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

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1 A new algorithm for the estimation of the instantaneous frequency of a signal 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 representations, mathematical morphology. Abstract Any time-frequency representation of a signal contains crucial information on the characteristics of the signal. In this paper we deal with the estimation of the instantaneous frequency of signals corrupted by additive noise. We propose a robust instantaneous frequency s estimation procedure based on the use of the computation of a time-frequency representation of the analyzed signal, followed by the computation of the skeleton of the image obtained. Two examples for the operation mode of the proposed method are presented. These examples prove the robustness of this estimation method. The signals analyzed have low signal to noise ratios. 1 The Estimation of the Instantaneous Frequency The estimation of the instantaneous frequency of a monocomponent signal, not corrupted by noise, is a problem already studied [1], [2]. When the useful signal is multicomponent or it is perturbed by additive noise the estimation problem is more complicated and the algorithms already reported generally don t work. This is the reason why we propose here a method based on the use of time-frequency representations. These distributions have two useful properties: 1. They have a very good concentration around the curve of the instantaneous frequency of the analyzed signal, [3]; 2. They realize a diffusion of the perturbation noise s power in the time-frequency plane. So, computing the time-frequency representation of the analyzed signal: x (t) = s (t) + n (t), where n (t) is the perturbation, we can obtain a good estimation of the ridges of the time-frequency representation of the signal s (t). Projecting these ridges on the time-frequency plane we obtain a good estimation of the instantaneous frequency of the signal s (t). There are many methods to estimate the ridges of a time-frequency representation. Some of them are presented in [4]. We propose here a new method based on the use of mathematical morphology. This paper has the following structure. The second paragraph analyzes the role of the time-frequency representations in the implementation of the method of instantaneous frequency estimation. The reasons for the selection of the Gabor transform are envisaged. The noise s spreading effect in the time-frequency plane is proved. The role of the mathematical morphology operators in the instantaneous frequency estimation process is analyzed in paragraph 3. The aim of this paper, the algorithm of the new estimation method is presented in paragraph 4. The fift paragraph is dedicated to the presentation of two simulation results. The paper s conclusion is presented in the last paragraph. 2 The role of the time-frequency representations The role of the time-frequency representation in our estimation method is to spread the noise in the time-frequency plane and to locate the ridges of the time-frequency representation of the useful signal. There are a lot of time-frequency representations: the Short-time Fourier transform, the wavelet transform (linear representations), the members of the Cohen class, etc. (bi-linear representations). Some of them realize a good localization of the ridges of the analyzed signal. A very good example is the Wigner-Ville distribution, [5]. For the monocomponent signal not perturbed by noise: s (t) = cos ( 2πt 2) the associated analytical signal is: j2π t2 s a (t) = e and the instantaneous frequency is: f i (t) = 2t The Wigner-Ville representation of the signal s a (t) is: T F W V s a (t, ω) = 2πδ (ω 4πt) So, this time-frequency representation is perfectly concentrated on the curve ω = 2πf i (t) of the instantaneous pulsation of the signal s (t). Hence for the estimation of the

2 instantaneous frequency of the signal s (t) the better timefrequency representation is the Wigner-Ville distribution. But for the instantaneous frequency estimation of multicomponent signals or of signals perturbed by noise, the linear time-frequency representations are more useful due to the presence of the interference terms of bilinear timefrequency representations. The good concentration around the instantaneous frequency law properties of the linear timefrequency representations of signals with double modulation of the form: s (t) = A (t) e jb(t) are proved in [2]. These time-frequency representations realize an important diffusion of the noise in the timefrequency plane. A good example for this second class of time-frequency representations is the short time Fourier transform. We have find by simulations that the best timefrequency representation for the estimation of the instantaneous frequency of a signal corrupted by additive noise is the Gabor transform. Perhaps this useful property of the Gabor time-frequency representation is due to the fact that this time-frequency representation realizes the better localization in the time-frequency plane (see the Heisenberg principle). 2.1 Spreading the noise in the time-frequency plane Let T F G x (t, ω) be the Gabor transform of the signal x (t). Because this is a linear time-frequency representation it can be written : T F G x (t, ω) = T F G s (t, ω) + T F G n (t, ω) where T Fs G (t, ω) represents the useful part of the timefrequency representation and T Fn G (t, ω) represents the noise in the time-frequency domain. So, in the timefrequency domain the perturbation is also additive, because the representation is linear. This is an important reason to use a linear time-frequency representation for the estimation of the instantaneous frequency. If n (t) is a stationary random signal with zero mean, the mean of T Fn G (t, ω) is equal with: E { T F G n (t, ω) } = E {n (τ)} g (τ t) e jωτ dτ = 0 where g (τ) is a Gaussian window. So the mean of the noise T F G 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 system in 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 signal u (t) is: u (t) 2 = n (t) e jωt 2 = n (t) 2 dt = E n n (t) e jωt n (t) e jωt dt = So, the energy of the signal u (t) is equal with the energy of the signal n (t). The Fourier transform of the signal at the output of the system in figure 1 is: Z (ω) = U (ω) G ( ω) So the energy of the output signal of the system in figure 1 is: But: So: E z = 1 1 2π 1 2π 2π Z (ω) 2 dω = U (ω) 2 G ( ω) 2 dω U (ω) 2 dω G ( ω) 2 dω = 1 E z E u G ( ω) 2 dω The proposition is proved. So the Gabor transform realizes a diffusion of the noise in the time-frequency plane. Adding the fact that only for the Gabor transform in the Heisenberg inequality appears the sign equal we finish the explanation of our selection of this time-frequency representation for the estimation of the instantaneous frequency of the signal s (t). 3 The role of the mathematical morphology operators We use some mathematical morphology operators to estimate the ridges of the time-frequency representation. There are two goals of this estimation procedure: - to de-noise the time-frequency representation; - to extract the skeleton of the time-frequency representation.

3 In the following we present a list of mathematical morphology operators useful for the estimation of the ridges of the time-frequency representation: - The conversion in the binary form. This operator realizes a thresholding of the time-frequency representation image. In fact this is a denoising procedure. A statistical analysis of this operator is presented in [6].So, the effect of the use of this operator is a denoising of the image of the timefrequency representation. - The R-h-maxima operator. Using this operator the peak regions of the time-frequency representation are extracted. So, this operator is useful for the detection of the ridges of the time-frequency representation. - The exclusive or operator. Taking into account the random nature of the noise T F G n (t, ω), the use of a boolean operator having for entries two slightly different versions of the image of the time-frequency representation (the image obtained after the conversion in the binary form and the image obtained after the application of the R-h-maxima operator) has a denoising effect. - The skeleton operator. Using this operator an estimate of the ridges of the time-frequency representation can be obtained. All these operators are very well described in [7]. 4 The new algorithm The algorithm that represents the aim of this paper has the following steps: 1. The Gabor transformation of the signal x (t) is computed. 2. After the conversion of this image in the binary form the image Im1 is obtained. 3. Using the R-h-maxima operator the regions where belong the peaks of the image Im1 are located. A new image Im2 is obtained. 4. Applying the exclusive or operator to the images Im1 and Im2 the details in the image Im1 caused by the noise are eliminated. A new image, containing only the peaks of the time-frequency representation T Fs G (t, ω) is obtained. 5. Applying the skeleton operator to the last image an estimation of the instantaneous frequency of the signal s (t) is obtained. This image represents the result of our estimation method. Figure 2: The signal x(t) (SNR=1.3). The Gabor type time-frequency representation of the signal 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 the result presented in figure 4 is obtained. Applying to the same image the R-h-maxima operator we have obtained the image in figure 5. Figure 4: After the conversion in binary form. 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 linear. It has constant amplitude. The perturbation n (t) is a train of noise pulses. This kind of perturbation appears frequently in practice. The value of the signal to noise ratio (SNR) of the analyzed signal is small. Figure 5: After the computation of R-h-maxima. Applying the exclusive or operator to the last two images the result presented in figure 6 is obtained. Finally after the application of the skeleton operator the estimation of the instantaneous frequency of the signal s (t),

4 Figure 6: The effect of the exclusive or operator. the deterministic part of the signal x (t), presented in figure 7, is obtained. Making the difference between the true instantaneous frequency of the signal s (t) and the result presented in the last figure we have obtained little deviations. So the estimation method proposed is precise and robust. Figure 9: Time-frequency representation. 5.2 Second Example Figure 7: The estimation result. In the second example we use the signal presented in figure 8. s (t) is a multicomponent signal representing the sum of two frequency modulated signals. The first one has a quadratic modulation law and the second a linear modulation law. These two signals have constant amplitudes. The perturbation n (t) is of the same kind like in the first example. The value of the signal to noise ratio of the analyzed signal is of Figure 10: The estimation result. The two types of modulation laws, quadratic and linear, can be easy recognized. The most important estimation errors appear at the intersection of the modulation laws (in the circled zone in figure 10). The relative error s value for the estimation of the instantaneous frequency in example 1, in the absence of perturbation is not bigger than a superior bound of 0.5 %. This value is affected in the presence of noise by a value smaller than 1 %. For the estimation presented in the second example the bigger value of the relative error is smaller than 5 %. This is remarkable because the identification of the two modulation laws using the figure 8 seems to be impossible. 6 Conclusion Figure 8: The analyzed signal. The time-frequency representation of Gabor type of this signal is presented in figure 9. The noise spreading and the time-frequency representation concentration around the instantaneous frequency law effects are obvious. The result of the instantaneous frequency estimation procedure proposed in this paper can be seen in figure 10. The instantaneous frequency estimation method proposed in this paper has performances similar with the methods proposed in [4], [8], [9] and [10]. This is a complete method, after the acquisition of the signal x (t), the plot of the instantaneous frequency of the signal s (t) is directly obtained. The method is quite universal, the SNR of the input signal can be very small and the result is not affected by the statistics of the perturbation n (t). For example similar results can be obtained for white Gaussian noise. For signals x (t) of constant amplitude, using the method proposed in this paper, the reconstruction can be achieved too. So, this method can be regarded like a denoising method for frequency modulated signals with constant amplitude, when it is equipped

5 with a signal generator too. Knowing the instantaneous frequency law the frequency modulated signal with constant amplitude can be synthesized very easy. For the case of the use of the time-frequency representation of continuous wavelet transform type a similar conclusion is reported in [8]. Such a method outperforms the majority of denoising methods for the frequency modulated signals. The instantaneous frequency estimation method can be extended to the use of bilinear time-frequency representations if a reallocation method [11], [12] for the rejection of the interference terms is used. This is the idea of a future work. The class of morphological operators used in this estimation method can be extended too. Another future research direction is the statistical analysis of the proposed method. An interesting work with this subject is [13]. Other useful references are [14]-[26]. The algorithm proposed in this paper is of empirical nature. We have not found yet the better explanations for the parameters selection required in every step of the implementation. The aim of this paper is only to propose an alternative method for the instantaneous frequency estimation, based on the conjoint use of two very modern theories, that of time-frequency representations and that of mathematical morphology. This connection is very important because the time-frequency representations are generally used for the processing of signals with only one dimension and the mathematical morphology is used to process images. Our proposition permits to use the image processing techniques to the analysis of monodimensional signals. This strategy permits the enhancement of the set of signal processing methods with the aid of some methods developed in the context of image processing. This contribution of the image processing theory to the development of the signal processing theory is very important tacking into account the fact that at the basis of the development of the image processing theory lies the signal processing theory. The estimation method proposed in this paper can be used in a lot of applications. Some of them, like radar, sonar, or telecommunications are already recognized as applications of the time-frequency representations theory. This method can be used in measurements, too. 7 Acknowledgment This work was realized under the Grant MCT 3019/98, of the Romanian Research and Technology Minister. The subject of this grant was the theory of time-frequency representations. The authors want to thanks to Professor Francoise Preteux for their instruction in the mathematical morphology field. She presented in their university a cycle of conferences on the subject of mathematical morphology. 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