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1 (a) DFE (b) DFE + SOM (c) DFE (d) DFE + SOM (e) DFE (f) DFE + SOM Figure 6: The start-up behaviour under moderate noise (S=N = db) conditions: undistorted channels (a) and (b), corner collapse distortions (c) and (d), and grid collapse distortions (e) and (f) Conventional DFE on the left and SOM assisted DFE on the right Self-Organizing Map algorithm n addition, the new structure has a slightly faster locking capability in "normal", linearly distorted channels As the improvement in performance can be achieved with no increase in the number of arithmetic operations, the new approach should be very interesting and may well lead to new receiver realizations ACKNOWLEDGEMENT This work was partially nanced by NOKA Telecommunications REFERENCES PROAKS, JG: 'Advances in Equalization for ntersymbol nterference', in Advances in Communication Systems Theory and Applications, Vol, Edited by AJ Viterbi, (Academic Press, 975) BENEDETTO, S, BGLER, E, and CASTELLAN, V: 'Digital transmission theory' (Prentice Hall, nc, New Jersey, 988) PALCOT, J: 'Equalisation of non-linear perturbations', (in french), in Treizieme Colloque Gretsi, Juan-Les- Pins, Sept 6-, 99 KOHONEN, T, RAVO, K, SMULA, O, VENT A, O, and HENRKSSON, J: 'Combining Linear Equalization and Self-Organizing Adaptation in Dynamic Discrete- Signal Detection', Proc JCNN, San Diego, CA, June 8-, 99, Vol, pp -8 5 KOHONEN, T, RAVO, K, SMULA, O, and HENRKS- SON, J: 'Performance Evaluation of Self-Organizing Map Based Neural Equalizers in Dynamic Discrete- Signal Detection', Proc CANN, Helsinki, Finland, June -8, 99, Vol, pp SU, S, GBSON, GJ, and COWAN, CFN,: 'Decision Feedback Equalisation Using Neural Network Structures and Performance Comparison with Standard Architecture', EE Proc, Vol 7, Pt, No, August 99, pp -5 7 MANABE, T, and KANEDA, R,: 'Adaptive Decision- Feedback Equalization of Digital Transmission Channels Using Forward-Only Counterpropagation Networks', Proc JCNN, Singapore, Nov 8-, 99, pp-5 8 KOHONEN, T: 'Self-Organization and Associative Memory', Series in nformation Sciences, Vol 8, (Springer-Verlag, rd Edition, 989) 9 KOHONEN, T: 'A Method for Adaptive Detection of uantized Signals', (in nnish), Patent No 8577, July 5, 99

2 (a) DFE (b) DFE + SOM (c) DFE (d) DFE + SOM (e) DFE (f) DFE + SOM Figure 5: The start-up behaviour under low noise (S=N = 8 db) conditions: undistorted channels (a) and (b), corner collapse distortions (c) and (d), and grid collapse distortions (e) and (f) Conventional DFE on the left and SOM assisted DFE on the right 5 RESULTS n the examples, the start-up behaviour of the two equaliser structures was compared using the above described twopath channel model with two dierent noise levels The simulations have been run times with the same parameter values The Bit-Error-Ratio (BER), shown in Figures 5 and 6, has been calculated by averaging the last from each run As the rst example, the start-up behaviour in low noise channel was studied The signal-to-noise ratio of the undistorted channel was 8 db in this case Figure 5 depicts the bit-error-ratios (BER) of the DFE (left) and the SOM assisted DFE (right) The undistorted, corner collapsed, and grid collapsed channels are compared in Figures 5 (a) and (b), (c) and (d), and (e) and (f), respectively n the second example the channel was more noisy, the signal-to-noise ratio of the undistorted channel being db The undistorted, corner collapsed, and grid collapsed channels are compared in Figures 6 (a) and (b), (c) and (d), and (e) and (f), respectively Again, the conventional 9-tap DFE is on the left and the SOM assisted -tap DFE on the right The simulations show that the DFE alone behaves well in the undistorted two-path channel, as expected The nal BER is about the same for both equalisers in this case Under non-linear distortions the BER is considerably smaller for the combined equaliser structure Longer simulation runs do not aect the nal BER t should be noticed that both distortion types used in simulation examples are relatively large Extreme values have been investigated to be able to demonstrate the advantages of the new structure Also, the curves show that in the undistorted channel the Self-Organizing Map is able to slightly speed up the convergence (Figures 5 and 6, (a) and (b)) t should be noticed that the computational complexity of both equaliser structures, ie the 9-tap DFE and SOM assisted -tap DFE, is about the same, only the structure of computation is dierent Thus, no extra cost has to be paid for the improved performance 6 CONCLUSONS The simulations have shown that the neural network assisted equalisers adapt very well to various channel conditions, including non-linear distortions This is due to the topology-preserving property of the Self-Organizing Map algorithm Especially in dicult channels, consisting of both linear multipath propagation and non-linear distortions, the new scheme, with equal computational complexity, outperforms the conventional equaliser The combined equaliser structures have a wider range of capability against distortions than the conventional presently used structures This is due to the learning ability of the

3 x(n) T T Self-Organizing Map (n) y(n) Control unit Figure : The block diagram of the cascade combination of a -tap DFE and the SOM and the Self-Organizing learning algorithm [] n these systems the Self-Organizing Map is used in cascade (or parallel) with the conventional equaliser The block diagram of a cascade combination of the DFE structure and SOM is shown in Figure The values of the Self-Organizing learning network are used for generating the decision levels in the detector of the equaliser The output error, "(n) = (n)? y(n), is needed in the adaptation of the equaliser The basic idea in this neural equaliser is that the Self-Organizing learning network adaptively compensates for non-linear distortions and the conventional equaliser is used to correct linear distortions The operation of the Self-Organization based adaptation is illustrated in Figure n non-linear distortion, eg in the corner collapse situation, the drifts in the signal values are followed by the SOM The decision levels between the cells are adaptively moved according to the received signal as shown by the dashed lines in Figure for one quarter of the map (a) (b) Figure : The adaptation of the decision levels in ideal situation (a) and corner collapse distortion (b) n practice, the corner collapse distortion may result from non-linear amplier characteristics; corners correspond to the signals with the largest amplitudes The various combined equaliser structures have been tested by imposing dierent kinds of articial non-linear distortions on the signal constellation [], [5] n this paper, we have concentrated on the situation, in which the start-up behaviour of the new equaliser structure is investigated The computational complexity of the Map algoriths only dependent on the number of signal states, while in transversal equalisers the number of arithmetic operations is dependent also on the degree of the lter The performance of the -tap neural equaliser has been compared with a 9-tap decision feedback equaliser (DFE) The decision feedback equaliser has ve forward and four feedback taps n both cases, taps are adjusted using gradient algorithm to full least mean square error criterion These structures have approximately equal computational complexity SMULATED SYSTEM The performance characteristics of the neural network assisted DFE has been compared to the conventional DFE structure by extensive simulations The communication channel was modelled by the complex two-path model The delay of the model was corresponding to one symbol interval The relative gain and phase of the delayed signal were xed to 5 and, respectively This channel model represents, due to the long delay, a highly dispersive channel Additive white Gaussian noise (AWGN) was used in the simulations n the beginning of each adaptation the central tap of the DFE was initialized to unity and the other taps to zero The parameters of the map were initialized to the ideal signal values n the beginning of each simulation run the channel is xed to its nal state For example, the non-linear distortions were present already in the beginning Both non-linear distortion types shown in Figure have been considered and compared to the undistorted channel The corner collapse distortion was 7 units the diagonal distance between the neighbouring signal values being units n the grid collapse distortion the deviation from rectangularity was 5 degrees

4 The adaptation is designed to maintain the two-dimensional parameters, as close as possible, to the current levels of the corresponding (; ) values, even when distortions are present The adaptive and time-varying signal identication The neighbourhood learning is usually applied symmetrically in each direction in the array of adaptive cells n general, this causes some bias in the values towards the group center of the parameter values because the cells near the edges of the array may not have neighbours in both directions This can, however, be compensated by eectively enlarging the input signal on the edges of the array The input signal x(n) is modied to b i x(n) where b i and d i are node-specic parameters This yields the modied adaptation equations: x ( n ) m c Figure : Two-dimensional Self-Organizing Map proceeds according to the following rules that are based on the original SOM algorithm: (i) At each discrete time instant n, determine the cell c with the best matching parameter (n) in respect to the current received signal sample x(n), ie, jjx(n)? m c (n)jj = min i N c fjjx(n)? (n)jjg () (ii) Adapt the parameters in the neighbourhood N c of the computed cell c (n + ) = (n) + [x(n)? (n)]; (n + ) = (n) + [x(n)? (n)]; i = c i N c ; i 6= c (n + ) = (n) i 6 N c () The topological neighbourhood N c consists of the best matching cell itself and its direct neighbours up to depth,, Since the problem now is to follow the drifts and other deteriorations in the quantized signal values we used xed values for the iteration coecients, and (and not monotonically decreasing functions of time as generally in the Self-organizing process [8]) As shown in the earlier studies [], [5], if the and values and the neighbourhood radius are selected properly, the values will trace reasonably well the time-varying x(n) values, ie, adaptively identify the received quantized signal Because the transmitted signal space has a well-dened topology, and due to neighbourhood learning, the mapping is topology-preserving, the adaptation is very eective This is true unless the distortions are so large that the linear topology of the quantized signal is destroyed (n + ) = (n) + [b i x(n)? (n)]; i = c (n + ) = (n) + [b i x(n)? (n)]; i N c ; i 6= c (n + ) = (n) i 6 N c () The algoriths able to follow up the distortions, if the distortions are such that the local order of the signal constellation is preserved n particular, it should be pointed out that the distortions need not be linear All kinds of rotations, grid collapses, etc, can be compensated provided that all mutually neighbouring signal levels remain in the same topological order as in the initial signal constellation Even when absolute signal levels change drastically, they can be eectively and steadily followed (a) (b) Figure : The corner collapse (a) and grid collapse (b) distortions with noise The non-linear distortion types mainly studied are shown in Figure The corner collapse situation of Figure (a) corresponds to the eects of non-linear amplitude distortion n the grid collapse distortion of Figure (b) the in-phase and quadrature channels deviate from rectangularity NEURAL NETWORK ASSSTED EUALSATON Conventional equalisers are mainly able to adapt to linearly distorted discrete signal levels, whereas the Self-Organizing learning networks readily compensate for non-linear distortions Nonlinear equalisers, like DFE, are in principle more ecient than their linear equivalents To gain the advantages of both systems, a dynamic adaptation method has been developed by combining the conventional equalisation

5 START-UP BEHAVOUR OF A NEURAL NETWORK ASSSTED DECSON FEEDBACK EUALSER N A TWO-PATH CHANNEL Teuvo Kohonen, Kimmo Raivio, Olli Simula, Jukka Henriksson + ) Helsinki University of Technology, Laboratory of Computer and nformation Science TKK-F, Rakentajanaukio C, SF-5 Espoo, Finland + ) Nokia Research Center, Transmission Systems PO Box 56, SF- Espoo, Finland ABSTRACT t has recently been shown that the performance of traditional transversal equalisers in adaptive discrete-signal detection can be improved with the aid of neural computation The performance characteristics of the neural network assisted decision-feedback-equaliser (DFE) has been investigated by extensive simulations using a two-path channel model and 6AM modulation This paper reports some of the simulation results on the start-up behaviour of the novel equaliser structure under distorted channel conditions The results have shown that especially in dicult channels, including both linear multipath and nonlinear distortions, the neural network assisted DFE structures are superior when compared to the traditional DFE with equal computational complexity NTRODUCTON Adaptive equalisers in the form of linear or nonlinear transversal lters have traditionally been used in digital transmission to rectify the deterioration caused by dispersive transmission media, eg [], [] Typical application areas are data transmission, xed radio links and mobile radio services n these cases the transmission path may cause, in addition to linear dispersion, also other unwanted eects like non-linear distortions or phase shifts Moreover, slightly non-ideal transmitters may include non-linearity and a phase error between the in-phase and quadrature channels The amount of these eects may vary in the production of equipment and these may also be slowly timevarying due to ageing These eects could be corrected adaptively in the receiver The conventional equalisers, however, do not respond particularly well to these other eects Recently dierent new approaches to solve the problems caused by non-linear distortions have been presented n [] a parallel bank of linear equalizers is proposed to solve the non-linear eects We have suggested a new equalizer structure employing neural computation having the capability of combatting the unwanted distortions described above [], [5] Neural network algorithms have also been used in updating the adaptive taps of conventional equalizer structures [6], [7] n our approach, a neural network algorithm, called Self- Organizing Map (SOM) algorithm [8], has been connected with a linear transversal equaliser (LE) or with a decision feedback equaliser (DFE) [] Patent applications have been made regarding the new equalising schemes [9] The analyses have been concentrating on quadrature amplitude modulation (AM) [], particularly on AM and 6AM because they have great interest in xed and mobile radio telecommunications 6AM has also been proposed for digital HDTV transmission in USA The present paper gives some results of computer simulations on the start-up behaviour of the combined SOM-DFE structure with 6AM A two-path channel model having non-linear distortions and additive white Gaussian noise has been used The simulations show that an enhanced performance can be achieved with no or relatively minor increase in the receiver complexity SELF-ORGANZNG MAPS Self-Organizing Maps are neural networks that produce localized responses to input signals and represent the topology of the input signal space over the network [8] n the adaptive detection method based on the SOM the learning framework in the algoriths a planar array of adaptive cells, each cell corresponding to a particular gridpoint in the discrete signal constellation Each cell of the SOM is characterized by an adaptive parameter f the ideal and distorted signal values are two-dimensional, also the values are two-dimensional vectors Before transmission, the adaptive parameter vectors of each cell are initialized to the ideal (; ) values During the transmission, the learning framework of the receiver receives a replica of the transmitted discrete signal x(n) which in general contains the original signal and possibly some noise and distortion The SOM corresponding to (; ) values is shown in Figure The parameter of the cell nearest to the received signal sample is denoted by m c

Exploring QAM using LabView Simulation *

OpenStax-CNX module: m14499 1 Exploring QAM using LabView Simulation * Robert Kubichek This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 2.0 1 Exploring