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Instantaneous usually means the value of something right at this time. I think what this means in your case is that the function $f(t)$ is complex and you are taking the argument (finding the angle): …

As an example of what I mean, let's have a look at a simple sinusoid (though I see you're actually assuming double gamma functions; I may rework this later to use those). This models the signals as: …

The "Nyquist plot" you have is really just the Fourier transform of the vibration data. That the plot is "circular" doesn't mean much: it just means that the phase of the signal is continually increa …

So, let's check. For a signal to be periodic with period $P$, we need: $$ x(t) = x(t+P) $$ for all $t$. Here $$ x(t) = 1/2(\cos(4\pi t))$$ and $$ x(t+P) = 1/2(\cos(4\pi [t+P]))$$ so for such a $P$ …

Precisely the form that the $\mathbf{H}$ matrix takes depends on your signal model. The nice thing about positional models is that we have a very clear relationship between acceleration, velocity, an …

When we say: let $f(t)=\sin⁡(\omega t)$ be a signal, do we mean a function or the set of output from that function i.e. is signal a function or the output? As Gilles says in his answer, when we t …

The Earth gives you an acceleration of $ g = 9.8 \mbox{m/s}^2$ in the $z$ direction. That is what the accelerometer is measuring.

The usual way I'd think about approaching this sort of problem is to apply a Kalman filter. The first step in applying a Kalman filter is knowing what the model for your signal is: Can you write down …

As Hilmar says, you can get pretty loose bounds. Might be better just to grind it out: import numpy as np from matplotlib import pyplot as plt def calculate_signal(A, omega, phi, t): if isinstanc …

Jim's answer sent me to think about how to test this statistically. This led me to the Durbin-Watson autocorrelation test. The generalization of it is to form: $$ DW(\tau) = \frac{\displaystyle \su …

The in-phase and quadrature components are easy to break out. You just need to demodulate using in-phase and quadrature local oscillators. See the diagram below. The received signal $r(t,k)$ is mix …

Jim Clay's answer is good, but there is a caveat: you want the sampling frequency low, but not too low. If you have real-valued data (as opposed to complex-valued data), the you should try to avoid t …

I'd just take a KIS (keep it simple) approach as a first step. Define your (unknown) signal as $$ s(t) = k A(t) + m B(t) + n(t)$$ and then just define the error: $$ e(\tilde{k}, \tilde{m}) = \sum_{\f …

When I was working on SONAR data, the interesting SNR regime was -30dB to -20dB. For this specific paper, we were trying to first find the shape of the towed hydrophone array, and then use that estim …

I would have thought the place to start was option 4: at the peaks of the acceleration curve. Just before your foot hits the ground, it is going down as fast as possible. Just after it hits the gro …