Research paper
The discrete wavelet transform has been seed to detect and analyze power quality disturbances. The disturbances of interest include sag, swell, outage and transient. Waveforms at strategic points can be obtained for analysis, which includes different power quality disturbances. The wavelet has been chosen to perform feature extraction. The outputs of the feature extraction are the wavelet coefficients representing the power quality disturbance signal. Wavelet coefficients at different levels reveal the time localizing information about the variation of the signal.
A wavelet packet transform (WEPT) method is introduced as a useful LOL for detecting, classifying and quantifying the ARMS of testing waveform and harmonic ratio value. The proposed approach can measure the distribution of the ARMS of testing waveform with respect to individual frequency bands directly from the wavelet transform coefficients. The harmonic ratio value relative to basic frequency band can be calculated as well. The method is evaluated by its application to both analytical waveform and actual testing waveform data.
Keywords: wavelet packet transform (WEPT), Discrete Fourier Transform (DAFT), Pouf Erie Series (FSP), Discrete Wavelet Packet Transform (DEPT), Multi-Resolution Analysis (MR.) . Introduction Mathematical transformation are applied to signals to obtain a further information from that signal that is not readily available in the raw signal Raw signal – Time domain signal Processed Signal – Signal that has been “Transformed” by any of the available mathematical transformation as a processed signal Transformation tools : Pouf Erie transformation Fast Fourier transformation Wavelet transformation 1. Wavelets Wavelets are a set of non-linear bases. When projecting (or approximating) a function in terms of wavelets, the wavelet basis functions is chosen according to the function being proximate. Hence, unlike families of linear bases where the same, static set of basic functions are used for every input function, wavelets employ a dynamic set of basic functions that represents the input function in the most efficient way. Thus wavelets are able to provide a great deal of compression and are therefore very popular in the fields of image and signal processing. 1. Wavelet Transform Wavelets are mathematical functions that cut up data into different frequency components, and then study each component with a resolution matched to its scale. They have advantages over traditional Fourier methods in analyzing hysterical situations where the signal contains discontinuities and sharp spikes. Wavelets were developed independently in the fields of mathematics, quantum physics, electrical engineering, and seismic geology. Interchanges between these fields during the last ten years have led to many new wavelet applications such as image compression, turbulence, human vision, radar, and earthquake prediction.
This introduces wavelets to the interested technical person outside of the digital signal processing field. It describe the history of wavelets beginning with Fourier, compare wavelet transforms with Fourier transforms, state properties and other special aspects of wavelets, and finish with some interesting applications such as image compression, musical tones, and De-noising noisy data. 2. HARMONIC ANALYSIS USING DISCRETE WAVELET TRANSFORM Harmonic analysis is done by Wavelet Transform. The Wavelet is a powerful signal processing tool that is particularly useful for the analysis Of non stationary signals.
Wavelets are short duration oscillating waveforms which is zero mean and fast decay to zero amplitude at both ends which are dilated and shifted to vary their time frequency resolution. The wavelet is irregular in have and compactly supported. It is these properties of being irregular in shape and compactly supported make the Wavelets an ideal tool for analyzing signals of a non stationary nature. The harmonics and inter harmonics estimation of non stationary signal is carried out by Wavelet Transform.
The two approaches of Wavelet Transform are Wavelet Packet Transform and Discrete Wavelet Transform. Discrete Wavelet Transform is more suitable for analyzing power system waveforms due to its less computational burdens 2. 2 The Hear Wavelet Transform and Noise Filters The wavelet transform could be use to filter out noise. The Idea behind noise filtering is to remove the noise while leaving the important detail, while this is fine as an ideal it is hard to realize in practice. Wavelets do provide a powerful way to analyze non – stationary time series(e. . , time series were the Fourier transform is not successful). While there is a great deal of material on wavelets, there is less accessible material on wavelets, there is less accessible material on wavelet filtering. The definition of noise turns out to be somewhat arbitrary. One technique that has been proposed for filtering wavelet coefficients that are within an error range of the mean. However this may remove important information. 2. SHORT TIME FOURIER TRANSFORM AND WAVELET ANALYSIS The Wavelet transform provides the time-frequency representation. There are other transforms which give this information too, such as short time Fourier transform, Wagner distributions, etc. ) Often times a particular spectral component occurring at any instant can be of particular interest. In these cases it may be very beneficial to know the time intervals these particular spectral components occur. For example, in Eggs, the latency of an event- related potential is of particular interest (Event-related potential is the espouse of the brain to a specific stimulus like flash-light, the latency of this response is the amount of time elapsed between the onset of the stimulus and the response).
Wavelet transform is capable of providing the time and frequency information simultaneously, hence giving a time-frequency representation of the signal. Functioning of wavelet transform completely different , and should be explained only after short time Fourier Transform (STET) . The WTG was developed as an alternative to the STFW. Lot suffices at this time to say that the WTG was developed to overcome some resolution related problems of the STFW. To make a real long story short, we pass the time- domain signal from various high pass and low pass filters, which filters out either high frequency or low frequency portions of the signal.
This procedure is repeated, every time some portion of the signal corresponding to some frequencies being removed from the signal. Here is how this works: Suppose we have a signal which has frequencies up to 1 000 Hz’s. In the first stage we split up the signal in to two parts by passing the signal from a high pass and a low pass filter (filters should satisfy some certain conditions, so-called admissibility condition) which results in two efferent versions of the same signal: portion of the signal corresponding to 0-500 Hz’s (low pass portion), and 500-1000 Hz’s (high pass portion).
Then, we take either portion (usually low pass portion) or both, and do the same thing again. This operation is called decomposition. Assuming that we have taken the low pass portion, we now have 3 sets of data, each corresponding to the same signal at frequencies 0-250 Hz’s, 250-500 Hz’s, 500-1000 Hz’s. Then we take the low pass portion again and pass it through low and high pass filters; we now have 4 sets of signals corresponding to 0-125 Hz’s, 125-250 Hz’s,250-500 Hz’s, and 500-1000 Hz’s.
We continue like this until we have decomposed the signal to a pre-defined certain level. Then we have a bunch of signals, which actually represent the same signal, but all corresponding to different frequency bands. We know which signal corresponds to which frequency band, and if we put all of them together and plot them on a 3-D graph, we will have time in one axis, frequency in the second and amplitude in the third axis.
This will show us which frequencies exist at which time ( there is an issue, called “uncertainty principle”, which states that, we cannot exactly know what frequency exists at what time instance , but we can only know what frequency bands exist at what time intervals The uncertainty principle, originally found and formulated by Heisenberg, states that, the momentum and the position of a moving particle cannot be known simultaneously. This applies to our subject as follows: The frequency and time information of a signal at some certain point in the time-frequency plane cannot be known.
In other words: We cannot know what spectral component exists at any given time instant. The best we can do is to investigate what spectral components exist at any given interval of time. This is a problem of resolution, and it is the main reason why researchers have switched to WTG from STET. SST FT gives a fixed resolution at all times, whereas WTG gives a variable resolution as follows: Higher frequencies are better resolved in time, and lower frequencies are better resolved in frequency. This means that, a certain high frequency component can be located better in time (with less relative error) than a low frequency component.
On the contrary, a low frequency component can be located better in frequency compared to high frequency component. Take a look at the following grid interpret the above grid as follows: The top row wows that at higher frequencies we have more samples corresponding to smaller intervals of time. In other words, higher frequencies can be resolved better in time. The bottom row however, corresponds to low frequencies, and there are less number of points to characterize the signal, therefore, low frequencies are not resolved well in time.
In discrete time case, the time resolution of the signal works the same as above, but now, the frequency information has different resolutions at every stage too. Note that, lower frequencies are better resolved in frequency, where as higher frequencies are not. Note how the spacing between subsequent frequency components increase as frequency increases. Below, are some examples of continuous wavelet transform: Let’s take a sinusoidal signal, which has two different frequency components at two different times. Note the low frequency portion first, and then the high frequency.
We’ve seen that Fourier analysis is an useful mathematical approach to the analysis of the frequencies of a signal. Nevertheless, Fourier analysis is not the best way, for many aspects, to solve the problem of analysis and reconstruction. Which is the main problem with Fourier analysis. The answer can be summed up in a magic word: lack of localization. Let me explain what is meant by localization. Sinusoidal waves are equal all over the space domain (they are periodic, hence translation-invariant): this means they are not lazed. So, well, what does non-localization imply?
Let me explain with a typical example. Suppose we have a simple signal which is quite flat all over the (time-)line, except for a sudden peak around a point, with a certain frequency phi. When we decompose this signal on the standard base of sinusoidal waves, we get not only large coefficients relative to regencies near phi, but we get also significantly large coefficients in a wide spectrum. This is due to the fact that once we’ve got the frequencies around phi, we’re with a sinusoidal wave, which maybe approximates well the peak, but certainly doesn’t look flat far from it.
This implies that, since, for the reconstruction formula, we must have back our original signal from the anti- transform, the oscillations afar the peak are to be cancelled to recreate the flat signal. How does this happen? By adding higher and higher frequencies, each partially canceling the others. So we get a transform big over many frequencies, and it is not easy to read from it information’s about the space- signal. This is why the classical Fourier transform is so ineffective in handling with peaks and sharp edges (in images).
It is very useful, nevertheless, in studying signals that statistically show the characteristics all over the time, whose behavior doesn’t vary a abruptly intimae. The problem of localization is solved by wavelets. A family of wavelets (the classical ones, but there are many others with different characteristics and arising in different ways) is obtained by orientations and dilation’s (generally by a factor of 2, but the case in which the factor is an integer greater than 2 is studied) of a single wavelet (often called the “mother wavelet”).
SSI(x;j,m) = Index j gives the level Of dilation, m the translation. The factor AAA(j/2) is introduced so that all the wavelets have the same energy. By translating we obtain localization in space, by dilating we obtain better and better resolutions. We can parameterize the family of wavelets with two parameters as above, one indicating where (in spatial sense) the wavelet is centered, the other indicating at what resolution the wavelet is analyzing. The two parameters are independent.
