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Application of time-frequency spectral analysis methods.

UDC 621.391:517.951

Introduction. The spectral analysis through Fourier transform obtained an extensive use, especially when switching to discrete and fast Fourier transforms.

But this analytical method has some limitations, the main one consisting in requirement of signals stationary characteristic with the same range during whole time period from zero to infinity. Only at that precondition the spectrum frequency components are determined quite accurately, i.e. the signal spectral density is represented as delta functions with harmonics zero width.

The spectral components do loss their delta functions characteristic [delta]([omega] - n[OMEGA]) even while the stationary signal existing not infinitely but during a given lifespan 0 ... [T.sub.c]. At that the signal spectral density is extended with multiplier [gamma](j[omega]), and the spectral lines obtain their final width 2[DELTA]f = 2/[T.sub.c].

Literature review. Generic feature of modern analytical methods for non-stationary signals and dynamic modes consists in the use of two-dimensional characteristics that allow analysis implemented not by ordinary planes of time or frequency, but within a complex time-frequency plane that extends the analysis possibilities in practical applications [1 ... 4].

It is known that the Fourier transform weakness is manifested in lack of response to local changes in the signal, and impossibility to identify this signal by its frequency spectrum [4]. More convenient is to use time-frequency wavelet transform that can reflect local changes in the signal and being suitable for complex non-stationary processes analysis. Unlike the Fourier transform, the time-frequency wavelet-method allows to separate time and frequency components of non-stationary signals [2].

The real signals are continuously changing, i.e. their stationary characteristics disappears. For that consideration taking into account the time-frequency analysis methods are developed [1 ... 4].

The need to implement the time-frequency analysis methods is illustrated by the following example. Let a signal has two discrete components. If they are present at all times, in the frequency plane the signal has two harmonics. But the same spectrum is appropriate to a signal whose frequencies are successively mutually replaced [3, 4]. In such a case, we observe a time-frequency spectrum that depends now on two parameters: frequency and time. Classical Fourier transform does not allow identification of these signals to distinguish them.

Thus, there arises a need for transformation that has flexible time-frequency window able to narrow automatically when signal high areas considering and to expand when low areas study.

Aim of Research consists in comparative analysis using time-frequency methods of signals (circuits) spectral analysis to provide reasonable guidelines as to those methods further use.

Main Body. For further development of signals and circuits spectral analysis methods especially in a dynamic state, the transition to windowed Fourier transform (WFT) is made [3]. There exist numerous windows which must satisfy the requirement of window width overlapping the signal stationary area across which its parameters remain unchanged.

A good example of such a window represents the Gaussian function


where a--parameter assigning the window width;


Then the signal x(t) depends upon two parameters


where x(t)--given signal;

T--time of function (1) shift along the time axis from 0 to signal end [T.sub.c];

V(t)--window function;

[omega]--angular velocity;

*--mark identifying the complex mating.

The WFT disadvantage is that the window width remains the same at all signal spectrum harmonics frequencies. In such a way the time-frequency transformation gets assigned time intervals and the spectrum width. Based on the Heisenberg uncertainty principle [3] these values must be reciprocally proportional. Moreover, with window introduction there appears dynamics in determining the signal,

i.e. a signal amplitude growth time characteristic. On the basis of calculations (2) obtained are two-dimensional time-frequency functions for different values of a parameter at the above considered example of harmonic signal having two frequencies. Fig. 1 shows an image of signal with a narrow Gaussian window (a = 0,001), and Fig. 2 represents a wide window picture (a = 0,0001).



From Fig. 2 we do clearly observe a gradual increase in amplitude, i.e. the transition process. At that the frequency evaluation becomes more accurate, but the time intervals are not clearly defined. Fig. 1, on the contrary, shows the fast increase in amplitude, but the frequency band does significantly expand. At the same time, clearly defined are the time intervals.

Another method of time-frequency analysis is this one of dynamic coefficient of transition K(j[omega], t) (DCT) [1], using Fourier transform technique applied to the pulse response with variable upper limit


where h(t)--linear circuit impulse characteristic.

It is advisable to use DCT when harmonic oscillations amplitude bouncing.

Fig. 3 shows the envelope fluctuations time-dependencies when the second order circuits analysis. The parameter [xi] does mean the signal deviation from the resonant frequency range.



Fig. 4 represents the circuit resonance parameters family obtained over time at various parameter metering scheduled instances.

Both Fig. 3 and 4 refer to a fixed time value

y = t/[[tau].sub.c],

where [[tau].sub.c]--circuit time constant.

Further developing the DCT we can analyze fluctuations at phase and frequency alternations.

The second form of DCT method represents the Turbovich method [1], serving in basis to analyze the linear frequency modulation (LFM)


where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]--circuit transition stationary coefficient;

[K.sup.II] (j[omega]), [K.sup.IV] (j[omega])--respectively its 2nd and 4th derivatives by frequency;

[beta]--rate of LFM signal frequency readjustments.

Fig. 5 represents the family of second order circuit dynamic resonance parameters at various rate of LFM signal frequency readjustments [beta] = 2[delta][omega]/[[tau].sub.i], where 2[DELTA][omega]--width of frequency readjustment at impulse of [[tau].sub.i], duration. The resonance curve 1 refers to the stationary mode (frequency without changes). Readjustment velocity increased the curves are shifted to the readjustment side, expanding, and their height decreases.


Results. The most developed from among frequency-time analysis methods is this one of wavelet transform where due to scaling (window width value introduced) a the WFT imperfections are eliminated. At that not only the window shifting for [tau] interval takes place, but also modified is the frequency-dependent scale at the expense of a coefficient. The wavelet transform spectrum is


where [??](t - [tau]/a) I is called the parent wavelet.

There exist numerous parent wavelets [2], and their selection depends onto a specific given problem to resolve.

Conclusions. Thus, presently the radio signals and circuits analysis when a dynamic state may can be implemented using different methods. The method selection depends on what parameters are required to be analyzed: frequency or time. In stationary signals analysis more convenient is the method of dynamic coefficient of transfer. Analysing non-stationary signals, in a narrow frequency range the preference should be given to the windowed Fourier transform, and when wide frequency range, to the wavelet transformations.

DOI 10.15276/opu.1.45.2015.23


[1.] Boashash, B. (2003). Time Frequency Signal Analysis and Processing: A Comprehensive Reference. Amsterdam; Boston: Elsevier.

[2.] Goswami, J.C. and Chan, A.K. (2011). Fundamentals of Wavelets: Theory, Algorithms, and Applications (2nd Ed.). Hoboken, N.J.: Wiley.

[3.] Filipsky, Yu.K. and Semenyuk, S.V. (2012). Analysis of the dynamic characteristics of analog and digital filters of arbitrary orders. Odes'kyi Politechnichnyi Universytet. Pratsi, 1, 163-0169.

[4.] Pavlov, A.N., Filatova, A.E. and Hramov, A.E. (2011). Time-frequency analysis of nonstationary processes: Concepts of wavelets and empirical modes. Izvestiya VUZ. Applied Nonlinear Dynamics, 19(2), 141-157.


Received November 19, 2014

Yu.K. Filipsky, DEng, Prof., A.R. Agadzhanyan, PhD, I.V. Svyryd, Master, Odessa National Polytechnic University
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Author:Filipsky, Yu. K.; Agadzhanyan, A.R.; Svyryd, I.V.
Publication:Odes'kyi Politechnichnyi Universytet. Pratsi
Article Type:Report
Date:Mar 1, 2015
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