Arbitrary-Length Walsh-Jacket Transforms

Due to the efficiency in implementation, the Walsh (Hadamard) transform plays an important role in signal analysis and communication. Recently, Lee generalized the Walsh trans- form into the Jacket transform. Since the entries of the Jacket transform can be ±2 k , it is more flexible than the Walsh trans- form. Both the Walsh transform and the Jacket transform are defined for the case where the length N is a power of 2. In this paper, we try to extend the Walsh transform and the Jacket transform to the case where N is not a power of 2. With the "folding extension algorithm" and the Kronecker product, the arbitrary-length Walsh-Jacket transform can be defined success- fully. As the original Walsh and Jacket transforms, the proposed arbitrary-length Walsh-Jacket transform has fast algorithms and can always be decomposed into the 2-point Walsh-Jacket transforms. We also show the applications of the proposed arbi- trary-length Walsh-Jacket transforms in step-like signal analysis and electrocardiogram (ECG) signal analysis.