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## Hot answers tagged time-frequency

34 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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29 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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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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11 votes

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

‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎ ‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎ ‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎ ‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎ ‎‎‎‎‎‎‎‎
• 8,984
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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9 votes

• 2,710
7 votes
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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, $W1_i \... • 8,984 7 votes Accepted ### Why does a signal with constant frequency have spots that changes colors at a specific value of scale (and so frequency) in the scalogram? Re: real part There are oscillations because that's what the wavelet transform is - a decomposition into zero-mean, localized oscillations. CWT is convolution (rather, cross-correlation) of signal ... • 8,984 6 votes Accepted ### Motivation of time-frequency analysis What a tricky question to overlook. Indeed I'm one of those who would immedieately press that Fourier transforms do lose time localization of the events as the comments stated. Yet it's certainly (... • 28.3k 6 votes ### Motivation of time-frequency analysis Fourier transforms generally yield complex spectral data. Under some technical conditions, they are bijections. From the Fourier transform, you can uniquely recover one single signal. However, when ... 6 votes Accepted ### Uncertainty principle - Duration bandwidth principle An important theorem, known as Weyl's, 1931, is: if function$s(t)$and related functions$ts(t)$,$s'(t)$are in$L^2$(square integrable) with the related$\|\cdot\|L_2$norm symbol then: $$\|... 6 votes Accepted ### How to interpret the effect of different windows in short time fourier transform? A window w[n] truncates and weights (tapers) an input signal x[n], to produce v[n] = x[n]. w[n]., for subsequent spectral analysis of x[n]. A windows's effect on the input signal's true ... • 28.3k 6 votes ### Comparison of Linear Convolution and N Point DFT The result {4,1,2,3} is the circular convolution of {1,2,3,4} and {0,1,0,0} which you correctly get by taking the inverse DFT of the product of the DFTs of the two sequences. We can check this by ... • 52.3k 5 votes ### Motivation of time-frequency analysis An FT (being invertible) does preserve all transient event time information, however this time locality information is usually preserved by distributed it as varying changes to the entire phase ... • 35.4k 5 votes Accepted ### Infinite extent of spectrum, but also in time in Oppenheim's Discrete Time Signal Processing? Not at all. The Uncertainty Principle says that a function cannot be both limited in time and limited in frequency. More specifically, the product of the signal's widths in time and in frequency (i.e.,... • 90.5k 5 votes ### Stability of system with poles inside unit circle - conflict with differential equation You're conflating the discrete-time definition of a system with the continuous-time representation of a system. Your discrete-time$$Y(z)\cdot\big(z-\frac{1}{2}\big)=X(z)\cdot z$$does not ... • 25.9k 5 votes Accepted ### Wavelet "center frequency" explanation? Relation to CWT scales? The "frequency" in "center frequency" does refer to Fourier frequency - rate of sinusoidal oscillation - but measures (and interpretations) can vary depending on the wavelet. Said ... • 8,984 5 votes Accepted ### Impulse response of Time Varying Channel In the context of wireless communications, the channel impulse response (CIR) is often estimated indirectly via the time-varying transfer function (TVTF) H(t, f), defined by:$$ H(t, f) = \mathcal ... • 2,222 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,... • 15.3k 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 ... • 90.5k 5 votes ### Amplitude extraction using STFT Firstly, STFT is fundamentally a time-frequency transform: convolutions with windowed complex sinusoids (i.e. bandpass filtering). You aren't going to "frequency", and "windowed Fourier ... • 8,984 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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