Chen, S.

Liu, Y.

Luk, B. L.

English

Southampton (e-Prints Soton)

References 1  GOODMAN, D.J.,  HENRY, P.s.,  and  PRABHU, v.K.:  ‘Frequency-hopped multilevel FSK for mobile radio’, Bell Syst. Tech. J., 1980, 59, (7), pp. 1251-1275 2  YAMAMURA, H.,  KOHNO, R.? and IMAI, H.: ‘Performance of multihop fast frequency hopping/spread spectrum multiple access system in a selective fading channel’, IEZCE Trans.,  1991, E-74, (5), pp. 1145- 1153 3  PARK, c., and LEE, J.-H.:  ‘Dual FHiMFSK  system over a  Rayleigh fading channel’, Electron. Lett., 1996, 32,  (20), pp. 1852-1854 4  SCHWARTZ,  M.,  BENNETT,  W.R.,  and  STEIN,  s.:  ‘Communication systems and techniques’ (McGraw-Hill, 1966) 5  VITERBI, A.J.:  ‘A processing satellite transponder for multiple access by  low rate mobile users’. Digital Satellite Commun. Conf., 23-25 September 1978, Montreal, pp. 16C174 Application of adaptive simulated annealing to blind channel identification  with HOC S. Chen, B.L. Luk and Y. Liu A higher  order cumulant (HOC) fitting approach is  a powerful technique for blind identification of no”m phase channels. Owing to the fact that HOC cost functions are multimodal, global optimisation  techniques  offer  the  important  advantage  of avoiding local minima. An  adaptive simulated annealing (ASA) algorithm  is  applied  to  optimise  an  HOC  cost  function.  A simulation study demonstrates that this ASA is  accurate and has a  fast  convergence  rate  compared  with  standard  simulated annealing (SA). Introduction: An important class of blind identification techniques for  nonminimum phase channels is  based  on  the  higher  order cumulant (HOC) fitting method [l, 21.  HOC cost  functions are, however,  multimodal, and  conventional gradient methods  [3,  41 may converge to give ‘wrong’ solutions. To overcome the problem of  local minima, a simulated annealing (SA) algorithm has been applied to optimise an HOC cost function [5]. Although SA repre- sents a general global optimisation technique, it suffers from the serious drawback of  being too slow. An  improved version of SA, known as adaptive simulated annealing (ASA) [6, 71,  provides a significant improvement in convergence speed. We use this ASA algorithm for blind channel identification. Simulation results con- firm that this ASA achieves faster convergence in the HOC fitting process. We will assume a real-valued channel and a PAM symbol con- stellation. The channel is modelled as a finite impulse response fd- ter with an additive white Gaussian noise: n, [6, 71  is a very fast version of  SA. We apply ASA to optimise the cost function (eqn. 2). A flow chart of the algorithm is depicted in Fig.  1. I  I  initialisation . JI- I generate a new & accept or reject 6 I  I ?-I1 reannealing I  I Fig. 1 Flow chart of adaptive simulated annealing In the initialisation, an initial i3  is randomly generated; the tem- perature of the acceptance probability function T, is set to the ini- tial  value  of  the  cost  function,  and  the  temperatures  of  the parameter generating probability functions, T,  0 2 i 5 i,,  are set to 1.0. Two control parameters in annealing, the coefficient c and the quenching factor Q, are also given, and the annealing times, k, for 0 2 i 2 ria and k,, are all set to 0. The algorithm generates a  new  point in  the parameter space with: Lire, =  &;ld  +  yZ(B, -  A,)  and  Lire,  E [A  -  2, B -  21 for 0 5 i 5 6, (3) where A, and B, are the lower and upper bounds for a,, respec- tively; yz  is calculated as Z=O Blind identification refers to the determination of  the channel a = [ao  a, . . a,J  using only the noisy received signal {~(k)}  and some knowledge of the statistical properties of s(k). Method: In HOC fitting, the channel model is  obtained by  opti- mising  an HOC cost function. The fourth-order cumulant cost function adopted in our application is defmed by /  min{i?,,+i.-T}  \2  and  U, is  a uniformly distributed random variable in  10,  I]. The new point iW  is accepted or rejected according to the acceptance probability function where k4?(7,  T,  T)  is the time estimate of  the diagonal slice of  the fourth-order cumulant for r(k),  is the kurtosis of s(k), ria is an estimated channel length, and L is the channel estimate. HOC cost functions are well known as being multimodal, and gradient optimisation methods may  fail to work. Using  a global optimisation method, such as SA, has the advantage of guarantee- ing that a global optimal channel estimate will be found. Global optimisation algorithms however, usually, suffer from the draw- backs  of  slow  convergence  and  high computational cost. ASA After  every  N,,,,,  acceptance point,  the  algorithm  performs reannealing by rescaling k, according to the sensitivity aJ/aa, and resetting k, to the current optimal value of  the cost function (see [6,  71  for  details). The  derivatives  dJ/da, are actually calculated using the fist-order difference approximations. After every N,,,,,, generated points, annealing takes place with k, = k, + 1 for 0 5 i 5 ria and k, = k, + 1: 234  ELECTRONICS LETTERS  5th February 1998  Vol. 34  No. 3 T,(ka)  =  T,(O)exp(-ckp'(''""))  0 5 r 5 6,  (6) and T&)  = TJO)  exp,(-c:'('+fi"))  (7) The  algorithm  is  terminated  if  either  the  parameters  have remained unchanged for a few successive reannealings, or a preset maximum number of function evaluations has been reached. Table 1: Blind identitication results averaged over 100 runs n, = 4 4.21 4.50 0.72 0.36 0.21 - I  True  I  Estimate (mean f  standard deviation)  I ?io = 5 4.2090 f  0.0441 -0.5042 f  0.0305 0.7186 f  0.0164 0.3708 f  0.0441 0.21 19 k  0.0409 4.0084 f  0.0574 Results:  The above ASA algorithm was used in the blind identifi- cation of the following channel with 8-PAM data symbols and a signal-to-noise  ratio of 20dB. In practice, the true channel length n, = 4 is unknown and, therefore, an estimated  Vi,  = 5 was  assumed  in  simulation. Figs.  2 and  3 show the evolutions of the cost function and the mean tap error (MTE), defined as MTE = 111  -  allz, averaged over  10 different ms,  respectively.  Table  1  summarises  the  blind  identification results. It can be  seen that ASA achieved convergence after 8000 function  evaluations.  The  conventional  SA  employed  in  [5] required > 20000 function evaluations. a  = [-0.21  -  0.50  0.72 0.36  0.21IT  (8) 4 10 c 0  .- L 03 c  5 10 c 8 t  1 2 10 2600  4600  6600  8dOO  lOb0  12000  14hOO number of function evaluations Fig. 2  Cost function against number of function evaluations averaged over  IO runs 0 10  ,  I  I  I L a,  E IO' a E!  t:: c E -3 10 0  2000  4000  6000  8000  10000  12000  14000 number of function evaluations 197213 Fig. 3 Mean tap error against number of function evaluations averaged over IO runs ELECTRONICS LETTERS  5th February 1998  Vol. 34 Conclusions:  An ASA  algorithm has been  implemented for blind identification of  nonminimum phase channels based on the HOC fitting approach. ASA is a global optimisation method that is con- siderably faster  than  standard  versions  of  SA.  Our  simulation study has confirmed that ASA offers a viable global optimisation tool for obtaining an accurate channel estimate using only noisy received signals. 0  IEE 1998 Electronics Letters Online No: 19980235 S. Chen, B.L. Luk and Y. Liu (Department of Electrical and Electronic Engineering,  University of Portsmouth, Anglesea Building, Portsmouth POI 3DJ, United Kingdom) 22 October 1997 References MENDEL, J.M.:  'Tutorial  on higher-order statistics (spectra) in signal processing  and  system  theory:  theoretical  results  and  some applications',  Proc. ZEEE,  1991, 79, (3), pp. 278-305 NIKIAS, c.L., and PETROPULU, A.P.:  'Higher-order  spectra analysis - a nonlinear  signal processing framework'  (Prentice-Hall, Englewood Cliffs, NJ, 1993) ZHENG, F.-c., MCLAUGHLIN, s., and MULGREW,  B.:  'Blind equalization of  nonminimum  phase  channels:  higher  order  cumulant  based algorithm',  IEEE Trans. Signal Proces., 1993, 41, pp. 681-691 TUGNAIT, J.K.:  'Blind  equalization  and  estimation  of  digital communication  FIR  channels  using  cumulant  matching',  IEEE Trans. Commun., 1995, 43, pp. 1240-1245 ILOW, J.,  HATZINAKOS, D.,  and  VENETSANOPOULOS, A.N.:  'Blind equalizers  with  simulated  annealing  optimization  for  digital communication systems', Int. J. Adaptive  Control Signal  Process., 1994, 8, pp. 501-522 INGBER, L.,  and  ROSEN, B.:  'Genetic  algorithms  and  very  fast simulated  reannealing:  a  comparison',  Math.  Comput.  Model., 1992, 16, (Il), pp. 87-100 INGBER, L.: 'Simulated  annealing:  practice  versus  theory',  Math. Comput. Model., 1993, 18, (II), pp. 29-57 Chaotic sequences for multiple access D. Sandoval-Morantes and D. Mufioz-Rodriguez The use of chaotic signals as generators of binary and multilevel sequences suitable for code division multiple access (CDMA) is proposed. The  number  of  obtained sequences  is  found  to  be greater than the number of m-sequences of the same length. These new chaotic sequences are used as spreading codes in a CDMA system. Simulation results are presented and compared with those obtained for m-sequences in terms of BER performance. Introduction:  Chaotic signals have been  proposed in communica- tions systems in different contexts. In some systems, the synchro- nisation properties of some chaotic systems have been used in an additive form in order to hide information [l]. Other works pro- pose the use of chaos for spread spectrum [2, 31  using time discrete chaotic maps, which produce continuous range sequences. In this Letter, binary and four-level chaotic sequences suitable for multi- ple access are proposed. Since  the  performance  of  a  code  division  multiple  access (CDMA) system depends, among other parameters, on the spread- ing properties of the code sequences, in order to guarantee an ade- quate  performance, the  proposed  digital chaotic sequences  are selected as satisfying a suitable correlation criterion. Chaotic signals:  The  generation  of  orthogonal sequences  is  of utmost  importance in  multi-code CDMA  systems,  in  order  to increase the spectrum efficiency in multirate communications sys- tems  [4].  In CDMA, sets of non-correlated sequences with  good autocorrelation and  crosscorrelation properties  are  required  in order to provide low  interference between  users  [5]. It  is  also known that traditionally used m-sequences exhibit  a poor cross- correlation performance. However, the noise-like performance of the chaotic solutions allows us to obtain sequences with desirable properties for CDMA. No. 3  235 

Application of adaptive simulated annealing to blind channel identification with HOC fitting

Abstract

Similar works

Full text

Available Versions

Southampton (e-Prints Soton)