The paper by Jianqin Zhou explores the discrete cosine transform (DCT) and its variants, focusing on their orthogonality properties. The DCT, introduced by Ahmed, Natarajan, and Rao, is widely used in digital signal processing, data compression, and information hiding. Zhou proposes a generalized DCT with three parameters and proves its orthogonality for certain cases. Additionally, a new type of DCT is introduced, and its orthogonality is demonstrated. The paper also generalizes the discrete W transform with three parameters and proves its orthogonality for new cases. The author provides detailed mathematical proofs and discusses the applications of these transforms in various fields. The research is supported by grants from the Zhejiang Natural Science Foundation and the Chinese Natural Science Foundation.The paper by Jianqin Zhou explores the discrete cosine transform (DCT) and its variants, focusing on their orthogonality properties. The DCT, introduced by Ahmed, Natarajan, and Rao, is widely used in digital signal processing, data compression, and information hiding. Zhou proposes a generalized DCT with three parameters and proves its orthogonality for certain cases. Additionally, a new type of DCT is introduced, and its orthogonality is demonstrated. The paper also generalizes the discrete W transform with three parameters and proves its orthogonality for new cases. The author provides detailed mathematical proofs and discusses the applications of these transforms in various fields. The research is supported by grants from the Zhejiang Natural Science Foundation and the Chinese Natural Science Foundation.