This paper proposes a new nonlinear signal processing by using a three-layered network which is trained with self-organized clustering and supervised learning. The network consists of three layers, i.e., a self-organized layer, an evaluation layer, and an output layer. Since the evaluation layer is designed as a simple perceptron network and the output layer is designed as the fixed weight linear nodes, the training complexity is the same as the self-organized clustering and a simple perceptron network. In other words, quite high speed training can be realized. Generally speaking, since the data range usually used in signal processing is arbitrarily large, the network output should also cover this range. However, it may be difficult for only one node in the network to output these data. Instead of this technique, if this dynamic range is covered by using several nodes, the complexity of each node is reduced and the associated range is also quite limited. This results in a higher performance of this network than the conventional ones. As one of the objectives, this paper introduces the spectrum envelope estimation of speech waveforms. It is shown that accurate spectrum envelopes can be obtained.