windows and signals whose IFs have nonzero high order derivatives is slow. Otherwise, polynomial phase signals and narrow windows result in fast convergence of the algorithm (in case of the polynomial phase signals the number of iterations is equal to the half of the polynomial degree) [1]. The algorithm is limited to the signals with no discontinuities at some time instants (where the derivative of the phase goes to ± ). The results of the proposed method are the estimated IF and the matched TFR of the signal. III. IF ESTIMATION OF POLYNOMIAL FM SIGNALS IN ADDITIVE GAUSSIAN NOISE A. IF Estimation Using the Polynomial Wigner-Ville Distribution Consider the noisy polynomial FM signal found in various technical fields, such as radar, sonar and telecommunications [1]: where: y(t) = z(t) + ɛ(t) = Ae jφ(t) + ɛ(t) (7) = Ae j P k= a kt k + ɛ(t), t = [, T], (8) z(t) is the noise-free signal, ɛ(t) is the complex Gaussian noise,

3 3 A is constant signal amplitude, φ(t) is signal phase, P is the order of polynomial signal phase, a k are corresponding polynomial coefficients, T is signal duration. The IF of the signal z(t) according to (1) is: f i (t) = 1 dφ(t) = 1 2π dt 2π P ka k t k 1, t = [, T]. (9) k=1 In case of complex sinusoidal (P = 1), the signal has linear phase whose IF is constant and time independent. Its Fourier transform is peaked having signal energy concentrated around the IF and dispersed noise energy. For P > 1 signal z(t) has non-linear phase and time-varying IF. In order to efficiently apply the Fourier transform to such signals we need an operator to transform the non-linear phase signal into a signal with linear phase, the frequency of which is equal to the IF for the given time instant [1]. This can be done using approach proposed in [9] and finding the IF as the peak of polynomial WVD Wz P (t, f) of the signal z(t) [9]: W P z (t, f) = where K P z (t, τ) is [9]: K P z (t, τ) = K P z (t, τ)e j2πfτ dτ, (1) I z(t+c i τ) ki z (t c i τ) ki, t = [, T], (11) i=1 where I is the number of coefficients of transform c i, and k i determines the order of transformation q, q = I i=1 2k i, obtained as in [1]. The polynomial WVD was shown to be efficient for polynomial IF estimation of noisy signals of high signal-to-noise ratio (SNR), while for low SNRs the iterative procedure was suggested [11]. The initial IF estimate ˆf i (t) is obtained as the peak of the short-time Fourier transform. It is then used to reconstruct the estimate of the noisy observation y(t), denoted as ŷ(t), both of which are used in the cross polynomial Wigner-Ville distribution (XPWVD) ( [1] and [11]): XW P y (t, f) = where XKy P (t, τ) is [11]: XK P y (t, τ)e j2πfτ dτ, (12) XK P y (t, τ) = y(t + c 1 τ) k1 ŷ (t c 1 τ) k1 I ŷ(t + c i τ) ki i=2 ŷ (t c i τ) ki, t = [, T]. (13) The IF value is then updated using the peak of XPWVD and the procedure is repeated iteratively until the obtained ŷ(t) closely matches y(t) from the previous iteration. In order to estimate the signal phase coefficients a k, the procedure proposed in [12] includes: 1) Defining the matrix Wy P (f) containing P polynomial WVD slices from the discrete observation y[n], 2) Estimation of a vector of IF estimates ˆf i as: ˆfi = arg max f {W P y (f)}, (14) where arg max is the vector argument of a vector (row by row) maximization, 3) Initial coarse estimates for the â i = [â i1 â i2 â ip ] T obtained as â i = 2πX 1ˆf i, 4) Calculation of y d [n] = y[n]e j(âi1n+âi2n2 + +â ipn p), 5) y [n] obtained by decimation of y d [n] by factor 1/B, where B is bandwidth of ideal low-pass filter y d [n], 6) Calculation of â d = (G T G) 1 G T V, 7) Finally, calculation of â f = â i +â d B ( â d B denotes element by element multiplication of â d and B), where ( [1] and [12]): X = 1 2n 1 3n Pn1 P 1 1 2n 2 3n Pn2 P n P 3n 2 P... Pn P 1 P, (15) V = [V ()V (1) V (NB 1)] T, (16) B = [1B B 2 B P ] T, (17) P G = P NB 1... (NB 1) P. (18) B. IF Estimation Using Higher Order Ambiguity Functions Another way to detect polynomial phase of signal (proposed in [13], [14]) in its initial step estimates parameter a P using the Fourier transform. Then the P th order polynomial phase signal is transformed into a signal with linear phase, the frequency of which is proportional to the a P using the higher order ambiguity functions (HAFs), defined for discrete noisy signal y(n) = z(n) + ɛ(n) as: P P [y; ω, τ] = N (p 1)τ 1 n= P P [y(n); τ]e jωn, (19) where p 1 P P [y(n); τ] = [y ( k) (n + (p 1 k)τ)] (p 1 k ), (2) y ( k) (n) = k= { y(n), k even y (n), k odd. (21) The parameter τ is a possitive integer, and ω [ π, π]. In the next step, the signal is reduced to (P 1) th order polynomial phase signal. This procedure is then repeated for estimation of a P 1 and every other a k parameter. The IF is calculated from the obtained a k parameters using (9). The method was combined with an iterative method for low SNRs [15]. In case of multicomponent signals, HAFs are first used to estimate the parameters of the highest amplitude component. It is then removed and the next highest amplitude component is analyzed. The procedure is repeated for all signal components [16].

4 4 IV. IF ESTIMATION OF FM SIGNALS IN MULTIPLICATIVE NOISE In some technical applications, such as fading in wireless communications, fluctuating targets in radar, structural vibration of a spacecraft during launch and atmospheric turbulence, the amplitude of the analyzed non-stationary signal cannot be assumed to be constant, but rather randomly modulated having properties of multiplicative noise [1]. The Wigner-Ville spectrum (WVS) was shown to be optimal, in respect to energy concentration for a linear FM signals corrupted by multiplicative noise [17]. For higher-order polynomial FM signals, the optimal representations for the IF estimation using its peaks were obtained using polynomial WVD [18]. Let us now consider the signal y(t) obtained as: y(t) = a(t)v(t) = a(t)e jφ(t) = [µ a + a (t)]e jφ(t), (22) where a(t) is a real-valued stationary noise, µ a its non-zero mean value, and a (t) zero-mean noise, the autocorrelation of which is R a. The autocorrelation of the signal y(t) is [1]: K y (t, τ) = E[y(t τ/2)y (t+τ/2)] = R a (τ)k v (t, τ). (23) The WVS, which is defined as the Fourier transform of the signal autocorrelation, can be written as [19]: W y (t, f) = W v (t, f) f S a (f), (24) where f is the convolution in the frequency domain and the WVD for linear FM is W v (t, f) = δ(f f i (t)) [19]. Therefore, the WVS is [1]: W y (t, f) = µ 2 aδ(f f i (t)) + S a (f f i (t)). (25) As it can be seen from (25), the WVS localizes the IF for the linear FM signals when µ a. In case of µ a =, the IF estimate variance tends towards infinity and the WVS based estimator breaks down [17]. In case of nonlinear polynomial FM signals, the WVS disturbs the signal representation and WVS peak based IF estimation becomes inefficient, so the use of polynomial WVD is proposed [1]. However, for µ a = the polynomial WVD based estimator also breaks down since the IF estimate variance tends towards infinity. V. ADAPTIVE IF ESTIMATION USING THE TFD MAXIMA IF estimation from the maxima of the TFRs was shown to have variance and bias highly affected by lag window width [2]. The optimal window width can be found by minimizing the estimation error if some signal and noise parameters are known in advance. However, it is not always so in practice, especially for the estimation bias. This is the reason why the IF estimation method with adaptive lag window width selection, which does not need the knowledge of estimation bias, based on the intersection of the confidence intervals (ICI) rule, was proposed by Stanković and Katkovnik [2], [21] and [22]. The abovementioned approach considers the signal of form in (7) embedded in additive white complex-valued Gaussian noise of a known variance σ. The IF is obtained as the maxima of the TFR ρ y (t, f) [2]: ˆf i (t) = arg{max ρ y (t, f)}. (26) f For each time instant t the mean squared error (MSE), E{( ˆf i (t)) 2 }, (the minimum of which is dependent upon the window length) was used as the measure of the estimation quality, where ˆf i (t) = f i (t) ˆf i (t) is the estimation error, and f i (t) = φ (t)/2π is the true IF value. The ˆf i,h (t), obtained using window width h, was shown in [2] to be random variable with the bias bias(t, h) and standard deviation σ(h). Then the following inequity ( [1] and [2]): f i (t) ( ˆf i,h (t) bias(t, h)) κσ(h), (27) holds with probability P(κ). The parameter κ is such that P(κ) 1 as κ grows. The method then introduces a sequence of dyadic window widths h H, H = {h s h s = 2h s 1, s = 1, 2,, J} and calculates a sequence of corresponding TFRs for each window width. Then the IF ˆf i,hs (t) is calculated for each window width from H using (26). The best estimate ˆf i,hs + (t) for each time instant t is obtained using the ICI rule which calculates J confidence intervals D s = [U s, L s ] for each time instant, where U s is upper and L s lower confidence interval limit obtained as [2]: U s = ˆf i,hs (t) + (κ + κ)σ(h s ), (28) L s = ˆf i,hs (t) (κ + κ)σ(h s ). (29) It was shown in [2] that for small window widths h s (s s + ) the bias is small, D s 1 D s and f i (t) D s with the probability P(κ) 1. For s s + the bias is large, but the variance is small and D s 1 and D s do not necessary intersect. The s + can be found as the largest s for which the pair-wise intersection of confidence intervals exists [2]. Three methods for selecting the algorithm parameters in dependence of reliability of our knowledge about the variance and bias behavior were proposed [2]. So, if that knowledge is unreliable, a heuristic approach setting P(κ) = 1 (consequently, κ = 2.5 and κ = 1) was shown to be applicable in most of the practical applications [2]. On the other hand, if that knowledge is reliable, the procedure for selecting the method s parameters based on the first three confidence intervals was described in [2]. Another way for the (κ + κ) selection is based on the statistical nature of confidence intervals and a posteriori check of the fitting quality [1] and [2]. The ICI rule applied to the IF estimation chooses wider window widths in case of small changes in IF in order to reduce estimation variance, while close to the fast variations in the IF the narrow windows are used. This adaptive method outperforms the IF estimate obtained using the best constantwindow TFR (which also is not known in advance). However, the method is highly dependant on the initial window width set H selection.

5 5 VI. IF ESTIMATION OF MULTICOMPONENT SIGNALS A. Monocomponent IF Estimation Methods Extended to Multicomponent Signals IF Estimation Signals found in practical technical applications often consist of several components embedded in additive noise, where each component may have different time varying IF and amplitude. Various signal processing methods of such multicomponent signals often require detection of the number of components M (what can be done using Huang-Hilbert transform [23] or Renyi entropy [24]), the IF f i,m (t) and the amplitude a m (t) for each component m, m = 1, 2,, M. This can be done by representing the noisy signal y(t) in time-frequency domain and recovering each component using time-frequency filtering techniques described in [19]. Another approach upgrades the existing monocomponent IF estimation techniques to multicomponent IF estimation by extending the methods to simultaneously track the various IF components [2], [25]. Both of this approaches use TFRs. Consider a multicomponent signal embedded in additive noise: y(t) = M z m (t) + ɛ(t) = m=1 M a m (t)e jφm(t) + ɛ(t), (3) m=1 where a m (t) and φ m (t) are the mth component amplitude and phase respectively, while ɛ(t) is complex-valued Gaussian noise of independent and identically distributed real and imaginary parts with the total variance σɛ 2. The IF of each component is [19]: f i,m (t) = 1 2π dφ m (t), m = 1,, M. (31) dt As already mentioned, one of the conventional ways to estimate IF is to find the peaks of TFR of y(t) [1]: ˆf i,m (t) = arg[max ρ y,m (t, f)] f f f s 2, (32) where ρ y,m (t, f) is the mth peak of TFR, and f s the sampling frequency [1]. The advantages of using the spectrogram for the TFR in this multicomponent signal IF estimation approach are its simplicity and absence of cross-terms. However, its main disadvantage is poor time-frequency resolution, especially if the signal s components are close or one of them has significantly smaller power. However, this time-frequency resolution can be improved using other TFRs such as the WVD which is optimal for IF estimation using its peaks in case of linear FM signals and high to moderate SNRs [26]. As shown in [27], in case of small SNRs, the unwanted cross-terms peaks may be mistaken for the IF estimate. If the signal FM is polynomial, the polynomial WVD should be used instead of the WVD [18]. However, in case of the multicomponent signals both WVD and polynomial WVD contain the unwanted cross-terms which may be mistaken for the IF estimate. To avoid this, the crossterms can be reduced using RIDs [28], but the IF estimation using the peaks of RIDs results in significantly biased IF. The cross-terms, although reduced by RIDs, effect the weak signal components. In order to solve this problem, the MBD which is a special purpose RID with suppressed cross-terms, improved time-frequency resolution and reduced IF bias was developed [2], [3]. In general, quadratic TFRs of the signal y(t) can be expressed as [19]: where: ρ y (t, f) = F τ f [ G(t, τ) (t) K y (t, τ) ], (33) G(t, τ) is the time-lag kernel, K y (t, τ) = y(t + τ 2 )y (t τ 2 ) is the signal kernel or the instantaneous autocorrelation function (IAF), (t) denotes time convolution, F τ f denotes Fourier transform. TFR smoothing and localization is done by applying a real-valued symmetric window with unity length w h (τ) = t h w( τ 2h ) on K y(t, τ) (where t is sampling interval, h is window width, and w(t) = for t > 1 2 ) [1]. The TFR s dependence on window width h can be represented as [1]: ρ y,h (t, f) = F τ f [ wh (τ)g(t, τ) (t) K y (t, τ) ]. (34) Furthermore, TFR ρ y,h (t, f) discretized over time, lag, and frequency can be written as [1]: ρ y,h (n, k) = N s 1 l= N s N s 1 m= N s w h (m t) K y (l t, 2m t) km j2π G(n t l t, 2m t) e 2Ns, (35) with the number of samples being 2N s. So, the signal IF estimate is [1]: ˆf i,h (t) = arg[max ρ y,h (f, t)] f f f s 2. (36) For IF estimation of multicomponent FM signals in additive Gaussian noise using quadratic TFRs, it was shown in [2] that those TFRs need to satisfy the following criteria: Be a high resolution TFRs with reduced cross-terms, Have an optimized lag window size, which ensures a good IF estimation variance to bias tradeoff, Allow estimation of each component amplitude needed for the IF estimation variance calculation [2], [8], [25] and signal reconstruction. Some TFRs, like the Choi-Williams distribution (CWD) and the spectrogram satisfy the first two criteria, but they do not allow for a direct amplitude estimation, and hence they are not the best choice for the IF estimation of multicomponent signals. However, the B-distribution (BD) was shown to outperform other fixed-kernel TFRs in terms of cross-terms reduction and resolution improvement [29]. Its kernel was then modified in order to allow for the component amplitude estimation leading to the MBD, the kernel of which is [2]: G(t, τ) = G β (t) = k β / cosh 2β (t), (37) where β is a real positive number and k β = Γ(2β)/(2 2β 1 Γ 2 (β)), Γ denotes the gamma function [1]. For the sum of two complex sinusoids y(t) = z 1 (t)+z 2 (t) =

6 6 a 1 e j(2πf1t+θ1) + a 2 e j(2πf2t+θ2), the obtained cross-terms for the MBD are 2a 1 a 2 γ β (t)δ[f (f 1 + f 2 )/2], where γ β (t) = Γ(β + jπ(f 1 f 2 )) 2 cos(2π(f 1 f 2 )t + θ 1 θ 2 )/Γ 2 (β) [1]. Hence, although the MBD cross-terms depend on the frequency separation between the signal components as with other quadratic TFRs, the MBD outperforms most of those TFRs even for f 1 and f 2 being relatively close [2]. The MBD supports IF estimation by peak localization causing the high IF estimation bias in case of nonlinear FM signals with rapid changes in IF [2]. It was shown that the IF estimation bias is a continuously decreasing function, while the bias is continuously increasing function of window width [2]. The adaptive algorithm (described in Section V) for monocomponent FM signals proposed in [2] converges at optimum window length, giving a good bias-variance tradeoff. In order to apply this algorithm to IF estimation of multicomponent signals, the confidence intervals D s,m should be calculated for each signal component m and each window width h s. The appropriate window width is the one for which the current confidence interval D s,m and the previous confidence interval D s 1,m have at least one point in common [2], [25]. However, for the efficient application of this algorithm, tracking of the maxima in the time-frequency plane for each component while ignoring the local maxima caused by the cross-terms and windowing is needed. B. Multicomponent Signal IF Estimation Based on Evolutionary Spectrum Another approach to estimate IF of multicomponent signals is based on dividing the TFR in small regions and calculating local IF of signal components in each region [3]. Adequate time-frequency energy density is obtained using optimal combination, in terms of least squares, of evolutionary spectra calculated by multi-window Gabor expansion for a finite support signal (or a periodic signal of period N) x(n) [31]: x(n) = 1 I I 1 M 1 K 1 i= m= k= a i,m,k hi (n ml)e jω kn, (38) where h i (n) is a periodic extension of h i (n) generated by contracting a mother Gabor window h(n), ω k = 2πk/K (L < K), I is the number of different scales used, M and K are the number of analysis samples in time and frequency, respectively, L is the time step, and a i,m,k are the Gabor coefficients calculated as in [31]. The optimal weights are obtained by minimizing the squared error between the combination of evolutionary spectra and a reference TFR (which is taken to be the WVD of the signal). The method was shown to maintain good properties of multiple-window evolutionary spectral analysis and high resolution TFRs, providing non-negative and high resolution timevarying spectral estimates [3]. However, the local IF can be extracted only for signal components being well separated. Thus, the TFR resolution plays the crucial role in IF estimation of multicomponent signals using this approach. C. Method Based on Component Linking Algorithm Image processing techniques, including those based on the modified versions of the Hough transform, were applied to TFRs [32], [33] and [34]. However, those Hough transform based approaches require knowledge of signal components class and do not perform well if all components are not from the same class (for example, if the signal contains both linear and sinusoidal component). This is a significant setback of Hough transform based IF estimation algorithms. Various IF estimation techniques based on TFR peak detection require a threshold in the time-frequency domain (the value of which is application and signal dependent) in order to suppress local maxima caused by spurious crossterms [35]. Besides, the method described in [35] requires the knowledge of the ratio between the signal component amplitudes and assumes that the component amplitudes are constant. The method is not efficient for low SNRs or signals with time-varying amplitudes. For those case, the use of pattern recognition techniques in determining IF components was proposed [36]. In order to avoid the thresholding in the time-frequency domain (value of the threshold is often selected manually), a component linking procedure was proposed [37]. The IF estimation using this method starts with 1D signal transformation to 2D TFR using RIDs, extended with a preprocessing step called time-frequency peak filtering in case of low SNRs, and followed by a component linking algorithm. The TFR was transformed to the binary image, the value of which is zero everywhere except at the location of local peaks. In order to extract those signal components from the obtained binary image, the standard 8-connected and 1-connected neighboring and an application dependent threshold for determining correctly or falsely connected pixels were used. The method classifies a multicomponent signal as either timefrequency separable (whose components are not too close and do not intersect) or as time-frequency nonseparable. It has been applied to time-frequency separable multicomponent signals, as most physiological signals are (for example, newborn EEG and heart rate variability). VII. RECENT IF ESTIMATION APPROACHES A. IF Estimation Based on Cauchy s Integral Formula Discretization A recently proposed IF estimation method based on Cauchy s integral formula discretization, was improved by increasing the number of discretization points on the circle and multiple successive integrating over the circle [38]. A novel general forms of TFRs were obtained by increasing the number of discretization points, while the multiple successive integrating led to the so called L-forms of the signal z(t) = ae jφ(t) TFRs [38]: ) GCD L N=2(t, ω) r,b = z (t L(r+jb) τ + 2L(r + jb) ) z (t L(r+jb) τ e jωτ dτ, (39) 2L(r + jb) where L is the scaling factor, and τ/(2(r+jb)) and τ/(2(r+ jb)) two points on the circle located symmetrically around

7 7 the time instant t. The method was shown to outperform the Wigner distribution, the L-Wigner distribution and the smoothed pseudo Wigner distribution in terms of the IF estimation accuracy, both for monocomponent and multicomponent signals. B. IF Estimation Based on S-Transform The S-transform based IF estimator for signals embedded in additive white Gaussian noise was proposed in [39]. The S- transform (used in various fields as geophysics, cardiovascular time series analysis, pattern recognition and signal processing for mechanical systems [39]) combines short-time Fourier analysis and wavelet analysis. The S-transform for the signal x(t) is defined as [39]: S c (t, ω; w(τ, ω)) = e jωt x(t+τ)w(τ, ω)e jωτ dτ, (4) where the window w(τ, ω) is [39]: w(τ, ω) = ω τ 2 ω 2 (2π) 1.5 e 8π 2. (41) For a discrete signal, the S-transform is [39]: S d (t, ω; w(nt, ω)) = e jωt T x(t + nt) n= w(nt, ω) e jωnt. (42) Variable window widths used in the S-transform (narrow timedomain windows used for higher frequencies and wider timedomain windows for lower frequencies) have improved the signal frequency-domain localization. The IF estimated as the peak of the considered adaptive S-transform TFR was shown to significantly outperform (in terms of the MSE) the one obtained using the short-time Fourier transform or the pseudo WVD [39]. C. IF Estimation Based on Directionally Smoothed Pseudo- Wigner-Ville Distribution Bank Most of the existing IF estimation methods based on the quadratic TFRs do not perform well for low SNRs (timefrequency points not belonging to the signal component are often contained in the IF estimate). In order to improve the IF estimation in low SNRs, a family of joint distributions (which use pointwise adaptive weight averaging) from the directional smoothed pseudo Wigner-Ville distribution (DSPWVD) bank with different directions was introduced [4]. The general form of the smoothed pseudo Wigner-Ville distribution (SPWVD), obtained by smoothing the pseudo Wigner-Ville distribution (PWVD) along the time axis for the signal x(t), can be expressed as [4]: SPWVD x (n, f; h, g) = m g(m)pwv D x (n+m, f, h), (43) where h is the lag window and g the time smoothing window. The DSPWVD is obtained by smoothing the pseudo WVD along the direction θ [4]: DSPWVD θ x (n, f; h, g) = = m = m g(m)pwv D x (n + m, f + m tanθ; h) k 4πjkm tan θ g(m)h(k)e x(n + m + k)x (n + m k)e 4πjkf. (44) The optimal lag and smooth window were designed by maximizing the ratio of the average magnitude of the DSPWVD of the signal at IF points to the distribution standard deviation [4]. Then the joint distribution was constructed from DSPWVDs by pointwise adaptive weighted combination. The method based on the joint distribution was proven to outperform the IF estimation method using the PWVD, SPWVD and adaptive optimal kernel distribution (OAKD) in terms of the number of outliers contained in the estimated IF and thus in the IF estimation accuracy. VIII. CONCLUSION Selection of the appropriate method for signal IF estimation depends on the signal class and the method s limitations. In case of the signals with no discontinuities in time, the iterative TFR based method can be used. The convergence of the method for the polynomial phase signals and narrow windows is fast, while for the wide windows and signals whose IFs have nonzero high order derivatives convergence is slow. For the signal embedded in additive noise of high SNR, the polynomial WVD should be used, while for low SNRs the iterative procedure based on XPWVD was proposed. Another way to estimate IF from the noisy signal is adaptive method based on the maxima of pseudo WVD and ICI rule (used for adaptive varying window width selection). The IF can also be estimated as the peak of the adaptive S-transform, which was shown to significantly outperform the IF obtained using the short-time Fourier transform or the pseudo WVD in terms of the MSE. Most of the IF estimation methods based on the quadratic TFRs do not perform well for low SNRs when the family of joint distributions (which use pointwise adaptive weight averaging) from the DSPWVD bank with different directions can be used. This method was proven to outperform the IF estimation approach using the PWVD, SPWVD and OAKD in terms of the number of outliers contained in the estimated IF. For multiplicative noise and linear FM signals, the WVS based method was proposed, while for the nonlinear polynomial FM signals, polynomial WVD was used. The method is limited to the non-zero mean value noise in order to avoid IF estimate variance tending towards infinity. Recently, the method based on the Cauchy s integral formula was shown to outperform the Wigner distribution, the L-Wigner distribution and the smoothed pseudo Wigner distribution in terms of the IF estimation accuracy, both for monocomponent and multicomponent signals. The IF estimation methods of multicomponent signals are highly dependent on the selected TFR and presence of crossterms which may be mistaken for the IF estimate. Thus, for the IF estimation using TFR peaks in case of the linear FM signals and high to moderate SNRs, the WVD was proposed. If the signal FM is polynomial, the polynomial WVD should be used. However, both WVD and polynomial

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