In this paper we study from an algorithmic perspective the problem of finding the peak value of a bandlimited signal. This problem plays an important role in the design and optimization of communication systems. We show that the peak value problem, i.e., computing the peak value of a bandlimited signal from its samples, can be solved algorithmically if oversampling is used. Without oversampling this is not possible. There exist bandlimited signals, for which the sequence of samples is computable, but the signal itself is not. This problem is directly related to the question whether there is a link between computability in the digital domain and the analog domain, and hence to a fundamental signal processing problem. We show that there is an asymmetry between continuous-time and discrete-time computability. Further, we study the decay behavior of computable bandlimited signals, which describes the concentration of the signals in the time domain, and, for locally computable bandlimited signals, we analyze if it is always possible to decide algorithmically whether the peak value is smaller than a given threshold.

中文翻译：

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带限信号峰值问题的可解性及其应用

在本文中，我们从算法的角度研究了寻找带宽受限信号的峰值的问题。这个问题在通信系统的设计和优化中起着重要的作用。我们表明，如果使用过采样，则可以通过算法解决峰值问题，即从其样本中计算带宽限制信号的峰值。没有过采样，这是不可能的。存在带宽受限的信号，其采样序列是可计算的，但是信号本身不是。这个问题与数字域和模拟域中的可计算性之间是否存在联系的问题直接相关，因此与基本信号处理问题有关。我们证明了连续时间和离散时间可计算性之间存在不对称性。进一步，