The Cascade Multiplier

A new principle and circuit for the generation of the product of two variables is described in which the required result is a linear combination of readily produced secondary variables. The product of two variables can be expanded in a series in which each term is derived from the preceding one and from an auxiliary variable by linear operations and by maximum and minimum selection. Two ``converted´´ variables are associated with each term of the expansion. The product is equal to a linear combination of the converted variables plus the product of the last two converted variables. The range of variation of successive converted signals decreases by half so that the series converges by factors of ¿ per term. The exrpansion stands in direct correspondence with electronic multipliers which use linear elements and diode selection circuits. Only two sign-changers are required for four-quadrant operation. When used in conjunction with an ancillary four-quadrant multiplier, the cascade multiplier increases its accuracy by factors of 4, 16, 64,..., if 1, 2, 3, ..., stages are used. Alternatively, a sufficient number of stages provides a complete multiplier even without the use of an ancillary multiplier. Accuracies of the order of ±0.1 per cent are readily obtained and speed of response is high.