A real-arithmetic implementation of the constant modulus algorithm

An earlier paper [1] presented the Constant Modulus Algorithm, an adaptive filtering algorithm which can be employed to remove interference and equalize propagation anomolies that impair communications via constant envelope signals, such as FM and QPSK. The technique presented in that paper employs complex waveforms and a FIR adaptive filter with complex coefficients so that signal properties such as the instantaneous amplitude (modulus) can be measured. Even so many practical applications of such equalization or interference reduction involve real signals and adaptive filters implemented with real rather than complex arithmetic. This paper presents a version of the constant modulus algorithm (CMA) which employs real signals and real arithemtic. The algorithm is developed in an evolutionary form from the version based on complex arithmetic. A key technical result is that certain error terms can be highly simplified due to the smoothing intrinsic to a gradient search algorithm.