# Tag Info

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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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### 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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### 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, ...

### 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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### 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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### 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 (...

### 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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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: $$\|... 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 ... 6 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 ... 6 votes Accepted ### Show Equivalence Between Multiplication in Time Domain to Convolution in Frequency Domain Let's assume you have 2 signals: vX and vY. So: ... 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 ... 5 votes Accepted ### 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 ... 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.,... 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 ... 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. ... 5 votes Accepted ### 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[ ... 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 ...

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,...

### 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} ...

### 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 ...
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 ...