Reconstruction Error in Nonuniformly Sampled Approximately Sparse Signals

With its aim to reduce the amount of sensed data and to improve the energy efficiency, compressive sensing (CS) is recently witnessing a growing research interest in remote-sensing applications. The Fourier transform domain plays a significant role as a signal-processing tool and the sparsity domain for the CS-reconstruction methods. A generalized expression for the error in the reconstruction of nonuniformly sampled, approximately sparse, or nonsparse, noisy signals in the Fourier domain is presented in this letter. This expression holds for a wide range of practically important nonuniform signal-sampling strategies, covering the uniform and completely random sampling as the special cases. Additive noise and noise-folding effects are included in the analysis. Statistical examples and two real-world examples validate the presented theory.