  An input pair $(A,B)$ is triangular input normal if and only if $A$ is
triangular and $AA^* + BB^* = I_n$, where $I_n$ is theidentity matrix. Input
normal pairs generate an orthonormal basis for the impulse response. Every
input pair may be transformed to a triangular input normal pair. A new system
representation is given: $(A,B)$ is triangular normal and $A$ is a matrix
fraction, $A=M^{-1}N$, where $M$ and $N$ are triangular matrices of low
bandwidth. For single input pairs, $M$ and $N$ are bidiagonal and an explicit
parameterization is given in terms of the eigenvalues of $A$. This band
fraction structure allows for fast updates of state space systems and fast
system identification. When A has only real eigenvalues, one state advance
requires $3n$ multiplications for the single input case.
