No Relaxation: Guaranteed Recovery Of Finite-Valued Signals From Undersampled Measurements

This paper considers the problem of recovering a unipolar finite-valued signal from compressive measurements of its convolution with a known finite impulse response filter. We show that owing to the finite-value constraint the problem remains identifiable if the downsampling factor is smaller than the filter length. We develop a new computationally efficient decoding algorithm that can operate at the optimal downsampling factor under mild conditions on the filter. This allows us to explicitly impose the finite value constraint (no relaxation) without compromising on the computational tractability.