Reconstruction is possible so long as NOLA is obeyed - which is an easier criterion (on synthesis information) to meet than what you seek (analysis information). To discriminate temporal variations ...

Time scattering by design yields timeshift-invariant representations, with control over amount of desired invariance. An advantage is ability to compare within a frequency range of interest rather ...

Clarifying for signal processing, I looked into this from VVT's answer and it appears this has nothing to do with FFT of some time shifted waveform but rather the physical consequence of doing so in ...

The full question probes as far as "what is science?", so I'll try simplifying. Fourier Transform is a tool. A mathematical construct. The goal is to accurately describe reality. Suppose a ...

Confusion seems rooted in notation: In $s = f_0 / (f \cdot \text{sampling rate})$, $f_0$ is the center frequency of mother $\psi$, and $f$ is the center frequency of the scaled $\psi$, i.e. $\psi_s$. ...

The mel spectrogram additionally includes a step of projecting (power of) STFT bins onto Mel-frequency bins via a Mel filterbank; I don't have access to path so I made demo on exponential chirp: You ...

Depending on the chirp, if you seek a closed form solution, there may not be one. For empirical estimation, one can use non-stationary methods. Take exponential with frequency ranging from 0.005 to ....

We take a wavelet transform over other wavelet transforms, which involves multiple wavelets. E.g.: First-order uses 100 wavelets to tile the frequency domain and produce 100 coefficients Second-order ...

PyWavelets' cwt is flawed, and so is scipy's; use ssqueezepy.cwt. a low pass for the lower frequencies not covered using the mother and daughter wavelets ... with the short time Fourier transform of ...

unbuffer in Python; helpful post. Nice visuals in MATLAB docs (bottom).

Relevant visual, and on FFT meaning. If you append a zero to a perfect cosine, you increase lengths of all FFT bases (real cosines, imag sines) by 1, so none of them perfectly correlate (multiply &...

You can use a time-frequency transform like CWT or STFT and focus it via reassignment for component extraction - example, synchrosqueezing (full post): From there you run basic math on the relevant ...

My example isn't in fact below $f_s/2$; I blundered with $f_s=N$. Adjusting this eliminates aliasing, for linear, exponential, and hyperbolic chirps - and, in fact, for any signal with $\phi '(t) < ... View answer 0 votes Accepted answer is wrong: DWT (actually CWT) plot y-axis must read frequency not scales; the two are inversely related. CWT and STFT aren't equivalentlish-ly similar as suggested; the same plot ... View answer 0 votes Summary: the dual wavelet's role is analogous to that of$e^{j\omega t}$; it undoes the wavelet's convolving with the signal (integrated inner product). The main intricacy's indeed in normalization; ... View answer Accepted answer 0 votes Verdict: I conclude scipy's higher leftmost peak is due to both pywt's wavelets' lower amplitude at lower scales and scipy's wavelets' stronger correlation with lower frequencies at lower scales. ... View answer Accepted answer 0 votes First see "Naive Breakdown" in a below section. Onto PyWavelets: the algorithm was found on Github to stem from an old MATLAB implementation, but it provides no details on coding the wavelet ... View answer Accepted answer 0 votes The normalization is indeed by 1 / sqrt(scale), and it's an L2-norm; the trick's in the scale wavelet. I'll use wavelet='morl' throughout. Prior to integrating, we can inspect the wavelet here; it's ... View answer 0 votes WINNER: Olli's solution: $$F_{N-1}[k] = X_{N-1}[k]\big(e^{j2\pi k / (N - 1)} - 1\big) + (x[N-1] - x)e^{j2\pi k/(N-1)}$$ Code + Demo: def dft(x): return np.fft.fft(x[:-1]) def d_idft(coef, x):... View answer Accepted answer -1 votes Note: Originally I failed to mention the "complex" clarification to follow (and at bottom), hence the downvotes (which now are undue). The coefficients give amplitude and phase of a complex ... View answer -1 votes It's$N/f_s$. Because consider what happens if it's$(N-1)/f_s$instead; take a sequence$x$of 6 samples, sampling once per second. Going by latter, defining "duration" as that between ... View answer -1 votes The textbook is wrong, or very misleading depending on interpretation. The frequency response is square - or, constant over the swept frequency. Assuming an ideal chirp system which can scale up$f$... View answer -2 votes They do have a 1Khz component, but not via a Fourier lens. To see what's happening, let's superimpose the AM signals with pure carrier tones: Left plot reveals the key difference between the two AM ... View answer -3 votes Post's structured addressing an originally different formulation of the question, "agree" and "disagree" responding to "can a bandlimited signal amplitude-alias?" - but ... View answer Accepted answer -4 votes TL;DR No, they don't. But in a specific sense, their continuous-time interpolation will be zero-phase if symmetric about$t=n=0\$ (i.e. in periodic extension). In discrete case, however, symmetric will ...