A new derivation is presented of a previously known relationship for phasing N equal-amplitude, equally spaced (in frequency) sinusoids so that the peak-to-average power ratio of their sum is 2.6 dB. The method of derivation provides intuitive insight into why this result is possible, and shows that the 2.6 dB figure is obtained independently of N as long as N is large. Any phase or frequency modulation can be applied simultaneously to all N sinusoids without altering this result.>

A computationally feasible parametric procedure for unsupervised learning has been given by Agrawala [1]. The procedure eliminates the computational difficulties associated with updating using a mixture density by making use of a probabilistic labeling scheme. Shanmugam [2] has given a similar parametric procedure using probabilistic labeling for the more general problem of imperfectly supervised learning. Both procedures assume known class probabilities. In this correspondence a computationally feasible parametric procedure using probabilistic labeling is given for imperfectly supervised learning when the class probabilities are among the unknown statistical parameters.

A decision-directed error correcting scheme for non-paramnetric imperfectly supervised learning is proposed. The proposed procedure is based on a nearest neighbor rule with a reject option and combines the knowledge accumulated by the learning scheme with that provided by the supervision. An asymptotic analysis of the error correction scheme shows that relabeling of training measurements can have a lower probability of error than the supervision provided Bayes' probability of error is less thani the supervisions probability of error. Computer simulations are used to compare the performance of the proposed scheme with the performance of the k nearest neighbor rule without error correction.