Abstract: A modulation and demodulation system is provided. The system uses n sub-carrier frequencies and an oversampling positive integer and includes a modulation circuit and a demodulation circuit. The modulation circuit includes a modulation ROM that stores columns of elements independently in a modulation ROM matrix. The demodulation circuit includes 2n numbers of a ROM1 that stores elements of a combined matrix. The combined matrix is generated by combining a number of inverse matrices where the inverse matrices are generated from the modulation ROM.