Adaptive Fourier Decomposition Based Time-Frequency Analysis

Li-Ming Zhang

1. Introduction

The main objective of time-frequency analysis is to provide the energy density of a signal simultaneously in time and frequency domains, which is of particular importance in dealing with non-stationary signals[1]-[5]. In the time- frequency literature, frequency is defined as the derivative of the phase of the signal[1]. The physical meaning of frequency is the number of the vibrations during a unit time duration. It should be a non-negative quantity varying along with time. A sophisticated but delicate way of defining time-varying amplitude-phase representation, and instantaneous frequency as well, is through an application of Hilbert transformation. Such defined instantaneous frequency (IF) is called analytic IF[3],[6].

The power of Fourier analysis is that it can decompose a signal into individual frequency components and establish the relative intensity of each component[7]. It builds a transformation bridge for a signal between the time domain

L.-M. Zhang is with the Faculty of Science and Technology, University of Macau, Macau, China. (Corresponding author e-mail: lmzhang@umac.mo).

Color versions of one or more of the figures in this paper are available online at http://www.intl-jest.com.

Digital Object Identifier: 10.3969/j.issn.1674-862X.2014.02.012 and frequency domain. In Fourier analysis, each exponential has a constant frequency. The spectra of the frequencies can be easily calculated. This provides a broad overview of the nature of the signal, which is important for theoretical studies.

Although the Fourier transform is widely used, some restrictions have been well noted. There are two major restrictions.

· For a given signal, the Fourier transform always expresses it into a linear combination of simple but fixed sine and cosine functions. Such expansions are argued to be impossible to represent local characteristic properties,especially the time-varying frequencies of the signal. The Fourier analysis suffers from absence of time localization.In this situation, IF in the Fourier analysis is interpreted as the average frequency of the component signal. In the Fourier analysis, the spectrum allows us to determine which frequencies exist and how much is the overall quantity over the whole time range, but it does not allow us to determine which frequencies exist at a particular time moment and how much is the quantity[1].

· For a given signal, the Fourier decomposition starts with finding the spectra in the order of sin x and cosx,sin2x and cos2x, ⋅⋅, sinnx and cosnx, and so on.The decomposition may lead to slow convergence due to the fact that the principal sine and cosine components contributing significant shares of the total energy of the original signal may arrive late. That is, it would result in a large iteration number n to reach the required approximation.

A novel signal decomposition method, adaptive Fourier decomposition (AFD), was proposed recently by Qian et al with proven mathematical foundation[6],[8]. AFD, built upon harmonic and analytic function theory, belongs to the category of mono-component decomposition[9]that contains various Fourier type decompositions as particular cases. It breaks through the two mentioned restrictions of the Fourier decompositions. AFD has the following characteristic properties:

· For a given signal, AFD decomposes it into a linear combination of analytic signals of which each has a non-negative analytic IF function in the whole time range(mono-component). In this situation, the decomposition by AFD allows to determine both the existing frequencies and the densities at any time moment. So, AFD provides a time-varying amplitude-frequency representation of the signal.

· For a given signal, the AFD algorithm starts with finding a mono-component that is in the energy sense closest to the given signal. Furthermore, at every consecutive step, it applies the same energy principle to find a closest mono-component to the reduced remainder.The principle is similar with the “maximal selection principle”[10]. It is the reason why the decomposition is regarded as adaptive. The decomposition usually leads to fast convergence. That is, it needs a small number of iterations to reach the required approximation accuracy.

The detailed AFD algorithm was presented in [11]. This article mainly makes comparisons between AFD and the Fourier transform, and AFD and the short-time Fourier transform in different aspects. The short-time Fourier transform is among the most commonly used methods to study time-varying signals[12]. However, there are two disadvantages in applying it. First, there exist natural and man-made signals whose spectral contents change rapidly so that finding an appropriate short-time window is problematic. In such cases, there may not be any time interval in which the signals is more or less stationary.Secondly, decreasing the width of the window function to locate events in time usually results in reducing the frequency resolution[12]. Through AFD, a time-varying spectrum is obtained out of the composing monocomponents. Meanwhile, it does not cause parameter selection problems, such as window size selection in the short-time Fourier transform.

The purpose of this article is to address some time-frequency distribution aspects of AFD that have not been concerned in the literature. Experiments are conducted and compared with the Fourier transform in the respect of frequency spectrum, and with the short-time Fourier transform in the respect of spectrogram. This paper is organized as follows. The AFD theory foundation is briefly introduced in Section 2. The principles of the AFD based time-frequency analysis are presented in Section 3. The experiment results are shown in Section 4. The conclusions are drawn in Section 5.

2. Brief Theory Foundation of AFD

AFD was motivated by the fact that Gabor’s[13]method does not always produce analytic signals of non-negative IF[9]. To stick on the analytic signal idea, one has to accept that not all analytic signals have non-negative analytic phase derivative, or IF. This suggests seeking ways to approximate the given signal by appropriate basic signals with non-negative analytic IF, viz., mono-components[9].Progress has been made along this direction in which AFD is a core concept in the theory and practice. The brief explanation of AFD is provided below. The detailed theoretical foundation can refer to [6], [8], [11], and [14].

AFD decomposes any given analytic signal of finite energy into a linear combination of a type of basic normal mono-component signals called modified Blaschke products. For a real-valued signal f defined on the unit circle, AFD is to be applied to its Hardy space projection:

where Hf is the circular Hilbert transform of f, i denotes the complex number, and0c is a constant.

where 〈⋅〉 means the inner product and

The algorithm stops at the step N such that

for the first time. The approximation to the original real-valued function f is

where θk( t) represents the polar coordinators, and

Practically we take the initial value1=0.a

This article makes use the algorithm code in the homepage of Qian: http://www.fst.umac.mo/en/staff/fsttq.html. There are two AFD algorithms. The algorithm used in this paper is the one with the main function:find_poles.

3. AFD Based Time-Frequency Analysis

AFD has two salient characteristics. One is the welldefined IF in each composing mono-component; the other is the time-frequency distribution that is the densities represented in the time and frequency simultaneously. AFD based time-frequency analysis includes three aspects[15].

3.1 IF Analysis of Signal

The first aspect is the analysis of the well defined time varying non-negative frequency. Let f be a given realvalued signal and the series (2) is its AFD. The instantaneous amplitude ρk( t) is explicitly given in (9),and then we have

The mean IF at the time moment t is the weighted average of all the θk′(t):

It is, in fact, a marginal distribution.θ′(t) is time varying,which can be considered as a candidate for IF of the original f. In the experimentθ′(t) is to be compared with the mean of the Fourier frequencyl = ∑ l | c |2∑|c |2ll which is a constant.

3.2 Frequency Spectrum Analysis of Signal

The frequency spectrum is defined to be

It accumulates all the amplitude values corresponding to a given frequency. This is, in fact, the second marginal distribution. The analysis in frequency spectrum is similar to that of the Fourier frequency spectrum. In the latter, ω has to be integers, and corresponding to each admissible ω,there is only one spectrum that is a constant.

3.3 Spectrogram or Time-Frequency Distribution Analysis

Combining the IF and frequency spectrum, we get the time-frequency distribution. It is to be compared with the short-time Fourier transform. As an example, now we treat a frequency band analysis. For a given band of frequency(ω1,ω2) , the time band of the original signal corresponding to the given frequency band can be computed. The fraction of the energy in the corresponding frequency and time band can also be worked out.

The kth mono-component in AFD is denoted by

non-overlap, and lkis the number of the intervals. The time band is given by

∞

in which and only in which the original signal f has its frequency in (ω1,ω2) We can extract out the mono-components together with the time intervals A, which produces the corresponding frequencies

where

We will call it the constructed fraction for the frequency bbaanndd ((ω11, , ω22)).. CCoommppaarraattiivveellyy tthhee rreessttrriiccttiioonn ( 2oo)ff tthhee original function to the time band T(ω1, ω2) iscalled the restricted fraction for the time band T(ω1,ω2) .

For each k, due to the non-overlap property of(,), the energy ration (ER) is

where for k=1, then ρ1cosθ1is replaced by c0. The ER for f(ω1,ω2)is reasonably defined to be the weighted average of the above energy ratio values depending on the k’s value, namely,

4 Experiment Results

The signal chosen for the experiment is composed of two sinusoid components.

4.1 IF Analysis of the Signal

The original signal s and the AFD based reconstructed signal AFD are illustrated in Fig. 1. The original signal s and Fourier decomposition based reconstructed signal FD are shown in Fig. 2. In both the algorithms the signal is expanded into its 45 partial sums. The energy differences between the original and the reconstructed signals are,respectively, 0.002 in AFD and 0.027 in the Fourier method.It is an example to exhibit that AFD performs better in approximation than the Fourier method. The convergence rate of AFD is 10 times faster than that of Fourier decomposition.

After the 45 times iterations with AFD, the corresponding IFs graphs are plotted and shown in Fig. 3.The mean IF is illustrated in Fig. 4 with the comparison with the mean of Fourier frequency (FD in Fig. 4) shown as a constant.

4.2 Frequency Spectrum Analysis of Signal

The AFD and Fourier decomposition based frequency spectra are illustrated in Fig. 5 and Fig. 6, respectively.Note that there are two frequency peaks around 10 and 40 in both figures, and the peak at around 40 is about 2 times of that at around 10.

Fig. 1. Original and AFD based reconstructed signals.

Fig. 2. Original and Fourier reconstructed signals.

Fig. 3. IF of all mono-components.

Fig. 4. Mean IF in AFD and the mean frequency in Fourier transform.

Fig. 5. Frequency spectrum based on AFD decomposition.

Fig. 6. Frequency spectrum based on Fourier decomposition.

Fig. 7. Spectrogram of AFD.

Fig. 8. Spectrogram of short-time Fourier transform.

4.3 Spectrogram Analysis

Fig. 9. AFD-constructed fraction signals for the frequency band from 30 to 50.

The time-frequency distribution under AFD and that under short-time Fourier transform are illustrated in Fig. 7 and Fig. 8, respectively. Both figures demonstrate two major frequencies around 10 and 40 that have stronger intensities. Comparatively, the intensities is more concentrated by using AFD than those by using the short-time Fourier transform. Choosing a frequency range 30 to 50, the corresponding time band crosses the whole time range. In this situation, the restricted fraction is identical with the original signal. The constructed fraction f(30,50)through the mono-components containing the frequency band from 30 to 50 is illustrated in Fig. 9. The constructed fraction in the figure is the approximation for 2cos(40.5)t.

5 Conclusions

An AFD based time-frequency analysis method is presented for the IF analysis, frequency spectrum analysis,and spectrogram analysis. A function with two sinusoid components is chosen for the experiment so that the kind of frequencies can be explicitly viewed. The sinusoid case is also suitable for the Fourier transform. So in the experiment,the Fourier analysis methods are comparable to the AFD based one. The experiment results show that the AFD based time-frequency analysis performs better in all signal analysis aspects. Due to the page limitation there is no experiment on non-sinusoid signals reported in this paper. It has been observed that AFD-based methods have great advantages in the analysis to non-sinusoid signals and real data.

[1] L. Cohen, Time-Frequency Analysis, Upper Saddle River:Prentice Hall, 1995.

[2] B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal―part 1: Fundamentals,” Proc. of the IEEE, vol. 80, no. 4, pp. 520-538, 1992.

[3] B. Picinbono, “On instantaneous amplitude and phase of signals,” IEEE Trans. on Signal Processing, vol. 45, no. 3,pp. 552-560, 1997.

[4] L. Stankovic, “A method for time-frequency analysis,” IEEE Trans. on Signal Processing, vol. 42, no. 1, pp. 225-229,1994.

[5] B. Lovell, R. Williamson, and B. Boashash, “The relationship between instantaneous frequency and time-frequency representations,” IEEE Trans. on Signal Processing, vol. 41,no. 3, pp. 1458-1461, 1993.

[6] T. Qian, “Intrinsic mono-components decomposition of functions: An advance of Fourier theory,” Math. Meth. Appl.Sci., vol. 33, no. 7, pp. 880-891, 2010.

[7] P. Butzer and R. Nessel, Fourier Analysis and Approximation, Volumn 1: One-dimensional Theory, New York: Birkhauser, Basel, and Academic Press, 1971.

[8] T. Qian and Y. Wang, “Adaptive Fourier series―A variation of greedy algorithm,” Advances in Computational Mathematics, vol. 34, no. 3, pp. 279-293, 2011.

[9] T. Qian, “Mono-components for decomposition of signals,”Math. Meth. Appl. Sci., vol. 29, no. 10, pp. 1187-1198,2006.

[10] S. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. on Signal Processing, vol. 41, no. 12, pp. 3397-3415, 1993.

[11] T. Qian, L. M. Zhang, and Z. Li, “Algorithm of adaptive Fourier transform,” IEEE Trans. on Signal Processing, vol.59, no. 12, pp. 5899-5906, 2011.

[12] L. Cohen, “Time-frequency distributions―a review,” Proc.of the IEEE, vol. 77, no. 7, pp. 941-981, 1989.

[13] D. Gabor, “Theory of communication,” Journal of the IEEE,vol. 93, pp. 429-457, 1946.

[14] T. Qian, L.-M. Zhang, and H. Li, “Mono-components in signal decomposition,” Int. Journal of Wavelets.Multiresolution and Information Processing, vol. 6, no. 3, pp.353-374, 2008.

[15] B. Boashash. TFSA 6.0, Time-Frequency Signal Analysis Package. UQ Centre for Clinical Research, The University of Queensland, Queensland, 2010.