OverLordGoldDragon
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## 146 Answers

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Doable. But only for odd $N$. And it will have infinite solutions. Example: \begin{align} x &= [x_0, x_1, 0] \\ x_0 &= 0.5 \left(1 + \frac{1}{\sqrt{3}}\right) + j 0.5\left(1 - \frac{1}{\... View answer Accepted answer 2 votes This can be accomplished with an analytic time-frequency representation, like CWT or STFT. Goal must be known precisely to attain desired result, however, as time and frequency are coupled and ... View answer 2 votes 2D wavelet transform is well suited. It's an extension of 1D CWT where we correlate wavelets of different center frequencies and "scales" (widths in time domain). Wavelets can be calibrated ... View answer 2 votes The DFT is an orthogonal transform. This means it preserves the input with zero redundancy. If length is: longer than input: redundancy shorter than input: loss of information It's also equivalent ... View answer 2 votes Applying ssq_cwt with extract_ridges, I obtain below. Improvable with better windowing, more samples. Smoothing can be applied on amplitude plot to make it more interpretable without losing much ... View answer Accepted answer 2 votes 'Imaginary' is a misnomer; they 'exist' as much as reals do. As for the envelope: it provides a meaningful representation of phase and amplitude over time, not otherwise doable with reals alone. To ... View answer Accepted answer 2 votes The key is units, and understanding wavelet behavior in context of application (in this case CWT). Full implementations for all discussed here is available at squeezepy. This answer assumes analytic ... View answer 2 votes Complementing, and based on C. Dawg's answer, discarding the slope addition, the effect on magnitude and phase are \begin{aligned} |X[k]| & \rightarrow M|X[k]| \\ \angle{(X[k])} &...

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2 votes

Short version: DFT's bin indices are input length-dependent; "center frequency" is measured relative to the function generating the wavelet. Generated length can vary, so must be accounted ...

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1 votes

The choice of $\phi$ affects: Time-shift invariance: slower decay in time will increase it Time-warp stability: slower decay in time will increase it mainly for deformations along time (but not only ...

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1 votes

It's indeed only a statement on proportionality. Wavelets aren't compat, so any notion of "support" invokes a heuristic (engineered criterion). However, it's not applicable to all wavelets: ...

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1 votes

It keeps the scale of the largest scale wavelet $\leq T$ while still tiling the entire frequency axis. using a single low pass filter built with a single scaling function would not achieve this: $\... View answer 1 votes Worth looking into the log Fourier transform (and FFTLog), which I know nothing about except that its abstract reads exactly like what you seek: We present an exact and analytical expression for ... View answer 1 votes What I'd do: Transform data into a representation that maximizes "similarity" of chunks that are otherwise "alike" but have large Euclidean distance in terms of raw waveforms ... View answer Accepted answer 1 votes JTFS is an extension of Wavelet Scattering that exploits time-frequency structure, adding sensitivity to frequency-dependent time shifts, invariance to frequency transposition, and stability against ... View answer Accepted answer 1 votes This is subject of ridge analysis. The "quality" of a representation can be quantified as follows: Component extraction Ability to separate intrinsic modes / independent time-frequency ... View answer 1 votes CQT by general definition is a constraint on ratio of center frequency to frequential width; no anti-wavelet criterion baked in. Note it's not sufficient to have an exponentially distributed center ... View answer 1 votes Fundamentally: DWT is orthogonal, CWT is redundant. Former packs the most information per sample, latter spreads out its decomposition. As will be explained: CWT yields vastly superior analysis ... View answer Accepted answer 1 votes Transform to time domain Center Plot real & imag separately, or use complex colormap, or take modulus Results below. Python code -- more examples -- other examples Minimal code import numpy as ... View answer 1 votes TP9 and AF7 are definitely not brain waves. AF8 is more realistic, but still too clean; I suspect both it and TP10 were heavily filtered. Possibly TP9 and AF7 had negligible activity and were ... View answer 1 votes OP's time vector is What I'd do: Treat it as piecewise-lienar, i.e. ignore that the time vector isn't uniformly spaced except for jumps. This should work reasonably - but if greater accuracy is ... View answer Accepted answer 1 votes noverlap = nperseg - 1 provides maximum possible information - it is the 'ideal' configuration. A spectrogram is$|\text{STFT}|$, and$\text{STFT}$is input convolved with windowed complex sinusoids. ... View answer Accepted answer 1 votes CWT/STFT/Synchrosqueezing Scattering -- lecture MFCC Learning references I recommend against FFT and its simple manipulations; it produces at best a non-robust representation with weak representative ... View answer 1 votes If superposition works, then independent mode/component extraction is of interest. Synchrosqueezing is well-suited for this task. Extracted features can ten be fed to an anomaly detection system - ... View answer 1 votes That's not spectral density, it's just real DFT. If you have a$60 Hz$sine that lasts for 1 sec, its DFT will be half as large as for the same sine that lasts for 2 secs, since there's half the ... View answer Accepted answer 1 votes A) Fourier transform of a finite sequence is periodic and infinite. The FT of$x(t)$is is$1/(2 + j\omega)$, which decays permanently and power zeros at infinity; for$x[n]$, an approximation (see C) ... View answer 1 votes If there no overlap, then invertibility is lost, and so is information. If there is overlap, but it is little, then analysis information is lost (but not synthesis, which is more fundamental). Namely, ... View answer Accepted answer 1 votes Algorithm To encode a signal$x_m$in a carrier with frequency$f_c$, we proceed as: $$y(t) = \cos(\phi(t)), \\ \phi(t) = 2\pi \cdot \left(f_c t + f_\Delta \int_0^t x_m(\tau)d\tau \right)$$ where$...

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1 votes

0.11Hz is 110mHz Vertical lines are due to the zoom; they're the bottom row ... zoomed: This is a purely visual effect, no data is added. To make the data zoom between 0 and 2mHz, increase nfft, then ...

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1 votes

-30 dB is still very noisy. If you've had success with EMD, I'd try an inspired transform that's improved on it: synchrosqueezing. Whether it's best to denoise before classifying depends on amount of ...

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