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We investigate the upper and lower bounds on the quantization distortions for independent and identically distributed continuous sources in the finite block-length regime. We derive a lower bound on the quantization distortion, as well as an upper bound on the quantization distortion based on random quantization codebooks. Moreover, we apply the obtained bounds to the continuous Gaussian source. For the Gaussian source, we propose a computationally tractable method to numerically compute the upper and lower bounds, for both bounded and unbounded quantization codebooks. Numerical results show that the gap between the upper and lower bounds is small for the block length of several hundreds, and the upper and lower bounds are tighter than those proposed in the existing works. [ABSTRACT FROM AUTHOR]

Sampling above the Nyquist rate is at the heart of sigma–delta modulation, where the increase in sampling rate is translated to a reduction in the overall (mean-squared-error) reconstruction distortion. This is attained by using a feedback filter at the encoder, in conjunction with a low-pass filter at the decoder. The goal of this paper is to characterize the optimal trade-off between the per-sample quantization rate and the resulting mean-squared-error distortion under various restrictions on the feedback filter. To this end, we establish a duality relation between the performance of sigma–delta modulation and the performance of differential pulse-code modulation when applied to (discrete-time) band-limited inputs. As the optimal trade-off for the latter scheme is fully understood, the full characterization for sigma–delta modulation, as well as the optimal feedback filters, immediately follows. [ABSTRACT FROM AUTHOR]

It is revealed that there is a link between the quantization approach employed and the dimension of the vector parameter which can be accurately estimated by a quantized estimation system. A critical quantity called inestimable dimension for quantized data (IDQD) is introduced, which does not depend on the quantization regions and the statistical models of the observations but instead depends only on the number of sensors and on the precision of the vector quantizers employed by the system. It is shown that the IDQD describes a quantization-induced fundamental limitation on the estimation capabilities of the system. To be specific, if the dimension of the desired vector parameter is larger than the IDQD of the quantized estimation system, then the Fisher information matrix for estimating the desired vector parameter is singular, and, moreover, there exist infinitely many nonidentifiable vector parameter points in the vector parameter space. Furthermore, it is shown that under some common assumptions on the statistical models of the observations and the quantization system, a smaller IDQD can be obtained, which can specify an even more limiting quantization induced fundamental limitation on the estimation capabilities of the system. [ABSTRACT FROM AUTHOR]

We consider the problem of recovering a high-dimensional structured signal from independent Gaussian linear measurements each of which is quantized to $b$ bits. The focus is on a specific method of signal recovery that extends a procedure originally proposed by Plan and Vershynin for one-bit quantization to a multi-bit setting. At the heart of this paper is a characterization of the optimal trade-off between the number of measurements m$ and the bit depth per measurement b$ given a total budget of B = m \cdot b -error in estimating the signal. It turns out that the choice b = 1$ is optimal for estimating the unit vector (direction) corresponding to the signal for any level of additive Gaussian noise before quantization as well as for a specific model of adversarial noise, whereas in a noiseless setting the choice b = 2$ is optimal for estimating the direction and the norm (scale) of the signal. Moreover, Lloyd–Max quantization is shown to be an optimal quantization scheme with respect to -estimation error. Our analysis is corroborated by the numerical experiments showing nearly perfect agreement with our theoretical predictions. This paper is complemented by an empirical comparison to alternative methods of signal recovery. The results of that comparison point to a regime change depending on the noise level: in a low-noise setting, the approach under study falls short of more sophisticated competitors while being competitive in moderate- and high-noise settings. [ABSTRACT FROM PUBLISHER]

Asymptotic formulas are derived for the mean-squared error (MSE) distortion of N-level uniform scalar quantizers designed to be MSE optimal for one density function, but applied to another, as N \to \infty . These formulas, which are based on the Euler–Maclaurin formula, are then applied with generalized gamma, Bucklew–Gallagher, and Hui–Neuhoff density functions as the designed-for and applied-to densities. It is found that the mismatch between the designed-for and applied-to densities can disturb the delicate balance between granular and overload distortions in optimal quantization, with the result that, generally speaking, the granular or overload distortion dominates, respectively, depending on whether the applied-to density function has a lighter or heavier tail than the designed-for density. Specifically, in the case of generalized gamma densities, a variance mismatch makes overload distortion dominate for an applied-to source with a slightly larger variance, whereas a shape mismatch can tolerate a wider variance difference while retaining the dominance of the granular distortion. In addition, for the studied density functions, the Euler–Maclaurin approach is used to derive asymptotic formulas for the optimal quantizer step size in a simpler, more direct, way than previous approaches. [ABSTRACT FROM PUBLISHER]