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I am looking for some interesting and physically meaningful applications of different signal models. I am currently working with complex analytic signal model given below, but I couldn't come up with some applications of it, where it is meaningful to understand the model better.

\begin{equation} x[n] = \sum_{k=1}^{K} (a_k e^{(j\phi_k)})(e^{\{(- \alpha_k + j2\pi f_k)T\}n})u(nT), \quad n = 0,1,...,N-1 \end{equation}

By "physically meaningful" I mean the following case: Let us consider an object falling in a resistive medium and its velocity is given by: \begin{equation} v(t) = \frac{g}{k}(1-e^{-kt}) \end{equation} where $g$ is acceleration due to gravity, here $-9.81 m/s^2$ and $k$ is the resistance factor of the medium. Similarly, the distance of an object $s$, moving with acceleration $a$ and initial velocity $u$ is defined by quadratic time series model as: $$ s = \frac{1}{2}at^2 + ut $$

I would also like to get some meaningful examples for other time series models available like sinusoidal, ramp etc.

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    $\begingroup$ Where did you get that model from? Math without context is just a Rorschach blot for the helpful. It looks like the solution to an ordinary linear differential equation of order $K$, with no forcing function -- if, in fact, it is, then there's your physical application. $\endgroup$
    – TimWescott
    Jul 9, 2021 at 14:37

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Easy ones first:

Ramp: Position when moving through space at a constant rate. Or modulo position from a closest 1 foot (or 1 mile, or 1 inch, or 1 meter etc) boundary when moving through space at a constant rate for a repeating ramp.

Sine wave: The horizontal displacement of a spinning wheel at a fixed location. The shadow of a stick spinning in the air with a light above and perpendicular to it when the stick is horizontal.

For the first equation, let's consider it in parts, dropping the redundant $u(nT)$ that does not need to be included since $n$ is specified:

$$ x[n] = \sum_{k=1}^{K} (a_k e^{(j\phi_k)})(e^{\{(- \alpha_k + j2\pi f_k)T\}n}), \quad n = 0,1,...,N-1$$

$$ = \sum_{k=1}^{K} (a_k e^{j\phi_k})e^{- \alpha_k Tn} + \sum_{k=1}^{K} (a_k e^{(j\phi_k)})e^{j2\pi f_kTn}, \quad n = 0,1,...,N-1$$

For the quantity $a_k e^{j\phi_k}$, I assume $a_k$ and $\phi_k$ are real as would be the common conventions for the form $Ae^{j\phi}$ which represents a phasor of magnitude $A$ and angle $\phi$. So for that consider it a spoke on a bicycle wheel that has length $a_k$ (assuming $a_k$ is positive, otherwise it would be the spoke directly opposite), and rotated from the horizontal plane by an angle $\phi_k$. For the first term above in the summation given for each of the $N$ samples, we have $K$ of these phasors (bicycle spokes) with arbitrary lengths and arbitrary phase, and for each of those we multiply them over every sample $n$ by an amount that decreases with every sample $n$ and then sum that result. (assuming $\alpha_kT$ is a real positive number). So if we had $K$ spokes each with an arbitrary length and starting phase angle, we modify each length according to $e^{\alpha_k n}$ and then sum the total result which means placing each spoke on the end of the previous (imagine the hub of the wheel secured to the end of the modified spoke length) and then rotated each spoke to it's proper angle relative to the horizontal, the result would be the straight line from the initial origin to the end of the last length modified spoke (as a vector itself with magnitude and phase).

For the second term we are multiplying each of the $K$ spokes that each have an arbitrary magnitude and phase angle by a rotating phasor: $e^{j2\pi f_k T n}$. This phasor has a magnitude of 1 and a phase that changes for each sample $n$. But since this phase is dependent on each $k$ (given by $f_k$), without further information we assume $f_k$ is real and there is no relationship between any of the $f_k$'s for all values of $k$. This means the rotation that would occur for any given $k$ is arbitrary. So for each of the $n$ samples, we would rotate each of the $K$ modified length spokes by an angle given by whatever $2\pi f_k T n$ is and sum those modified spokes as described for the first term.

If there is more context on what this formula is for, or more specifically more details that would restrict what $f_k$, $\theta_k$, and $\alpha_k$ may be, there may be further physical insight or intuition as to what is occurring. This appears to be a finite length Discrete Laplace Transform in which $f_k$ itself would be a fundamental frequency that increases by integer $k$, but as written each of these values can be completely independent for each sample $k$, and as such it is difficult to provide any clearer physical meaning.

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  • $\begingroup$ Thank you. Do you have some examples in mind, regarding the complex analytic signal mentioned in the first equation? $\endgroup$
    – Neuling
    Jul 9, 2021 at 13:01
  • $\begingroup$ @Neuling I updated my answer. $\endgroup$ Jul 9, 2021 at 13:45

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