We consider ARMAX system representations and identification problems. Identifiability conditions in terms of the correlation function of the process are given. One of the conditions is persistency of excitation of an input component of the process and another one is a rank condition for a pair of Hankel matrices. We study the linear combinations of the process and its shifts that produce a process independent of the input. The set of all such linear combinations, called the orthogonalizers, has

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... a module structure and under identifiability conditions completely specifies the deterministic part of the ARMAX system. Computing a module basis for the orthogonalizers is a deterministic identification problem. We propose an ARMAX identification algorithm, which has three steps: first compute the deterministic part of the system via the orthogonalizers, then the AR part, which also has a module structure, and finally the MA part.