Analysis and Design of Two Stage Mismatch Quantizer for Laplacian Source

Two stage quantization is a well-known but still very popular model for signal processing. However, in a number of occasions we have information about a discrete entrance and we do not know the nature of the continuous signal which preceded it. Hence, information source is commonly modelled by using Laplacian or Gaussian distribution but designed quantizers often do not match entire signal range. A typical analysis for discretized input signal does not consider the changes of the continuous signal variance. The aims of this paper are providing an improved analysis by introducing a novel measure CDSVR, designing the second stage quantizer, as well as estimating system performance for mismatched variances. This way, we discuss the influence of A/D conversion on the signal variance and propose an improved model for performance estimation. DOI: http://dx.doi.org/10.5755/j01.eee.21.3.10380


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 (-∞, ∞).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 N1-level quantizer with Q1 and second N2-level quantizer with Q2 (N1 > N2).Quantizer Q2 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 (Q1 and Q2) 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 where is  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 Q1 converts analog signal to discrete samples, whereas Q2 performs additional data compression by using nonuniform quantization in the second stage.Thereby, decision thresholds and representational levels of Q1 can be defined with [11] where 0,..., / 2 i N  .
where 1,..., / 2 i N  . In ( 2) and ( 3) xmax 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 between continuous signal variance of original signal and variance of discretized signal after processing with Q1 we introduce a novel measure: continuous -to -discrete -signal -variance ratio (CDSVR).It is defined with In previous equation x 2 represents the variance of continuous signal whereas y 2 is the variance of discretized signal, obtained after processing with Q1.These two parameters can be defined with [9]: where reff 2 is the referent variance.Furthermore, Pi are probabilities of discrete input levels of the second quantizer where is for calculating y 2 in (6).
Design of the nonuniform quantizer Q2 is done in two steps.Firstly, we design the optimal compandor with N2 quantization levels for the unit standard deviation (σ = 1).After that we discuss range variations by introducing the parameter of proportionality k.Compandor's compressor function maps the range (-, ) to (-1, 1) and it can be defined with [12] .
Its decision thresholds and representational levels obtained in this way are defined with [12]: ' 2 )exp where is .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 Q2 are ' , where 2 1 j N   .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]: In (15) parameter ri denotes the number of input levels mapped with i whereas yij are output levels of Q1.Furthermore, 2 max N    in (16) whereas parameter s represents the number of output levels from Q1 that are not within designed input range of Q2.
In the end, total distortion is equal to .
Beside CDSVR, overall system performance will be measured by using SQNR that represents a common measure of performance [12], [14] .
Numerical results will be calculated for the standard model (SQNR(x 2 )) as well as for the proposed model (SQNR(y 2 )).

III. NUMERICAL RESULTS
In Fig. 1 is shown CDSVR in function of x 2 .Observing Fig. 1 we can conclude that values of y 2 do not match corresponding values of x 2 .Their difference, or in another words a CDSVR value, increases with increasing the continuous variance of input signal x 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 N1, that can be regarded as a reducing the loss of information occurred due to A/D conversion using Q1.Consequently, we can expect that proposed modelling that uses y 2 will achieve higher SQNR (SQNR(y 2 )), as it was confirmed experimentally [15].
In Fig. 2 and Fig. 3 the overall SQNR depending on various values of N1 in function of x 2 is shown.SQNR value is calculated in two ways -using the continuous variance of the input signal x 2 (SQNR(x 2 )) and using the variance of discretized signal y 2 (SQNR(y 2 )).
It can be noticed that system's performance calculated in both ways have good matching in the range of small variances (x 2 < 5 [dB]).With increasing the variance x 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 Q1 (SQNR(y 2 )).
In Fig. 4 and Fig. 5 SQNR depending on parameter of proportionality k for fixed N1 and various numbers of representational levels N2 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 x 2 it is obtained higher SQNR for lower k).
On the other hand, for x 2 >5 [dB], SQNR value decreases by decreasing parameter k.Changing this parameter affects the both SQNR(x 2 ) and SQNR(y 2 ) in the same way.In Table I average SQNR (SQNR(x 2 ) and SQNR(y 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 x 2  [-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 SQNRav of the observed system.
Table II shows average SQNR (SQNRav(x 2 ) and SQNRav(y 2 )) for various sub-ranges of input continuous variance (x 2 ) in function of parameter k.We can conclude that parameter k affects overall SQNRav 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 (x 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 x 2 > 5 [dB].Consequently, modelling should deal with a variance of discretized signal (y 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 (Rb).

TABLE I .
AVERAGE SQNR FOR VARIOUS VALUES OF THE BOTH PARAMETER K AND N2 (N1 = 256).

TABLE II .
AVERAGE SQNR FOR DIFFERENT SUB-RANGES OF INPUT SIGNAL VARIANCE IN THE FUNCTION OF PARAMETER K (N1 = 256; N2 = 32).x