Wavelet Transform Method for the Signature Extraction of a Polynomial FM Signal Covered With Noise
Nicolae Drăghiciu, Simona Castrase, Nistor Daniel Trip, Adrian Şchiop, Cornelia Gordan, Lucian Morgoş
Department of Applied Electronics
University of Oradea
5 Armatei Romane Str., 410087, Oradea,
ROMANIA
Abstract: - The paper present some of the results obtained by the authors using an original algorithm for the instantaneous frequency IF (ridges) estimation of 4 degree polynomial frequency modulated FM signals. The method is the following: first of all is obtained the wavelet time-frequency representation of the signal. Then this image is processed by Matlab program in order to extract the local maximum. At the end, is used also Matlab to interpolate the points for the maximum and to calculate the errors characterising the wavelet representation. The 4-degree polynomial FM signals were covered with a low-pass filtered Gaussian noise.
Key-Words: - wavelet, time-frequency representation, noise variance, instantaneous frequency
1 Introduction
Wavelet theory provides a unified framework for a number of techniques, which had been developed independently for various signal-processing applications. For example, multiresolution signal processing (used in computer vision), subband coding (developed for speech and image compression) and wavelet series expansions (developed in applied mathematics), have been recently recognised as different views of a single theory.
In particular the "wavelet” transform (WT) is of interest for the analysis of non-stationary signals, because it provides an alternative to the classical Short-Time Fourier Transform (STFT) or Gabor transform. The basic difference is as follows: in contrast with the STFT, which uses a single analysis window, the WT uses short windows at high frequencies and long windows at low frequencies. The WT is also related to time-frequency analysis based on Wigner-Ville distribution.
WT of a signal contain very important informations concerning the regions from the time-frequency plane where the signal's energy is maximum. It was demonstrated that the ridges of the module of any time-frequency representation correspond to the maximum values of the signal's energy, which form the skelet of the analysed transform [3], [4]. These maxims are localised around the instantaneous frequency (IF) of the signal, which means that the detection of the ridges offers the possibility to estimate the IF and to reconstruct the original signal.
The importance of the instantaneous frequency concept stems from the fact that in many applications the signal analyst is confronted with the task of processing signals whose spectral characteristics (in particular the frequency of the spectral peaks) are varying with time. These signals are often referred to as "nonstationary". For these signals, the ridge is an important characteristic, because it is a time-varying parameter that defines the location of the signal's spectral peak as it varies with time.
2 Extraction method
The analysing method used in this paper, to estimate the instantaneous frequency of a 4 degree polynomial FM signal covered by a low-pass filtered Gaussian noise, is an original one. The practical results obtained in this case can be compared with those included in some papers mentioned in the references and which were published in IEEE Transactions and in other scientific journals [1], [2].
At the beginning, the authors used this method to estimate the IF of some linear and parabolic modulated chirps, covered or not by a zero-mean Gossip noise. The results and the conclusions were published in other papers [5]. In these cases the noise was characterised by a wide frequency band. Then the noise was filtered in order to limit its band.
The 4-degree polynomial FM signal, covered by a low-pass filtered Gaussian noise, is generated with Matlab. Then it is processed using the “wavelet” toolbox of the same program. This way can be obtained the “wavelet” time-frequency representation. In order to obtain the time-frequency localisation of the analysed signal is used an original program written also in Matlab, which eliminates the interference terms and gives a concrete, thin line approximation of the time-frequency dependence (ridges). This algorithm is based on the following assumptions:
- The time-frequency dependence is represented as a grey level image;
- The highest grey level gives the best time-frequency localisation;
- Only the major component is important because the other components represent the interference terms;
- The best time-frequency approximation is the “Water-Shed” of the major component from the image.
This original algorithm can extract efficiently the time-frequency dependence from a time-frequency signature. Also, the interference terms will be filtered out. The advantage of the presented algorithm comparing with the morphological “Water-Shed” is that the interference terms do not have a bad influence on the time-frequency localisation.
After the image segmentation and the filtering of the interference terms can be notified that the described algorithm seems to be the SKIZ (Skeleton by Influence Zone) algorithm. The main difference is the implementation of the algorithm. While the SKIZ algorithm use some increasing circles to find the central point that describes the SKIZ line, the algorithm proposed here use repeated conditional erosion. Theoretically both algorithms should provide the same result but in practice, when are used discrete images (pixel images), it becomes obvious that the implementation of a given circle will not be accurate because of the image pixel representation. So, for small width surfaces the SKIZ algorithm can provide some zigzag errors. The proposed algorithm will not suffer from this symptom.
Then, every time-frequency representation is processed using an original program written in Matlab. It localises the maximum values of the signal’s time-frequency signature. Finally, by the help of another program, developed by the authors, is obtained the curve corresponding to the detected maximum values of the analysed curve, having the width of a pixel, and which, represents the ridges of the signal.
In the end, the parameters corresponding to the curve are processed with a program that calculates the coefficients of the polynomial, which interpolates the given curve. The program develops a comparison between the analysed and the interpolated curves, and makes possible the calculus of the errors of the interpolation polynomial coefficients. These errors are calculated after a comparison between the interpolation coefficients and their initial values used to generate the signal.
3 Problem Solution
The instantaneous frequency variation law of a 4-degree polynomial frequency modulated signal is also a polynomial described by the relation:
f(t) = at3 + bt2 + ct + d (1)
Using the following definition:
(2)
can be easily obtained the phase of the FM signal, which is defined by a 4-degree polynomial:
s(t) = cos [×at4 + ×bt3 + ×ct2 + 2p ×dt ] (3)
The parameters included in relation (3) were chosen as follows: a == 18.52, b = -= -166.67, c =500×= 333.33, d == 250. In these conditions the signal is 6 seconds long, the frequency band is [50,500] Hz and the sampling frequency is equal to 2 kHz. This signal was generated using the Matlab program.
In this paper the instantaneous frequency of the signal defined with relation (3) is estimated using the wavelet time-frequency transform for the case when the signal is covered with a low-pass filtered Gaussian noise, generated also by Matlab. For this experiment was used an elliptic low-pass filter with a critic frequency of 500 Hz.
First of all, using the Matlab were obtained the Daubechies, Discrete Meyer and Haar wavelet representations of the signal covered by noise, for different values of the noise variance s2: 0.01 0.04, 0.09, 0.16 and 0.36. These representations characterised by s2=0.09 are presented in figure 1.
Using the detection algorithm, developed by the authors, were obtained the instantaneous frequency variation curves of the signal s(t) for both types of wavelets and for every value of the noise variance s2. The curves estimated for s2=0.09 are presented in figure 2. For every detected curve were also estimated the interpolation polynomials and were calculated the coefficients of these polynomials and the errors which characterise them.
In tables 1, 2, 3, 4 and 5 are presented the interpolation coefficients and their errors for the three types of wavelets and for the five different values of the noise variance mentioned above.
Table 1. s2= 0.01
Wavelet / a / b / c / d / er.a[%] / er.b
[%] / er.c
[%] / er.d
[%]
Dau -bechies / 18.57 / -167.04 / 334.08 / 250.01 / 0.279 / 0.224 / 0.223 / 0.005
Discrete
Meyer / 18.53 / -166.70 / 333.74 / 250.13 / 0.083 / 0.022 / 0.122 / 0.053
Haar / 18.49 / -166.71 / 333.88 / 250.22 / -0.165 / 0.023 / 0.165 / 0.089
Table 2. s2= 0.04
Wavelet / a / b / c / d / er.a[%] / er.b
[%] / er.c
[%] / er.d
[%]
Dau -bechies / 18.58 / -167.24 / 335.61 / 250.15 / 0.312 / 0.345 / 0.683 / 0.060
Discrete
Meyer / 18.54 / -166.72 / 333.92 / 250.87 / 0.105 / 0.030 / 0.176 / 0.348
Haar / 18.55 / -167.18 / 334.20 / 250.66 / 0.161 / 0.309 / 0.260 / 0.265
Table 3. s2= 0.09
Wavelet / a / b / c / d / er.a[%] / er.b
[%] / er.c
[%] / er.d
[%]
Dau -bechies / 18.57 / -167.41 / 335.01 / 251.28 / 0.281 / 0.445 / 0.503 / 0.510
Discrete
Meyer / 18.55 / -166.79 / 334.27 / 251.14 / 0.162 / -0.558 / 0.281 / 0.455
Haar / 18.48 / -166.01 / 332.21 / 251.75 / -0.216 / -0.392 / -0.336 / 0.701
Table 4. s2= 0.16
Wavelet / a / b / c / d / er.a[%] / er.b
[%] / er.c
[%] / er.d
[%]
Dau -bechies / 18.62 / -167.63 / 335.68 / 251.39 / 0.542 / 0.578 / 0.702 / 0.557
Discrete
Meyer / 18.57 / -167.58 / 334.35 / 251.40 / 0.423 / 0.641 / 0.305 / 0.561
Haar / 18.60 / -167.33 / 333.87 / 252.39 / 0.434 / 0.398 / 0.161 / 0.954
Table 5. s2= 0.36
Wavelet / a / b / c / d / er.a[%] / er.b
[%] / er.c
[%] / er.d
[%]
Dau -bechies / 18.66 / -168.43 / 336.61 / 248.59 / 0.761 / 1.055 / 0.983 / -0.56
Discrete
Meyer / 18.65 / -168.13 / 335.27 / 252.29 / 0.697 / 0.879 / 0.581 / 0.917
Haar / 18.69 / -168.93 / 337.58 / 245,89 / 0.950 / 1.356 / 1.275 / -1,64
a) Daubechies.
b) Discrete Meyer
c) Haar
Figure 1. Wavelet representations of s(t) when s2=0.09.
a) Daubechies
b) Discrete Meyer
c) Haar
Figure 2. Instantaneous frequency curves of s(t) estimated for s2=0.09
4 Conclusion
Analysing the results presented in Tables 1, 2, 3, 4 and 5 it has to be noticed that the errors characterising the coefficients are very small (less then 2.5%) for all three types of wavelets even when s2=0.09. Comparing the results, which were very good in all cases, the conclusion was that Discrete Meyer has the best behaviour.
It means that for the ridge extraction of the polynomial frequency modulated signals the wavelet representation has a very good behaviour, even in the case when the signal is covered with a zero-mean Gaussian noise characterised by a noise variance s2 = 0,36.
Similar conclusions were obtained by different researchers from all over the world, some of them being presented in the papers mentioned in the references.
References:
[1] Boashash B., Estimating and Interpreting the Instantaneous Frequency of a Signal -Part 1:
Fundamentals, Proceedings of the IEEE, vol 80, no.4, 1992.
[2] Boashash B., Estimating and Interpreting the Instantaneous Frequency of a Signal -Part 2:
Algorithms and Applications, Proceedings of the IEEE, vol 80, no.4 1992.
[3] Carmona R.A., Hwang L.W., Torrésani B., Multi-Ridge Detection and Time- Frequency
Reconstruction, Tech.Rep.,Princeton University, Princeton NJ,1995.
[4] Carmona R.A., Torrésani B., Characterisation of Signals by the Ridges of Their Wavelet
Transforms, IEEE Trans. signals processing, vol.45, no.10, oct.1997.
[5] Gordan C., PhD Thesis, Oradea 1999.
[6] Kadambe S., Orr S.R., Instantaneous Frequency Estimation Using the Cross-Term Deleted
Wigner Representation (CDWR) ,TFTS’96, 1996.
[7] Isar A., Naforniţă I., Reprezentări Timp-Frecvenţă, Editura Politehnica Timişoara 1998.
[8] Ali Al - Haj, A Parallel Distributed Arithmetic Implementation of the Discrete Wavelet Transform, WSEAS MMACTEE-WAMUS-NOLASC 2003
[9] Cornelia Gordan, Romulus Reiz, Mathematical Morphology Elements Used for the Ridges Extraction of Linear AM-FM Modulated Signals, WSEAS ICOSMO-ICOSSIP 2003