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15

Imagine for one second, that you just plotted your daubechies-4 wavelet, as you can see here in red. Now imagine that you take this waveform in red, and simply do a cross-correlation with your signal. You plot that result. This will be the first row of your plot. This is scale-1. Next, you dilate your Daubechies-4 wavelet, (that is, you simply make it '...


14

All three transforms are inner product transforms, meaning the output is the inner product of a family of basis functions with a signal. The parametrization and form of the basis functions determine the properties of the transforms.The number of basis functions for a complete picture (i.e. a result that contains enough information to reconstruct the original ...


11

These plots were helpful for me to understand, coming from a STFT background: The complex Morlet (sinusoidal) wavelet looks and behaves like the complex kernel of a STFT (since it's derived from the Gabor transform, a type of STFT). When you "slide it past" a signal of the same frequency, it matches, no matter the phase of the signal you're measuring, ...


10

As Mohammad stated already the terms Continuous Wavelet Transforms (CWT) and Discrete Wavelet Transforms (DWT) are a little bit misleading. They relate approximately as (Continuous) Fourier Transform (the math. integral transform) to DFT (Discrete Fourier Transform). In order to understand the details it is good to see the historical context. The wavelet ...


10

The short-time Fourier transform doesn’t offer better analysis of data than the discrete Fourier transform, it offers a different kind of analysis. The DFT offers an exact decomposition of data to a frequency representation. The STFT offers an approximate decomposition to a time/frequency representation. Which is better depends on what you are after. The ...


10

Remember that Wavelet Transforms are nothing but time-localized filtering/correlation operations. The wavelet transforms provide a unified framework for getting around the Heisenberg Uncertainly Principle that the Fourier Transform suffers from. So when you ask "what should my settings be for bandwidth, and center frequency", you are asking for filter ...


9

Continuous wavelet transform is suitable for a scalogram because the analysis window can be sized and placed at any position. This flexibility allows for the generation of a smooth image in both the time in scale (analogous to frequency) directions. The continuous wavelet transform is a redundant transform because the analysis window can overlap. In fact ...


9

I think it is kind'a similar to soft and hard thresholding using in wavelet de-noising. Have you come across this topic? pywt has already an in-built function for this purpose. Please take a closer look at this code and try to play with it: import pywt import matplotlib.pyplot as plt import numpy as np ts = [2, 56, 3, 22, 3, 4, 56, 7, 8, 9, 44, 23, 1, 4, 6,...


9

In designing such transformations, one should take into account competing interests: fidelity to the human auditory system (that varies with people), including non-linear or even chaotic aspects (tinnitus) easiness of the mathematical formulation for the analysis part possibility to discretize it or allow fast implementations existence of a suitable stable ...


9

My answer is for real scale $a$ and the fact that wavelet transform is usually defined in $L_2$ with norm $$||\Psi(\tau)|| = \int_\mathbb{R} \Psi(\tau)\Psi^*(\tau)\mathrm{d}\tau $$ So $$||\Psi_{a,t}(\tau)|| = \int_\mathbb{R} \frac{1}{|a|}\Psi(\frac{\tau-t}{a})\Psi^*(\frac{\tau-t}{a})\mathrm{d}\tau$$ Set $\tau' = \frac{\tau-t}{a} \implies d\tau' = d\tau / ...


8

If you're familiar with Fourier transforms, I think the bridge between the Fourier worlds and the wavelet worlds is the Gabor transform (a Gaussian-windowed STFT) and the complex Morlet wavelet transform. This is historically how they developed, too. They are basically the same thing, breaking down a signal into "blips" of complex sinusoids: But the time-...


8

It is important to define the time and frequency widths $\Delta_t$ and $\Delta_{\omega}$ of a signal before discussing any special forms of the uncertainty principle. There is no unique definition of these quantities. With appropriate definitions it can be shown that only the Gaussian signal satisfies the uncertainty principle with equality. Consider a ...


7

You can consider anisotropic diffusion. There are many methods based on this technique. Generally spoken, it is for images. It is an adaptive denoising method which aims to smooth non-edge parts of an image, and preserve edges. Also, for Total variation minimization, you can use this tutorial. Authors provide MATLAB code also. They recognize the problem as ...


7

A moment is a generalization of the notion in physics of moment of a (point) mass about an axis being the product of the mass and the distance from the axis. For a continuous random variable $X$ with probability density function $f(x)$, the $n$-th moment is $$m_n = \int_{-\infty}^\infty x^n f(x)\,\mathrm dx.$$ The zero-th moment is $1$ (the area under ...


7

1) Create a 50 Hz sinusoid and then simply add it to your ECG signal. You can control the power of the 50 Hz noise by multiplying the sinusoid by some gain factor (can be less than or more than 1) before you add it to the ECG. 2) I'm not familiar with the Welch periodogram, but if it displays the power spectral density then it should do fine. I would just ...


7

Laplace of Gaussian The Laplace of Gaussian (LoG) of image $f$ can be written as $$ \nabla^2 (f * g) = f * \nabla^2 g $$ with $g$ the Gaussian kernel and $*$ the convolution. That is, the Laplace of the image smoothed by a Gaussian kernel is identical to the image convolved with the Laplace of the Gaussian kernel. This convolution can be further expanded, ...


6

Simply speaking both the const-Q-transform and the Gabor-Morlet wavelet-transform are just continuous wavelet transforms. Or, more precisely, approximations thereof, as there will always be discretization issues in real applications. A property of wavelet transforms is that they have build in the constant Q-factor property, or in other words logarithmic ...


6

Intuitively speaking, anything that is 'high frequency' is something that is 'rapidly changing in time'. Anything that is 'low frequency' is something that is 'slowly changing in time'. If you think about it, any time you have 'detail' in a signal or image, it means that you have, very quick, rapid variations in time or space. This then becomes the 'detail' ...


6

A very common yet unfortunate mis-conception in the field of wavelets has to do with the ill-coined terminology of "Continuous Wavelet Transforms". First thing's first: The Continuous Wavelet Transform, (CWT), and the Discrete Wavelet Transform (DWT), are both, point-by-point, digital, transformations that are easily implemented on a computer. The ...


6

I think "Introduction to Wavelets and Wavelet Transforms: A Primer" by Sidney Burrus (et al.) is a very good and practical book. It is very clear, has exercises, and contains some Matlab programs. EDIT: I forgot to mention that this paper is also a very nice introduction to wavelets.


6

I don't think there is any difference. The documentation for dwt2 says Single-level discrete 2-D wavelet transform The dwt2 command performs a single-level two-dimensional wavelet decomposition... While the documentation for wavedec2 says Multilevel 2-D wavelet decomposition The difference is that dwt2 is single-level (produces a single A, H, ...


6

The HUP follows directly from the properties of the Fourier Transform, because time and frequency are orthogonal bases in which we can expand the co-efficient sequence of our signal. In fact all pairs of orthonormal bases will have some kind of Uncertainty Principle associated with them. In traditional Fourier analysis, the either the time axis or the ...


5

It looks like there is a problem with your scaling. The scaling for the DWT has the same interpretation as it does for the CWT. For the DWT, the scale of the analyzing function (wavelet) is increased using a dyadic scale (increasing by factors of 2) in non-overlapping time intervals. Increasing scale corresponds to narrower frequency distributions in ...


5

you can have a look at the LTFAT's wavelet module http://ltfat.sourceforge.net/doc/wavelets/index.php it runs in Matlab/Octave with backend written in C. It has fairly large database of wavelet filters and new ones can be added easily. What exactly do you mean by 2) Ability to run in parallel - VERY IMPORTANT Should the computation itself be somewhat ...


5

If you approximate the Fourier transform $$X(f)=\mathcal F(x)(f)=\int_{-\infty}^\infty x(t)\,e^{-2\pi j\,ft}\,dt$$ by the discrete Fourier transformation for by sampling on the time segment $[-T,T]$ as $$X(f_n)\approx \sum_{k=-N}^{N-1} x(k\tau)\,e^{-2\pi j\,f_nk\tau}\,\tau=s[n]\,\tau$$ with $T=N\tau$, $f_n=n/(N\tau)=n/N*f_s=n/T$, $n=-N,...,N-1$, $s$ the ...


5

Since the bulk of R’s DSP capability comes from the signal package which was ported over from the open source project Octave (itself influenced by MATLAB), there's no intrinsic limitation of R. What you have picked up on, are ecosystem preferences. We learned MATLAB in university, picked up numpy/scipy/sklearn at work, so R isn't the first weapon of choice. ...


5

Yeah some of us can do it, you can speed up or slow down without affect the pitch, some guys call this applications of Time Stretch, there different ways to do it, you can do in frequency domain or time domain, you will need choose what is best for you, you will find some advantages and disadvantages of each. Time Domain: In Time Domain you can try some ...


5

The Ricker wavelet, the (isotropic) Marr wavelet, the Mexican hat or the Laplacian of Gaussians belong to be the same concept: continuous admissible wavelets (satisfying certain conditions). Traditionally, the Ricker wavelet is the 1D version. The Marr wavelet or the Mexican hat are names given in the context of 2D image decompositions, you can consider ...


5

Wavelet transforms can be more difficult to interpret than FFT at face value due to the various representations, nomenclature and output formats. I had to study more than 15 resources to get a good sense of the variety and which one is used by Pywavelets (which does not provide much theory or explanation in its documentation). In order to grasp the meaning ...


5

Yes indeed! In theory as long as the wavelet is orthogonal, the sum of the squares of all the coefficients should be equal to the energy of the signal. In practice, one should be careful that: the decomposition is not "expansive", i.e. the number of samples and of coefficients is the same. wavelet filter coefficients are not re-scaled, as happens in some ...


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