This paper introduces a generalized discrete cosine transform (DCT) with three parameters and proves its orthogonality for certain cases. The authors also propose a generalized discrete W transform with three parameters and prove its orthogonality for some new cases. The paper discusses various types of DCT, including DCT-I-E, DCT-II-E, DCT-III-E, and DCT-IV-E, and their generalizations. The authors also propose new types of discrete cosine transforms, discrete sine transforms, and discrete sine-cosine transforms, and prove their orthogonality. The paper also discusses the discrete W transform, which is used in digital signal processing, data compression, and information hiding. The authors generalize the discrete W transform with three parameters and prove its orthogonality for some new cases. The paper concludes with a discussion of the orthogonality of the generalized transforms and their applications. The research was supported by the Zhejiang Natural Science Foundation and the Chinese Natural Science Foundation.This paper introduces a generalized discrete cosine transform (DCT) with three parameters and proves its orthogonality for certain cases. The authors also propose a generalized discrete W transform with three parameters and prove its orthogonality for some new cases. The paper discusses various types of DCT, including DCT-I-E, DCT-II-E, DCT-III-E, and DCT-IV-E, and their generalizations. The authors also propose new types of discrete cosine transforms, discrete sine transforms, and discrete sine-cosine transforms, and prove their orthogonality. The paper also discusses the discrete W transform, which is used in digital signal processing, data compression, and information hiding. The authors generalize the discrete W transform with three parameters and prove its orthogonality for some new cases. The paper concludes with a discussion of the orthogonality of the generalized transforms and their applications. The research was supported by the Zhejiang Natural Science Foundation and the Chinese Natural Science Foundation.