Since high-order relationships among multiple brain regions-of-interests (ROIs) are helpful to explore the pathogenesis of neurological diseases more deeply, hypergraph-based brain networks are more suitable for brain science research. Unlike the existing hypergraph based brain network (brain hypernetwork), where hyperedges containing the same number of ROIs are assumed to have equal weights (to some extent, the network is unweighted), and the underlying structure is described only by an incidence/adjacency matrix, in this paper, we propose a framework for constructing a truly weighted brain hypernetwork described by an adjacency tensor. Considering the relationships among vertices within a hyperedge, we propose a novel hyperedge weight estimation method and convert the incidence matrix into a weighted adjacency tensor. On the basis of tensor decomposition, we apply hypergraph signal processing tools, such as hypergraph Fourier transform, to analyze and compare the spectrum between schizophrenia patients and normal controls. It is found that there are more high frequency components in the spectrum of patients than controls, and the average amplitude is significantly greater than that of controls. Instead of extracting some simple topological features from brain hypernetworks for classification, we innovatively use the hypergraph spectrum and the spectral signal as classification features, and the classification results on two public datasets demonstrate the effectiveness of our proposed method.