Relative Karhunen-Loeve transform

The Karhunen-Loeve transform (KLT) provides the best approximation for a stochastic signal under the condition that its rank is fixed. It has been successfully used for data compression in communication. However, since the KLT does not consider noise, its ability to suppress noise is very poor. For the optimum linear data compression in the presence of noise, we propose the concept of a relative Karhunen-Loeve transform (RKLT). It minimizes the sum of the mean squared error between the original signal and its approximation and the mean squared error caused by a noise under the condition that its rank is fixed. We also provide another type of RKLT. It minimizes the same sum under the condition that its rank is not greater than a fixed integer. Since the former type of RKLT does not always exist, we provide a necessary and sufficient condition under which it does exist. We also provide their general forms. The advantage of RKLTs is illustrated through computer simulations