A Genetic Adaptive Algorithm for Data Equalization - CiteSeerX
A Genetic Adaptive Algorithm for Data Equalization Michael S. White and Stuart J. Flockton
Abstract| A technique for adjusting the coecients of adaptive IIR equalizing lters using a genetic algorithm is presented. This technique is shown to be able to signi cantly reduce the intersymbol interference and noise created by imperfect communications channel transmission characteristics.
lar encoded (1) PCM pulses is illustrated in Figure 1. The communications channel is represented by a sixth order all-zero channel, detailed further in Section IV. On this graph the channel input and output are superimposed to show the extent of the distortion caused by ISI.
I. Introduction
Equalizing lters can signi cantly reduce the number of errors due to intersymbol interference over a digital communications channel. The advantages of IIR equalizers over their FIR counterparts have yet to be fully exploited due to the diculties encountered when adapting IIR lters. In this paper we put forward a method to successfully adapt IIR equalizing lters whilst maintaining lter stability. A brief introduction to data communications, intersymbol interference and equalization is given in Section II. The use of genetic algorithms for IIR system identi cation and data equalization is outlined in Section III and experimental results are presented in Section IV. The paper nishes with a brief summary and details further work. II. Intersymbol Interference and Data Equalization
Figure 1: Intersymbol interference
The authors are with the Department of Physics, Royal Holloway, University of London, Egham, Surrey TW20 OEX, UK.
The e ect of intersymbol interference can be reduced by equalization. An equalizing lter is designed to compensate for the imperfect transmission characteristics of the channel. A popular design of equalizing lter due to its ease of implementation is the transversal equalizer. This form of equalizing lter consists of a tapped delay line with the ouput of each of the taps weighted by a gain factor. The output of the transversal equalizer is dependent only upon present and previous values of the input, ie it is non-recursive and has an impulse response of nite duration (FIR). As a consequence, transversal equalizers are suitable only where the time dispersion of pulse signals is over a small number of symbol intervals or where, for longer intervals, the channel output is highly correlated. A lter with feedback, however, has an output which depends on previous
The most commonly employed technique for transmitting analogue information over a digital communications system is known as pulse code modulation, or PCM [Cou87, Str90]. In this pulse modulation technique, the amplitude of the analogue signal at regular sampling intervals is represented by digital words in a serial bit stream. As the PCM pulses are passed through a communications channel which has nite bandwidth, the individual pulses are elongated so that pulses corresponding to any one bit will smear into adjacent bit slots. This pulse smearing e ect is known as intersymbol interference (ISI) and can result in biterrors at the receiver. The e ect of intersymbol interference on a stream of randomly generated po-
values of the output as well as of the input. These recursive or in nite impulse response (IIR) lters do not su er from the same restrictions as their nonrecursive counterparts and require fewer weights ( lter coecients) to achieve the same quality of ISI minimization as an FIR equalizer. Unfortunately, due to their recursive nature IIR lters may, under certain conditions, become unstable. For medium speed modems used in computer communications operating over switched telephone lines at around 2400 bits/second, xed equalizers based on the average channel characteristics can be used. as the speed of communication increases, the variation in channel characteristics from one line to another is so great that the equalizer may need to be able to adjust its response to each individual channel. The weights of a transversal equalizer can be adjusted automatically using iterative techniques, either by adding or subtracting a xed increment, to each tap weight depending on an error signal generated from the thresholded channel output or by minimizing the mean square error (MSE) between the channel output sequence and a known input sequence. Preset equalizing lters make use of a test bit sequence (known variously as a learning or training sequence or preamble) transmitted prior to the data. Adaptive equalizers, however, use the data signal itself to adjust the lter coecients. III. GAs for System Identification and Data Equalization
Conventionally, adaptive algorithms for IIR ltering have been less well developed than those for FIR lters. Individually they su er from a number of drawbacks, including convergence to sub-optimal solutions (especially in the presence of noise) or becoming stuck in local minima when the error surface used for cost minimization is multimodal. System identi cation is a related task to equalization requiring an adaptive lter. Work in the eld of natural algorithms (ie those based on such natural processes as evolution or simulations of many-particle bodies) used for adaptive ltering has tended to concentrate in this area. In the case of system identi cation, the task of the adaptive algorithm is to adjust the coecients of an adaptive lter in order to match its response to that of some unknown system. A degree of success has been achieved using genetic algorithms [EHC82, NTM92, KD92, FW93], evolutionary strategies [NM93] and simulated annealing [WF93] to perform this adaptation. It is intended that this work be followed up with a comparison of the performance of adaptive simulated annealing [Ing89] for IIR adaptive equalization but just the genetic algorithm (GA) results are presented
here. Figure 2 gives block diagrams of adaptive lters in both system identi cation and adaptive equalization con gurations.
Figure 2: System identi cation and adaptive equalization con gurations In adapting equalizer lter coecients, the genetic algorithm maintains a xed-size population of structures, each one representing a set of lter coecients (feedback and feedforward). Every iteration, or generation of the adaptive algorithm, a proportion of these structures undergo mutation and recombination. The tness of each lter structure is then derived from from the mean square error calculated from the input to the channel and the output from the equalizer. The best structures are probabilistically selected according to their ability to reduce the distorting e ects of the channel, in order to form the next population. The adaptation is halted after a predetermined number of generations have been evaluated. In order to ensure that the equalizers are stable, each lter is implemented as an IIR lattice structure. Then, simply by constraining the magnitude of the feedback lter coecients to be less than unity, stability can be guaranteed. IV. Experimental Results
A number of simulation experiments have been carried out in order to determine the capabilities of a genetic adaptive algorithm for data equalization. In all cases, a sixth order IIR equalizing lter was used and the task set was to reduce the distorting effect of the channel to the point where the output of the receiver was error-free, or nearly so. Schraudolph and Grefenstette's GAucsd1.4 function optimization package [SG92], a traditional bit-string GA, was used throughout to adjust the coecients of the equalizer. The MSE was used as the criterion
for minimization and the mutation and crossover rates were set to 0.01 and 0.9 respectively. The rst set of results presented are those for a sixth order all-zero channel with complex-conjugate zero pairs shown in the pole-zero plot given in Figure 3 and in the equations below.
Figure 4: Equalization of an all-zero channel Figure 3: Pole-zero plot of sixth order channel z1 = 0:6 exp j 6
z2 = 0:4 exp j 2
2 3 A sequence of 100 pseudo-randomly generated polar values (the same input as shown in Figure 1) was input to the channel. Figure 4 shows the percentage of bit-errors (mean of ten runs) plotted against the generation number. With no equalization, 32% of the bits output from the channel are incorrect but using the GA driven equalization lter the bit-error rates drops to zero after 55 generations have elapsed. Figure 5 gives a graphical representation of the output of both the channel and best equalizer found after 100 generations. Although the output of the equalizer produces values other than 1, when this is passed through the thresholding device, or slicer (Figure 2) the input sequence can be perfectly reconstructed. In this case because a polar encoding is used, the slicer maps positive values output from the equalizer to +1 and negative values to ?1. The di erence between the slicer output and channel input bit sequence provides the bit-error count for the equalizing lter under evaluation. The second simulation uses an all-pole channel with complex-conjugate pole pairs in the same locations as the zeroes in the previous experiment. The z3 = 0:7 exp j
Figure 5: All-zero channel and equalizer outputs percentage bit-errors is plotted against the generation number in Figure 6. Again, with zero equalization, a large proportion (26%) of the values output from the channel are erroneous. However, after 48 generations, the bit-error rate has once more fallen to zero. In Figure 7 the results of the third set of experiments are presented. Here, the channel has both poles and zeroes, which are located at the following positions: z1 = 0:8 exp j
6
Figure 6: Equalization of all-pole channel
Figure 7: Equalization of a pole-zero channel
2 3 5 z3 = 0:7 exp j 6 p1 = 0:9 exp j 4
particular Ingber and Rosen's Adaptive Simulated Annealing). The work will also be extended to channels of higher order and non-minimum phase.
z2 = 0:75 exp j
References
[Cou87] Leon W. Couch. Digital and Analog Communication Systems. Macmillan Publishp2 = 0:8 exp j 2 ing Company, 1987. 7 [EHC82] D. M. Etter, M. J. Hicks, and K. H. Cho. p3 = 0:4 exp j 8 Recursive adaptive lter design using an adaptive genetic algorithm. In ProceedIn addition, pseudo-random gaussian noise has ings of IEEE International Conference on been added at a signal-to-noise ratio of 17dB in orAcoustics, Speech and Signal Processing der to see how the equalizing lter copes with ad(ICASSP 82) , pages 635{638. IEEE, May ditive channel noise. As can be seen from the g1982. ure, this channel poses more of a challenge for the equalizer and many more generations are required [FW93] Stuart J. Flockton and Michael S. White. Pole-zero identi cation using genetic alin order to obtain a best bit-error rate of 3%. gorithms. In Stephanie Forrest, editor, V. Conclusions Genetic Algorithms: Proceedings of the The adaptive equalization technique described in Fifth International Conference (GA93), this paper has shown itself capable of reducing the San Mateo, July 1993. Morgan Kaufe ects of intersymbol interference caused by the nonmann. ideal transmission characteristics of the channel in [Ing89] Lester Ingber. Very fast simulated anneala digital communications system. In addition, the ing. Mathematical and Computer Modequalizer is able to signi cantly decrease the numelling, 12(8):967{973, 1989. ber of bit-errors at the receiver due to channel noise. [KD92] Kristinn Kristinsonn and Guy A. DuStability of the equalizer, a major problem when mont. System identi cation and control utilising higher order IIR lters, is ensured due to using genetic algorithms. IEEE Transthe lattice representation used to implement the equalactions on Systems, Man and Cybernetization lters. Areas of further work which will be ics, 22(5):1033{1046, September/October undertaken in the future are a direct comparison be1992. tween this genetic adaption technique, conventional [NM93] R. Nambiar and P. Mars. Adaptive IIR adaptive methods and other natural algorithms (in ltering using natural algorithms. In Nat
ural Algorithms in Signal Processing, volume 2, pages 20/1{20/7. IEE Electronics Division in association with the IEEE UK and RI Signal Processing Chapter, IEE, November 1993. [NTM92] R. Nambiar, C. K. K. Tang, and P. Mars. Genetic and learning automata algorithms for adaptive digital lters. In Proc. of IEEE International Conference on Acoustics, Speech and Signal Processing, 1992. [SG92] Nicol N. Schraudolph and John J. Grefenstette. A users guide to gaucsd1.4. Technical Report CS92-249, University of California, San Diego, CSE Department, UC San Diego, La Jolla, CA 9209-0114, July 1992. [Str90] Ferrel G. Stremler. Introduction to Communication Systems. Addison Wesley, 1990. [WF93] Michael S. White and Stuart J. Flockton. A comparative study of natural algorithms for adaptive IIR ltering. In Natural Algorithms in Signal Processing, volume 2, pages 22/1{22/8. IEE Electronics Division in association with the IEEE UK and RI Signal Processing Chapter, IEE, November 1993.
Abstract| A technique for adjusting the coecients of adaptive IIR equalizing lters using a genetic algorithm is presented. This technique is shown to be able to signi cantly reduce the intersymbol interference and noise created by imperfect communications channel transmission characteristics.
lar encoded (1) PCM pulses is illustrated in Figure 1. The communications channel is represented by a sixth order all-zero channel, detailed further in Section IV. On this graph the channel input and output are superimposed to show the extent of the distortion caused by ISI.
I. Introduction
Equalizing lters can signi cantly reduce the number of errors due to intersymbol interference over a digital communications channel. The advantages of IIR equalizers over their FIR counterparts have yet to be fully exploited due to the diculties encountered when adapting IIR lters. In this paper we put forward a method to successfully adapt IIR equalizing lters whilst maintaining lter stability. A brief introduction to data communications, intersymbol interference and equalization is given in Section II. The use of genetic algorithms for IIR system identi cation and data equalization is outlined in Section III and experimental results are presented in Section IV. The paper nishes with a brief summary and details further work. II. Intersymbol Interference and Data Equalization
Figure 1: Intersymbol interference
The authors are with the Department of Physics, Royal Holloway, University of London, Egham, Surrey TW20 OEX, UK.
The e ect of intersymbol interference can be reduced by equalization. An equalizing lter is designed to compensate for the imperfect transmission characteristics of the channel. A popular design of equalizing lter due to its ease of implementation is the transversal equalizer. This form of equalizing lter consists of a tapped delay line with the ouput of each of the taps weighted by a gain factor. The output of the transversal equalizer is dependent only upon present and previous values of the input, ie it is non-recursive and has an impulse response of nite duration (FIR). As a consequence, transversal equalizers are suitable only where the time dispersion of pulse signals is over a small number of symbol intervals or where, for longer intervals, the channel output is highly correlated. A lter with feedback, however, has an output which depends on previous
The most commonly employed technique for transmitting analogue information over a digital communications system is known as pulse code modulation, or PCM [Cou87, Str90]. In this pulse modulation technique, the amplitude of the analogue signal at regular sampling intervals is represented by digital words in a serial bit stream. As the PCM pulses are passed through a communications channel which has nite bandwidth, the individual pulses are elongated so that pulses corresponding to any one bit will smear into adjacent bit slots. This pulse smearing e ect is known as intersymbol interference (ISI) and can result in biterrors at the receiver. The e ect of intersymbol interference on a stream of randomly generated po-
values of the output as well as of the input. These recursive or in nite impulse response (IIR) lters do not su er from the same restrictions as their nonrecursive counterparts and require fewer weights ( lter coecients) to achieve the same quality of ISI minimization as an FIR equalizer. Unfortunately, due to their recursive nature IIR lters may, under certain conditions, become unstable. For medium speed modems used in computer communications operating over switched telephone lines at around 2400 bits/second, xed equalizers based on the average channel characteristics can be used. as the speed of communication increases, the variation in channel characteristics from one line to another is so great that the equalizer may need to be able to adjust its response to each individual channel. The weights of a transversal equalizer can be adjusted automatically using iterative techniques, either by adding or subtracting a xed increment, to each tap weight depending on an error signal generated from the thresholded channel output or by minimizing the mean square error (MSE) between the channel output sequence and a known input sequence. Preset equalizing lters make use of a test bit sequence (known variously as a learning or training sequence or preamble) transmitted prior to the data. Adaptive equalizers, however, use the data signal itself to adjust the lter coecients. III. GAs for System Identification and Data Equalization
Conventionally, adaptive algorithms for IIR ltering have been less well developed than those for FIR lters. Individually they su er from a number of drawbacks, including convergence to sub-optimal solutions (especially in the presence of noise) or becoming stuck in local minima when the error surface used for cost minimization is multimodal. System identi cation is a related task to equalization requiring an adaptive lter. Work in the eld of natural algorithms (ie those based on such natural processes as evolution or simulations of many-particle bodies) used for adaptive ltering has tended to concentrate in this area. In the case of system identi cation, the task of the adaptive algorithm is to adjust the coecients of an adaptive lter in order to match its response to that of some unknown system. A degree of success has been achieved using genetic algorithms [EHC82, NTM92, KD92, FW93], evolutionary strategies [NM93] and simulated annealing [WF93] to perform this adaptation. It is intended that this work be followed up with a comparison of the performance of adaptive simulated annealing [Ing89] for IIR adaptive equalization but just the genetic algorithm (GA) results are presented
here. Figure 2 gives block diagrams of adaptive lters in both system identi cation and adaptive equalization con gurations.
Figure 2: System identi cation and adaptive equalization con gurations In adapting equalizer lter coecients, the genetic algorithm maintains a xed-size population of structures, each one representing a set of lter coecients (feedback and feedforward). Every iteration, or generation of the adaptive algorithm, a proportion of these structures undergo mutation and recombination. The tness of each lter structure is then derived from from the mean square error calculated from the input to the channel and the output from the equalizer. The best structures are probabilistically selected according to their ability to reduce the distorting e ects of the channel, in order to form the next population. The adaptation is halted after a predetermined number of generations have been evaluated. In order to ensure that the equalizers are stable, each lter is implemented as an IIR lattice structure. Then, simply by constraining the magnitude of the feedback lter coecients to be less than unity, stability can be guaranteed. IV. Experimental Results
A number of simulation experiments have been carried out in order to determine the capabilities of a genetic adaptive algorithm for data equalization. In all cases, a sixth order IIR equalizing lter was used and the task set was to reduce the distorting effect of the channel to the point where the output of the receiver was error-free, or nearly so. Schraudolph and Grefenstette's GAucsd1.4 function optimization package [SG92], a traditional bit-string GA, was used throughout to adjust the coecients of the equalizer. The MSE was used as the criterion
for minimization and the mutation and crossover rates were set to 0.01 and 0.9 respectively. The rst set of results presented are those for a sixth order all-zero channel with complex-conjugate zero pairs shown in the pole-zero plot given in Figure 3 and in the equations below.
Figure 4: Equalization of an all-zero channel Figure 3: Pole-zero plot of sixth order channel z1 = 0:6 exp j 6
z2 = 0:4 exp j 2
2 3 A sequence of 100 pseudo-randomly generated polar values (the same input as shown in Figure 1) was input to the channel. Figure 4 shows the percentage of bit-errors (mean of ten runs) plotted against the generation number. With no equalization, 32% of the bits output from the channel are incorrect but using the GA driven equalization lter the bit-error rates drops to zero after 55 generations have elapsed. Figure 5 gives a graphical representation of the output of both the channel and best equalizer found after 100 generations. Although the output of the equalizer produces values other than 1, when this is passed through the thresholding device, or slicer (Figure 2) the input sequence can be perfectly reconstructed. In this case because a polar encoding is used, the slicer maps positive values output from the equalizer to +1 and negative values to ?1. The di erence between the slicer output and channel input bit sequence provides the bit-error count for the equalizing lter under evaluation. The second simulation uses an all-pole channel with complex-conjugate pole pairs in the same locations as the zeroes in the previous experiment. The z3 = 0:7 exp j
Figure 5: All-zero channel and equalizer outputs percentage bit-errors is plotted against the generation number in Figure 6. Again, with zero equalization, a large proportion (26%) of the values output from the channel are erroneous. However, after 48 generations, the bit-error rate has once more fallen to zero. In Figure 7 the results of the third set of experiments are presented. Here, the channel has both poles and zeroes, which are located at the following positions: z1 = 0:8 exp j
6
Figure 6: Equalization of all-pole channel
Figure 7: Equalization of a pole-zero channel
2 3 5 z3 = 0:7 exp j 6 p1 = 0:9 exp j 4
particular Ingber and Rosen's Adaptive Simulated Annealing). The work will also be extended to channels of higher order and non-minimum phase.
z2 = 0:75 exp j
References
[Cou87] Leon W. Couch. Digital and Analog Communication Systems. Macmillan Publishp2 = 0:8 exp j 2 ing Company, 1987. 7 [EHC82] D. M. Etter, M. J. Hicks, and K. H. Cho. p3 = 0:4 exp j 8 Recursive adaptive lter design using an adaptive genetic algorithm. In ProceedIn addition, pseudo-random gaussian noise has ings of IEEE International Conference on been added at a signal-to-noise ratio of 17dB in orAcoustics, Speech and Signal Processing der to see how the equalizing lter copes with ad(ICASSP 82) , pages 635{638. IEEE, May ditive channel noise. As can be seen from the g1982. ure, this channel poses more of a challenge for the equalizer and many more generations are required [FW93] Stuart J. Flockton and Michael S. White. Pole-zero identi cation using genetic alin order to obtain a best bit-error rate of 3%. gorithms. In Stephanie Forrest, editor, V. Conclusions Genetic Algorithms: Proceedings of the The adaptive equalization technique described in Fifth International Conference (GA93), this paper has shown itself capable of reducing the San Mateo, July 1993. Morgan Kaufe ects of intersymbol interference caused by the nonmann. ideal transmission characteristics of the channel in [Ing89] Lester Ingber. Very fast simulated anneala digital communications system. In addition, the ing. Mathematical and Computer Modequalizer is able to signi cantly decrease the numelling, 12(8):967{973, 1989. ber of bit-errors at the receiver due to channel noise. [KD92] Kristinn Kristinsonn and Guy A. DuStability of the equalizer, a major problem when mont. System identi cation and control utilising higher order IIR lters, is ensured due to using genetic algorithms. IEEE Transthe lattice representation used to implement the equalactions on Systems, Man and Cybernetization lters. Areas of further work which will be ics, 22(5):1033{1046, September/October undertaken in the future are a direct comparison be1992. tween this genetic adaption technique, conventional [NM93] R. Nambiar and P. Mars. Adaptive IIR adaptive methods and other natural algorithms (in ltering using natural algorithms. In Nat
ural Algorithms in Signal Processing, volume 2, pages 20/1{20/7. IEE Electronics Division in association with the IEEE UK and RI Signal Processing Chapter, IEE, November 1993. [NTM92] R. Nambiar, C. K. K. Tang, and P. Mars. Genetic and learning automata algorithms for adaptive digital lters. In Proc. of IEEE International Conference on Acoustics, Speech and Signal Processing, 1992. [SG92] Nicol N. Schraudolph and John J. Grefenstette. A users guide to gaucsd1.4. Technical Report CS92-249, University of California, San Diego, CSE Department, UC San Diego, La Jolla, CA 9209-0114, July 1992. [Str90] Ferrel G. Stremler. Introduction to Communication Systems. Addison Wesley, 1990. [WF93] Michael S. White and Stuart J. Flockton. A comparative study of natural algorithms for adaptive IIR ltering. In Natural Algorithms in Signal Processing, volume 2, pages 22/1{22/8. IEE Electronics Division in association with the IEEE UK and RI Signal Processing Chapter, IEE, November 1993.
