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Concept# Signal reconstruction

Résumé

In signal processing, reconstruction usually means the determination of an original continuous signal from a sequence of equally spaced samples.
This article takes a generalized abstract mathematical approach to signal sampling and reconstruction. For a more practical approach based on band-limited signals, see Whittaker–Shannon interpolation formula.
General principle
Let F be any sampling method, i.e. a linear map from the Hilbert space of square-integrable functions L^2 to complex space \mathbb C^n.
In our example, the vector space of sampled signals \mathbb C^n is n-dimensional complex space. Any proposed inverse R of F (reconstruction formula, in the lingo) would have to map \mathbb C^n to some subset of L^2. We could choose this subset arbitrarily, but if we're going to want a reconstruction formula R that is also a linear map, then we have to choose an n-dimensional linear subspace of L^2

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Reconstructing continuous signals based on a small number of discrete samples is a fundamental problem across science and engineering. We are often interested in signals with "simple" Fourier structure - e.g., those involving frequencies within a bounded range, a small number of frequencies, or a few blocks of frequencies i.e., bandlimited, sparse, and multiband signals, respectively. More broadly, any prior knowledge on a signal's Fourier power spectrum can constrain its complexity. Intuitively, signals with more highly constrained Fourier structure require fewer samples to reconstruct. We formalize this intuition by showing that, roughly, a continuous signal from a given class can be approximately reconstructed using a number of samples proportional to the statistical dimension of the allowed power spectrum of that class. We prove that, in nearly all settings, this natural measure tightly characterizes the sample complexity of signal reconstruction. Surprisingly, we also show that, up to log factors, a universal non-uniform sampling strategy can achieve this optimal complexity for any class of signals. We present an efficient and general algorithm for recovering a signal from the samples taken. For bandlimited and sparse signals, our method matches the state-of-the-art, while providing the the first computationally and sample efficient solution to a broader range of problems, including multiband signal reconstruction and Gaussian process regression tasks in one dimension. Our work is based on a novel connection between randomized linear algebra and the problem of reconstructing signals with constrained Fourier structure. We extend tools based on statistical leverage score sampling and column-based matrix reconstruction to the approximation of continuous linear operators that arise in the signal reconstruction problem. We believe these extensions are of independent interest and serve as a foundation for tackling a broad range of continuous time problems using randomized methods.

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