Sequency-ordered generalized Walsh–Fourier transform

A new transform family, called the sequency-ordered generalized Walsh–Fourier transform (SGWFT), is proposed in this paper. Using the kernel matrix generation process and the controllable phase quantization parameter, the Walsh–Hadamard transform (WHT), the sequency-ordered Hadamard transform (SCHT), and the discrete Fourier transform (DFT) become special cases of the SGWFT. The SGWFT can be adjusted by a single parameter to become the WHT, the SCHT, and the DFT. In addition, the SGWFT also has the radix-2 and the split-radix fast algorithms. Compared with the WHT and the SCHT, the properties and the performance of the SGWFT are more similar to those of the DFT. On the other hand, compared with the DFT, the number of multiplications in the SGWFT is less. We also show that the proposed SGWFT has better performance in the applications of DS-CDMA sequence design and transform coding.

Graphical abstractFigure optionsDownload full-size imageDownload as PowerPoint slideHighlights► The SGWFT generalize the existing transforms such as the DFT and WHT. ► We can switch the SGWFT behavior by a single controllable parameter. ► We can implement the SGWFT by the radix-2 and split-radix fast algorithms. ► The SGWFT saves the multiplication number compared with the DFT. ► The SGWFT has better performance when applied to DS-CDMA sequence design.