34
votes
Accepted
Synchrosqueezing Wavelet Transform explanation?
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 ...
- 8,815
27
votes
Accepted
Wavelet Scattering explanation?
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, ...
- 8,815
17
votes
Accepted
Why LTI system cannot generate new frequencies?
One of the definitive features of LTI systems is that they cannot generate any new frequencies which are not already present in their inputs.
One way to see why this is so, comes by observing the ...
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11
votes
A case that zero padding increase real resolution and extract more info than naive DFT?
- 8,815
10
votes
Why LTI system cannot generate new frequencies?
You can make a simple algebraic argument, given the premise that you provided. If:
$$
Y(\omega) = X(\omega) H(\omega)
$$
where $X(\omega)$ is the spectrum of the input signal and $H(\omega$) is the ...
- 24.5k
9
votes
Doppler shift in time domain?
The term Doppler Shift is actually a bit of a misnomer. The frequencies are not actually shifted but they are scaled (see http://fourier.eng.hmc.edu/e101/lectures/handout3/node2.html for definition of ...
- 42.6k
9
votes
A system that perfoms Fourier Transform operation - is it an LTI system?
The Fourier transform operator $\mathscr{F}$ is a linear one; i.e.,
$$\mathscr{F}\{x(t)\}=X(f) ~,~ \mathscr{F}\{y(t)\}=Y(f) \implies \mathscr{F}\{\alpha x(t) + \beta y(t) \} = \alpha X(f) + \beta Y(...
- 28k
9
votes
Discrete wavelet transform; how to interpret approximation and detail coefficients?
Wavelet transforms can be more difficult to interpret than FFT at face value due to the various representations, nomenclature and output formats. I had to study more than 15 resources to get a good ...
- 191
9
votes
Synchrosqueezing Wavelet Transform explanation?
Low-level intuition can be obtained by inspecting the phase transform, visually. Answer complements and is complemented by this one. (-- Answer code)
We consider a pure sinusoidal tone; ideas extend ...
- 8,815
8
votes
Accepted
How to validate a wavelet filterbank (CWT)?
Wavelets isn't just sampling with scales from some min to max - but it is what many implementations do, including scipy and <...
- 8,815
8
votes
Accepted
Equivalence between "windowed Fourier transform" and STFT as convolutions/filtering
This answer isn't fully developed (but is correct), I may update in the future.
Indeed, STFT is strided convolutions, and there's many practical caveats, discussed in this answer.
Standard vs improved ...
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7
votes
Accepted
Stability of system with poles inside unit circle - conflict with differential equation
What you are missing is that this is about a discrete-time system, because we're talking about poles and zeros in the complex $z$-plane and about poles inside or outside the unit circle. So there is ...
- 88.9k
7
votes
Accepted
Fourier Transform with both Time Delay and Frequency Shift
If you are ever unsure, just go back to the definition and work out the Fourier Transform property for the specific situation:
$$\begin{align*}\mathscr{F}\left\{x\left(t-t_0\right)e^{j2\pi f_0\left(t-...
- 2,700
7
votes
Accepted
Wavelet Scattering properties & implementation?
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 \...
- 8,815
7
votes
Accepted
Why does a signal with constant frequency have spots that changes colors at a specific value of scale (and so frequency) in the scalogram?
Re: real part
There are oscillations because that's what the wavelet transform is - a decomposition into zero-mean, localized oscillations. CWT is convolution (rather, cross-correlation) of signal ...
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6
votes
What's the difference between the Gabor and Morlet wavelets?
For those looking for a compact description, I found this nice docstring while inspecting the kymatio GitHub repository:
A Morlet filter is the sum of a Gabor ...
- 160
6
votes
Accepted
Motivation of time-frequency analysis
What a tricky question to overlook. Indeed I'm one of those who would immedieately press that Fourier transforms do lose time localization of the events as the comments stated. Yet it's certainly (...
- 28k
6
votes
Motivation of time-frequency analysis
Fourier transforms generally yield complex spectral data. Under some technical conditions, they are bijections. From the Fourier transform, you can uniquely recover one single signal. However, when ...
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6
votes
Accepted
Uncertainty principle - Duration bandwidth principle
An important theorem, known as Weyl's, 1931, is:
if function $s(t)$ and related functions $ts(t)$, $s'(t)$ are in $L^2$ (square integrable) with the related $\|\cdot\|$ $L_2$ norm symbol then:
$$ \|...
- 31.7k
6
votes
Accepted
How to interpret the effect of different windows in short time fourier transform?
A window $w[n]$ truncates and weights (tapers) an input signal $x[n]$, to produce $v[n] = x[n]. w[n]$., for subsequent spectral analysis of $x[n]$. A windows's effect on the input signal's true ...
- 28k
6
votes
Comparison of Linear Convolution and N Point DFT
The result {4,1,2,3} is the circular convolution of {1,2,3,4} and {0,1,0,0} which you correctly get by taking the inverse DFT of the product of the DFTs of the two sequences.
We can check this by ...
- 48.9k
5
votes
Accepted
Infinite extent of spectrum, but also in time in Oppenheim's Discrete Time Signal Processing?
Not at all. The Uncertainty Principle says that a function cannot be both limited in time and limited in frequency. More specifically, the product of the signal's widths in time and in frequency (i.e.,...
- 88.9k
5
votes
Motivation of time-frequency analysis
An FT (being invertible) does preserve all transient event time information, however this time locality information is usually preserved by distributed it as varying changes to the entire phase ...
- 35.2k
5
votes
Stability of system with poles inside unit circle - conflict with differential equation
You're conflating the discrete-time definition of a system with the continuous-time representation of a system.
Your discrete-time
$$Y(z)\cdot\big(z-\frac{1}{2}\big)=X(z)\cdot z$$
does not ...
- 25.2k
5
votes
Accepted
Wavelet "center frequency" explanation? Relation to CWT scales?
The "frequency" in "center frequency" does refer to Fourier frequency - rate of sinusoidal oscillation - but measures (and interpretations) can vary depending on the wavelet.
Said ...
- 8,815
5
votes
Accepted
Impulse response of Time Varying Channel
In the context of wireless communications, the channel impulse response (CIR) is often estimated indirectly via the time-varying transfer function (TVTF) $H(t, f)$, defined by:
$$
H(t, f) = \mathcal ...
- 2,212
5
votes
A case that zero padding increase real resolution and extract more info than naive DFT?
What zero padding cannot do is help you resolve frequencies that cannot be resolved without it.
Without zero-padding, the resolution is 1 Hz and the DFT can in fact resolve the tones at 440 and 441 Hz,...
- 14.9k
5
votes
Accepted
"Instantaneous impulse response" in a linear time-varying system
The problem is that there are two common definitions of the impulse response of an LTV system, resulting in the following input-output relations:
$$ y(t)=\int_\tau h_1(t,\tau)x(\tau)d\tau\tag{1}$$
and
...
- 88.9k
4
votes
How to Map CWT to Synchrosqueezed wavelet transform?
Let me explain the intuition briefly. The authors of the paper you've cited assume that the signal $x(t)$ can be written in the form
\begin{align*}
x(t) &= \sum_{k=1}^K a_k(t) \exp(2\pi\mathrm{i} ...
- 41
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