Strong divergence of reconstruction procedures for the Paley–Wiener space PW(1_π) and the Hardy space H^1

Previous results on certain sampling series have left open if divergence only occurs for certain subsequences or, in fact, in the limit. Here we prove that divergence occurs in the limit. We consider three canonical reconstruction methods for functions in the Paley–Wiener space PW^1_π. For each of these we prove an instance when the reconstruction diverges in the limit. This is a much stronger statement than previous results that provide only lim sup divergence. We also address reconstruction for functions in the Hardy space H^1 and show that for any subsequence of the natural numbers there exists a function in H^1 for which reconstruction diverges in lim sup. For two of these sampling series we show that when divergence occurs, the sampling series has strong oscillations so that the maximum and the minimum tend to positive and negative infinity. Our results are of interest in functional analysis because they go beyond the type of result that can be obtained using the Banach–Steinhaus Theorem. We discuss practical implications of this work; in particular the work shows that methods using specially chosen subsequences of reconstructions cannot yield convergence for the Paley–Wiener Space PW^1_π

On representing signals using only timing information

It is well known that only a special class of bandpass signals, called real-zero (RZ) signals can be uniquely represented (up to a scale factor) by their zero crossings, i.e., the time instants at which the signals change their sign. However, it is possible to invertibly map arbitrary bandpass signals into RZ signals, thereby, implicitly represent the bandpass signal using the mapped RZ signal’s zero crossings. This mapping is known as real-zero conversion (RZC). In this paper a class of novel signal-adaptive RZC algorithms is proposed. Specifically, algorithms that are analogs of well-known adaptive filtering methods to convert an arbitrary bandpass signal into other signals, whose zero crossings contain sufficient information to represent the bandpass signal’s phase and envelope are presented. Since the proposed zero crossings are not those of the original signal, but only indirectly related to it, they are called hidden or covert zero crossings (CoZeCs). The CoZeCs-based representations are developed first for analytic signals, and then extended to real-valued signals. Finally, the proposed algorithms are used to represent synthetic signals and speech signals processed through an analysis filter bank, and it is shown that they can be reconstructed given the CoZeCs. This signal representation has potential in many speech applications

The role of zero crossings in speech recognition and processing

Imperial Users onl

Analog Signal Buffering and Reconstruction

Wireless sensor networks (WSNs) are capable of a myriad of tasks, from monitoring critical infrastructure such as bridges to monitoring a person\u27s vital signs in biomedical applications. However, their deployment is impractical for many applications due to their limited power budget. Sleep states are one method used to conserve power in resource-constrained systems, but they necessitate a wake-up circuit for detecting unpredictable events. In conventional wake-up-based systems, all information preceding a wake-up event will be forfeited. To avoid this data loss, it is necessary to include a buffer that can record prelude information without sacrificing the power savings garnered by the active use of sleep states.;Unfortunately, traditional memory buffer systems utilize digital electronics which are costly in terms of power. Instead of operating in the target signal\u27s native analog environment, a digital buffer must first expend a great deal of energy to convert the signal into a digital signal. This issue is further compounded by the use of traditional Nyquist sampling which does not adapt to the characteristics of a dynamically changing signal. These characteristics reveal why a digital buffer is not an appropriate choice for a WSN or other resource-constrained system.;This thesis documents the development of an analog pre-processing block that buffers an incoming signal using a new method of sampling. This method requires sampling only local maxima and minima (both amplitude and time), effectively approximating the instantaneous Nyquist rate throughout a time-varying signal. The use of this sampling method along with ultra-low-power analog electronics enables the entire system to operate in the muW power levels. In addition to these power saving techniques, a reconfigurable architecture will be explored as infrastructure for this system. This reconfigurable architecture will also be leveraged to explore wake-up circuits that can be used in parallel with the buffer system