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 ...

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

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, ...

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 ...

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 ...

STFT is frequency-shift equivariant - same absolute shift has same effect on representation regardless of original frequency${}^1$: $$\hat x(\omega) \rightarrow \hat x(\omega - c) \Leftrightarrow \... View answer Accepted answer 4 votes Scattering overview provided in this answer. Computational structure Fig 4, Deep Scattering Spectrum In steps: (First order begins) x convolves with \psi1_i --> W1_i Modulus, W1_i \... View answer Accepted answer 4 votes Perfect recovery is one thing, niceness is another. Sampling above x2 Nyquist is sufficient for perfect recovery, after which we can FFT-upsample to make it look nice - which is more efficient than ... View answer Accepted answer 3 votes They aren't equivalent; "stability" is used differently in each context. (1) guarantees a stable inverse. If A=0, we lose information. Existence of such A and B ensure the ... View answer Accepted answer 3 votes The effect can be alleviated with appropriate padding, which imposes a 'statistical prior' (i.e. assumption). No padding is equivalent to periodic padding, meaning signal's right joins its left, and ... View answer 3 votes Observe the magnitude of frequency response (rescaled to 0 to 1): where$$ \begin{align} H(\omega) &= e^{-j 0\omega} - e^{-j 1\omega} + e^{-j 2\omega} \\ &= 1 - e^{-j 1\omega} + e^{-j 2\...

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 ...

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

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 ...

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'...

It's convention, they're equivalent: $$\exp{\left(j2 \pi \frac{N}{2}n/N \right)} = \exp{\left(j2\pi \frac{-N}{2}n/N\right)} \\ \Rightarrow e^{j\pi n} = e^{-j \pi n} \Rightarrow \cos(\pi n) = \cos(-\... View answer 3 votes The key is to understand what the DFT says, vs what we seek. Consider a cosine, where we change f \text[Hz], N, and t and observe the effect on DFT: : DFT "sees" 1 cycle in the &... View answer 3 votes FFT convolution is certainly scalable, but what you really ask is if it's faster when one of inputs is small (<1000) or input lengths differ greatly. Then indeed FFT convolution can be slower, as ... View answer Accepted answer 3 votes Let's follow the math from incubation to delivery. It begins with psi, a rescaled morlet2 (as shown previously) at a scale a=64, and \sigma=5:$$ \psi = \psi_{\sigma}(t/a) = e^{j\sigma (t/a)} e^{-(...

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 "...

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 ...

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 $\Leftrightarrow$...

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. ...

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

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 &...

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 ...

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, $M$ ...

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. $f_2 / f_1 > 5$), ...