A real-time time-frequency based instantaneous frequency estimator

Signal Processing 93 (2013) 1392–1397

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A real-time time-frequency based instantaneous frequency estimator Ljubiˇsa Stankovic´ a,n, Miloˇs Dakovic´ a, Thayananthan Thayaparan b a b

Elektrotehnicˇki fakultet, University of Montenegro, 81000 Podgorica, Montenegro R&D, Ministry of Defence, Ottawa

a r t i c l e in f o

abstract

Article history: Received 5 July 2012 Received in revised form 9 October 2012 Accepted 8 November 2012 Available online 17 November 2012

Commonly used time-frequency representations, like the short-time Fourier transform, the Wigner distribution and the higher order polynomial distributions, estimate the instantaneous frequency at the middle of the time-interval used in the analysis. For real time applications, like for example in radar signal processing, where the target parameters are estimated in the same way as the instantaneous frequency, the delay of a half of the considered interval may be unacceptable. Here, we propose a distribution that inherently estimates the instantaneous frequency at the end of the considered time interval. With a presented procedure for on-line implementation it can outperform other time-frequency representations for real time instantaneous frequency estimation. & 2012 Elsevier B.V. All rights reserved.

Keywords: Time-frequency Instantaneous frequency Estimation Real-time

1. Introduction Estimation of the instantaneous frequency (IF) is of great importance in many applications. For example, in radar signal processing it produces information about target’s range and cross-range position. Many IF estimators are based on the time-frequency representations [1–7]. Most commonly used are the short time Fourier transform (STFT), the Wigner distribution (WD) and other quadratic reduced interference distributions, defined by the Cohen class. Higher order spectra and distributions are introduced in order to improve the IF estimation, as well. In most of them the IF estimate is obtained for the mid point of the lag interval. Therefore, the IF estimate is delayed for a half of the lag interval. In radars, for example, it means that the estimate of target data is done with a delay corresponding to the half of lag interval (coherence integration time—CIT ). This delay can be significant and unacceptable in many cases. Some efforts

have been made in the IF extrapolation, in order to deal with this problem. In this paper we will present a distribution that inherently estimates the IF value at the ending point of the lag window. The distribution preserves property of the WD that it is fully concentrated if the phase variations of signal are up to the quadratic order. Bias and variance analysis of the proposed IF estimator, in the case of signals with non-linear IF, are done. Simulations show the efficiency of the presented distribution, when the current instant for the IF estimate is considered as relevant, rather than the one delayed for half of the lag interval. A procedure for reduced cross-terms realization of the proposed distribution is described. This procedure can be used in the multicomponent signals analysis. It improves performance in the case of high noise, as well.

2. Definition n

Corresponding author. E-mail addresses: [email protected], [email protected] (L. Stankovic´). 0165-1684/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2012.11.005

In recent research, an interest to the highly concentrated distributions has been again increased. For a frequency modulated signal xðtÞ ¼ A expðjjðtÞÞ the model

L. Stankovic´ et al. / Signal Processing 93 (2013) 1392–1397

of a fully concentrated distribution, along the IF 2

2 jj0 ðtÞt

TFðt, oÞ ¼ 2pA dðoj0 ðtÞÞ ¼ FT t ½A e



ð1Þ

is set again as the basic reference model in sparse timefrequency signal processing (and other recent developments). Assume that the phase function could be considered as quadratic, within the lag time interval. Our goal here is to find the simplest distribution, that is fully concentrated at j0 ðtÞ, at the current instant t. In contrast to the other distributions derived in a similar way, here we impose the constraint that only the past signal values are available for calculation. This kind of constraint is especially important in real time applications where an IF estimation, delayed by a half of the lag interval, may not be acceptable. In general, time-frequency distributions are defined as the Fourier transform (FT) of generalized local-autocorrelation functions. Here, in the definition of the local auto-correlation function, we use only the current and past instants as rðt, tÞ ¼ ej½a0 jðtÞ þ a1 jðtt=2Þ þ a2 jðttÞ ¼ ejj ðtÞt : 0

ð2Þ

Without loss of generality, a signal normalized in amplitude A¼1 is used. We also assume that the lag values are of the form t and t=2, so that they can be sampled with sampling interval corresponding to the WD calculation, i.e., with Dt=2. Therefore, an interpolation of discrete signal is not required. After an expansion of the phase functions jðtt=2Þ and jðttÞ into a Taylor series around t, the coefficients should satisfy the following system: a0 þa1 þ a2 ¼ 0; a1 =2a2 ¼ 1 and a1 =8 þa2 =2 ¼ 0: It produces a0 ¼ 3, a1 ¼ 4 and a2 ¼ 1, with  t ð3Þ rðt, tÞ ¼ x3 ðtÞxn4 t xðttÞ: 2 A time-frequency representation (TFR) of a continuous signal x(t), then, is Z 1  t Dðt, oÞ ¼ x3 ðtÞxðttÞxn4 t ejot dt: ð4Þ 2 0 Integration is from 0 to 1 since this distribution is defined to be causal. Notation xnn ðtÞ means the n-th power of complex conjugate of a signal. Since the causal from is used, the representation of a signal xðtÞ ¼ A expðjjðtÞÞ, with j000 ðtÞ being negligible, is j2o Dðt, oÞ ¼ pA2 dðoj0 ðtÞÞ þ oj0 ðtÞ

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In order to analyze performance of (6) as an IF estimator, let us consider a single component frequency modulated signal xðtÞ ¼ AðtÞejjðtÞ

ð8Þ

with small amplitude changes over the considered time interval, i.e., AðttÞ ffi AðtÞ for 0 r t rh and jðtÞ is a continuous and differentiable function of time. The IF of this signal is

oi ðtÞ ¼

djðtÞ ¼ j0 ðtÞ dt

ð9Þ

with the phase in the local auto-correlation function of the form  t 1 000 ¼ j0 ðtÞt 3jðtÞ þ jðttÞ4j t j ðtÞt3 þ    12 2 ð10Þ ¼ j0 ðtÞt þ Djðt, tÞ: The IF estimation is based on 000 ðtÞ 3

Dðt, oÞ ffi A8 ðtÞWðoj0 ðtÞÞno FTfejð1=12Þj

t þ  g,

ð11Þ 2

where WðoÞ is the FT of wðtÞ. Maximal value of 9Dðt, oÞ9 is reached at o ¼ j0 ðtÞ with a possible bias caused by the third and higher order derivatives of phase. Note that in the WD case the maximal value would be reached at oWD ¼ j0 ðth=2Þ. This distribution has been defined with the aim to be real-time instantaneous frequency estimator, in the first place. It also satisfies some other time-frequency representation properties. It preserves shift in time and frequency. For yðtÞ ¼ xðtt 0 Þ expðjoo tÞ, Dy ðt, OÞ ¼ Dx ðtt 0 , oo0 Þ: In 2

addition for a signal yðtÞ ¼ xðtÞejat =2 , Dy ðt, oÞ ¼ Dx ðt, oatÞ. R1 8 Time marginal is 1 Dðt, oÞ do=2p ¼ 9xðtÞ9 . For a scaled qffiffiffiffiffiffiffi version of the signal yðtÞ ¼ 9a9xðatÞ, aa0, this distribution reads Dy ðt, oÞ ¼ Dx ðat, o=aÞ: The time constraint is satisfied, as well, since Dðt, oÞ ¼ 0, when xðtÞ ¼ 0: 3. Estimator performance A discrete-time form definition of (6), at an instant t ¼nT, reads Dðt, oÞ ¼

N 1 X

wh ðkTÞxðt2kTÞxn4 ðtkTÞej2okT :

ð12Þ

k¼0

ð5Þ

with second term, which does not change the nature of 0 distribution begin concentrated at o ¼ f ðtÞ. Its pseudo form reads Z h  t wðtÞx3 ðtÞxðttÞxn4 t ejot dt: ð6Þ Dðt, oÞ ¼ 2 0 The past time interval ½th,t is used for Dðt, oÞ calculation. A lag window is denoted by wðtÞ for t 2 ½0,h. Note that the pseudo WD definition, by using signal values from the same interval, would be of the form       Z h h h t n h t jot x t  e wðtÞx t þ dt: WD t , o ¼ 2 2 2 2 2 0 ð7Þ

The lag independent part x3 ðtÞ is omitted, since it will not influence the IF analysis. Consider a noisy signal xðnTÞ ¼ sðnTÞ þ eðnTÞ ¼ AejjðnTÞ þ eðnTÞ,

ð13Þ

where eðnTÞ is a complex i.i.d. white Gaussian stationary noise with zero mean and variance s2e , and jðnTÞ are discrete time samples of the differentiable continuous function jðtÞ, with bounded derivatives. The IF, at a time instant t ¼ nT, is estimated by

o^ ðtÞ ¼ arg max9Dðt, oÞ92 ¼ argmax Fðt, oÞ, o

o

ð14Þ

where Fðt, oÞ ¼ Dðt, oÞDn ðt, oÞ: In order to analyze the estimators performance, we will linearize @Fðt, oÞ=@o around the stationary point where @Fðt, oÞ=@oj0 ¼ 0, with respect to the estimation

L. Stankovic´ et al. / Signal Processing 93 (2013) 1392–1397

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^ , noise and higher order phase terms [8] error Do 2

@Fðt, oÞ @ Fðt, oÞ @Fðt, oÞ @Fðt, oÞ ^ þ ffi Do d þ de : @o 90 @o 90 DjðtÞ @o 90 t @o2 90 ð15Þ

Having in mind that the variance of the real part of a complex zero mean random variable X can be expressed as var½2Re X ¼ E½X 2 þX n2 þ2XX n , after some direct, but lengthy, calculations, with o ¼ j0 ðtÞ and Djðt, tÞ ¼ 0, for the stationary point, we obtain   @Fðt, oÞ var det ¼ 2s2e A18 ð48U 1 V 1 V 0 þ 68W 0 V 21 @o 90

Approach to the analysis of this expression is similar to the one applied in the WD case [8]. However, since the distribution Dðt, oÞ is not real-valued, it is not straightforward. Thus, we will provide few basic steps for this derivation in the case of complex-valued distributions. The first derivative in (15) is calculated as   @Fðt, oÞ @Dðt, oÞ n ¼ 2Re D ðt, oÞ : ð16Þ @o @o

where

The second derivative @2 Fðt, oÞ=@o2 , at the stationary point (with eðnTÞ ¼ 0, o ¼ j0 ðtÞ and Djðt, tÞ ¼ 0, [8]), is obtained in the form

Ui ¼

ðkTÞi w2h ðkTÞ,

ð26Þ

k¼0 N=21 X

ðkTÞi wh ðkTÞwh ð2kTÞ:

ð27Þ

k¼0

^ ¼ var Do

s2e

1

A2 32½V 2 V 0 V 21 2

ð12U 1 V 1 V 0 þ 17W 0 V 21

þ17W 2 V 20 34W 1 V 1 V 0 þ 8U 0 V 21 þ4U 2 V 20 Þ: i

ðkTÞ wh ðkTÞ:

The theoretical variance is checked against the statistically obtained one, within the example. For a rectangular window wh(nT), a simple form of the variance follows:

ð19Þ

^ ¼ var Do

The estimation error follows from (15): @Fðt, oÞ @Fðt, oÞ de þ d @o 90 t @o 90 DjðnTÞ 10

8A

½V 2 V 0 V 21 

:

The estimation bias is ^ ¼ bias Do

1 8A10 ½V 2 V 0 V 21 

@Fðt, oÞ d , @o 90 DjðnTÞ

ð20Þ

where 1 X @Fðt, oÞ 3 ðkÞ ð2Þk 4ð1Þk d ¼ A10 j ðtÞ ðV k þ 1 V 0 V k V 1 Þ k! @o 90 DjðnTÞ 4 k¼3

 A10

j000 ðtÞ 2

ðV 1 V 3 V 4 V 0 Þ,

ð21Þ

assuming small Djðt,kTÞ, expðjDjðt,kTÞÞ ffi1 þ jDjðt,kTÞ, eðnTÞ ¼ 0 and o ¼ j0 ðtÞ. Its final form is ^ ¼ bias Do

V 1 V 3 V 4 V 0 16ðV 2 V 0 V 21 Þ

j000 ðtÞ:

ð22Þ

The estimation variance is ^ ¼ var Do

1 ð8A10 ½V 2 V 0 V 21 Þ2

var

@Fðt, oÞ de @o 90 t

ð23Þ

with @Fðt, oÞ de @o 90 t

"

N 1 X

¼ 2Re

ðj2kTÞwh ðkTÞsðt2kTÞsn4 ðtkTÞej2okT

k¼0



N 1 X

wh ðkTÞsn ðt2kTÞs4 ðtkTÞej2okT

k¼0



N1 X

ðj2kTÞwh ðkTÞxðt2kTÞxn4 ðtkTÞej2okT

k¼0



N1 X k¼0

 wh ðkTÞxn ðt2kTÞx4 ðtkTÞej2okT :

ð28Þ

ð18Þ

k¼0

^ ¼ Do

N 1 X

ð17Þ

where Vi ¼

Wi ¼

ð25Þ

Finally, the estimator variance can be written as

@2 Fðt, oÞ ¼ 8A10 ½V 2 V 0 þ V 21 , @o2 90

N 1 X

þ68W 2 V 20 136W 1 V 1 V 0 þ 32U 0 V 21 þ 16U 2 V 20 Þ,

ð24Þ

s2e 9

5N þ 3

A2 2 T 2 NðN þ 1ÞðN 2 1Þ



s2e

45

A2 2 T 2 N3

:

ð29Þ

In terms of the window length h ¼ NT, the variance is ^  22:5s2e T=ðA2 h3 Þ: For the WD with the same var Do ^ WD  rectangular window, we have [8], var Do 3 6s2e T=ðA2 h Þ, meaning that the presented estimator has a higher variance than the WD, applied to the same signal. Note that the WD provides estimation at the central point of the considered lag interval resulting in estimation delay of h=2 with respect to the current time-instant t. Time delay in the WD based estimation can produce significant bias and increase total MSE. Its value is ^ Þ2WD þ ðj0 ðtÞj0 ðth=2ÞÞ2 þ 6s2e T=ðA2 h3 Þ: MSEWD ¼ ðbias Do Thus, the proposed distribution will produce better results, if the variation of the IF within the lag window 3 is such that ðj0 ðtÞj0 ðth=2ÞÞ2 4 16:5s2e T=ðA2 h Þ: For the 2 2 SNR of 10 dB, A =se ¼ 10 and quite small number of samples (producing high estimation variance) h ¼16, T¼1 and N ¼16 it follows that the proposed IF estimator is still better if 9j0 ðtÞj0 ðth=2Þ9 4 0:006om , where om is the maximal frequency. Increasing the number of samples will additionally favor the proposed estimator. Thus, we may conclude if any time variation of the IF is expected the proposed estimator will produce better estimate, at the current time instant t, than the WD for any reasonable window width. It is also well known that all higher order approaches require high values of SNR for successful estimation. In order to illustrate this effect and propose a method for its reduction, consider a very simplified case of a sinusoid x(n) with a frequency k0. Assume that its FT is XðkÞ ¼ Adðkk0 Þ þ E1 dðkk1 Þ where E1 dðkk1 Þ is a disturbing term. Then, we will have the correct estimate of frequency k0 ¼ argfmaxk XðkÞ g if the disturbing term is

L. Stankovic´ et al. / Signal Processing 93 (2013) 1392–1397

lower than the desired signal A 4 maxfE1 g. Now consider a higher order distribution, requiring the powers of x(n), for example x2 ðnÞ. Its FT is XðkÞnk XðkÞ ¼ A2 dðk2k0 Þ þ E21 dðk2k1 Þ þ 2AE1 dðkk0 k1 Þ. It is clear that if A 4max fE1 g then A2 4maxfE21 g, i.e., the first two terms in XðkÞnk XðkÞ, representing the auto-terms of signal and disturbing term in convolution, will not influence the estimation, in this sense. However, the cross-term 2AE1 dðkk0 k1 Þ is introduced, as well. For the correct estimation, here we must have A4 maxf2E1 g. It corresponds to the well known 6 dB worsening with each higher order degree. Thus, our idea will be to remove, or reduce the cross-terms, as much as possible (in an ideal case to get Cross-Terms-Reduced fXðkÞnk XðkÞg ¼ A2 dðk2k0 Þ þ E21 dðk2k1 Þ) and to improve the estimation limit for low SNR, keeping higher order representations property of improving concentration for non-stationary signals. 4. The S-method based realization Based on (6) we can write

p

k=2L rp rk=2 þ L:

ð31Þ

oL

This may be considered as a windowed convolution, corresponding to the S-method [9]. It will reduce all cross-terms . The cross-terms between components being at least 2oL apart will be completely removed. Now we will present discrete form for the proposed distribution (6) realization based on this method, that will reduce the effect of high noise to the proposed distribution and also provide possibility of its direct application to the multicomponent signals. Let us consider a single time instant t and discrete samples xt(n) of a continuous signal xðttÞ sampled along t. We will assume that a rectangular window function w(n) of length N is used. Step 1: Calculate a discrete signal wt ðnÞ as samples of xn ðtt=2Þ. Its Fourier transform is STFT 1 ðt,kÞ ¼ FT½wt ðnÞ for N=2 r k rN=21. Step 2: We will now calculate STFT 2 ðt,kÞ ¼ FT½w2t ðnÞ by convolution X STFT 2 ðt,kÞ ¼ STFT 1 ðt,pÞSTFT 1 ðt,kpÞ: ð32Þ

ð33Þ

Component in STFT 1 ðt,kÞ, corresponding to k0 , is located in STFT 2 ðt,kÞ within the range ½2k0 2L,2k0 þ 2L. Step 3: The second convolution is performed in the same manner as in Step 2, to obtain X STFT 2 ðt,pÞSTFT 2 ðt,kpÞ: ð34Þ STFT 4 ðt,kÞ ¼ FT½w4t ðnÞ ¼ p

Similar analysis, as in Step 2, leads to the summation limits ðk=2Þ2Lr p r ðk=2Þ þ 2L. Note that component corresponding to k0in STFT 1 ðt,kÞ is located in STFT 4 ðt,kÞ within the range ½4k0 4L,4k0 þ4L. The convolution STFT 4 ðt,kÞ contains 4N3 frequency samples in total. Step 4: We will now calculate STFT x ðt,kÞ ¼ FT½xt ðnÞ. For component corresponding to k0 in STFT 1 ðt,kÞ the corresponding component in STFT x ðt,kÞ is located within ½2k0 4L,2k0 þ 4L. It is obvious, since STFT 1 ðt, oÞ ¼ FT½xn ðtt=2Þ and STFT x ðt, oÞ ¼ FT½xðt þ tÞ. The final distribution, with local calculation, is

ð30Þ

where wðtÞ ¼ xn ðtt=2Þ and STFT w4 ðt, oÞ ¼ STFT w ðt, oÞ no STFT w ðt, oÞno STFT w ðt, oÞno STFT w ðt, oÞ. Let us now assume that components in STFT w ðt, oÞ are localized, i.e., that a component centered at any o0 is spread in frequency, but only over a region ½o0 oL , o0 þ oL . Then STFT w ðt,2oÞno STFT w ðt,2oÞ Z 1 1 ¼ STFT x ðt, o þ xÞSTFT x ðt, oxÞ dx p 1 Z oL 1 ¼ STFT x ðt, o þ xÞSTFT x ðt, oxÞ dx:

The limits on p in (32) are k0 Lr p r k0 þL and k0 L r kp rk0 þ L. Eliminating unknown k0 we get

Dðt,kÞ ¼

X STFT x ðt,kÞSTFT 4 ðt,kpÞ,

ð35Þ

p

where limits for p are k4Lr p r kþ 4L. Note that we can calculate convolutions in different order and obtain similar results. The signal dependent S-method realization does require any distance between auto-terms . It would here produce a cross-terms free form of the D-distribution if the components do not overlap in the STFT. Presented theory is illustrated by an example with a noisy signal xðnÞ ¼ expðj10pn2 =N2 Þ þ eðnÞ, where the SNR is changed within 10 r SNR r 20, N ¼256 and T¼1. The value of Dðt,kÞ is calculated by definition. Its reduced interference form is calculated by using the proposed procedure with L¼7. The IF is estimated, with additional fine tuning using the interpolation and displacement technique. The MSE is calculated in 1000 realizations. The same is done for the WD. The MSE in WD is limited by a large bias, being equal to the change of the IF within a half of the lag interval. The proposed calculation procedure significantly improves the estimation for low and 104 Wigner distribution

102 MSE

Dðt, oÞ ¼ x3 ðtÞSTFT x ðt, oÞno STFT w4 ðt,2oÞ,

1395

100

D-distribution Proposed calculaton

D-distribution Direct calculation

Theory 10-2

p

Let us assume that components in STFT 1 ðt,kÞ are localized, i.e., that a component centered at any k0 is spreaded over the region ½k0 L,k0 þ L. It means that only frequency range ½kL,k þL will be used for calculation of STFT 2 ðt,kÞ for all k since k0 is not known.

10-4 -10

SNR [dB] -5

0

5

10

15

20

Fig. 1. The estimation MSE, in frequency steps, for a noisy signal: (o)—the Wigner distribution, (þ)—the proposed D distribution, calculated by definition, ()—the proposed D distribution, calculated by the proposed procedure and (- -)—theoretically obtained MSE values.

L. Stankovic´ et al. / Signal Processing 93 (2013) 1392–1397

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multicomponent signals, since the presented calculation procedure removes (reduces) cross-terms . A realization of the STFT, the WD and the proposed Ddistribution (by definition and by using the S-method based calculation procedure) on a two-component signal

moderate SNR. The theoretically obtained MSE (29) for Dðt,kÞ is presented, as well, Fig. 1. The STFT can be realized in a recursive (on-line) manner [9]. Then, with a few multiplications and additions the function Dðt,kÞ is obtained, for each k, in a numerically efficient way. It makes real time application of the proposed distribution complete (with possible hardware, on-line, implementation). In theory, the value of 2L should be equal to the signal component width, but in practice just a few samples around each frequency will significantly improve the concentration. The method will work well in time-frequency representation of

xðnÞ ¼ expðj230pðn=NÞ þ j24pðn=NÞ2 j10ðn=NÞ3 Þ þexpðj230pðn=NÞj20pðn=NÞ2 Þ

ð36Þ

with N ¼256 and on a mono-component noisy signal xðnÞ ¼ expðj48pðn=NÞ2 Þ þ eðnÞ

ð37Þ

with SNR ¼ 5 dB, L¼7 is presented in Figs. 2 and 3. Note that the IF representation in the D-distribution corresponds to the current instant (last available instant in calculation), while the IF in the WD is significantly delayed with respect to this instant, Fig. 2. In the spectrogram we can see that the IF, corresponding to the current instant, is at the position of one of the ending frequencies in the wide auto-term . In the Wigner distribution the IF is in the middle of the STFT’s auto-term . In the noisy case, the proposed, reduced interference realization (for noise, inspired by the analysis in the last paragraph of Section 3) significantly improves the D-distribution performance. However, it is still a higher order distribution, sensitive to noise. In many applications, when noise is not extremely high, this is the price that is not significant, as compared to the benefit of being able to accurately estimate the current IF. It is not delayed for a half of the lag of window, as in the case of other time-frequency based estimators. The delay of N=2 in the IF estimation can be easily seen in Fig. 3.

-500

5. Conclusion

-400

A distribution that inherently estimated the IF at the last instant of a considered interval is proposed. This distribution, along with a procedure for its reduced interference realization, provides an efficient tool for the IF estimation in real time, when a delay of the half of the lag interval is not acceptable, like for example in radar signal processing, for target localization and identification. The obtained distribution, with the procedure for its calculation, combines good properties of high concentration in higher order distributions and low noise sensitivity of the STFT. It can be directly applied to the multicomponent signals analysis, as well.

time

-300 -200 -100 -100

-50

0

frequency

50

100

time

Fig. 2. Time-frequency representation of a two-component signal: The Spectrogram (top), the Wigner distribution (middle) and the D-distribution with proposed cross-terms free (reduced) realization (bottom). Vertical dashed lines indicate true instantaneous frequencies, at the initial time instant, for each component.

200

200

200

100

100

100

0

0

0

-100

-100

-100

-200

-200

-200

-100

-50

0

50

frequency

100

-100

-50

0

50

frequency

100

-100

-50

0

50

100

frequency

Fig. 3. Time-frequency representation of a mono-component noisy signal: the spectrogram (left), the Wigner distribution (alias free realization), (middle) and the D-distribution with proposed cross-terms free (reduced) realization (right). Signal to noise ratio is SNR¼ 5 dB.

L. Stankovic´ et al. / Signal Processing 93 (2013) 1392–1397

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[5] S. Chandra Sekhar, T.V. Sreenivas, Auditory motivated level-crossing approach to instantaneous frequency estimation, IEEE Transactions on Signal Processing 53 (April (4)) (2005) 1450–1562. [6] E. Sejdic, I. Djurovic, J. Jiang, Time-frequency feature representation using energy concentration: an overview of recent advances, Digital Signal Processing 19 (1) (2009) 153–183. [7] J. Lerga, V. Sucic, Nonlinear IF estimation based on the pseudo WVD adapted using the improved sliding pairwise ICI rule, IEEE Signal Processing Letters 16 (11) (2009) 953–956. [8] L. Stankovic, V. Katkovnik, Instantaneous frequency estimation using the higher order L-Wigner distributions with the data driven order and window length, IEEE Transactions on Information Theory 46 (January) (2000) 302–311. [9] L. Stankovic´, A method for time-frequency analysis, IEEE Transactions on Signal Processing 42 (January) (1994) 225–229.