OverLordGoldDragon
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Synchrosqueezing Wavelet Transform explanation?
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15 votes

Synchrosqueezing is a powerful reassignment method. To grasp its mechanisms, we dissect the (continuous) Wavelet Transform, and how its pitfalls can be remedied. Physical and statistical ...

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Wavelet Scattering explanation?
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7 votes

Wavelet Scattering is an equivalent deep convolutional network, formed by cascade of wavelets, modulus nonlinearities, and lowpass filters. It yields representations that are time-shift invariant, ...

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Python audio analysis: which spectrogram should I use and why?
6 votes

I recommend a Synchrosqueezed Continuous Wavelet Transform representation, available in ssqueezepy. Synchrosqueezing arose in context of audio processing (namely speaker identification), and there's ...

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Why the value of Gaussian curve drop to 1/19 at 2 standard deviation?
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5 votes

You've computed the most general case and have shown it's always 2/15, thus 1/19 is incorrect, at least if interpreting "1/x-th of value" as $f(\text{value})/f_\text{max}$. This simulation ...

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Synchrosqueezing Wavelet Transform explanation?
5 votes

Low-level intuition can be obtained by inspecting the phase transform, visually. Answer complements and is complemented by this one. (-- Answer code) We consider a pure sinusoidal tone; ideas extend ...

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Advantage of STFT over wavelet transform
4 votes

STFT is frequency-shift equivariant - same absolute shift has same effect on representation regardless of original frequency${}^1$: $$ \hat x(\omega) \rightarrow \hat x(\omega - c) \Leftrightarrow \...

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Wavelet Scattering properties & implementation?
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4 votes

Scattering overview provided in this answer. Computational structure Fig 4, Deep Scattering Spectrum In steps: (First order begins) $x$ convolves with $\psi1_i$ --> $W1_i$ Modulus, $W1_i \...

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Why does twice the sampling rate (Nyquist Theorem) seem inadequate?
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4 votes

Perfect recovery is one thing, niceness is another. Sampling above x2 Nyquist is sufficient for perfect recovery, after which we can FFT-upsample to make it look nice - which is more efficient than ...

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Understanding stability in frame theory. Wavelets
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3 votes

They aren't equivalent; "stability" is used differently in each context. $(1)$ guarantees a stable inverse. If $A=0$, we lose information. Existence of such $A$ and $B$ ensure the ...

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How to alleviate the edging effect of the Hilbert transform?
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3 votes

The effect can be alleviated with appropriate padding, which imposes a 'statistical prior' (i.e. assumption). No padding is equivalent to periodic padding, meaning signal's right joins its left, and ...

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Why is $y(n)=x(n)-x(n−1)+x(n-2)$ a low-pass filter?
3 votes

Observe the magnitude of frequency response (rescaled to 0 to 1): where $$ \begin{align} H(\omega) &= e^{-j 0\omega} - e^{-j 1\omega} + e^{-j 2\omega} \\ &= 1 - e^{-j 1\omega} + e^{-j 2\...

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How does the scale of a wavelet relate to the Fourier frequency (or period) under CWT?
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3 votes

The "frequency" in "center frequency" does refer to Fourier frequency - rate of sinusoidal oscillation - but measures (and interpretations) can vary depending on the wavelet. Said ...

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Compared with the real-valued continuous wavelet, what are the advantages of the complex-valued continuous wavelets?
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3 votes

Complex, or analytic wavelets enable: Instantaneous frequency, amplitude, and phase extraction - detailed post. Robust feature extraction for classification, stable against time-warping deformations (...

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Nyquist frequency isn't working
3 votes

Your approach confuses frequency with phase; the correct formulation is $$ \sin(2\pi \phi(t)) $$ where $\phi(t) = \int \omega(t)dt$. Related post. I derived the most general form for a linear chirp ...

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FFT for long waveform
3 votes

If compute time doesn't matter, you can overcome the memory problem easily with a dual for-loop implementation. But there's an FFT alternative with caveats: break up signal into M seqments, take M FFT'...

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In the context of DFT, Where Does the Nyquist Frequency Sample Belong In a Double Sided Frequency Spectrum (Positive / Negative Side)?
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3 votes

It's convention, they're equivalent: $$ \exp{\left(j2 \pi \frac{N}{2}n/N \right)} = \exp{\left(j2\pi \frac{-N}{2}n/N\right)} \\ \Rightarrow e^{j\pi n} = e^{-j \pi n} \Rightarrow \cos(\pi n) = \cos(-\...

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Interpreting N in DFT as the Number of Points vs. Number of Intervals
3 votes

The key is to understand what the DFT says, vs what we seek. Consider a cosine, where we change $f \text[Hz]$, $N$, and $t$ and observe the effect on DFT: [1]: DFT "sees" 1 cycle in the &...

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Can FFT convolution be faster than direct convolution for signals of large sizes?
3 votes

FFT convolution is certainly scalable, but what you really ask is if it's faster when one of inputs is small (<1000) or input lengths differ greatly. Then indeed FFT convolution can be slower, as ...

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PyWavelets CWT: normalization? Vs Scipy?
Accepted answer
3 votes

Let's follow the math from incubation to delivery. It begins with psi, a rescaled morlet2 (as shown previously) at a scale $a=64$, and $\sigma=5$: $$ \psi = \psi_{\sigma}(t/a) = e^{j\sigma (t/a)} e^{-(...

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find phase greater than 2*pi in MATLAB
3 votes

You can, if you increase phase between samples slowly enough, using unwrap(angle(signal)). "Slowly enough" means the phase doesn't jump by more than $2 \pi$; unwrap works by tracking "...

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Role of window length and overlap in uncertainty principle?
2 votes

Overlap is and isn't related to time resolution: in sense of the uncertainty principle, only the window width plays a role. However, any overlap other than maximum (hop_size = len(window) - 1) will ...

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What is the importance of the translational invariance of the CWT?
2 votes

CWT is translation-invariant in feature sense: translating a pattern translates its representation but not modify it. In coefficient sense, it is translation equivariant: shift signal $\Leftrightarrow$...

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Why are there copies of a signal in the frequency domain?
2 votes

It's the spectrum of a discrete signal: sampling in time $\Leftrightarrow$ periodizing in frequency - explained in detail here. Overlap means there's aliasing, and we require a higher sampling rate. ...

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Computable Time-Frequency Distribution without Cross-Terms
2 votes

Synchrosqueezing. Diminishes Wigner-Ville interaction disadvantages as described here.

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Is it possible to "determine"/"evaluate" the perceived quality of a music audio/video record by using software in an automatic way?
2 votes

Is it possible to analyze a recording in a software way ... to "determine"/"evaluate" its subjective quality Yes, very possible; all one needs is to define, mathematically, what &...

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What exactly is meant by "translation invariant dictionaries/wavelets"?
Accepted answer
2 votes

It's rather translation equivariant: $$ \text{CWT}_{s, t}x(t - t_0) = \text{CWT}_{s, t - t_0}x(t) \tag{1} $$ and $$ \langle x(t−t_0),\psi(t) \rangle= \langle x(t), \psi(t+t_0)\rangle \tag{2} $$ That ...

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Which of the following sampling methods can be used to sample x(t) such that this signal can be uniquely recovered from its samples?
Accepted answer
2 votes

B): $f_s \geq 2B = 2\cdot(2015-1612) = 2\cdot 403 = 806$. "Second-order" bandpass sampling is described in this paper (or older, here). It's sampling $x(t)$ at a lower sampling rate, $M$ ...

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Difference between single tone and dual tone signals?
2 votes

It's either, depending how we look at it: $$ \sin(f_1t) + \sin(f_2t) = 2 \cos(.5(f_1 - f_2)t)\sin(.5(f_1 + f_2)t) \tag{1} $$ A common empirical rule is that if $f_1 \ll f_2$ (e.g. $f_2 / f_1 > 5$), ...

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Can Goodness-of-fit increase while noise also increases?
2 votes

The figure means to illustrate a consequence of overfitting: if we fit train data too well, we'll fit noise too, and thus generalize poorer. "Noise > regularity" can be thought of as low ...

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Why does my amplitude change upon inverse Fourier Transform when I am only randomizing the phase of the fourier transform using Python numpy?
Accepted answer
2 votes

OP seeks to randomize phase while keeping spectrum magnitude unchanged - and has achieved it. All that remains is to plot the spectrum. Just add this code: def plot(x0, x1, title): plt.plot(x0) ...

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