Abstract
The current methods used to convert analogue signals into discrete-time sequences have been deeply influenced by the classical Shannon–Whittaker–Kotelnikov sampling theorem. This approach restricts the class of signals that can be sampled and perfectly reconstructed to bandlimited signals. During the last few years, a new framework has emerged that overcomes these limitations and extends sampling theory to a broader class of signals named signals with finite rate of innovation (FRI). Instead of characterising a signal by its frequency content, FRI theory describes it in terms of the innovation parameters per unit of time. Bandlimited signals are thus a subset of this more general definition. In this paper, we provide an overview of this new framework and present the tools required to apply this theory in neuroscience. Specifically, we show how to monitor and infer the spiking activity of individual neurons from two-photon imaging of calcium signals. In this scenario, the problem is reduced to reconstructing a stream of decaying exponentials.
Highlights
The world is analogue, but computation is digital
Over the last six decades, our understanding of the conversion of continuous-time signal in discrete form has been heavily influenced by the Shannon–Whittaker–Kotelnikov sampling theorem (Shannon 1949; Whittaker 1929; Kotelnikov 1933; Unser 2000) which showed that the sampling and perfect reconstruction of signals are possible when the Fourier bandwidth or spectrum of the signal is finite
They introduced a new class of signals called signals with finite rate of innovation (FRI) which includes both bandlimited signals and the non-bandlimited functions discussed so far. They went on showing that classes of FRI signals can be sampled and perfectly reconstructed using an appropriate acquisition device
Summary
The world is analogue, but computation is digital. The process that bridges this gap is known as the sampling process. Over the last six decades, our understanding of the conversion of continuous-time signal in discrete form has been heavily influenced by the Shannon–Whittaker–Kotelnikov sampling theorem (Shannon 1949; Whittaker 1929; Kotelnikov 1933; Unser 2000) which showed that the sampling and perfect reconstruction of signals are possible when the Fourier bandwidth or spectrum of the signal is finite. They introduced a new class of signals called signals with finite rate of innovation (FRI) which includes both bandlimited signals and the non-bandlimited functions discussed so far. They went on showing that classes of FRI signals can be sampled and perfectly reconstructed using an appropriate acquisition device.
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