Abstract: In this paper we consider the model fitting and spectral density estimation problem und rer the criterion of mutual information for observed data y_t=x_t+n_t, where n_t is an MA q) noise independent of x_t. Suppose that the MA ( q ) equation of n_t is as (3), for given covariances \left\r_k\right\ 8 of y_t, the solution of the Yule-Walker equation is as (15), (16), then the following theorems hold: Theorem 1. Suppose that the p+1 real values r_0, r_1, \cdots, r_p of the covariance function of a nonsingular stationary Gaussian series y_t are given, the MA( q ) equation (2) is known, and \forall m>0, b_0>0, \boldsymbolB_m is determined by (18), (9), (12), (14) is nonnegative definite. Then there exists an ARMA (p, p+q) Gaussian stationary series \xi_t, ^s uch that a. \xi_t is independent of n_t b. the covariance function of the output \eta_t=\xi_t+n_t satisfies B_\eta \eta(\tau)=r_\tau, 0 \leqslant \tau \leqslant p c. for any Gaussian stationary input series \tilde\xi_t and output \tilde\eta_t=\tilde\xi_t+n_t which satisties conditions a. \&.b., for \forall k>0, the mutual information quantity (21) between \tilde\boldsymbol\xi(k+p)=\left(\tilde\xi_1, \cdots, \tilde\xi_k+p\right) and \tilde\eta(k+p)=\left(\tilde\eta_1, \cdots, \tilde\eta_p+k\right) is maximized by I(\xi(k+p), \eta(k+ p )). Theorem 2. Suppose that \left\r_t\right\ \delta and n_t are the same as iin Th. 1. In order to get an ARMA (p, p+q) Gaussian stationary series \xi_t satisfying oonditions a., b., o. of Th. 1, with I(\xi(k+p), \eta(k+p)) determined by (40),(40 \mathrma),(30)^\prime,(31), the neoessary conditions are b_0>0, \forall m>0, \boldsymbolB_\mathrmm \geqslant 0. Theorem 3. Suppose that \left\r_t\right\_0, n_t are the same as in Th. 1. Let \xi_t be any Gaussian stationary series satisfying a., b., o. of Th. 1. Then \xi_t must be an ARMA (p, p+q) series and the coefficients of the model are uniquely determinod by \left\r_i\right\_i of y_t and \left\a_k\right\_0 of n_t. Theorem 4. Suppose that y_t is a Gaussian stationary regular series satisfying the conditions (a), (b) of (45) and the spectral density of x_4 satisfies (46). Let y_1, \cdots, y_n be the N sample data of y_t ;\left\\hatr_j^(N)\right\_0^p_x, P_N are determined by (47), (47)'. Then the optimum input ARMA \left(P_N, P_N \div q\right) series \xi_t^(N) satisfying a., b., o. of Th. 1 exists almost surely when N is sufficiently large and \sup _\lambda\left|f_\infty(\lambda)-f_\mu^(N)(\lambda)\right|=0(1) \quad \text a.s. where f\left(k^\prime \prime(\lambda)\right. is the spectral density of \xi^(N). The practioal calculation procedure and the Monte-Carlo simulation are given in the paper. The author is much indebted to Professor Chiang T'sepei for his valuable suggestions in the writting of this paper.