IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 40, NO. 7, JULY 1992 1813 order to use the FFT and IFFT. As b normally has structure, we may wish to model it using an affine model. Following Starer and Nehorai [8], constrain b to be of the form b = Tf + c. Here T is a (p + 1) x /-dimensional constraint matrix with full column rank, c is a real (p + l)-dimensional vector, and/is a real /-dimensional vector. Both T and c are chosen to enforce the structure desired of b. The vector/contains the / degrees of freedom of b and is the vector for which we solve-. Given: 6,_,, T, c, e, Y Compute: b t d= FFKtA/l, 0 • • • 0]) r = 0 for; = 0 • • • (N - 1) d, = \d,\ 2 if dj < t then g(r) = i r = r + 1 d, = 4+1 endif d, = dj x endfor q = IFFT(rf) Form a, 0, and y from q Ml = 7 - /3* X aT 1 X 0 if r > 0 Form E 2 using g Ml = Ml X £ 2 Ml = Ml + M2 X (/ - £ 2 * x M2)~' X M2* endif M3 = ((r* x ?*) x Ml) x Y Solve the system - (real(M3 x T)) x b = real(M3 X c) for b b, = T x / + c. V. COMPUTATION COUNTS AND CONCLUSION We are now at a point where we can compare the complexity of the new algorithm with those of other methods. It is easily shown that the new method requires O((p + I + r)N 2 ) multiplications and 0(/> 3 + / 3 + r 3 ) computations for the two matrix inversions and the solution of the system of linear equations. This represents an improvement over the method of Kumaresan et al. [1] which re- quires Q((2pf + / 3 ) computations for the matrix inversions. Fur- ther, the new method is always stable. The Steiglitz-McBride al- gorithm requires ©((/> + /)iV 2 ) multiplications and 0 ( p 3 + / 3 ) computations for matrix inversions. Thus it is cheaper than the new method only when r =£ 0. On machines with sufficient precision this is a low probability event. Therefore, with our improvement, the computation required by IQML is almost always the same as that required by the Steiglitz-McBride algorithm. Thus, neither Steiglitz-McBride nor IQML should be preferred. This is intui- tively pleasing since the iterations have been shown to produce equivalent estimates [4]. REFERENCES [1] R. Kumaresan, L. L. Scharf, and A. K. Shaw, "An algorithm for pole- zero modeling and spectral analysis," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 637-640, June 1986. [2] Y. Bresler and A. Macovski, "Exact maximum likelihood parameter estimation of superimposed signals in noise," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 1081-1089, Oct. 1986. [3] A. G. Evans and R. Fischl, "Optimal least squares time-domain syn- thesis of recursive digital filters," IEEE Trans. Audio Electroacoust., vol. AU-21, pp. 61-65, Feb. 1973. [4] J. H. McClellan and D. Lee, "Exact equivalence of the Steiglitz- McBride iteration and IQML," IEEE Trans. Signal Processing, vol. 39, pp. 509-512, Feb. 1991. [5] K. Steiglitz and L. E. McBride, "A technique for the identification of linear systems," IEEE Trans. Automat. Contr., vol. AC-10, pp. 461- 464, Oct. 1965. [6] R. T. Behrens, "Subspace signal processing in structured noise," Ph.D. dissertation, Univ. Colorado, Boulder, CO, 1990. [7] R. A. Roberts and C. T. Mullis, Digital Signal Processing. Reading, MA: Addison-Wesley, 1987. [8] D. Starer and A. Nehorai, "Maximum likelihood estimation of expo- nential signals in noise using a Newton algorithm," in Proc. IEEE ASSP Workshop Spectral Estimation Modeling (Minneapolis, MN), Aug. 1988, pp. 240-245. On Fast Evaluation of Bivariate Polynomials at Equispaced Arguments S. C. Dutta Roy and Shailey Minocha Abstract—The initial value problem arising in the recursive evalua- tion of a 2-D polynomial at equispaced points is treated in detail; the results facilitate efficient implementation of Bose's recursive algo- rithm. A comparison has been made of the computational complexity with that involved in a direct computation, and some general obser- vations have been made for an alternative scheme proposed by Nie and Unbehauen. I. INTRODUCTION In many situations in general computing and in digital signal processing, one is frequently interested in the evaluation of an n-dimensional (n - D, n > 1) polynomial at a large number of equispaced arguments. Several recursive algorithms have been pro- posed for the 1-D case (see, e.g., [1] and [2]). Nuttall [3] gener- alized the results of [1] and [2], while Bose [4] extended Nuttall's approach to the n — D(n > 1) case, using n = 2 for illustration. However, neither [3] nor [4] treated the problem of computing the initial conditions required for the recursion to commence. That this is an important aspect has been demonstrated recently for the 1-D case [5]. In this correspondence, we treat the initial value problem (IVP) in detail for the evaluation of a 2-D polynomial; the results allow efficient implementation of Bose's recursive algorithm. We have also compared the computational complexity of this algorithm with that of direct evaluation. Finally, we have carried out an algo- rithmic analysis of the alternative scheme of [6] for solving the IVP, and compared its performance with that of the scheme pro- posed here. II. COMPUTING THE COEFFICIENTS OF Qo(nt, n2) A 2-D polynomial of degree (m,, m 2 ) P(x t , X 2 ) = Piuh x " x 'l (1) Manuscript received August 12, 1990; revised January 31, 1992. The authors are with the Department of Electrical Engineering, Indian Institute of Technology, Delhi, New Delhi-110 016, India. IEEE Log Number 9100248.