Efficient Half-Storage Generation of Frequency-Ordered Walsh Functions uUsing XoraAnd Cyclic Shift

Walsh functions play a critical role in modern digital signal processing, telecommunications, image compression, error correction, and hardware-accelerated computations due to their orthogonality, simplicity, and efficient transform properties. This paper presents an efficient half-storage algorithm for the generation of frequency-ordered Walsh functions using only XOR and cyclic shift operations. Unlike conventional approaches that store the entire Walsh matrix, the proposed method requires storing only the first half of the basis vectors. The second half is derived in constant time by combining a pre-computed alternating mask with the stored vectors. This approach significantly reduces memory usage while maintaining high generation speed, making it particularly suitable for low-power and memory-constrained DSP and IoT applications. Theoretical analysis and experimental results demonstrate the proposed algorithm’s computational efficiency, scalability, and potential for hardware implementation.