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

Question

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?

Answer 1

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.