Appendix 2

Porous III-V Semiconductors by I. Tiginyanu, S. Langa, H. Föll and V. Ursachi

Wavelets Transformation

The basis set for Fourier analysis consists of harmonic (i.e. perfectly monochromatic) waves which extend to infinity. If we consider the 1D case, where the function being analyzed is a time signal, this is a process of decomposing the function f(t) into its frequency components. Many physical problems involve waves and can be analyzed in this way. However, although we have gained information about the frequency content of f(t) we have lost all temporal information. Alternatively we can consider the function f(t) to be an expansion in terms of a basis set of Dirac functions - in other words the basis set (and hence the information gained) is localized in time but not in frequency.

In many cases we would wish to examine both time and frequency information simultaneously. This can be accomplished by expanding f(t) in terms of functions that are both oscillatory and localized in time. These functions are known as wavelets, and this leads to wavelet transformation.

Let us consider the continuous wavelet transform first. A function is called a mother wavelet if it has certain properties: in particular, its integral equals zero. This implies that the function has some oscillatory characteristics. From the mother function a family of daughter wavelets, the so-called basis set, by scaling and shifting can be obtained. The scaling is performed in a self-similar manner. From this basis set one can construct an integral wavelet transform by taking the inner product between f(t) and the various daughter wavelets.

In oder to calculate the inner product a special procedure was included in the SiPor program for pore etching. The wavelet procedure uses the so-called Morlet function as a 'mother' function. In practice, the Morlet wavelet is defined as the product of a complex exponential wave and a Gaussian envelope:


Equation 65(65)


where Ψo is the wavelet value at non-dimensional time η , ω0 is the wave number. This is the basic wavelet function, but now one needs in some way to change the overall size as well as to slide the entire wavelet along the time axis. Thus the 'scaled wavelet' is defined as:


Equation 66(66)


where s is the 'dilatation' parameter used to change the scale, and n is the translation parameter used to slide the function in time. Se(- 1/2) is a normalization factor, in order to keep the total "area" of the scaled wavelet constant.




For a time series X, with values of xn, at time index n (each value is separated in time by a constant time interval dt) the wavelet transform Wn(s) is just the inner product (or convolution) of the wavelet function with this time-series:


Equation 67(67)


where the asterisk (*) denotes complex conjugate. The above sum can be evaluated for various values of the scale s, usually taken to be multiples of the lowest possible frequency, as well as for all values of n between the start and end data. Then, a two-dimensional picture can be obtained by plotting the wavelet amplitude, i.e. equation 48, for different n;s pairs. Different colors can be used in the map in order to distinguish between different amplitudes.

The main information which one can get by analyzing such kind of two dimensional maps is the change in time of the dominant frequencies, e.g. the one with the highest amplitude of the wavelet. Therefore, this is an ideal tool to investigate the frequency change with time of the self-induced voltage oscillations observed during pore formation in InP.



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