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Filter by Lables. Message 1 of 8. Surface Tutorial Excersize 1 dwt file missing. Excersize 1 requires us to open the surface. I cannot find this file anywhere in the computer or even in the help Tutorial files. Is there a place to get this template from? Message 2 of 8. The first surface tutorial I have refers to opening this file: Surface-1A.
Message 3 of 8. Message 4 of 8. The size and orientation of the blocks indicate how small the features are that we can distinguish in the time and frequency domain.
The original time-series has a high resolution in the time-domain and zero resolution in the frequency domain. This means that we can distinguish very small features in the time-domain and no features in the frequency domain. Opposite to that is the Fourier Transform, which has a high resolution in the frequency domain and zero resolution in the time-domain. The Short Time Fourier Transform has medium sized resolution in both the frequency and time domain. In other words, the Wavelet Transforms makes a trade-off; at scales in which time-dependent features are interesting it has a high resolution in the time-domain and at scales in which frequency-dependent features are interesting it has a high resolution in the frequency domain.
The Fourier Transform uses a series of sine-waves with different frequencies to analyze a signal. That is, a signal is represented through a linear combination of sine-waves. The Wavelet Transform uses a series of functions called wavelets, each with a different scale.
The word wavelet means a small wave, and this is exactly what a wavelet is. In Figure 3 we can see the difference between a sine-wave and a wavelet. This allows the wavelet transform to obtain time-information in addition to frequency information.
Since the Wavelet is localized in time, we can multiply our signal with the wavelet at different locations in time. We start with the beginning of our signal and slowly move the wavelet towards the end of the signal. This procedure is also known as a convolution. After we have done this for the original mother wavelet, we can scale it such that it becomes larger and repeat the process. This process is illustrated in the figure below.
As we can see in the figure above, the Wavelet transform of an 1-dimensional signal will have two dimensions. This 2-dimensional output of the Wavelet transform is the time-scale representation of the signal in the form of a scaleogram.
Above the scaleogram is plotted in a 3D plot in the bottom left figure and in a 2D color plot in the bottom right figure. PS: You can also have a look at this youtube video to see how a Wavelet Transform works.
So what is this dimension called scale? Since the term frequency is reserved for the Fourier Transform, the wavelet transform is usually expressed in scales instead. That is why the two dimensions of a scaleogram are time and scale. For the ones who find frequencies more intuitive than scales, it is possible to convert scales to pseudo-frequencies with the equation. We can see that a higher scale-factor longer wavelet corresponds with a smaller frequency, so by scaling the wavelet in the time-domain we will analyze smaller frequencies achieve a higher resolution in the frequency domain.
And vice versa, by using a smaller scale we have more detail in the time-domain. So scales are basically the inverse of the frequency. PS : PyWavelets contains the function scale2frequency to convert from a scale-domain to a frequency-domain. Another difference between the Fourier Transform and the Wavelet Transform is that there are many different families types of wavelets.
The wavelet families differ from each other since for each family a different trade-off has been made in how compact and smooth the wavelet looks like. This means that we can choose a specific wavelet family which fits best with the features we are looking for in our signal. The PyWavelets library for example contains 14 mother Wavelets families of Wavelets :. Each type of wavelets has a different shape, smoothness and compactness and is useful for a different purpose.
Since there are only two mathematical conditions a wavelet has to satisfy it is easy to generate a new type of wavelet. The two mathematical conditions are the so-called normalization and orthogonalization constraints:. A wavelet must have 1 finite energy and 2 zero mean. Finite energy means that it is localized in time and frequency; it is integrable and the inner product between the wavelet and the signal always exists.
The admissibility condition implies a wavelet has zero mean in the time-domain, a zero at zero frequency in the time-domain. This is necessary to ensure that it is integrable and the inverse of the wavelet transform can also be calculated. Below we can see a plot with several different families of wavelets. The first row contains four Discrete Wavelets and the second row four Continuous Wavelets.
Figure 5. Several families of Wavelets. In the first row we see discrete wavelets and in the second row we see several continuous wavelets. PS: To see how all wavelets looks like, you can have a look at the wavelet browser. Within each wavelet family there can be a lot of different wavelet subcategories belonging to that family.
You can distinguish the different subcategories of wavelets by the number of coefficients the number of vanishing moments and the level of decomposition. Figure 6. The Daubechies family of wavelets for several different orders of vanishing moments and several levels of refinement. In the first column we can see the Daubechies wavelets of the first order db1 , in the second column of the second order db2 , up to the fifth order in the fifth column.
PyWavelets contains Daubechies wavelets up to order 20 db The number of the order indicates the number of vanishing moments.
So db3 has three vanishing moments and db5 has 5 vanishing moment. The number of vanishing moments is related to the approximation order and smoothness of the wavelet. If a wavelet has p vanishing moments, it can approximate polynomials of degree p — 1. When selecting a wavelet, we can also indicate what the level of decomposition has to be. By default, PyWavelets chooses the maximum level of decomposition possible for the input signal.
The maximum level of decomposition see pywt. As we can see, as the number of vanishing moments increases, the polynomial degree of the wavelet increases and it becomes smoother.
And as the level of decomposition increases, the number of samples this wavelet is expressed in increases. As we have seen before Figure 5 , the Wavelet Transform comes in two different and distinct flavors; the Continuous and the Discrete Wavelet Transform. The values of the scaling and translation factors are continuous, which means that there can be an infinite amount of wavelets. You can scale the mother wavelet with a factor of 1. When we are talking about the Discrete Wavelet Transform, the main difference is that the DWT uses discrete values for the scale and translation factor.
The scale factor increases in powers of two, so and the translation factor increases integer values. To be able to work with digital and discrete signals we also need to discretize our wavelet transforms in the time-domain. In practice, the DWT is always implemented as a filter-bank. This means that it is implemented as a cascade of high-pass and low-pass filters.
This is because filter banks are a very efficient way of splitting a signal of into several frequency sub-bands. Below I will try to explain the concept behind the filter-bank in a simple and probably oversimplified way.
It is necessary in order to understand how the wavelet transform actually works and can be used in practical applications. To apply the DWT on a signal, we start with the smallest scale. As we have seen before, small scales correspond with high frequencies. This means that we first analyze high frequency behavior. At the second stage, the scale increases with a factor of two the frequency decreases with a factor of two , and we are analyzing behavior around half of the maximum frequency.
At the third stage, the scale factor is four and we are analyzing frequency behavior around a quarter of the maximum frequency. And this goes on and on, until we have reached the maximum decomposition level.
What do we mean with maximum decomposition level? To understand this we should also know that at each subsequent stage the number of samples in the signal is reduced with a factor of two. At lower frequency values, you will need less samples to satisfy the Nyquist rate so there is no need to keep the higher number of samples in the signal; it will only cause the transform to be computationally expensive.
Due to this downsampling, at some stage in the process the number of samples in our signal will become smaller than the length of the wavelet filter and we will have reached the maximum decomposition level. To give an example, suppose we have a signal with frequencies up to Hz. In the first stage we split our signal into a low-frequency part and a high-frequency part, i. At the second stage we take the low-frequency part and again split it into two parts: Hz and Hz.
At the third stage we split the Hz part into a Hz part and a Hz part. This goes on until we have reached the level of refinement we need or until we run out of samples. We can easily visualize this idea, by plotting what happens when we apply the DWT on a chirp signal. A chirp signal is a signal with a dynamic frequency spectrum; the frequency spectrum increases with time. The start of the signal contains low frequency values and the end of the signal contains the high frequencies.
This makes it easy for us to visualize which part of the frequency spectrum is filtered out by simply looking at the time-axis. Figure 7. The approximation and detail coefficients of the sym5 wavelet level 1 to 5 applied on a chirp signal, from level 1 to 5.
On the left we can see a schematic representation of the high pass and low pass filters applied on the signal at each level. In Figure 7 we can see our chirp signal, and the DWT applied to it subsequently. There are a few things to notice here:. So now we have seen what it means that the DWT is implemented as a filter bank; At each subsequent level, the approximation coefficients are divided into a coarser low pass and high pass part and the DWT is applied again on the low-pass part.
As we can see, our original signal is now converted to several signals each corresponding to different frequency bands. Later on we will see how the approximation and detail coefficients at the different frequency sub-bands can be used in applications like removing high frequency noise from signals, compressing signals, or classifying the different types signals.
PS: We can also use pywt. This functions takes as input the original signal and the level and returns the one set of approximation coefficients of the n-th level and n sets of detail coefficients 1 to n-th level. So far we have seen what the wavelet transform is, how it is different from the Fourier Transform, what the difference is between the CWT and the DWT, what types of wavelet families there are, what the impact of the order and level of decomposition is on the mother wavelet, and how and why the DWT is implemented as a filter-bank.
We have also seen that the output of a wavelet transform on a 1D signal results in a 2D scaleogram. Such a scaleogram gives us detailed information about the state-space of the system, i. The el-Nino dataset is a time-series dataset used for tracking the El Nino and contains quarterly measurements of the sea surface temperature from up to In order to understand the power of a scaleogram, let us visualize it for el-Nino dataset together with the original time-series data and its Fourier Transform.
Figure 8. In Figure 8 we can see in the top figure the el-Nino dataset together with its time average, in the middle figure the Fourier Transform and at the bottom figure the scaleogram produced by the Continuous Wavelet Transform. In the scaleogram we can see that most of the power is concentrated in a year period.
The increase in power can also be seen in the Fourier transform around these frequency values. The main difference is that the wavelet transform also gives us temporal information and the Fourier Transform does not. For example, in the scaleogram we can see that up to there were many fluctuations, while there were not so much between — We can also see that there is a shift from shorter to longer periods as time progresses. This is the kind of dynamic behavior in the signal which can be visualized with the Wavelet Transform but not with the Fourier Transform.
This should already make clear how powerful the wavelet transform can be for machine learning purposes. But to make the story complete, let us also look at how this can be used in combination with a Convolutional Neural Network to classify signals.
In section 3. We have seen that applied on the el-Nino dataset, it can not only tell us what the period is of the largest oscillations, but also when these oscillations were present and when not. Such a scaleogram can not only be used to better understand the dynamical behavior of a system, but it can also be used to distinguish different types of signals produced by a system from each other. If you record a signal while you are walking up the stairs or down the stairs, the scaleograms will look different.
ECG measurements of people with a healthy heart will have different scaleograms than ECG measurements of people with arrhythmia. Or measurements on a bearing, motor, rotor, ventilator, etc when it is faulty vs when it not faulty. The possibilities are limitless! So by looking at the scaleograms we can distinguish a broken motor from a working one, a healthy person from a sick one, a person walking up the stairs from a person walking down the stairs, etc etc.
One way to automate this process is to build a Convolutional Neural Network which can automatically detect the class each scaleogram belongs to and classify them accordingly. What was the deal again with CNN? In previous blog posts we have seen how we can use Tensorflow to build a convolutional neural network from scratch. And how we can use such a CNN to detect roads in satellite images. In the next few sections we will load a dataset containing measurement of people doing six different activities , visualize the scaleograms using the CWT and then use a Convolutional Neural Network to classify these scaleograms.
Let us try to classify an open dataset containing time-series using the scaleograms and a CNN. The Human Activity Recognition Dataset UCI-HAR contains sensor measurements of people while they were doing different types of activities, like walking up or down the stairs, laying, standing, walking, etc.
There are in total more than The training set contains signals where each signal has measurement samples and 9 components. The signals from the training set are loaded into a numpy ndarray of size , , 9 and the signals from the test set into one of size , , 9. Since the signal consists of nine components we have to apply the CWT nine times for each signal. Below we can see the result of the CWT applied on two different signals from the dataset. The left one consist of a signal measured while walking up the stairs and the right one is a signal measured while laying down.
Figure 9. Each signal has nine different components. On the left we can see the signals measured during walking upstairs and on the right we can see a signal measured during laying. Since each signal has nine components, each signal will also have nine scaleograms. So the next question to ask is, how do we feed this set of nine scaleograms into a Convolutional Neural Network? There are several options we could follow:.
Below we can see the Python code on how to apply the CWT on the signals in the dataset, and reformat it in such a way that it can be used as input for our Convolutional Neural Network. The total dataset contains over As you can see above, the CWT of a single signal-component samples results in an image of by pixels. So the scaleograms coming from the signals of the training dataset are stored in an numpy ndarray of size , , , 9 and the scaleograms coming from the test signals are stored in one of size , , , 9.
Now that we have the data in the right format, we can start with the most interesting part of this section: training the CNN! For this part you will need the keras library , so please install it first. As you can see, combining the Wavelet Transform and a Convolutional Neural Network leads to an awesome and amazing result!
Much higher than you can achieve with any other method. In Section 2. In this section, let us see how we can use PyWavelets to deconstruct a signal into its frequency sub-bands and reconstruct the original signal again. Figure A signal together with the reconstructed signal. Above we have deconstructed a signal into its coefficients and reconstructed it again using the inverse DWT.
The second way is to use pywt. In the previous section we have seen how we can deconstruct a signal into the approximation low pass and detail high pass coefficients. If we reconstruct the signal using these coefficients we will get the original signal back.
But what happens if we reconstruct while we leave out some detail coefficients? Since the detail coefficients represent the high frequency part of the signal, we will simply have filtered out that part of the frequency spectrum.
If you have a lot of high-frequency noise in your signal, this one way to filter it out. Leaving out of the detail coefficients can be done using pywt. This is a dataset containing high frequency sensor data regarding accelerated degradation of bearings. Click the thumbnail for a larger image. Once you save the page as a DWT, one editable region is created for the doctitle.
You will need to create additional editable regions, for the content area, your keywords and description meta tags. NOTE : The editable region name must be all one word and in lower case. You can call it anything you want, however relating it to the placement on the page helps keep the various regions straight. Step 4 : To create an editable region for your keyword and description meta tags, Copy and Paste the just before your description meta tag.
Copy and Paste the tag right after the keywords meta tag. Double click the 2nd occurrence of the doctitle tag and change it to headsection. Click thumbnail for larger image.
You can now see that the text on your DWT page is surrounded by a yellow border and titled maincontent. Step 6 : If your navigation is going to change from section to section, you will need to create an editable region for that area too. Once finished, save your page. Step 7 : Now that you have created your DWT, you need to check and make sure it works correctly. Click on site.
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