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Very often, random signals are modeled as autoregressive moving-average (ARMA) processes in engineering and scientific applications. The acquisition of the second-order statistics or the calculation of the autocorrelation function for an ARMA process is very important in those applications. In this paper, we discuss two kinds of ARMA-autocorrelation computation approaches, namely the recursive approach and the direct approach. To overcome the problems of memory usage and computational burden, we design a new ARMA-autocorrelation calculation algorithm which belongs to the direct approach. Our novel algorithm originates from the partial fractional decomposition for the rational trigonometric functions and the associated definite integrals. For comparison, we also provide the theoretical analysis of computational complexity for our new scheme and the conventional recursion-based algorithm. From our studies, it can be shown that our new ARMA-autocorrelation calculation method is more efficient than the conventional algorithm when the underlying autocorrelation function is long or the corresponding power spectrum possesses the narrowband characteristics. On the other hand, the conventional algorithm is more efficient than our new algorithm when the model orders of the ARMA process are large and the underlying autocorrelation function is short (the corresponding power spectrum is wideband). Since our proposed new scheme involves the root-finding of a polynomial, the effect of the error-tolerance in solving polynomials on the induced complexity is also discussed in this paper.