9 Facta Universitatis ser.: Elect. and Energ. vol. 11, No.3 è1998è this paper we have considered shaping gain for two interesting quantization procedu

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1 FACTA UNIVERSITATIS èniçsè Series: Electronics and Energetics vol. 11, No.3 è1998è, NONLINEAR TRANSFORMATION OF ONEíDIMENSIONAL CONSTELLATION POINTS IN ORDER TO ERROR PROBABILITY DECREASING Zoran Periçc Abstract. This paper presents the original gradient method for the determination of the optimal signal constellation points disposition after quantization, when the error probability is minimized under the power constraint. This method consists of the nonlinear transformation and rescaling. The method diæers from the method presented in ë1ë, which can be used for uniform constellations with great numbers of points èwhere the continual approximation can be appliedè, since it can be applied for generating the nonequiprobable nonuniform constellations containing any number of points. 1. Introduction The paperë1ëis concerned with the performance of nonlinear encoding in a Gaussian noise environment èwhere the equiprobable constellation points are discussed è.it is interesting to compare scaling gain with shaping gain obtained by nonequiprobable signaling on the Gaussian channel. One cost of shaping gain is constellation expansion. The optimal nonlinear encoder is not the one that results in Gaussian distribution at the channel input. One similarity with shaping is that the warped and rescaled constellation ènonlinear transformation constellationè has been expanded. In ë1ë, the optimal nonlinear transformation èwarping transformationè is done using the continuous approximation for representing a signal constellation. The input distribution that achieves the capacity of the Additive White Gaussian Noise èawgnè channels with an average power constraint is also Gaussian. This paper considers the performance of nonequiprobable nonuniform oneídimensional signal constellations, obtained by shaping and nonlinear transformation when used on the Gaussian channel. In the ærst part of Manuscript received Jule 14, The author is with Faculty of Electronic Engineering, Beogradsa 14, Niçs, Yugoslavia, peric@elfa.ni.ac.yu. 91

2 9 Facta Universitatis ser.: Elect. and Energ. vol. 11, No.3 è1998è this paper we have considered shaping gain for two interesting quantization procedures of the Gaussian source. The Gaussian source is quantized applying the method given in ë3ë èthe ærst methodè, which gives the equiprobable appearance of signal constellation points. After that, the same source is quantized using the Lloyd-Max's iterative quantization method ë4ë èthe second methodè. In the second more important part of the paper a new gradient method for determining the conditional minimum of the average error probability per symbol under the average power constraint is presented ènonlinear transformation constellation pointsè. Using this method, it is possible to determine the optimal arrangement of constellation points, under the average power constraint, for transmission over the Gaussian channel. This signal constellation transformation method is independent ofthe quantization method and information source, and adapts the given source signal constellation to the transmission channel.. Shaping gain determination for diæerent quantization method The saving in average power over equiprobable transmission using a oneí dimensional constellation is referred to as the shaping gain. The maximal shaping gain is achieved when the Gaussian source is applied. In this section, a Gaussian source quantization, using two diæerent methods of nonlinear quantization, is performed. The analysis is done on the basis of the shaping gain. The Gaussian source is quantized applying the method given in ë3ë, which gives the equiprobable appearance of signal constellation points. After that, the same source is quantized using the LloydíMax's iterative quantization method. The numerical analysis is done for the sixteen points of a oneídimensional signal constellation. Using the ærst method èiè, the decision levels èr è and reconstruction levels èm è are determined by: 1 L = p 1 r+1 Z x exp, çp P dx r è1è m = p L r+1 Z ç exp ç, x dx; =1; ;::: ;L; èè çp P r where L is a number of levels and P is an average power.

3 Z. Periçc: Nonlinear transformation of oneídimensional The second method èiiè is Max's iterative quantization which leads to r and m given by ë4ë: r ;opt = 1 èm ;opt+m,1;optè; =; 3;::: ;L; r 1;opt =,1; r L+1;opt = 1; m ;opt = r +1;opt R r ;opt R r +1;opt r ;opt rp r èrèdr è3è ; =1; ;::: ;L è4è p r èrèdr where p r èrè is a probability density function of the source. The shaping gain èg S è is deæned as diæerence between SNR èsignal to noise ratioè for nonuniform constellation èsnr n è, and SNR for uniform constellation èsnr u è èsimilarly as in ë5ëè, when the error probabilities èp e è are equal and the bit rates are almost equal: G S èp e è=snr u èp e è, SNR n èp e è è5è The shaping gains for the previous two quantization methods are shown in Fig. 1. Figure 1. Shaping gain èg S è for two diæerent quantization methods.

4 94 Facta Universitatis ser.: Elect. and Energ. vol. 11, No.3 è1998è The shaping gain depends very much on the quantization method choice. From Fig. 1. it is evident that the ærst method has the negative shaping gain value in the wide range of the error probability, so in this case it is better to use the equiprobable transmission with uniform constellation. In the next section we will show how to compensate the negative shaping gain using the appropriate choice of nonlinear transformation and rescaling. 3. Algorithm for determination of optimal signal constellation points disposition The problem of minimizing the average error probability èp e è under the average power constraint is the nonlinear programming tas where the goal function is P e and constraint isp èaverage powerè. The average error probability and power of oneídimensional signal constellation can be calculated as 3 LX Zy ;l P P e 6, r Z+1, r = p e çn dr + e ç 7 n dr ççn 5 ; P = LX P m ; 4,1 where çn is an average noise power, m is the íth representation level, L is anumber of levels P = Z r +1 p r èrèdr; y ;r è6è r p r èrè is a probability density function of the Gaussian source, and r are the decision levels. y l and y r are deæned in the following way èsee Fig..è y l = m, m,1 y 1l =,1; ç n + ln m, m,1 P P,1 ; =1; ;::: ;L; y r = m +1, m y Lr =+1: + çn ln P ; m +1, m P +1 =1; ;::: ;L; In our case the variables in the goal function are mutual dependent, so the solution in closed form using the Lagrange multiplicator method is impossible.the problem is in fact convex programming problem ëë, in which

5 Z. Periçc: Nonlinear transformation of oneídimensional P e is a convex function and inequality constraint is a convex function èthe constraints form a convex setè. Using the gradient method it is necessary to begin from the starting point M 0 èm è0è 1 ;mè0è ;::: ;mè0è è and to go towards the optimum M L i in each iterative step. The next value of m is determined by m èi+1è = m èiè + çèiè hèiè, where m èi+1è and m èiè represent the íth representation point at the i + 1íth and iíth iterative step, respectively. ç èiè is the weight coeæcient, and h èiè is a step. Figure. Oneídimensional signal constellation and a decision region example. The algorithm for determination of nonequiprobable nonuniform constellation points disposition is as follows: 0è step: The initialization. The point disposition obtained by described quantization methods application is used as the starting values. 1è step: The average error probability gradient rp e e =, P p ç n ç, y l ç n ; =;::: ;L; where æ = y,1;r + y l. è step: Increasing i.e. decreasing distance between two neighbouring points èæ è implies shifting the íth point tolíth point for some small value æh 1 in order to eep the relation among remaining points constant m ;èiè = m èiè +h 1 ; m ;;èiè = m èiè,h 1 ; = ; +1;::: ;L; =1; ;::: ;L: 3è step: The calculation of the changed power values due to æ value

6 96 Facta Universitatis ser.: Elect. and Energ. vol. 11, No.3 è1998è variation X L P ;èiè = av P èm ;èiè è ; X L P ;;èiè = av P èm ;;èiè è ; =1; ;::: ;L: 4è step: The coeæcients ç èiè è = 1; ;::: ;Lè are chosen so that the distances among the constellation points decrease i.e. increase, proportionally to the speed of error probabilitychanging and the power changing ç èiè e P 0 èiè, P av e P 0 èiè av, P av NP + NP P "èiè av, P P "èiè av, P : 5è step: The calculation of new signal point position value m èi+1è = m èiè + ç èiè hèiè, where h èiè is monotonously decreasing function of the ordinal number iterative step. 6è step: The determination of scaling constant, and based of this m èi+1è rescaling value çp èi+1è av = P av ; m èi+1è rescal = p çm èi+1è ; =1; ;::: ;L: 7è step: If condition max ç èiè æ hèiè ç"; where " is required to be accurate, is satisæed the procedure is ænished or if this condition is not satisæed rescaled values are set as the new signal points position values m èiè = m èi+1è rescal ; = 1; ;::: ;L and step 1è is applied. The nonlinear transformation gain is deæned as a diæerence between SNR before èsnr before è and SNR after èsnr after è the nonlinear transformation ègè of the signal constellation, when the error probabilities èp e è are

7 Z. Periçc: Nonlinear transformation of oneídimensional equal GèP e è=snr before èp e è, SNR after èp e è = 10 log P av ç, 10 log P av opt çopt è! P av çopt = 10 log P av opt ç = G scaling + G n ; è7è where G scaling =10 logèp av =P av opt è is a scaling gain and G n =10 logèç opt=ç è is a gain due to the nonlinear transformation, which corresponds to the deænition given in ë1ë èg = G scaling + G n è. The error probability decreasing obtained by using the proposed method, for the ærst quantization method, is illustrated in Fig. 3. Figure 3. Error probability per symbol before and after nonlinear transformation and rescaling for the ærst quantization method. The gain obtained by the nonlinear transformation and rescaling of the signal constellation points initially generated by the ærst and the second quantization method of the Gaussian source are shown in Fig. 4. The signal constellation design is done separately for each SNR value, and the constellation having the maximal gain is obtained. The aim of paper ë1ë is the analysis of nonlinear transformation named warping transformation of the uniform constellation for equiprobable trans-

8 98 Facta Universitatis ser.: Elect. and Energ. vol. 11, No.3 è1998è mission where the solution is obtained using the continual approximation. An increase in nonlinear transformation gain with a decrease of SNR for equiprobable constellations may be observed as in ë1ë. On the contrary, with nonequiprobable nonuniform constellations, the gain decreases with SNR decreasing. The total gain is equal to the sum of the shaping gain and gain caused by the nonlinear transformation and rescaling. Figure 4. Gain ègè after nonlinear transformation and rescaling. 4. Conclusion The original gradient method for generating optimal signal constellations minimizes the error probability under the power constraint.the method consists of the nonlinear transformation and rescaling. This signal constellation transformation method is independent of the quantization method and information source, and adapts the given source signal constellation to the transmission channel. Also, we presented a procedure for accurate calculation of shaping gain for any signal constellation. REFERENCES 1. W. Betts, A.R. Calderban and Raiv Laroia: Performance of Nonuniform Constellations on the Gaussian Channel. IEEE Transaction on Information Theory, Vol. 40, pp. 1633í1638, September 1994.

9 Z. Periçc: Nonlinear transformation of oneídimensional J.J. Petriçc, S. Zlobec: Nonlinear Programming. Nauçcna niga, Belgrade, èin Serbianè. 3. F. Sun and H.C.A. Tilborg: Approaching Capacity by Equiprobable Signaling on the Gaussian Channel. IEEE Transaction on Information Theory, Vol. 39, pp. 1714í1716, September N.S. Jayant, P. Noll: Digital Coding of Waveforms. Englewood Cliæs, NJ: PrenticeíHall, F.R. Kschishong, S. Pasupathy: Optimal Nonuniform Signaling for Gaussion Channels. IEEE Transaction on Information Theory, Vol. 39, pp. 913í99, May 1993.