Frequency-domain nonlinear microwave circuit simulation using the arithmetic operator method

Electro-Thermal Behavioral Meeting

“…As an illustrative example, consider for a moment that Y(a, + (0 2 ) is limited to just terms in k = 1 and k = 2 . Referring to the product terms in equations (1.18) and (1.19), then, the matrix form of equation (1.21) would be extended to add extra columns to the matrix and extra rows to the vector: r(fl),+fl) 2 )| t=1 , 2 This formulation can be generalized by adding two rows to the matrix for each unique output frequency in Y((0) , and it can be generalized for each additional input frequency by adding two columns to the matrix and two rows to the vector element. The matrix thus formulated has been termed the Spectrum Transform Matrix by Chang and Steer [2], In matrix algebra notation, the spectrum transform matrix is denoted T,, the spectral vector of frequency components for z(t) is denoted as Z , and the basic operation…”

“…Expanding this by substituting equations (1.14) and (1.15) into equation (1.16), we have: 2 Consider, for example, a case where x and z arc both composed of tones of 10 different frequencies. This corresponds to 21 frequency domain locations (after including DC) that must be convolved with 21 other frequencies, or computational complexity on the order of 21 2 = 441.…”

“…•■: X-, (1.19) Consider the frequency component ft), + C0 2 , which is indicated by any terms containing 8[C0-(ft), +C0 2 )] in equations (1.18) and (1.19). From equation (1.18), there is a term atö), + ft) 2 with the coefficient y X,Z 2 , while from equation (1.19), there is a term at ft), + ft) 2 with the coefficient 1 X 2 Z l .…”

“…This corresponds to 21 frequency domain locations (after including DC) that must be convolved with 21 other frequencies, or computational complexity on the order of 21 2 = 441. Restricting the frequency domain to the right hand side, we need convolve only 11 frequency domain components for a computational complexity on the order of 11 2 , or 121. The ratio is just under 4:1 or 2 2 :1, indicating the geometric reduction in computation.…”

“…The Arithmetic Operator Method (AOM) was pioneered by Chang and Steer and has previously been described in two publications, [1] and [2]. It has been successfully applied to electronic c.rcuit behavioral modeling by Gard et al [3] and by de Carvalho and Pedro [4], [5].…”