A fast-transform basis with hysteretic features

Abstract

Hysteretic distortion is the common feature of signals generated by magnetic quantum-and nano-systems. In light of recent advances in nano-electronics, particularly the assent of memristance, it is expected that hysteretic features will become the trademark of next-generation electronics. Consequently, signal processing tasks, such as compression or denoising, will benefit from the use of bases which adapt to the hysteretic distortion. In this paper I describe a basis that consists of periodic functions with an inherent hysteretic feature. These functions originate from a certain nonlinear eigenvalue problem which, it is demonstrated here, relates to oscillators with memristance.

Signals generated in memristive circuits

“…It should be emphasized, however, that the hysteretic distortion may arise in circuits with other M = M( ι), e.g. an alternative example is analyzed in [24] and [23]. is clearly different than in the subsequent cycles.…”

“…The construction below follows the method established in [24] and [26], and will only be outlined. First, consider the set { ( )}, = 1 2 , as a candidate for a basis in H h .…”

“…We emphasize that ( ) need not necessarily descend from a DAEP. However, it is demonstrated in [26] that the bases arising from the specific DAEP considered in [24] are in fact Riesz. The Riesz property is of interest from the point of view of applications as it guaranties numerically stable decomposition of a signal in the given basis.…”

“…It has been established that such bases have a very unusual property -namely, they furnish fast, O(N log N), change-of basis transforms, [24]. In fact, the change of basis can be carried out numerically with extreme efficiency via a lifting schema, [25].…”

“…This becomes a hindrance when one wishes to carry out, say, signal denoising or compression in a way that does not spoil the hysteretic features. Section 3 outlines an approach to this problem, which utilizes a novel fast transform, [24][25][26]. The discussion culminates in Section 4, where we argue that hysteretic distortion may also be simulated by certain fast transforms.…”