Consistent Signal Reconstruction from Streaming Multivariate Time Series

TL;DR

The paper addresses online, zero-delay consistent signal reconstruction from streaming multivariate time series, identifying a gap relative to offline methods. It proposes a spline-based, policy-driven framework that updates in real time while enforcing consistency through hyperslab constraints and exploiting spatiotemporal dependencies via multivariate modeling and RNN-based policies. The key contributions include a formal definition of online consistency, a zero-delay reconstruction method with closed-form update rules, and empirical evidence showing favorable error-rate decay and gains from leveraging spatial correlations. This approach enables real-time, low-latency, consistent reconstruction in high-speed acquisition and streaming applications, with practical impact for digital-to-analog conversion and related online signal processing tasks.

Abstract

Digitalizing real-world analog signals typically involves sampling in time and discretizing in amplitude. Subsequent signal reconstructions inevitably incur an error that depends on the amplitude resolution and the temporal density of the acquired samples. From an implementation viewpoint, consistent signal reconstruction methods have proven a profitable error-rate decay as the sampling rate increases. Despite that, these results are obtained under offline settings. Therefore, a research gap exists regarding methods for consistent signal reconstruction from data streams. Solving this problem is of great importance because such methods could run at a lower computational cost than the existing offline ones or be used under real-time requirements without losing the benefits of ensuring consistency. In this paper, we formalize for the first time the concept of consistent signal reconstruction from streaming time-series data. Then, we present a signal reconstruction method able to enforce consistency and also exploit the spatiotemporal dependencies of streaming multivariate time-series data to further reduce the signal reconstruction error. Our experiments show that our proposed method achieves a favorable error-rate decay with the sampling rate compared to a similar but non-consistent reconstruction.

Figures (8)

• Figure 1: Illustration of the considered signal acquisition-reconstruction process spanning from time $0$ to $X$. Stationary spatial relationships are represented as edges in a graph with black dotted lines.
• Figure 2: Visual representation of a consistent signal estimate and a consistent signal reconstruction.
• Figure 3: Snapshot of an ongoing signal reconstruction using the smoothing policy variation.
• Figure 4: Conceptualization of the hyperplane projection $P_{\hat{H}^{(n)}_t}$ in the inner product space $\left( \mathbb{R}^\rho , \langle\cdot , \cdot\rangle_{\bm{D}^{(n)}_t} \right)$.
• Figure 5: Representation of a tth forward pass of our RNN-based policy. CAT refers to a vector concatenation layer. Evaluating the policy consists of solving a convex optimization problem with respect to $\bm{a}$, see \ref{['eq:policy']} and Sec. \ref{['ssec:policy_evaluation']}. Therefore, its last step can be seen as a convex optimization layer agrawal2019differentiable.