Abstract
The influence of finite-length registers and the corresponding quantization effects on the reconstruction of sparse and approximately sparse signals from a reduced set of measurements is analyzed in this paper. For the nonquantized measurements, the compressive sensing (CS) framework provides highly accurate reconstruction algorithms that produce negligible errors when the reconstruction conditions are met. However, hardware implementations of signal processing algorithms inevitably involve finite-length registers and quantization of the measurements. A detailed analysis of the effects related to the measurement quantization, with an arbitrary number of bits, is provided in this paper. A unified novel mathematical model to characterize the influence of the quantization noise and the signal nonsparsity on the CS reconstruction is introduced. Using this model, an exact formula for the expected error energy in the CS-based reconstructed signal is derived, while in the literature its bounds have been reported only. The theory is validated through various numerical examples with quantized measurements, involving scenarios with approximately sparse signals, noise folding effect, and floating-point arithmetics.
Highlights
Compressive sensing (CS) theory provides a rigorous mathematical framework for the reconstruction of sparse signals, using a reduced set of measurements [1]–[10]
Since the establishment of CS, phenomena related to the reduced sets of measurements and sparse signal reconstruction have been supported by the fundamental theory and well-defined mathematical framework, while the performances of the reconstruction processes have been continuously improved by newly introduced algorithms, often adapted to perform well in a particular context, or to solve some specific problems [11]–[19]
PROBABILITY OF MISDETECTION The results for the mean squared error (MSE) are derived under the condition that the quantization does not influence the reconstruction condition and that the signal can be uniquely recovered from the available set of measurements
Summary
Compressive sensing (CS) theory provides a rigorous mathematical framework for the reconstruction of sparse signals, using a reduced set of measurements [1]–[10]. The quantization noise was studied in [33], where the lower and upper bound for the ratio of the reconstruction SNR and measurements SNR are derived and related to the noise folding effects in CS on the signal acquisition systems. The presented theory is unified by exact relations for the expected squared reconstruction error, derived to take into account all the studied effects. We have performed reconstructions with various numbers of bits, different sparsities, including approximately sparse signals, and noise folding effects. 2) calculation of the unknown coefficient values X (k) at the detected nonzero positions
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