Optimal Uniform Quantization Research Articles

Optimization of the 24-Bit Fixed-Point Format for the Laplacian Source

The 32-bit floating-point (FP32) binary format, commonly used for data representation in computers, introduces high complexity, requiring powerful and expensive hardware for data processing and high energy consumption, hence being unsuitable for implementation on sensor nodes, edge devices, and other devices with limited hardware resources. Therefore, it is often necessary to use binary formats of lower complexity than FP32. This paper proposes the usage of the 24-bit fixed-point format that will reduce the complexity in two ways, by decreasing the number of bits and by the fact that the fixed-point format has significantly less complexity than the floating-point format. The paper optimizes the 24-bit fixed-point format and examines its performance for data with the Laplacian distribution, exploiting the analogy between fixed-point binary representation and uniform quantization. Firstly, the optimization of the 24-bit uniform quantizer is performed by deriving two new closed-form formulas for a very accurate calculation of its maximal amplitude. Then, the 24-bit fixed-point format is optimized by optimization of its key parameter and by proposing two adaptation procedures, with the aim to obtain the same performance as of the optimal uniform quantizer in a wide range of variance of input data. It is shown that the proposed 24-bit fixed-point format achieves for 18.425 dB higher performance than the floating-point format with the same number of bits while being less complex.

Full-Duplex Cell-Free Massive MIMO Systems: Analysis and Decentralized Optimization

Cell-free (CF) massive multiple-input-multiple-output (mMIMO) deployments are usually investigated with half-duplex nodes and high-capacity fronthaul links. To leverage the possible gains in throughput and energy efficiency (EE) of full-duplex (FD) communications, we consider a FD CF mMIMO system with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">practical limited-capacity fronthaul links</i> . We derive closed-form spectral efficiency (SE) lower bounds for this system with maximum-ratio combining/maximum-ratio transmission processing and optimal uniform quantization. We then optimize the weighted sum EE (WSEE) via downlink and uplink power control by using a two-layered approach: the first layer formulates the optimization as a generalized convex program, while the second layer solves the optimization decentrally using the alternating direction method of multipliers. We analytically show that the proposed two-layered formulation yields a Karush-Kuhn-Tucker point of the original WSEE optimization. We numerically show the influence of weights on the individual EE of the users, which demonstrates the utility of the WSEE metric to incorporate heterogeneous EE requirements of users. We show that low fronthaul capacity reduces the number of users each AP can support, and the cell-free system, consequently, becomes user-centric.

Uplink Spectral and Energy Efficiency of Cell-Free Massive MIMO With Optimal Uniform Quantization

This paper investigates the performance of limited-fronthaul cell-free massive multiple-input multiple-output (MIMO) taking account the fronthaul quantization and imperfect channel acquisition. Three cases are studied, which we refer to as Estimate&Quantize, Quantize&Estimate, and Decentralized, according to where channel estimation is performed and exploited. Maximum-ratio combining (MRC), zero-forcing (ZF), and minimum mean-square error (MMSE) receivers are considered. The Max algorithm and the Bussgang decomposition are exploited to model optimum uniform quantization. Exploiting the optimal step size of the quantizer, analytical expressions for spectral and energy efficiencies are presented. Finally, an access point (AP) assignment algorithm is proposed to improve the performance of the decentralized scheme. Numerical results investigate the performance gap between limited fronthaul and perfect fronthaul cases, and demonstrate that exploiting relatively few quantization bits, the performance of limited-fronthaul cell-free massive MIMO closely approaches the perfect-fronthaul performance.

Max–Min Rate of Cell-Free Massive MIMO Uplink With Optimal Uniform Quantization

Cell-free Massive multiple-input multiple-output (MIMO) is considered, where distributed access points (APs) multiply the received signal by the conjugate of the estimated channel, and send back a quantized version of this weighted signal to a central processing unit (CPU). For the first time, we present a performance comparison between the case of perfect fronthaul links, the case when the quantized version of the estimated channel and the quantized signal are available at the CPU, and the case when only the quantized weighted signal is available at the CPU. The Bussgang decomposition is used to model the effect of quantization. The max-min problem is studied, where the minimum rate is maximized with the power and fronthaul capacity constraints. To deal with the non-convex problem, the original problem is decomposed into two sub-problems (referred to as receiver filter design and power allocation). Geometric programming (GP) is exploited to solve the power allocation problem whereas a generalized eigenvalue problem is solved to design the receiver filter. An iterative scheme is developed and the optimality of the proposed algorithm is proved through uplink-downlink duality. A user assignment algorithm is proposed which significantly improves the performance. Numerical results demonstrate the superiority of the proposed schemes.

Energy Efficiency of the Cell-Free Massive MIMO Uplink With Optimal Uniform Quantization

A cell-free Massive multiple-input multiple-output (MIMO) uplink is considered, where the access points (APs) are connected to a central processing unit (CPU) through limited-capacity wireless microwave links. The quantized version of the weighted signals are available at the CPU, by exploiting the Bussgang decomposition to model the effect of quantization. A closed-form expression for spectral efficiency is derived taking into account the effects of channel estimation error and quantization distortion. The energy efficiency maximization problem is considered with per-user power, backhaul capacity and throughput requirement constraints. To solve this non-convex problem, we decouple the original problem into two sub-problems, namely, receiver filter coefficient design, and power allocation. The receiver filter coefficient design is formulated as a generalized eigenvalue problem whereas a successive convex approximation (SCA) and a heuristic sub-optimal scheme are exploited to convert the power allocation problem into a standard geometric programming (GP) problem. An iterative algorithm is proposed to alternately solve each sub-problem. Complexity analysis and convergence of the proposed schemes are investigated. Numerical results indicate the superiority of the proposed algorithms over the case of equal power allocation.

OPTIMAL UNIFORM QUANTIZATION OF PARAMETERS OF CONVOLUTIONAL NEURAL NETWORKS

State of the art convolutional neural networks provide high accuracy in solving a wide range of problems. Usually it is achieved by a significant increasing their computational complexity and the representation of the network parameters in single-precision floating point numbers. However, due to the limited resources, the application of networks in embedded systems and mobile applications in real time is problematic. One of the methods to solve this problem is to reduce the bit depth of data and use integer arithmetic. For this purpose, the network parameters are quantized. Performing quantization, it is necessary to ensure a minimum loss of recognition accuracy. The article proposes to use an optimal uniform quantizer with an adaptive step. The quantizer step depends on the distribution function of the quantized parameters. It reduces the effect of the quantization error on the recognition accuracy. There are also described approaches to improving the quality of quantization. The proposed quantization method is estimated on the CIFAR-10 database. It is shown that the optimal uniform quantizer for CIFAR-10 database with 8-bit representation of network parameters allows to achieve the accuracy of the initial trained network.

Asymptotic Formulas for Mismatched Fixed-Rate Minimum MSE Laplacian Quantizers

This letter derives asymptotic formulas for the mean square error (MSE) distortion and the signal-to-noise ratio of a mismatched fixed-rate minimum MSE Laplacian quantizer. These closed-form formulas are expressed in terms of the number N of quantization points, the mean displacement, and the ratio p of the standard deviation of the source to that for which the quantizer is optimally designed. Numerical results show that the principal formula is accurate in that, for rate R = log2 N ges 6, it predicts signal-to-noise ratios within 1 % of the true values for a wide range of variance and mean mismatch. The new findings herein include the fact that, for heavy variance mismatch of p > 3/2, the signal-to-noise ratio increases at the rate of 9/p dB/bit, which is slower than the usual 6 dB/bit, and the fact that an optimal uniform quantizer, though optimally designed, is slightly more than critically mismatched to the source. The derived formulas can be useful in quantization of speech or music signals, which are modeled well as Laplacian sources and have changing short-term variances.

ANALYSIS OF OPTIMAL UNIFORM TWO-DIMENSIONAL QUANTIZATION FOR THE LAPLACE SOURCE OVER BENNETT’S INTEGRAL

In this paper, the analysis of Bennett's integral is given for uniform quantization for two-dimensional Laplace sources using granular distortion Dg. The goal of this paper is to find a simple equation for distortion by solving Bennett's integral for uniform two-dimensional quantization. The analysis for optimal uniform quantization for two-dimensional Laplace sources is done.

Successive refinement lattice vector quantization

Lattice Vector quantization (LVQ) solves the complexity problem of LBG based vector quantizers, yielding very general codebooks. However, a single stage LVQ, when applied to high resolution quantization of a vector, may result in very large and unwieldy indices, making it unsuitable for applications requiring successive refinement. The goal of this work is to develop a unified framework for progressive uniform quantization of vectors without having to sacrifice the mean- squared-error advantage of lattice quantization. A successive refinement uniform vector quantization methodology is developed, where the codebooks in successive stages are all lattice codebooks, each in the shape of the Voronoi regions of the lattice at the previous stage. Such Voronoi shaped geometric lattice codebooks are named Voronoi lattice VQs (VLVQ). Measures of efficiency of successive refinement are developed based on the entropy of the indices transmitted by the VLVQs. Additionally, a constructive method for asymptotically optimal uniform quantization is developed using tree-structured subset VLVQs in conjunction with entropy coding. The methodology developed here essentially yields the optimal vector counterpart of scalar "bitplane-wise" refinement. Unfortunately it is not as trivial to implement as in the scalar case. Furthermore, the benefits of asymptotic optimality in tree-structured subset VLVQs remain elusive in practical nonasymptotic situations. Nevertheless, because scalar bitplane- wise refinement is extensively used in modern wavelet image coders, we have applied the VLVQ techniques to successively refine vectors of wavelet coefficients in the vector set-partitioning (VSPIHT) framework. The results are compared against SPIHT and the previous successive approximation wavelet vector quantization (SA-W-VQ) results of Sampson, da Silva and Ghanbari.