Fast convolution algorithm for state space models

Abstract

We present an unconditionally stable algorithm for applying matrix transfer function of a linear time invariant system (LTI) in time domain. The state matrix of an LTI system used for modeling long range dependencies in state space models (SSMs) has eigenvalues close to $1$. The standard recursion defining LTI system becomes unstable if the $m\times m$ state matrix has just one eigenvalue with absolute value even slightly greater than 1. This may occur when approximating a state matrix by a structured matrix to reduce the cost of matrix-vector multiplication from $\mathcal{O}\left(m^{2}\right)$ to $\mathcal{O}\left(m\right)$ or $\mathcal{O}\left(m\log m\right).$ We introduce an unconditionally stable algorithm that uses an approximation of the rational transfer function in the z-domain by a matrix polynomial of degree $2^{N+1}-1$, where $N$ is chosen to achieve any user-selected accuracy. Using a cascade implementation in time domain, applying such transfer function to compute $L$ states requires no more than $2L$ matrix-vector multiplications (whereas the standard recursion requires $L$ matrix-vector multiplications). However, using unconditionally stable algorithm, it is not necessary to assure that an approximate state matrix has all eigenvalues with absolute values strictly less than 1 i.e., within the desired accuracy, the absolute value of some eigenvalues may possibly exceed $1$. Consequently, this algorithm allows one to use a wider variety of structured approximations to reduce the cost of matrix-vector multiplication and we briefly describe several of them to be used for this purpose.