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 the identity 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.