1. A method of approximating an FIR filter with low-order linear-phase IIR filters by the rational Arnoldi algorithm with adaptive orders containing the following steps: a. initialize the first vector of the Krylov sequence for each expansion point; b. in the jth iteration of the algorithm, choosing an expansion frequency such that the frequency gives the greatest difference between the (j+1)st-order output moment of the original FIR filter H(z) and that of the lower-order IIR filter Ĥ(z); c. after the choosing the expansion point in jth iteration being determined, the single-point Arnoldi method applied at the expansion point to generate the new orthnormal vector; and d. determine a new residual at each expansion point for next iteration; whereby, after the giving total iteration number of the algorithm, outputting the resulting orthogonal projection matrix.