Effect of Companding in the Reconstruction of Nonsparse Signals

In this paper, the error of companding quantization is analyzed in the reconstruction of nonsparse (approximately sparse) signals. Compressive sensing (CS) algorithms are used for the reconstruction of these signals from a reduced set of available observations. The standard CS theory assumes that the available and reconstructed samples are represented with unlimited number of bits. However, hardware implementation demands that only a finite number of bits can be used, thereby introducing the quantization effect into the CS framework. The reconstruction based on uniformly quantizatized measurements showed promising results. However, nonuniform quantization introduces additional improvements in the reconstruction performance. Companding is a way to nonuniformly quantize signals. It is based on compression of the signal, quantization, and subsequent expansion of the quantized signal. In this paper, we present a theoretical error characterizing the reconstruction process based on quantized measurements, which is further verified on series of numerical examples.