18 votes
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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 ...
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17 votes
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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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13 votes
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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, ...
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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 ...
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10 votes

A case that zero padding increase real resolution and extract more info than naive DFT?

‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎ ‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎ ‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎ ‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎ ‎‎‎‎‎‎‎‎
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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 ...
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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(...
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  • 26.5k
8 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 ...
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  • 181
7 votes
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How does time shift correspond to phase change in a discrete signal?

If you have a signal $$f[n]=\cos(\Omega_0n)$$ and you apply a time shift of $n_0$ you get $$f[n+n_0]=\cos(\Omega_0(n+n_0))=\cos(\Omega_0n+\Omega_0n_0)=\cos(\Omega_0n+\phi)$$ where $\phi=\...
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  • 79.2k
7 votes

Time-frequency representation of sound signal using Matlab

I believe that this "color graph" you are looking for is a spectrogram (although it looks to me more like a scalogram, but you did not mentioned wavelets). Let me give you an example in MATLAB of ...
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7 votes
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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-...
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  • 2,390
6 votes
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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 (...
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  • 26.5k
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
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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: $$ \|...
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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 ...
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6 votes
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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 ...
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6 votes
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Show Equivalence Between Multiplication in Time Domain to Convolution in Frequency Domain

Let's assume you have 2 signals: vX and vY. So: ...
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  • 39k
6 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 ...
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5 votes
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Continuous Wavelet Transform with Scipy.signal: what is parameter "widths" in cwt() function? How do time-frequency?

complex morlet was added Aug 10, 2007 ricker and cwt were added Sep 20, 2011 There's no indication that cwt is meant to be compatible with ...
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5 votes
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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.,...
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  • 79.2k
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 ...
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  • 33.8k
5 votes

Extracting Peak Frequencies Using FFT vs. Time Domain Peak Finding

You basically have a slowly changing Sine signal where the parameter which changes is its frequency. The right approach in my opinion would be to use Frequency Modulation processing of the signal. ...
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  • 39k
5 votes
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Play with a Gaussian Random Set in the Frequency Domain to Obtain Desired Effect in the Time Domain

You're asking to do a localized operation in time using the Frequency Domain. It's going to be not elegant, really. Here what you can do: Define the input signal in frequency domain as $ X \left[ ...
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  • 39k
5 votes
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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 ...
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  • 2,104
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,...
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  • 13.5k
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} ...
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  • 41
4 votes

Time-varying "impulse response"

In writing $h(\tau,t)$ with $t$ is time and $\tau$ is delay, we are in the model that $\tau$ varies "differently" from $t$. Or in other words, they are different notions in spite of the fact that they ...
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4 votes
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Confused on the difference between the frequency spectrum of an entire song, and the frequency spectrum of a point in time

As far as my logic goes, I wont ge able to tell any of musical features just of an amplitude value in the time domain. That's wrong. The amplitude-over-time actually is the way the song is played by ...
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4 votes
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How to interpret the effect of different windows in short time fourier transform?

A widnow $w[n]$ 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 ...
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