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Let's say we have an ARMA (Auto-regressive Moving Average Model) process where the transfer function is a minimum phase system (Namely Invertible). By Wold's Decomposition it is guaranteed to have MA representation of that system (Infinite order). My question is could it be also represented by AR system (Finite or infinite order)? If so, how could the coefficients be calculated? |
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Ok, The answer is as following, any ARMA process which is Minimum Phase system could be represented as MA or AR system. The MA part is easy by Wold's Decomposition. Since the filter in Wold's Decomposition is Invertible (Minimum Phase by itself), Shall be called $ H \left( z \right) $, It is easy to build the optimal linear predictor from it, $ P \left( z \right) = 1 - \frac{1}{H \left( z \right)} $, which a linear combination of the past of the process -> AR Model. It is all coming from the property of the Minimum Phase system being analytic in the unit circle. |
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