When to apply wavelet analysis to real-time temporal signals, where the future cannot be accessed, it is essential to base all the steps in the signal processing pipeline on computational mechanisms that are truly time-causal. This paper describes how a time-causal wavelet analysis can be performed based on concepts developed in the area of temporal scale-space theory, originating from a complete classification of temporal smoothing kernels that guarantee non-creation of new structures from finer to coarser temporal scale levels. By necessity, convolution with truncated exponential kernels in cascade constitutes the only permissable class of kernels, as well as their temporal derivatives as a natural complement to fulfil the admissibility conditions of wavelet representations. For a particular way of choosing the time constants in the resulting infinite convolution of truncated exponential kernels, to ensure temporal scale covariance and thus self-similarity over temporal scales, we describe how mother wavelets can be chosen as temporal derivatives of the resulting time-causal limit kernel. By developing connections between wavelet theory and scale-space theory, we characterize and quantify how the continuous scaling properties transfer to the discrete implementation, demonstrating how the proposed time-causal wavelet representation can reflect the duration of locally dominant temporal structures in the input signals. We propose that this notion of time-causal wavelet analysis could be a valuable tool for signal processing tasks, where streams of signals are to be processed in real time, specifically for signals that may contain local variations over a rich span of temporal scales, or more generally for analysing physical or biophysical temporal phenomena, where a fully time-causal analysis is called for to be physically realistic.

翻译：当将小波分析应用于实时时序信号时，由于无法获取未来信息，信号处理流程中的所有步骤都必须建立在真正时间因果的计算机制之上。本文阐述了如何基于时间尺度空间理论领域发展的概念实现时间因果小波分析——该理论源于对时间平滑核的完整分类，这些核能保证从精细到粗糙时间尺度层级不会产生新结构。必然地，级联截断指数核的卷积构成了唯一允许的核类别，其时间导数则作为自然补充以满足小波表示的容许性条件。为确保时间尺度协变性（即跨时间尺度的自相似性），我们采用特定方式选择级联无限截断指数核的时间常数，并说明如何将母小波选为所得时间因果极限核的时间导数。通过建立小波理论与尺度空间理论之间的联系，我们刻画并量化了连续尺度特性如何转移到离散实现中，证明所提出的时间因果小波表示能够反映输入信号中局部主导时间结构的持续时间。我们认为这种时间因果小波分析概念可成为信号处理任务中的有力工具，特别适用于需要实时处理信号流、且信号可能包含跨多时间尺度的局部变化的情形，更广泛而言，适用于需要完全时间因果分析以保持物理真实性的物理或生物物理时序现象研究。