Recently, it has been shown that radix-2 DFT algorithms can be designed based on vector representation of data providing several advantages. In this paper, an index mapping approach is used to derive these algorithms. This approach makes the derivation simpler and provides a better insight into the functioning of the algorithms. The computational complexity and the performance of the software implementation of the algorithms are also presented.

The discrete Walsh and Hadamard transforms are often used in image processing tasks such as image coding, pattern recognition, and sequency filtering. A new discrete Walsh transform (DWT) algorithm is derived in which a modified form of the DWT relation is decomposed into smaller-sized transforms using vectorized quantities. A new sequency-ordered discrete Hadamard transform (DHAT) algorithm is also presented. The proposed approach results in more regular algorithms requiring no independent data swapping and fewer array-index updating and bit-reversal operations. An analysis of the computational complexity and the execution time performance are provided. The results are compared with those of the existing algorithms.