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You may filter frequency band you are interested in and then downsample,so you won't lose much information. If that is not sufficient you can use filterbanks to process different frequency bands independently.

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Okay, I am not quite understanding your question. Let's start with the definition of the Gaussian, aka the Bell Curve, in its general form. $$f(t) = \frac{1}{ \sigma \sqrt{2\pi}} e^{ -\frac{(t-\mu)^2}{2\sigma^2} }$$ $\mu$ is the mean, and represents where the peak occurs. $\sigma$ is the standard deviation, and identifies where the inflection points ...

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You can find a nice tutorial for time-frequency analysis in Numerical python by Johansson, chapter 17. link to github repository. You can also check the scipy.signal.spectrogram. import numpy as np from scipy import signal from scipy.fft import fftshift import matplotlib.pyplot as plt # Generate a test signal, a 2 Vrms sine wave whose frequency # is ...

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This follows from the orthonormality of shifted versions of $\phi(x)$: $$\langle \phi(x),\phi(x+n)\rangle =\delta[n]\tag{1}$$ where $\delta[n]$ is the unit impulse. Taking the discrete-time Fourier transform (DTFT) of $(1)$ gives the equation in your question. Note that the left-hand side of $(1)$ is the auto-correlation of $\phi(x)$ evaluated at integers ...

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Wavelets, as bases or frames, are linear like-decompositions. As such, signals and processes can be decomposed into a linear or weighed sum of coefficients multipling (wavelet) vectors: $$s[n] = \sum a_n e[n]$$ The indexing above is not wavelet-specific. Traditionally, there was some natural order: low $n$ denoting low-scales, or high frequencies, and vice-...

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