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The discrete-time signal $x_n$ is a sum of $K$ discrete-time complex damped exponential: \begin{equation} x_n = \sum_{k=1}^{K} \underbrace{(a_k e^{j\phi_k})}_{c_k} {\underbrace{e^{\{(- \alpha_k + j2\pi f_k )\Delta t\}}}_{z_k}}^n, \quad n = 0,1,...,N-1 \end{equation}

where $a_k$ - amplitudes, $\phi_k$ - phases, $\alpha_k$ - damping factors and $f_k$ - frequencies are the parameters of the model and $n$ is the time index and $\Delta t$ is the time interval.

\begin{equation} \label{sig_modf} x_n = \sum_{k=1}^{K} c_k z_k^n, \quad n = 0,1,...,N-1 \end{equation}

where $c_k$ is the complex amplitude including the phase of the $k$-th damped exponential and $z_k$ is the $k$-th signal pole.

Such a signal will satisfy the below recurrence relation: $x_n = p_1 x_{n-1} + p_2 x_{n-2} + ... + p_K x_{n-K} \tag 1$

For this system, I would like to know what is the state-space and Transfer-function model. I know that the companion matrix given by the coefficients $p$'s in the first row and 1's in the sub-diagonal is the state matrix $A$.

$A = \begin{bmatrix} -p_1 & -p_2 & -p_3 & \dots & -p_{K-1} & -p_K \\ 1 & 0 & 0 & \dots & 0 & 0\\ 0 & 1 & 0 & \dots & 0 & 0\\ 0 & 0 & 1 & \dots & 0 & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & \dots & 1 & 0 \\ \end{bmatrix} \tag 2$

Is this correct and are my B, C and D matrices 0? Also, what is the TF model for this system?

Given the signal sequence $x_n$, is it possible to find the state-space and transfer-function model using MATLAB lsim?

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  • $\begingroup$ Hi: googe for "AR(p) model in state space form" and a lot of things will come up. I didn't know which one to send or which one was most useful so I'll leave that to you. $\endgroup$
    – mark leeds
    Sep 27, 2021 at 14:59
  • $\begingroup$ "are my B, C and D matrices 0?" I don't know -- does your system have any inputs you didn't mention? Does it have any outputs you didn't mention? A transfer function for a system without either input or output is boring; it is just $H(z) = \emptyset$. $\endgroup$
    – TimWescott
    Sep 27, 2021 at 15:54
  • $\begingroup$ Thank you for your comments. I have a signal sequence that is computed from a mathematical model - sum of damped complex exponentials. Thus, they satisfy the recurrence relation given in (1). This is the only information I have. I was also thinking the tf = 0, but I wanted to make sure, whether it is feasible. $\endgroup$
    – Neuling
    Sep 27, 2021 at 16:45
  • $\begingroup$ Presumably you have the initial state of the system, or some measurements of the states, and you're interested in some output or the states -- if so, edit your question to say so, and detail what you have and what you're interested in. $\endgroup$
    – TimWescott
    Sep 27, 2021 at 17:40
  • $\begingroup$ I provided the background. I do not have information, other than the signal sequence $x_n$ and the mathematical model to generate it. I am interested in knowing the state space model of this system. My ultimate goal is to estimate the parameters $f,a,\alpha,\phi$ by determining the eigenvalues, which are the signal poles. $\endgroup$
    – Neuling
    Sep 27, 2021 at 18:07

1 Answer 1

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Since you don't have any inputs, $x_n$ is only a function of past values of itself, your B an D should indeed be zero. You don't have your output defined either. You could pick the most recent $x_n$ as output, but it could also be something else.

A transfer functions describe an input output relation. However, you don't have inputs, so also no input output relation thus no transfer function.

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  • $\begingroup$ Could you also comment on the C matrix? is it also zero? $\endgroup$
    – Neuling
    Sep 27, 2021 at 16:47
  • $\begingroup$ @Neuling look at my comments regarding the output of the system. Using your current description there is also no definition of what the C matrix should be. $\endgroup$
    – fibonatic
    Sep 27, 2021 at 17:15

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