Resolution map in quantum computing: signal representation by periodic patterns

In this paper, a new concept of the resolution map is presented to extract periodic structures that compose the signal. The resolution map is described by using the frequency–time representation of the signal, which is known as the paired transform that provides the frequency-time representation of signals. The sequential calculation of resolution maps over the signal components of large sizes allows for calculating the small periodic structures, or patterns, which can be used for signal processing, for instance filtration, and from which the signal can be reconstructed. The length of the signal is considered to be a power of two, a case that fits well with qubit processing in quantum computing. The following new results are described: (1) the quantum scheme for the 1D discrete paired transform, (2) the quantum circuit for calculating the signal resolution map, (3) the quantum circuit for signal reconstruction from the resolution map, (4) different schemes for resolution maps for processing signals, and (5) the convolution of signals by their periodic patterns.