GB1338397A - Generation of non-linear functions - Google Patents

Abstract

1338397 Arbitrary function generators NATIONAL RESEARCH DEVELOPMENT CORP 30 Sept 1971 [6 July 1970 2 March 1971] 32749/70 and 5807/71 Heading G4G An arbitrary function generator comprises means for interpolating between known values of the function by forming the sum of a number of polynomials (e.g. hagrangian polynomials) each weighted according to the value of the function at a respective reference point and each of order less than the number of such reference points. If the number of points is, e.g. five, the polynomials may be of order four whereby complete accuracy is obtainable for functions of order four or less. Fig. 1 shows the function A having known values v at x = 1 ... 5, and the hagrangian polynomial of order four having value unity at x = 2 and value zero at all other points. The value of x is fed to power generators 12, 14, 16 and thence to a polynomial generator 18 which also receives a constant signal MU and forms the required polynomials by multiplying the powers of x by the coefficients a denoted on the drawing by a self-explanatory system of subscripts. The values of the polynomials are formed in adders 20-24. Each polynomial has value unity at one of the reference points and value zero at the others; e.g. the adder 21 forms the polynomial B for which v = 1 when x = 2. The outputs of the adders 20-24 are weighted by the respective values v 1 ... v 5 of the function at the reference points in multipliers 28-32, and these weighted values are summed in adder 34. Thus for an input value x = 2, the output of all adders 20-24 is zero except for adder 21 which has output unity. Hence, the adder 34 yields the output v 2 . For values of x between reference points, all adders 20-24 have non-zero outputs and hence the interpolation is dependent upon all reference points. In a modification (Fig. 4, not shown), the polynomial generator is fed with the coefficients of the first derivatives of the hagrangian polynomials and the apparatus yields interpolated values of F<SP>1</SP>(x). A differentiating interpolator of this kind may share power generators and reference value inputs with an interpolator of the ordinary kind (Fig. 2). The Specification also describes an interpolator for values of a function w of two variables x, y. Fig. 5 shows a regular distribution of reference values of and y in the xy plane and Fig. 6 shows a polynomial generator 64 for powers of x feeding interpolators 68-72 each supplied with sets of reference values of w respectively corresponding to different values of y. Since the same reference values of x are used for each level of y, only one x polynomial generator is necessary. The outputs of interpolators 68-72 are functions of x and are fed to respective multipliers 74, 78 which also receive respective outputs of generator 66 of polynomials in y, e.g. the multiplier 74 is connected to the output of generator 66 associated with the polynomial having value 1 at y = 1, and so on. The value of w = F(x,y) is derived from adder 82. An interpolator for a function of three variables using 125 reference points requires five units of the Fig. 6 kind each producing a function of two variables corresponding to a different level of the third variable. The components 60, 64, 68-72 of the Fig. 6 apparatus may be used to generate a function of a single variable which is known at 25 reference points divided into five sets of five, each set being associated with a segment of the curve. Each interpolator 68-72 receives reference values from a different set, the appropriate set being chosen by a comparator circuit (not shown) responsive to the input variable. The sets of five points may overlap. In the case of the two-variable function generator, the effect of tilting the xy plane may be obtained by applying appropriate values of bias to the reference signals.