A cost function for the equalization of constant amplitude signals based on statistical reference

A Cost Function for Constant Amplitude Signals based on Statistical Reference tt .losep Sala-Alvarez Department or Signal Theory and Communications (GPS) elliversitat Politecnica de Catalunya 0/ Gran Capita sin, Modu! D5 08034 Barcelona, Spain Tel, +34-3-401 64 40 ;Fax: +34-3-401 64 47 E-mail: [email protected] type schemes. An alternative cost function as described in [3] can be used. This cost funct.ion, which we reproduce here below, always leads to minimization of tile Kullback-Leibler information measure, Its utilization is precluded here because the target constant amplitude distribution is not discrete but continuous. The evaluation of the outer expectation, 15 AQ is thus difficult to attain as it involves the pvaluation of integrals dependent on the data. For the casc of discrcte distributions, this can be easily undert.aken as the integrals reduce to summations. Back to (3) and using the constant amplitude property of this cost functioIl, we arrive at the final expression, ABSTRACT The equalization of constant amplitude signals is considered in the scope of this paper. Acriterion based Oil tile probability density function (pdf) of the signal of interest is proposed. The objective is to derive asuitable soft-decision scheme, more robust than the classical C1IA algorithm that ensures r€covcrability of the signal. I Introduction Constant amplitude signals are widely used in digital cOIllImmicatiolls for the equalization of phasemodulated signals. In CMA algorithms one tries to make use of the constant amplitude property of the signals of interest. The most straightforward way is to utilize acost function bWled on amplitude errors. Here we propose one based on the probability density function of the signal of interest. In this way we use all aprioristical knowledge we have of the wanted distribution. We will show in terms of the error function of the adaptive algorithm that this is more robust than dassical CMA. { I(, ') (2A I)} J(w) =Ez ui Izi +A-ln 10 a~ Iz (5) where 1 00 stands for the modified Bessel function of the first kind and order zero. We can sec from the power series developmcnt of 1 0 (-) that the behaviour of Jfor Izi close to Ais similar to the following error norm, (6) (7) () -'II' 'I' .len", w - l!.,'z Iz -A 1 , J(w)", ,E z (1zl ~ A) ", The conventional CMA cost funeti'lll is denoted instead as, The robustness of this algorithm can be justified when we observe its behaviour for large values of the input data modulus, 14 The classical CMA function is biquadratic. Hence, its gradient behaves as the third power of the data. This lIlay easily lead the algorithm to divergence if the coefficient vector is far from its optimum setting. When the gradient of the classical CMA algorithm is normalized, we get an error function very close in shape to tbat obtained from statistical reference. The pdf of a ronstant amplitude signal is giycn in the followinll; terms in the complex plane C, I rAJx) ~ 2rrA o(lxl ~ A) (I) It is chosen to maximize the following niterion, 2 Statement 12 _ '" J(w) = -Hz ln EAo -., e1ja,I'-"QI (3) 1rai where we see that the cost function can be expressed ill tenus of anon-linear average of the error. The utilization of t.his cost function always leads to soft-decision J(w) = -EzInpAa+ N,(wHx) (2) where some uncertainty is modeled via trw AWGN term N1= i\T(O, ai). '0./e will see that this term is important for the performance of the adaptive algorithm. The subscripted expectation operator describes with respect to which random variable the expectation of the nonlinearity is realized. The final exprcssiOlI is given by, 3 Derivation where we have defined aregeneration function as, The coefficients are updated accorclillg to the gradient rule. The gradient of the cost function is calculated with respect to the Hermitian of the coefficient vector, ( 16) For the case of constant amplitude signals, this rege· neration function is expressed as, (8) (9) (17) If we define agencralized error function from the quality functions, the gradient can be exprcssed as follows, The appearance of the generalized error for the C::vlA case appears depicted in figure (3). Using thc following property, wc can express (11) in amore intuitive way. with 11 (.) the modified Bessel function of the first kind and order 1. Note that the regeneration function only depends OIl the modulus of the input data, Izl, which stems from the symmetry of the wanted distribution. The behaviour of this cost function is such as depicted in figure (I). It preserves the phase of the incoming vector, modifying only its modulus. Small values of jzl with respect to Alead the regeneratiou fuuction close to zero. Saturation is observed for those values larger than the reference amplitude A. It is useful now to compare the behaviour of the regeneration function with respect to that of the generalhed error. There exists one value of the amplitude of zfor which the generalized error goes to zero with positive slope. Around this point 0:0< A., the adaptive algorithm is at equilibrium. Note that the slope of the generalized error is linear in the modulus of z, while that of classical CMA is cubic. This causes that classical CMA be more sensitive. Asmaller stepsize must be used to keepciassical CMA from diverging, which is reflected in aslower convergence rate. Abehaviour similar to that of the generalized error can be obtained whcn the step-size is normalized to make the modified gradient linear with the modulus of the input signal. (12) (10) In the following, it will be understood that by Ez we mean that the expectation is carried not only with n:~spect to the random variable z, but with respect to all random variables depending all the data vector x, such as z=wHx. This last equation can be re-formulated in the follo""ing fashion, where the quality functions q(.,,) are defined, ll. c-l/<T;lz-<l~I~ '1 (z, ao)= ~---,-;-c';CC--:CloO EA~e If<T,lz <l~ These functions give an indication of how close one value of the input sequence is to the wanted distribution. The gradient is then exprcssed in terms of these functions as, EAuq(z,a o)=1 Then, the gradient becomes, (13) • l/a} Ez(z - Ekuoq(Z, ao))~ x (14) IjalE z (z-ii o )"x (15) Figure 2: Generalized error functions. <4 Adaptive Algorithm Figure LCMA cost functions. The adaptive algorithm is based on gradient techniques. The expect.ation operator Ez is dropped so that the averaging is done implicitly in the mefficient update equations, iftr.p step-size is small euough. The upgrade equations are defined as, 5 Annex. Derivation of the cost function Figure J: Generalized error functions for several ai's. (21 ) (20) (22) (26) (27) x2jj +xf; _(x2+n2)y = () Y=c1Jn(x) + C2 Kn(X) A(x) = I: e xcos8 d8 whose first and second derivatives arc expressed as, where in(x) and Kn(x) stand for the Bessell1lodified functions of the first and second kind, respectively. If we particularize for n=0 and divide by x2on both sides, we obtain (2S). It only remains now to calculate what the constants of the linear combination are. \Ve know that A(O) = 27f. Hence, given that Ko(O) tends to 00 and that 10(0) = I, we must have that Cl = 27f ami that C2 = O. Substituting the value of x, the cost function is elJual to that in equation (5). 6 Results A (.or) + I/xA(x) = I: (sin28+ cos~ B) e'"cu' Bri8 =A(;r) (25) Compared with the Bessel modified differential equation, A(x) =x I"" sin~Bexcm;l!d8 (24) Therefore, we get that Afulfils the followillg differential equation, A(x) = I: (;Qs8e.",,,u~Bd8,ii(x) = I: cos~ee;£wsade, (23) Integrating the first derivative by parts, it is easy to show that, From now on we will define the variable x= 2A/af. Placing the expectation operator in terms of the pdf of 04 0,we get, In the followinF; figures we compare the constellation obtained "rith the Cr .... lA algorithm and the Statistical Reference algorithm. Thf~ classical CMA algorithm usually shows unstability associated with it when the amplitude at the output of the equalizer, in the initial stages of acquisition is very dissimilar from the target amplituue. Very small step-size must be used to guarantee that the algorithm will not diverge. This is dile to tbe fact that the F;radient of the cost funetion is not (19) (18) ( IZII, (~lzl)) z· 1 0 (~lzl) Izi W/c - w*(z)x The choice of the appropriate a; parameter has important consequences On the convergence rate. To guarantee acquisition, the tentative variante parameter must not be chosen too small, so that the gradient does HOt deliver zero values when the valul:'~s of the data amplitudes are far from tbe wanted amplitude A. 1t. is important that the value of the tentative variance be itpproximately mat.ched t.o the noise variance at tlw system omput when cOllvergence has been readlCd. It is also possiLJle to use different variances for acquisition and tracking, but in general, we have fuund that. f(~aso nable choices of a; already guarantee fast convergence and good performance in tracking. The generalized error fUIlet.ions corresponding to several values of the tentative vari,mce have been represented in figure (3). Note that tbe slope of the error function deviates progressively from linearity in the lower amplitude range as the value of at increases. Also, the zero cross-point is shifted leftwards. This last effect is known as Constellation shrinkage. The more linear the error functiOll is, the more reliability we place on the data. In the statement of the rlassical CMA cost function, its associated error is not in alinear relationship with the amplitude error. This causes the usual problems of divergence for dissimilarities between the actual and the target amplitude. l'vlodificatiolls of the errOr function to make it ITlOTe IiIlear will always improve the behaviour of the algorithm. In t.his annex, we will calculate the closed expression for the cost function in the case of constant amplitude signals. Let us consider t.he argument of the natural logarithm (the pdf) and operate with the exponent of the Gaussian. linear with the true amplitude error. It behaves ills~ tead as the third power of the amplitude. Although this makes the C.\1A algorithm ~tablc, its cOllvergence rate is comparatively very large. In the Statistical Reference algorithm instead, comparatively larger step sizes can be used without making the algorithm ~table. The convergence rate is thus much faster. The gradient provides here 'true' undistorted estimates of the amplitude error. In many of the simulations we have carried out, it has not. been possible to make the convergence rate of classical C1JA equal that of statistical reference for large amplitude dissimilarities with respect to the target amplitude. Simulations with the Statistical Reference Algorithm are shown in figures (4) and (5). The following values have been chosen for the simulations: SNR =14dB, It =0.0007, number of coefficients of the fractionally-spaCf~d equalizer N=30 (around Ssymbols), channel response =[-0.1274,0.5542,-1.0073,- 0.7313,1.4047,-0.6202,0.2371,-1.5868,-0.4015,-0.7707]. Figure 4: Statistical Reference. Evolution of the inphase channel ,J>:':' .. ~~. ;-_ .... \~y D~ , - • statistical referellce C:vl A, this is 110t uccessary. If actual amplitude do llot differ much and are close to the target amplitude, performance of both algorithm is also similar. Aslif';htly higher misadjustment can be ohserved in C!",lA due to the non-lim-mr characteristic of the error under the same test conditions for both algorithms. References [1] Sala, J. 'Criterios de Teoria de la Informacion en Procesado Adaptativo de la Senal yAplicaciones'. PhD. Thesis. Dpt. of Signal Theory and Communications. Polytechnic University of Catalonia. June 1995. [21 Sala, J. Vazquez, G'Adaptive Blind equalization and Demodulation -without Channel alld Signal Parameter Extraction'. Proceedillgs of EUSIPCO'94. Edinburgh, 13-16 September [994. [31 5ala, .1. Vazquez, G .. 'A Statistical Reference Criterion for Adaptive Filtering'. Proceedings of the ICASSP'96, Atlanta, USA. 1996. [4] Nowlan, Steven .1.; Hinton, Geoffrey. 'A SoftDecision Directed U .... 1S algorithm for Blind Equalization'. IEEE Transactions on Communications. VOL. 41, No.2, February 1992. [5] Sala, J. Vazquez, G.'ACost Funcion for Blind Sigi1al Recovery based on an Implicit Cumulant Expansion'. Proceedings of the Joint IEEE Workshop on Higher Order Statistics. Begur, 1995. Spain. [6] Blahut, Richard E. 'Principles and Practice of Information Theory'. Adclison-Wesley 1987. [7] Haykin, S.. 'Adaptive Filter Theory'. Prentice-Hall Inc. 1991. " -1,5 .,L-,C--c---cC--c---c:'-:---c:'-! ~i _IS _D,S 0,5 IS Figure 5:·Statistical Reference. Output constellation. The convergence region of CMA iC! limited. If the amplitude of the signal is very dissimilar from the target amplitude, power normalization must be carried out to ensure convergence. Due to t.he special characteristic of