Companding Quantizer Research Articles

Performance improvement by an optimum companding-based Non-Uniform quantization scheme in low-bit resolution IMDD-UFMC system

In this paper, an optimum non-uniform quantization method is presented in the intensity-modulated direct detection universal filter multicarrier (IMDD-UFMC) system to achieve the system performance improvement with low-bit resolution digital-to-analog converter (DAC). By this specifically-designed companding quantizer without the probability density function (PDF) estimation, the number and interval size of the quantization interval allocated in the low probability region and the high probability region can be balanced with a modified optimal mapping relationship to obtain the optimal quantized level and then the low quantization noise is realized to greatly reduce the bit resolution of DAC in IMDD-UFMC system. Owing to this scheme is essentially a peak-to-average power ratio (PAPR) suppression technology of signal distortion, PAPR performance can be improved at the same time. Compared to the clustering and nonlinear programming schemes, the system performance in term of bit error rate (BER) can be improved and the difficulty of implementation can be reduced in the 25-Gb/s transmission system through 20-km fiber transmission. Moreover, a simplified method is presented by fitting the corresponding companding curve with error function and logistic function, which further decreases the complexity of the optimal non-uniform quantization. In this way, our scheme can offer a significant advancement in the design of IMDD-UFMC systems and has the potential to enhance the performance of future multi-carrier optical communication systems.

NOVEL EXPONENTIAL TYPE APPROXIMATIONS OF THE Q-FUNCTION

In this paper, we propose several solutions for approximating the Q-function using one exponential function or the sum of two exponential functions. As the novel Q-function approximations have simple analytical forms and are therefore very suitable for further derivation of expressions in closed forms, a large number of applications are feasible. The application of the novel exponential type approximations of the Q-function is especially important for overcoming issues arising in designing scalar companding quantizers for the Gaussian source, which are caused by the non-existence of a closed form expression for the Q-function. Since our approximations of the Q-function have simple analytical forms and are more accurate than the approximations of the Q-function previously used for the observed problem in the scalar companding quantization of the Gaussian source, their application, especially for this problem is of great importance.

Support region of μ-law logarithmic quantizers for Laplacian source applied in neural networks

The main aim of the paper is to provide effective and accurate solutions for the calculation of the support region of the μ-law logarithmic companding quantizers. A new solution for the starting point of iterative methods will be proposed, that provides very accurate value of the support region (being the main parameter needed for the design of the quantizer) only after one iteration of the iterative method. Based on this new starting point, an accurate closed-form approximate expression for the calculation of the support region will be derived, as one of the main contributions of the paper. To significantly simplify implementation of the μ-law companding quantizer, piecewise linearization is performed. A new linearization method is presented, based on the optimization of the last segments. Derivation of an accurate closed-form formula for the support region of the linearized quantizer is done, as an important contribution. The obtained linearized μ-law companding quantizer is very simple to design (due to closed-form formulas) and to implement (due to linearization), providing at the same time very high performance (due to optimization of the last segments). Due to these and other advantages (robustness, adjustability to the statistical distribution of the input signal), the proposed quantizer can be used in many topical applications, such as in receivers of 5G wireless systems or in neural networks for quantization of weights and activations. The paper provides an application of the designed quantizers for quantization of weights of a neural network, showing significant decreasing of the bit-rate compared to the standard full-precision representation (from 32 bits to just 5 bits), with the same prediction accuracy of the network.

Symmetric Quantile Quantizer Parameterization for the Laplacian Source: Qualification for Contemporary Quantization Solutions

In this paper, we consider the opportunities and constraints, which rest on quantization as a guiding principle for data representation and compression. In particular, we propose a novel model of Symmetric Quantile Quantizer (SQQ) and we describe in detail its parameterization. We suggest a simple method for offline precalculation of its parameters and we examine the inevitable loss of information introduced by SQQ, as an important part of bit optimization task at the traditional network level, which can be globally mapped out in many contemporary solutions. Our anticipation is that such precalculated values can be leveraged in deterministic quantization process. We highlight that this notice heavily relies on the fact that the values of interest are distributed according to the Laplacian distribution, which we consider in the paper. The basic difference of our SQQ and the previously established asymptotically optimal quantizer model, that is, Scalar Companding Quantizer (SCQ), is reflected in the fact that, in SCQ model, both decision thresholds and representation levels are determined in accordance with the specified compressor function, whereas in our SQQ model, a precedence of SCQ model for the straightforward decision thresholds calculation is used, while the representation levels are optimally determined for the specified decision thresholds and assumed Laplacian distribution. As a result, our SQQ outperforms SCQ in terms of signal-to-quantization noise ratio (SQNR). As stated in this paper, there are numerous indications to make us believe that appropriate quantizer parameterization will move us closer to an optimization in the amount of the transferred data in bits, which is strongly dependent on the amount of SQNR.

Circularly Symmetric Companding Quantization-Inspired Hybrid Constellation Shaping for APSK Modulation to Increase Power Efficiency in Gaussian-Noise-Limited Channel

In this paper we address hybrid probabilistic-geometric constellation shaping (HCS) of amplitude phase shift keying (APSK) constellation based on the reversed model of optimal companding quantization for circularly symmetric sources with the goal to increase the constellation power efficiency in Gaussian-noise-limited channel. To empirically optimize the proposed APSK constellation such that a symbol-error rate (SER) reaches its minimum under given constraints with respect to signal-to-noise ratio (SNR) and the prior probabilities of constellation points, for various settings of constellation parameters, we investigate SER dependence on SNR and determine the constellation parameters achieving minimum SER. The SER dependence on SNR we estimate theoretically by deriving approximate formula for SER of uncoded APSK constellation in Gaussian-noise-limited channel and practically by performing simulation. The obtained results are well-matched verifying the accuracy of approximate SER formula. The results also show that our APSK constellation outperforms some previous APSK and M-ary QAM constellations in terms of power efficiency. Thus, for SER equal to 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-6</sup> the gain in power efficiency amounts up to 2.35 dB, 2.23dB and 1.64 dB compared with the maximum mutual information-optimized 4+12-APSK, 4+12+16-APSK and 4+12+20+28-APSK constellations, respectively. This means that by employing APSK constellation we propose instead the traditional APSK constellations the transmitted signal power can be reduced by a third enabling lower power consumption. The improved power efficiency of our APSK constellation makes it suitable for application in power-limited communications such as fiber-optic communications, satellite communications, power-line communications, and multiple-input multiple-output wireless transmissions.

New adaptive compandor for LTE signal compression based on spline approximations

With the constant increase in network traffic, wireless operators are finding it more challenging to keep network hardware costs to a minimum. At the same time, the energy cost associated with operating a network has increased proportionally. Therefore, the search for higher network capacity is simultaneously accompanied by the search for a cost-efficient network deployment. In this paper, we show that a saving in transmitted signal energy can be achieved at the signal design level by deploying very specific signal processing techniques. Using an orthogonal frequency-division multiplexing signal for Long-Term Evolution networks as an example, we utilize a novel non-uniform companding quantizer to save a transmitted signal energy. Our results show that by using non-uniform quantization it is possible to further optimize 4G wireless networks.

Design of Compandor Based on Approximate the First-Degree Spline Function

In this paper, the approximation of the optimal compressor function using spline function of the first-degree is done. For the companding quantizer designed on the basis of the approximative spline function of the first-degree, the support region is numerically optimized to provide the minimum of the total distortion for the last segment. It is shown that the companding quantizer with the optimized support region threshold provides the signal to quantization noise ratio that is very close to the one of the optimal companding quantizer having an equal number of levels. DOI: http://dx.doi.org/10.5755/j01.eee.19.10.5911

Numerical Determination of the Optimal Value of Quantizer’s Segment Threshold Using Quadratic Spline Functions

In this paper, an approximation of the optimal compressor function using the quadratic spline functions has been presented. The coefficients of the quadratic spline functions are determined by minimizing the mean-square error (MSE). Based on the obtained approximative quadratic spline functions, the design for companding quantizer for Gaussian source is done. The support region of proposed companding quantizer is divided on segments of unequal size, where the optimal value of segment threshold is numerically determined depending on maximal value of the signal to quantization noise ratio (SQNR). It is shown that by the companding quantizer proposed in this paper, the SQNR that is very close to SQNR of nonlinear optimal companding quantizer is achieved.

Design of Fixed and Adaptive Companding Quantizer with Variable-Length Codeword for Memoryless Gaussian Source

The problem we address in this paper is the design of a quantizer that in comparison to the classical fixed-rate scalar quantizers provides more sophisticated bit rate reduction while restricting the class of quantizers to be scalar. We propose a switched variable-length code (VLC) optimal companding quantizer composed of two optimal companding scalar quantizers, the inner and the outer one, both designed for the memoryless Gaussian source of unit variance. Quantizers composing the proposed quantizer have a different codebook sizes and a different compressor functions. Particularly, we assume a smaller size of the inner quantizer's codebook in order to provide assignment of the shorter codewords to the high probability low amplitude speech samples belonging to the support region of the inner quantizer. We study the influence of codebook size of the inner and the outer quantizer on the Signal to Quantization Noise Ratio (SQNR). In such a manner the conclusion of the proposed quantizer significance in speech compression is distinctly shown in the paper. For the proposed quantizer model and its forward adaptive version the SQNR robustness analysis in a wide variance range is also presented in the paper. It is shown that our multi-resolution quantizer can satisfy G.712 Recommendation for high-quality quantization at the bit rate of 6.3 bit/sample achieving the compression of 1.7 bit/sample over the G.711 quantizer.

Designing of the forward adaptive companding quantizer with variable length codewords for stohastic measurement signals

This paper deals with the designing of the forward adaptive ?-law companding quantizer whose levels are coded with the Golomb-Rice code. The designing is performed for measurement signals with the Gaussian distribution and applied for the speech signal. The model satisfies the G. 712 standard and achieves the decreasing of the bit-rate for 1.34 bps (bits per sample) compared to the G. 711 standard.

Semi–logarithmic and hybrid quantization of Laplacian source inwide range of variances

A novel semilogarithmic hybrid quantizer for non-uniform scalar quantization of Laplacian source, which consist of uniform quantizer and companding quantizer is introduced. Uniform quantizer has unit gain in area around zero. Companding quantizer is defined with a novel logarithm characteristic. Also an analysis of classic semilogarithmic A-law for various values of A parameter is provided. Comparation with classic semilogarithmic A-law is performed. The main advantage of hybrid quantizer is that number of representation levels for both uniform and companding quantizer are not unambiguously determined function of the A parameter value, as it is the case with classic semilogarithmic A companding characteristic. It is shown that by using hybrid quantizer, average of signal-to-quantization noise ratio SQNR quality obtained by using classic A companding law can be overachieved for 0.47 dB. Numbers of representation levels of hybrid quantizer are adapted to the input signal variances, in order to achieve high SQNR in a wide range of signal volumes (variances). By using this adaptation higher average SQNR quality of 2.52 dB could be achieved compared to classic A companding law. Forward adaptation of hybrid quantizer is analyzed and obtained performances correspond to adaptive classic A companding law case but possible advantage arises in simpler practical realization of hybrid quantizers. Obtained performances correspond to classic A-law companding case, because during the adaptation process, optimal values of parameter A are chosen. For each other A parameter values proposed hybrid quantizer provides better results. For value of A = 50 hybrid model has higher SQNR value for 0.79 dB

Optimization of Quantizer’s Segment Threshold Using Spline Approximations for Optimal Compressor Function

In this paper, the optimization of quantizer’s segment threshold is done. The quantizer is designed on the basis of approximative spline functions. Coefficients on which we form approximative spline functions are calculated by minimization mean square error (MSE). For coefficients determined in this way, spline functions by which optimal compressor function is approximated are obtained. For the quantizer designed on the basis of approximative spline functions, segment threshold is numerically determined depending on maximal value of the signal to quantization noise ratio (SQNR). Thus, quantizer with optimized segment threshold is achieved. It is shown that by quantizer model designed in this way and proposed in this paper, the SQNR that is very close to SQNR of nonlinear optimal companding quantizer is achieved.

Simple Compression Algorithm for Memoryless Laplacian Source Based on the Optimal Companding Technique

This paper has two achievements. The first aim of this paper is optimization of the lossy compression coder realized as companding quantizer with optimal compression law. This optimization is achieved by optimizing maximal amplitude for that optimal companding quantizer for Laplacian source. Approximate expression in closed form for optimal maximal amplitude is found. Although this expression is very simple and suitable for practical implementation, it satisfy optimality criterion for Lloyd-Max quantizer (for R >= 6 bits/sample). In the second part of this paper novel simple lossless compression method is presented. This method is much simpler than Huffman method, but it gives better results. Finally, at the end of the paper, we join optimal companding quantizer and lossless coding method together in one generalized compression method. This method is applied on the concrete still image and good results are obtained. Besides still images, this method also could be used for compression speech and bio-medical signals.

Low-Complexity Source Coding Using Gaussian Mixture Models, Lattice Vector Quantization, and Recursive Coding with Application to Speech Spectrum Quantization

In this paper, we use the Gaussian mixture model (GMM) based multidimensional companding quantization framework to develop two important quantization schemes. In the first scheme, the scalar quantization in the companding framework is replaced by more efficient lattice vector quantization. Low-complexity lattice pruning and quantization schemes are provided for the $E_8$ Gossett lattice. At moderate to high bit rates, the proposed scheme recovers much of the space-filling loss due to the product vector quantizers (PVQ) employed in earlier work, and thereby, provides improved performance with a marginal increase in complexity. In the second scheme, we generalize the compression framework to accommodate recursive coding. In this approach, the joint probability density function (PDF) of the parameter vectors of successive source frames is modeled using a GMM. The conditional density of the parameter vector of the current source frame based on the quantized values of the parameter vector of the previous source frames is used to generate a new codebook for every current source frame. We demonstrate the efficacy of the proposed schemes in the application of speech spectrum quantization. The proposed scheme is shown to provide superior performance with moderate increase in complexity when compared with conventional one-step linear prediction based compression schemes for both narrow-band and wide-band speech.

Multidimensional companding quantization of the gaussian source

Multidimensional companding quantization is analyzed theoretically for the case of high-resolution and mean-squared-error distortion. The optimality of choosing the expander function to be the inverse of the compressor function is established first. Heuristic derivations of the point density and moment of inertia of companding are given, which combined result in the distortion formula by Bucklew. Further, the interaction between the lattice quantizer (LQ) and compressor function is studied. Optimality is achieved with a second moment optimal LQ, shaped by a compressor-dependent linear transform. High rate theory for a radial compander and spherically symmetric sources is reviewed. The result is used to evaluate the performance on an independent and identically distributed (i.i.d.) Gaussian source. The radial compander clearly outperforms both a product scalar quantizer and a spherical quantizer for dimensions higher than two. Radial companding is also generalized to the correlated Gaussian source. Finally, a comparison of theory and practical companders is made in a Gaussian framework.