A new algorithm for the estimation of the instantaneous frequency of asignal perturbed by noise

T. Asztalos, A. Marina, A. Isar

Electronics and Telecommunications Faculty,2 Bd. V. Parvan, 1900 Timisoara, Romania

isar@ee.utt.ro

Keywords: instantaneous frequency, time-frequency rep-resentations, mathematical morphology.

Abstract

Any time-frequency representation of a signal contains cru-cial information on the characteristics of the signal. In thispaper we deal with the estimation of the instantaneous fre-quency of signals corrupted by additive noise. We propose arobust instantaneous frequency’s estimation procedure basedon the use of the computation of a time-frequency represen-tation of the analyzed signal, followed by the computationof the skeleton of the image obtained. Two examples for theoperation mode of the proposed method are presented. Theseexamples prove the robustness of this estimation method.The signals analyzed have low signal to noise ratios.

1 The Estimation of the Instantaneous Fre-quency

The estimation of the instantaneous frequency of a mono-component signal, not corrupted by noise, is a problem al-ready studied [1], [2]. When the useful signal is multicompo-nent or it is perturbed by additive noise the estimation prob-lem is more complicated and the algorithms already reportedgenerally don’t work. This is the reason why we propose herea method based on the use of time-frequency representations.These distributions have two useful properties:

1. They have a very good concentration around the curveof the instantaneous frequency of the analyzed signal, [3];

2. They realize a diffusion of the perturbation noise’spower in the time-frequency plane.

So, computing the time-frequency representation of theanalyzed signal:x (t) = s (t) + n (t), wheren (t) is theperturbation, we can obtain a good estimation of the ridgesof the time-frequency representation of the signals (t). Pro-jecting these ridges on the time-frequency plane we obtaina good estimation of the instantaneous frequency of the sig-nals (t) . There are many methods to estimate the ridges of atime-frequency representation. Some of them are presentedin [4]. We propose here a new method based on the use ofmathematical morphology.

This paper has the following structure. The second para-graph analyzes the role of the time-frequency representationsin the implementation of the method of instantaneous fre-quency estimation. The reasons for the selection of the Gabortransform are envisaged. The noise’s spreading effect in thetime-frequency plane is proved. The role of the mathematicalmorphology operators in the instantaneous frequency estima-tion process is analyzed in paragraph 3. The aim of this pa-per, the algorithm of the new estimation method is presentedin paragraph 4. The fift paragraph is dedicated to the presen-tation of two simulation results. The paper’s conclusion ispresented in the last paragraph.

2 The role of the time-frequency representa-tions

The role of the time-frequency representation in our estima-tion method is to spread the noise in the time-frequency planeand to locate the ridges of the time-frequency representationof the useful signal. There are a lot of time-frequency rep-resentations: the Short-time Fourier transform, the wavelettransform (linear representations), the members of the Cohenclass, etc. (bi-linear representations). Some of them realize agood localization of the ridges of the analyzed signal. A verygood example is the Wigner-Ville distribution, [5].

For the monocomponent signal not perturbed by noise:

s (t) = cos(

2πt2)

the associated analytical signal is:

sa (t) = e−j2π t2

and the instantaneous frequency is:

fi (t) = 2t

The Wigner-Ville representation of the signalsa (t) is:

TFW−Vsa

(t, ω) = 2πδ (ω − 4πt)

So, this time-frequency representation is perfectly concen-trated on the curveω = 2πfi (t) of the instantaneous pul-sation of the signals (t) . Hence for the estimation of the

instantaneous frequency of the signals (t) the better time-frequency representation is the Wigner-Ville distribution.

But for the instantaneous frequency estimation of mul-ticomponent signals or of signals perturbed by noise, thelinear time-frequency representations are more useful dueto the presence of the interference terms of bilinear time-frequency representations. The good concentration aroundthe instantaneous frequency law properties of the linear time-frequency representations of signals with double modulationof the form:

s (t) = A (t) ejb(t)

are proved in [2]. These time-frequency representationsrealize an important diffusion of the noise in the time-frequency plane. A good example for this second classof time-frequency representations is the short time Fouriertransform. We have find by simulations that the best time-frequency representation for the estimation of the instanta-neous frequency of a signal corrupted by additive noise isthe Gabor transform. Perhaps this useful property of the Ga-bor time-frequency representation is due to the fact that thistime-frequency representation realizes the better localizationin the time-frequency plane (see the Heisenberg principle).

2.1 Spreading the noise in the time-frequency plane

Let TFGx (t, ω) be the Gabor transform of the signalx (t) .

Because this is a linear time-frequency representation it canbe written :

TFGx (t, ω) = TFG

s (t, ω) + TFGn (t, ω)

whereTFGs (t, ω) represents the useful part of the time-

frequency representation andTFGn (t, ω) represents the

noise in the time-frequency domain. So, in the time-frequency domain the perturbation is also additive, becausethe representation is linear. This is an important reason touse a linear time-frequency representation for the estimationof the instantaneous frequency. Ifn (t) is a stationary ran-dom signal with zero mean, the mean ofTFG

n (t, ω) is equalwith:

E{

TFGn (t, ω)

}

=∞∫

−∞E {n (τ)} g (τ − t) e−jωτdτ = 0

whereg (τ) is a Gaussian window. So the mean of thenoiseTFG

n (t, ω) is equal with zero.In figure 1 is presented a system that implements the Gabor

transform.We prove the following proposition:

Proposition 2.1 The energy of the output noise of the systemin figure 1 is inferior to the energy of the input noise.

Proof.

u (t) = n (t) e−jωt

Figure 1: The computation of the Gabor transform.

The energy of the signalu (t) is:

‖u (t)‖2 =∥

∥n (t) e−jωt∥

∥

2=

∞∫

−∞n (t) e−jωtn∗ (t) ejωtdt =

∞∫

−∞|n (t)|2 dt = En

So, the energy of the signalu (t) is equal with the energyof the signaln (t) .

The Fourier transform of the signal at the output of thesystem in figure 1 is:

Z (ω) = U (ω)G (−ω)

So the energy of the output signal of the system in figure1 is:

Ez = 12π

∞∫

−∞|Z (ω)|2 dω =

12π

∞∫

−∞|U (ω)|2 |G (−ω)|2 dω ≤

≤ 12π

∞∫

−∞|U (ω)|2 dω

∞∫

−∞|G (−ω)|2 dω

But:

∞∫

−∞|G (−ω)|2 dω = 1

So:

Ez ≤ Eu

The proposition is proved. So the Gabor transform realizesa diffusion of the noise in the time-frequency plane. Addingthe fact that only for the Gabor transform in the Heisenberginequality appears the sign equal we finish the explanationof our selection of this time-frequency representation for theestimation of the instantaneous frequency of the signals (t).

3 The role of the mathematical morphologyoperators

We use some mathematical morphology operators to esti-mate the ridges of the time-frequency representation. Thereare two goals of this estimation procedure:

- to de-noise the time-frequency representation;- to extract the skeleton of the time-frequency representa-

tion.

In the following we present a list of mathematical mor-phology operators useful for the estimation of the ridges ofthe time-frequency representation:

- The conversion in the binary form. This operator real-izes a thresholding of the time-frequency representation im-age. In fact this is a denoising procedure. A statistical anal-ysis of this operator is presented in [6].So, the effect of theuse of this operator is a denoising of the image of the time-frequency representation.

- The R-h-maxima operator. Using this operator the peakregions of the time-frequency representation are extracted.So, this operator is useful for the detection of the ridges ofthe time-frequency representation.

- The exclusive or operator. Taking into account the ran-dom nature of the noiseTFG

n (t, ω), the use of a boolean op-erator having for entries two slightly different versions of theimage of the time-frequency representation (the image ob-tained after the conversion in the binary form and the imageobtained after the application of the R-h-maxima operator)has a denoising effect.

- The skeleton operator. Using this operator an estimateof the ridges of the time-frequency representation can be ob-tained.

All these operators are very well described in [7].

4 The new algorithm

The algorithm that represents the aim of this paper has thefollowing steps:

1. The Gabor transformation of the signalx (t) is com-puted.

2. After the conversion of this image in the binary formthe image Im1 is obtained.

3. Using the R-h-maxima operator the regions where be-long the peaks of the image Im1 are located. A new imageIm2 is obtained.

4. Applying the exclusive or operator to the images Im1and Im2 the details in the image Im1 caused by the noise areeliminated. A new image, containing only the peaks of thetime-frequency representationTFG

s (t, ω) is obtained.5. Applying the skeleton operator to the last image an es-

timation of the instantaneous frequency of the signals (t) isobtained. This image represents the result of our estimationmethod.

5 Simulation results

5.1 First Example

In the first example we use the signal presented in figure 2.s (t) is a monocomponent signal. It’s modulation law is lin-ear. It has constant amplitude. The perturbationn (t) is atrain of noise pulses. This kind of perturbation appears fre-quently in practice. The value of the signal to noise ratio(SNR) of the analyzed signal is small.

Figure 2: The signal x(t) (SNR=1.3).

The Gabor type time-frequency representation of the sig-nal x (t) is computed and it’s image is represented in fig.3.

Figure 3: The time-frequency representation.

After the conversion of this image in the binary form theresult presented in figure 4 is obtained. Applying to the sameimage the R-h-maxima operator we have obtained the imagein figure 5 .

Figure 4: After the conversion in binary form.

Figure 5: After the computation of R-h-maxima.

Applying the exclusive or operator to the last two imagesthe result presented in figure 6 is obtained.

Finally after the application of the skeleton operator theestimation of the instantaneous frequency of the signals (t),

Figure 6: The effect of the exclusive or operator.

the deterministic part of the signalx (t) , presented in figure7, is obtained. Making the difference between the true instan-taneous frequency of the signals (t) and the result presentedin the last figure we have obtained little deviations. So theestimation method proposed is precise and robust.

Figure 7: The estimation result.

5.2 Second Example

In the second example we use the signal presented in fig-ure 8.s (t) is a multicomponent signal representing the sumof two frequency modulated signals. The first one has aquadratic modulation law and the second a linear modula-tion law. These two signals have constant amplitudes. Theperturbationn (t) is of the same kind like in the first exam-ple. The value of the signal to noise ratio of the analyzedsignal is of 0.75.

Figure 8: The analyzed signal.

The time-frequency representation of Gabor type of thissignal is presented in figure 9. The noise spreading and thetime-frequency representation concentration around the in-stantaneous frequency law effects are obvious.

The result of the instantaneous frequency estimation pro-cedure proposed in this paper can be seen in figure 10.

Figure 9: Time-frequency representation.

Figure 10: The estimation result.

The two types of modulation laws, quadratic and linear,can be easy recognized. The most important estimation er-rors appear at the intersection of the modulation laws (in thecircled zone in figure 10).

The relative error’s value for the estimation of the instanta-neous frequency in example 1, in the absence of perturbationis not bigger than a superior bound of 0.5 %. This value isaffected in the presence of noise by a value smaller than 1%. For the estimation presented in the second example thebigger value of the relative error is smaller than 5 %. This isremarkable because the identification of the two modulationlaws using the figure 8 seems to be impossible.

6 Conclusion

The instantaneous frequency estimation method proposed inthis paper has performances similar with the methods pro-posed in [4], [8], [9] and [10]. This is a complete method,after the acquisition of the signalx (t), the plot of the in-stantaneous frequency of the signals (t) is directly obtained.The method is quite universal, the SNR of the input signalcan be very small and the result is not affected by the statis-tics of the perturbationn (t). For example similar results canbe obtained for white Gaussian noise. For signalsx (t) ofconstant amplitude, using the method proposed in this pa-per, the reconstruction can be achieved too. So, this methodcan be regarded like a denoising method for frequency mod-ulated signals with constant amplitude, when it is equipped

with a signal generator too. Knowing the instantaneous fre-quency law the frequency modulated signal with constantamplitude can be synthesized very easy. For the case ofthe use of the time-frequency representation of continuouswavelet transform type a similar conclusion is reported in[8]. Such a method outperforms the majority of denoisingmethods for the frequency modulated signals. The instanta-neous frequency estimation method can be extended to theuse of bilinear time-frequency representations if a realloca-tion method [11], [12] for the rejection of the interferenceterms is used. This is the idea of a future work. The classof morphological operators used in this estimation methodcan be extended too. Another future research direction is thestatistical analysis of the proposed method. An interestingwork with this subject is [13]. Other useful references are[14]-[26]. The algorithm proposed in this paper is of empir-ical nature. We have not found yet the better explanationsfor the parameters selection required in every step of the im-plementation. The aim of this paper is only to propose analternative method for the instantaneous frequency estima-tion, based on the conjoint use of two very modern theories,that of time-frequency representations and that of mathemat-ical morphology. This connection is very important becausethe time-frequency representations are generally used for theprocessing of signals with only one dimension and the math-ematical morphology is used to process images. Our propo-sition permits to use the image processing techniques to theanalysis of monodimensional signals. This strategy permitsthe enhancement of the set of signal processing methods withthe aid of some methods developed in the context of imageprocessing. This contribution of the image processing theoryto the development of the signal processing theory is veryimportant tacking into account the fact that at the basis of thedevelopment of the image processing theory lies the signalprocessing theory.

The estimation method proposed in this paper can be usedin a lot of applications. Some of them, like radar, sonar, ortelecommunications are already recognized as applicationsof the time-frequency representations theory. This methodcan be used in measurements, too.

7 Acknowledgment

This work was realized under the Grant MCT 3019/98, of theRomanian Research and Technology Minister. The subject ofthis grant was the theory of time-frequency representations.The authors want to thanks to Professor Francoise Preteuxfor their instruction in the mathematical morphology field.She presented in their university a cycle of conferences onthe subject of mathematical morphology. The authors wantto thank to Professor Ioan Nafornita, the director of their re-search team, specialized in the field of time-frequency repre-sentations for his continuous help. Finally the authors wantto thank to Professor Mircea Sofonea for his important sup-port consisting in signaling and sending of bibliographic ref-erences.

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