Orthogonalization by principal components applied to CPM

Abstract

The orthonormal basis for the space spanned by a given signal set can be chosen in many different ways. However, when the basis is truncated to fewer dimensions, the quality of the approximated signals differs, depending on the choice of the original basis. We study two energy-related quality measures and show that the optimal lower-dimensional approximation is given by the principal components (PC) method, which is also a simple and efficient alternative to Gram-Schmidt techniques. In addition, we derive and bound the average decrease in squared Euclidean distances over one symbol interval caused by the PC method. This measure is relevant in serially concatenated continuous phase modulation, where a manifold of signal pairs contributes to the bit-error rate for low-to-medium signal-to-noise ratios. By a numerical evaluation, we find that for this measure, the decrease is lower than that of a previous method by J. Huber and W. Liu (see IEEE J. Select. Areas Commun., vol.7, p.1437-49, 1989). Finally, we compare the minimum squared Euclidean distance for error events, which is relevant for uncoded CPM systems. Here, the loss with the PC method is generally larger than with Huber and Liu's method, although examples of the opposite exist.