Conjugate symmetric sequency ordered Walsh Fourier transform

A new family of transforms, which is called the conjugate symmetric sequency-ordered generalized Walsh-Fourier transform (CS-SGWFT), is proposed in this paper. The CS-SGWFT generalized the existing transforms including the conjugate symmetric sequency ordered complex Hadamard transform (CS-SCHT) and the discrete Fourier transform (DFT) as the special cases of the CS-SGWFT. Like the CS-SCHT and the DFT, the spectrums of the CS-SGWFT for real input signals are conjugate symmetric so that we need only half memory to store the transform results. The properties of the CS-SGWFT are similar to those of the CS-SCHT and DFT, including orthogonality, sequency ordering, and conjugate symmetric. Meanwhile, the proposed CS-SGWFT has radix-2 fast algorithm. Finally, applications of the CS-SGWFT for image noise removal and spectrum estimation are proposed.