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Analysis and design of two stage mismatch quantizer for Laplacian source.

I. INTRODUCTION

Mismatch quantization has become very popular in recent years and its importance and appliance was researched in a number of papers [1]-[6]. It is generally considered that two stage quantization model is such that first quantizer has a smaller number of quantization levels in comparison to the second quantizer (i.e. required number of bits for transmission is higher for the second quantizer). This way, first quantizer determines the region and the second one determines the level within it [7]-[9]. The kind of two stage quantizer designed in this way is used for standard G. 711 [7]. On the other hand, the model that we discuss consists of two interconnected quantizers and it is performed in [8]. In [8] was concluded that such two stage quantization model should be designed so that second quantizer is described with at least 4 bits lower number of levels in comparison to the first quantizer. However, some systems (e.g. systems for image processing) have to be designed so that second quantizer does not meet this requirement. Here we propose an improved quantization model and performance estimation that will take into account aforementioned problem.

In this paper we analyse two interconnected quantizers with a different number of quantization levels. The first quantizer performs analog-to-digital (A/D) conversion with a large number of quantization levels [9], [10]. Its entrance deals with continuous signal that can take any real value from the infinite interval (-[infinity], [infinity]). We decided to choose fixed uniform quantizer for this task, since it has a low complexity but it still provides a high quality of reconstructed signal for high number of quantization levels. Furthermore, the second quantizer performs nonuniform quantization since it provides high quality for additional data compression with a small number of quantization levels. The output signal of the first quantizer represents the input signal at the second quantizer's entrance whereas the input samples at the both first and second quantizer's entrance are modelled with Laplacian distribution.

Let's denote the first [N.sub.1]-level quantizer with [Q.sub.1] and second [N.sub.2]-level quantizer with [Q.sub.2] ([N.sub.1] > [N.sub.2]). Quantizer [Q.sub.2] is intended to perform additional compression of previously discretized samples of limited amplitude [11]. In previous studies, we calculated theoretical results based on characteristics of continuous input signal. That way, we have obtained theoretical values of system's performance that have a small deviation in comparison to the experimental results. This study is aimed to determine how changes of signal characteristics, after A/D conversion, affect the performance of the whole system.

The paper is organized as follows. In Section II will be described two stage quantization system model--it will be shown design of the both fixed uniform and fixed nonuniform quantizers, as well as measures of the system's performance. Moreover, it will be introduced a novel measure that deals with variances of continuous and discretized signal. Next, numerical and graphical results will be shown in Section III. In the end, conclusions and ideas for future work will be presented in Section IV.

II. SYSTEM MODEL

This section's aim is to describe mismatch quantization model by using aforementioned quantizers ([Q.sub.1] and [Q.sub.2]) and to propose improved modelling that will provide higher accuracy. The improved model deals with a variance of discretized signal instead of continuous variance.

As it was said above, we discuss Laplacian information source with a memoryless property and a zero mean value

p(x) = 1/[square root of 2] [[sigma].sub.x] exp (- [square root of 2] [absolute value of x]/[sigma]), (1)

where is [sigma] standard deviation of the input signal. Primarily we have chosen Laplacian source since it is commonly used in systems for image processing.

Observed system model consists of two stages. In the first stage fixed uniform quantizer [Q.sub.1] converts analog signal to discrete samples, whereas [Q.sub.2] performs additional data compression by using nonuniform quantization in the second stage. Thereby, decision thresholds and representational levels of [Q.sub.1] can be defined with [11]

[x.sub.i] = i x [DELTA] = i x 2[x.sub.max]/[N.sub.1], (2)

where i = 0,...,N/2 .

[y.sub.i] = [x.sub.i-1] + [x.sub.i]/2 = 2i - 1/[N.sub.1] [x.sub.max], (3)

where i = 1,...,N/2 . In (2) and (3) [x.sub.max] denotes the maximal value of continuous signal amplitude which depends on the input signal range. Its optimal values, depending on the number of quantization levels, can be find in [11]. These two expressions define the positive range of fixed uniform quantizer. Since the quantizer is symmetric, in future consideration we will observe just the positive range.

In order to estimate a difference between continuous signal variance of original signal and variance of discretized signal obtained after processing with [Q.sub.1] we introduce a novel measure: continuous--to--discrete--signal--variance ratio (CDSVR). It is defined with

CDSVR = 10log [[sigma].sup.2.sub.x]/[[sigma].sup.2.sub.x] [dB]. (4)

In previous equation [[sigma].sub.x.sup.2] represents the variance of continuous signal whereas [[sigma].sub.y.sup.2] is the variance of discretized signal, obtained after processing with [Q.sub.1]. These two parameters can be defined with [9]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[[sigma].sup.2.sub.y] = 2 [N/2.summation over (i=1)] [y.sup.2.sub.i] x [P.sub.i], (6)

where [[sigma].sub.reff.sup.2] is the referent variance. Furthermore, [P.sub.i] are probabilities of discrete input levels of the second quantizer

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where is [sigma] = [[sigma].sub.x] for calculating [[sigma].sub.y.sup.2] in (6).

Design of the nonuniform quantizer [Q.sub.2] is done in two steps. Firstly, we design the optimal compandor with [N.sub.2] quantization levels for the unit standard deviation ([sigma] = 1). After that we discuss range variations by introducing the parameter of proportionality k. Compandor's compressor function maps the range ([infinity], [infinity) to (-1, 1) and it can be defined with [12]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Its decision thresholds and representational levels obtained in this way are defined with [12]:

[t'.sub.i] = 3/[square root of 2] log ([N.sub.2]/2([N.sub.2] - i) + (2i - [N.sub.2])exp (-[square root of 2]/3 [t.sub.max])), (9)

[[omega]'.sub.i] = 3/[square root of 2] log ([N.sub.2]/2([N.sub.2] - i) + 1 + (2i - 1- [N.sub.2])exp (-[square root of 2]/3 [t.sub.max])), (10)

where is [N.sub.2]/2 < i [less than or equal to] [N.sub.2]. Since that this type of quantizer is symmetric, it is valid

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where 0 < i [less than or equal to] N.sub.2] / 2.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where 1 < j [less than or equal to] [N.sub.2]/2. In (9)-(10) parameter tmax denotes the maximal amplitude of the optimal companding quantizer for the unit variance and its values, depending on the number of quantization levels, can be find in [12]. Finally, decision thresholds and representational levels of nonuniform quantizer [Q.sub.2] are

[t.sub.i] = [t'.sub.i] x k, (13)

where [N.sub.2] [less than or equal to] i [less than or equal to] [N.sub.2] / 2.

[[omega].sub.j] = [[omega]'.sub.j] x k, (14)

where 1 [less than or equal to] j [less than or equal to] [N.sub.2]. Since the compression process performed in this way enters some information lost, we measure granular and overload distortion that can be defined with [13]-[15]:

[D.sub.g] = 2 [[N.sub.2].summation over (i=[N.sub.2]/2)] [[r.sub.i].summation over (j= 1)] [([y.sub.ij] [[omega]'.sub.i]).sup.2] P([y.sub.ij]), (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

In (15) parameter r denotes the number of input levels mapped with [[omega].sub.i] whereas [y.sub.ij] are output levels of [Q.sub.1]. Furthermore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in (16) whereas parameter s represents the number of output levels from [Q.sub.1] that are not within designed input range of [Q.sub.2].

In the end, total distortion is equal to

[D.sub.t] = [D.sub.g] + [D.sub.0]. (17)

Beside CDSVR, overall system performance will be measured by using SQNR that represents a common measure of performance [12], [14]

SQNR([[sigma].sup.2]) = 10 [log.sub.10] ([[sigma].sup.2]/[D.sub.t]) [dB]. (18)

Numerical results will be calculated for the standard model (SQNR([[sigma].sub.x.sup.2])) as well as for the proposed model (SQNR([[sigma].sub.y.sup.2])).

III. NUMERICAL RESULTS

In Fig. 1 is shown CDSVR in function of [[sigma].sub.x.sup.2]. Observing Fig. 1 we can conclude that values of [[sigma].sub.y.sup.2] do not match corresponding values of [[sigma].sub.x.sup.2]. Their difference, or in another words a CDSVR value, increases with increasing the continuous variance of input signal [[sigma].sub.x.sup.2]. This means that our quantizer is a kind of a mismatch quantizer, since our signal's variance is varying. As a result, design of the second quantizer is not appropriate (it is designed for the unit standard deviation).

However, a CDSVR value is lower for a higher number of representational levels [N.sub.1], that can be regarded as a reducing the loss of information occurred due to A/D conversion using [Q.sub.1]. Consequently, we can expect that proposed modelling that uses [[sigma].sub.y.sup.2] will achieve higher SQNR (SQNR([[sigma].sub.y.sup.2])), as it was confirmed experimentally [15].

In Fig. 2 and Fig. 3 the overall SQNR depending on various values of [N.sub.1] in function of [[sigma].sub.x.sup.2] is shown. SQNR value is calculated in two ways--using the continuous variance of the input signal [[sigma].sub.x.sup.2] (SQNR([[sigma].sub.x.sup.2])) and using the variance of discretized signal [[sigma].sub.y.sup.2] (SQNR([[sigma].sub.y.sup.2])).

It can be noticed that system's performance calculated in both ways have good matching in the range of small variances ([[sigma].sub.x.sup.2] < 5 [dB]). With increasing the variance [sigma].sub.x.sup.2] there is a larger difference of calculated values of SQNR and system shows better performance in the case when we take into consideration information loss occurred due to A/D conversion while processing with [Q.sub.1] (SQNR([[sigma].sub.y.sup.2])).

In Fig. 4 and Fig. 5 SQNR depending on parameter of proportionality k for fixed [N.sub.1] and various numbers of representational levels [N.sub.2] is shown.

It can be seen that peaks of the curves are shifted left but their values remain approximately the same by changing the value of parameter k. In the range of small variances it can be noticed that SQNR is slower increasing by incrementing the value of parameter k (for the same [[sigma].sub.x.sup.2] it is obtained higher SQNR for lower k).

On the other hand, for [[sigma].sub.x.sup.2] >5 [dB], SQNR value decreases by decreasing parameter k. Changing this parameter affects the both SQNR([[sigma].sub.x.sup.2]) and SQNR([[sigma].sub.y.sup.2]) in the same way.

In Table I average SQNR (SQNR([[sigma].sub.x.sup.2]) and SQNR([[sigma].sub.y.sup.2])) for various values of parameter k is shown.

Obtained average values refer to corresponding graphically presented results in Fig. 4 and Fig. 5 for the range [[sigma].sub.x.sup.2] [member of] [-20 dB, 20 dB]. It can be concluded that in the aforementioned range, varying the parameter k does not have a significant influence on the overall [SQNR.sub.av] of the observed system.

Table II shows average SQNR ([SQNR.sub.av]([[sigma].sub.x.sup.2]) and [SQNR.sub.av]([sigma].sub.y.sup.2])) for various sub-ranges of input continuous variance ([[sigma].sub.x.sup.2]) in function of parameter k. We can conclude that parameter k affects overall [SQNR.sub.av] in all sub-ranges and its impact increases with decreasing the width of the range.

IV. CONCLUSIONS

In this paper we discuss limitations of two stage quantization modelling that uses just a variance of continuous signal ([[sigma].sub.x.sup.2]). Moreover, we introduce a new measure of system's performance (CDSVR). Obtained results show that aforementioned modelling does not provide a real estimation of SQNR for input variances [[sigma].sub.x.sup.2] > 5 [dB]. Consequently, modelling should deal with a variance of discretized signal ([[sigma].sub.y.sup.2]) for higher accuracy. In the end, we observe the impact of the input range choice (i.e. parameter k influence) on the system's performance.

Future work will be focused on applying the model to systems used for image processing. Also, we will research the difference of mismatch and proposed modelling for estimation of peak signal-to-quantization-noise ratio (PSQNR) and average bit-rate ([R.sub.b]).

http://dx.doi.org/10.5755/j01.eee.21.3.10380

Manuscript received January 05, 2015; accepted March 16, 2015.

This research is supported by Serbian Ministry of Education and Sciences (Projects III44006 and TR30245). This research was performed in cooperation with Mathematical institute of Serbian Academy of Sciences and Arts.

REFERENCES

[1] J. Nikolic, Z. Peric, A. Jovanovic, "Variance mismatch analysis of unrestricted polar quantization for Gaussian source", IEEE Signal Processing Letters, vol. 21, no. 5, pp. 540-544. [Online]. Available: http://dx.doi.org/10.1109/LSP.2014.2309093

[2] S. Na, "Asymptotic formulas for mismatched fixed-rate minimum MSE Laplacian quantizers", IEEE Signal Process. Lett., vol. 15, pp. 13-16, 2008. [Online]. Available: http://dx.doi.org/

10.1109/LSP.2007.910240

[3] S. Na, "Variance-mismatched fixed-rate scalar quantization of Laplacian sources", IEEE Trans. Inf. Theory, vol. 57, no. 7, pp. 4561-4572, 2011. [Online]. Available: http://dx.doi.org/10.1109/TIT.2011.2146390

[4] S. Na, "Asymptotic formulas for variance-mismatched fixed-rate scalar quantization of a Gaussian source", IEEE Trans. Signal Process., vol. 59, no. 5, pp. 2437-2441, 2011. [Online]. Available: http://dx.doi.org/10.1109/TSP.2011.2112354

[5] S. Na, D. L. Neuhoff, "On the support of MSE-optimal, fixed-rate, scalar quantizers", IEEE Trans. Inf. Theory, vol. 47, no. 7, pp. 2972-2982, 2001. [Online]. Available: http://dx.doi.org/10.1109/18.959274

[6] S. Na, D. L. Neuhoff, "Asymptotic MSE distortion of mismatched uniform scalar quantization", IEEE Trans. Information Theory, vol. 58, no. 5, 2012. [Online]. Available: http://dx.doi.org/10.1109/TIT.2011.2179843

[7] ITU-T Recommendation G.711.1 Wideband embedded extension for G.711 pulse code modulation, 2008.

[8] R. M. Gray, D. L. Neuhoff, "Quantizaton", IEEE Trans. on Information Theory, vol. 44, no. 6, pp. 2325-2383, 1998. [Online]. Available: http://dx.doi.org/10.1109/18.720541

[9] N. S. Jayant, P. Noll, Digital Coding of Waveforms, Prentice Hall Pb, 1984.

[10] Q. Yun, Shi, Huifnag Sun, Image and Video Compression for Multimedia Engineering, Taylor & Francis Group, 2008.

[11] M. Savic, Z. Peric, M. Dincic, "Design of forward adaptive uniform quantizer for discrete input samples for Laplacian source", Elektronika Ir Elektrotechnika, no. 9, pp. 73-76, 2010.

[12] Z. Peric, M. Petkovic, M. Dincic, "Simple compression algorithm for memoryless Laplacian source based on the optimal companding technique", Informatica, vol. 20, no. 1, pp. 99-114, 2009.

[13] M. Savic, Z. Peric, M. Dincic, "An algorithm for grayscale image compression based on the forward adaptive quantizer designed for signals with discrete amplitudes", Elektronika Ir Elektrotechnika, no. 2, pp. 13-16, 2012. [Online]. Available: http://dx.doi.org/10.5755/j01.eee.118.2.1166

[14] M. Savic, Z. Peric, M. Dincic, "Coding algorithm for grayscale images based on piecewise uniform quantizers", Informatica, vol. 23, no. 1, pp. 125-140, 2012.

[15] Z. Peric, M. Savic, N. Simic, "Design and implementation of nonuniform quantizers for discrete input samples", unpublished (submitted to Elektronika Ir Elektrotechnika).

Zoran H. Peric (1), Nikola Simic (1), Milan S. Savic (2)

(1) Department of Telecommunications, Faculty of Electronic Engineering, University of Nis, Aleksandra Medvedeva 14, 18000 Nis, Serbia (2) Mathematical Institute of Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000 Belgrade, Serbia zoran.peric@elfak.ni.ac.rs

TABLE I. AVERAGE SQNR FOR VARIOUS VALUES OF THE BOTH
PARAMETER K AND [N.sub.2] ([N.sub.1] = 256).

                          [N.sub.2] = 16

            [SQNR.sub.av]             [SQNR.sub.av]
       ([[sigma].sub.x.sup.2])   ([[sigma].sub.y.sup.2])
k               [dB]                      [dB]

1             11.00236                  12.60121
0.9           11.04911                  12.55164
0.8           11.10471                  12.49266
0.7           11.10435                  12.35924

                         [N.sub.2] = 32

            [SQNR.sub.av]             [SQNR.sub.av]
       ([[sigma].sub.x.sup.2])   ([[sigma].sub.y.sup.2])
k               [dB]                      [dB]

1             15.60852                  17.66160
0.9           15.74937                  17.67307
0.8           15.72813                  17.49095
0.7           15.37577                  16.98225

TABLE II. AVERAGE SQNR FOR DIFFERENT SUB-RANGES OF INPUT SIGNAL
VARIANCE IN THE FUNCTION OF PARAMETER K ([N.sub.1] = 256;
[N.sub.2] = 32).

                 [[sigma].sub.x.sup.2][member of] [-10,20] dB

k              [SQNR.sub.av]                  [SQNR.sub.av]
       ([[sigma].sub.x.sup.2]) [dB]    ([[sigma].sub.y.sup.2]) [dB]

1                16.39667                        19.09979

0.9              16.11152                        18.64390

0.8              15.63079                        17.95074

0.7              14.99498                        17.10880

                 [[sigma].sub.x.sup.2][member of] [-5,15] dB

               [SQNR.sub.av]                  [SQNR.sub.av]
       ([[sigma].sub.x.sup.2]) [dB]    ([[sigma].sub.y.sup.2]) [dB]

1                18.25484                        20.04838

0.9              17.70619                        19.42230

0.8              16.92263                        18.51965

0.7              16.00386                        17.48337

                [[sigma].sub.x.sup.2][member of] [-20,10] dB

k              [SQNR.sub.av]                  [SQNR.sub.av]
       ([[sigma].sub.x.sup.2]) [dB]    ([[sigma].sub.y.sup.2]) [dB]

1                18.09774                        18.44468

0.9              18.53039                        18.88257

0.8              18.75263                        19.09579

0.7              18.53101                        18.86237

                [[sigma].sub.x.sup.2][member of] [-15,5] dB

               [SQNR.sub.av]                  [SQNR.sub.av]
       ([[sigma].sub.x.sup.2]) [dB]    ([[sigma].sub.y.sup.2]) [dB]

1                20.09190                        20.13460

0.9              20.67117                        20.72220

0.8              20.98754                        21.04420

0.7              20.86813                        20.93039

                 [[sigma].sub.x.sup.2][member of] [-10,10] dB

k              [SQNR.sub.av]                  [SQNR.sub.av]
       ([[sigma].sub.x.sup.2]) [dB]    ([[sigma].sub.y.sup.2]) [dB]

1                20.44655                        20.94060

0.9              20.38928                        20.89165

0.8              20.04918                        20.53874

0.7              19.47140                        19.94448

                 [[sigma].sub.x.sup.2][member of] [-5,5] dB

               [SQNR.sub.av]                  [SQNR.sub.av]
       ([[sigma].sub.x.sup.2]) [dB]    ([[sigma].sub.y.sup.2]) [dB]

1                22.70497                        22.77518

0.9              22.68575                        22.77255

0.8              22.30115                        22.39910

0.7              21.62943                        21.73864
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Author:Peric, Zoran H.; Simic, Nikola; Savic, Milan S.
Publication:Elektronika ir Elektrotechnika
Article Type:Report
Date:Mar 1, 2015
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