# State-space model and/or TF for a $K$-th order linear system given the signal sequence

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

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.

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

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?

• 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. Sep 27, 2021 at 14:59
• "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 $H(z) = \emptyset$. Sep 27, 2021 at 15:54
• 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. Sep 27, 2021 at 16:45
• 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. Sep 27, 2021 at 17:40
• 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. Sep 27, 2021 at 18:07

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.