The decision-feedback equalizer optimization for Gaussian noise

Transcription

1 Journal of Theoretical and Alied Comuter Science Vol. 8 No ISSN (rinted (online htt:// The decision-feedback eualizer otimization for Gaussian noise Arkadiusz Grzbowski Jerz W. Kisilewicz Deartment of Sstems and Comuter Networks Facult of Electronics Wroclaw Universit of Technolog Wrocław Poland {arkadiusz.grzbowski jerz.kisilewicz}@wr.edu.l Abstract: The new method of decision-feedback arameters otimization for intersmbol interference eualizers is described in this aer. The error extension henomena is well known and investigated in the decision feedback eualizers in data transmission. The existing coefficient in decision feedback deends on the receive decision risk ualification. There is roved the bit error robabilit can be decreased b this method for an channel with single interference samle and small Gaussian noise. The exerimental results are resented for chosen te channels. The deendences of otimal feedback arameters on channel interference samle and on noise ower are resented too. Kewords: data communication decision feedback eualizers intersmbol interference error analsis. Introduction The intersmbol interference eualization roblem with the decision-feedback use has been described in man aers. This is the decision roblem concerned with incoming data receiving. It is done b taking the recognized binar data to the taed-dela line of the transversal filter and b using these data for incoming interference erasure. Since the data smbol is recognized its interference with incoming next data becomes known and can be subtracted from incoming signal. The decision-feedback eualizer works roerl if the decisions existing in its dela line are correct. In the case of received data error the wrong data is taken to the feedback and calculated incoming interference differs from the real one. For binar data the existing interference is rather dulicated instead of being cancelled. In this case no cancellation or artial cancellation is roosed b subtracting the less value than the calculated one from the signal. The artial cancellation is roosed onl when the receive decision is risk i.e. when the robabilit of receive error is high. Definition: The decision is ualified to be risk (risk decision if the signal value e k on decision unit inut is close to the decision level S * i.e. if e k S * <. If the robabilities of sending bit and sending bit are the same and the signal mean value is eual to zero the otimal decision level S *.

2 6 Arkadiusz Grzbowski Jerz W. Kisilewicz We assume the transmission ath is comosed of artiall eualized channel and decision feedback eualizer as is shown on Fig.. The channel discrete transfer function is Y(z z with the arameters >. The decision-feedback eualization with decision risk analsis is described in [3] [4] and [7]. The similar methods which do not take into eualization the interference calculated from received signal when the receive data were risk detected have been roosed b M. Chiani [4] and b K. Hacioglu [7]. These methods do not subtract the calculated interference from existing one if risk data have been used in calculations. This reduces the error extension henomena described in [] [5] [7] [8]. M. Chiani [4] examined the deendence of bit error rate (BER on risk threshold searching for the best values of for chosen channels to obtain the minimum BER. If the decision d k is risk the algorithm resented in this aer multilied the interference calculated from d k b the factor and after subtracts the result from signal. I.e. if the signal value e k on decision unit inut differs from the decision level S * less than so if e k < the incoming interference calculated from d k is multilied b (where < < and the result is subtracted from signal. In the case of error (d k a k the interference on decision unit inut is multilied b which is less than. The decision feedback contains an extra dela line which remembers the risk ualifications of the decisions existing in a normal dela line of this feedback. The otimal value of risk level as an analtic function of factor is given in this aer for channels with single interference samle. The grahic deendences of otimal and on arameters and on noise ower are resented. Noise z k a k Y(z z v k e k S * d k Data Risk dk Risk z d k z Figure. Assumed channel and eualizer. The risk level calculation Let assume the noise robabilit densit function to be even and the robabilit of binar data values in the transmitting seuences are eual.5. Let the bit zero be reresented b negative ulse (a k and the bit one be reresented b ositive ulse (a k. Let assume the noise robabilit densit function to be even and the robabilit of binar data values in the transmitting seuences are eual.5. Let the bit zero be reresented b negative ulse (a k and the bit one be reresented b ositive ulse (a k. The first ste is to find the deendence of bit error robabilit on arameters and for the sstem shown on Fig. with discrete transfer function given b euation Y(z z. (

3 The decision-feedback eualizer otimization for Gaussian noise 7 The samle of the channel resonse on inut signal in time tkt is eual v k a k a k z k where z k is the noise samle. If the ta gain of digital dela line is the value e k on the decision module inut is [6] when the decision d k is not ualified to be risk or e k a k (a k d k z k ( e k a k (a k d k z k (3 for risk decision d k ( < <. Let f z (z be the robabilit densit function of the noise z. We assume f z (z is the even function i.e. f z ( z f z (z the robabilities of a k and a k - are eual. Next we will assume the white Gaussian noise so f x ( z x e π. The sign of e k determines the decision d k. If e k > then the assumed d k otherwise d k -. If e k. < i.e. if e k is close to zero the decision d k is assumed to be risk and in the next time eriod the calculated interference is comensated carefull using (3 with coefficient < <. We will find the otimal values * and * giving minimum robabilit of error for assumed white Gaussian noise ower and for assumed channel factor /. Therefore we will obtain the robabilities of decision d k error (d k a k in the case of decision d k was correct (d k a k or errors (d k a k and was assumed risk ( e k < or not ( e k. So we consider the eualizer feedback as a first order Markov chain with four states as is shown on Fig. : S the decision ek is correct and non risk S the decision ek is false and non risk S3 the decision ek is correct and risk S4 the decision ek is false and risk. Each state S i (i 3 and 4 has: the stationar robabilit i that the eualizer is in state Si the robabilities ji (j 3 and 4 that the next state will be Sj the error robabilit Pei. The above described robabilities deend on / and on /. 3 S S S 3 S Figure. Four-state model of the eualizer feedback

4 8 Arkadiusz Grzbowski Jerz W. Kisilewicz For further investigations we assume the ositive values and. This assumtion doesn t limit the consideration generalit. For negative values and the aroriate signs receding the variables and in the next euations will change minus to lus and lus to minus giving the same final result (9. Let > > and x dt t x e ( π. So for S we have: P e. (4 For S we have: P e. (5 For S 3 we have: 3 ( ( 3 ( (

5 The decision-feedback eualizer otimization for Gaussian noise 9 ( ( ( ( 33 ( ( ( ( 43. ( ( P e (6 For S 4 we have: 4 ( ( 4 ( ( ( ( ( ( 34 ( ( ( ( 44. ( ( P e (7 The stationar robabilities i (i 3 and 4 are calculated to satisf the euations (8a 3 4. (8b The error robabilit P e (robabilit of d k a k is given b formula P e P e P e P e3 3 P e4 4 (9 where P e to P e4 are given b (4 to (7. The roblem is to find the otimal * ( / and * ( / that for assumed minimize the error robabilit P e given b (9. 3. Comutational exeriment The aim of the exeriment is to find the otimal values of * ( / and * ( / as a functions of channel arameters and to find how much the otimization of the decision feedback can decrease the bit error robabilit. The channels chosen in exeriment were described b two ulse resonse samles so its transfer function is given b (. These samles are normalized for their energ eual to one

6 Arkadiusz Grzbowski Jerz W. Kisilewicz so. Samle is the interference samle and samle is the main samle. If there is < then where <. First the value of was increased b some constant and next the was calculated. For such designed channels and the assumed noise ower the value ( < and the value ( < were chosen to find the oint ( * * which minimizes the bit error robabilit P e given b (4. The value * minimizing the bit error robabilit P e for was calculated too so the resented in this aer method can be comared with the method resented b Chiani [4]. 4. Results and conclusion The otimal values of * ( / and * ( / as well as the deendence of * and * on / are shown on Fig. 3 and Fig. 4 for SNR and 7 db. Otimal value of is between.3 and.4 for low interference ( / <. between. and.4 for medium interference (. < / <.6 and between. and.7 for high interference (.6 < / <.95. If the interference is high the otimal increases from about.5 for high SNR to.45 or.7 (SNR 6dB and decreases to about.4 for higher noise (SNR<dB. The otimal values of decreases from about.75 for SNRdB to about.3 for SNR5dB and to for high SNR. The high error extension is in case of low SNR and high interference. For this case the otimal feedback arameters are *.5 and. < * <.7. The error robabilit decrease is shown on Fig. 5 and it can be u to 4% for low noise or u to 5% for SNR<7dB. The otimization of and gives the best results for medium interference ( /.6 indeendentl of SNR. Following M. Chiani [4] we can use the otimal values * minimizing the bit error robabilit for. The error robabilit decrease given b otimization of and (P ot in comarition of onl otimization assuming (P can be u to 5% as is shown on Fig Alfa* / 6 db 8 db db db 4 db 7 db Figure 3. The otimal values of *(/ for SNR6 8 4 and 7 db

7 The decision-feedback eualizer otimization for Gaussian noise Beta* 6 db 8 db db db 4 db 7 db / Figure 4. The otimal values of * ( / for SNR and 7 db P ot /P / 6 db 8 db db db 4 db 7 db Figure 5. The error robabilit decrease given b otimal * and * for SNR6 8 4 and 7 db P ot /P db 8 db db db 4 db 7 db / Figure 6. The error robabilit decrease given b otimal * instead of for SNR and 7 db

8 Arkadiusz Grzbowski Jerz W. Kisilewicz 5. Summar There was roved for channels with discrete transfer functions Y(z z i.e. with one interference samle ( < and with white Gaussian noise. The decision feedback otimization using the risk threshold and erasure factor otimization decreases the bit error rate. The roosed otimization of the decision feedback gives the bit error rate (BER decreasing % to 4% deending on channel ( / and on noise ower. If the decision feedback eualizes more than one interference samle the feedback arameters can be otimized searatel for each samle i taking i / into consideration for i. References [] Altekar S. A. Beaulieu N. C.: Uer bounds on the error robabilit of decision feedback eualization. IEEE Transactions on Information Theor vol. 39 no Jan [] Beaulieu N. C.: Bounds on variances of recover times of decision feedback eualizers. IEEE Transactions on Information Theor vol. 46 no Set.. [3] Bergmans J. W. M. Voorman J. O. Wong-Lan H. W.: Dual decision feedback eualizer IEEE Transactions on Communications vol. 45 no Ma 997. [4] Chiani M.: Introducing erasures in decision-feedback eualization to reduce error roagation. IEEE Transactions on Communications vol. 45 no Jul [5] Cho W. W. Beaulieu N. C.: Imroved bounds for error recover times of decision feedback eualization. IEEE Transactions on Information Theor vol. 43 no Ma 997. [6] Grzbowski A. Kisilewicz J.: BER decreasing method using the receiving data reliabilit ualification for decision feedback eualizers. National Telecommunication Smosium 98 Bdgoszcz Set (in olish. [7] Hacioglu K. Amca H.: Decision feedback eualiser based on fuzz logic. Electronics Letters vol. 35 no Ar [8] Labat J. Laot C.: Blind adative multile-inut decision-feedback eualizer with a selfotimized configuration. IEEE Transactions on Communications vol. 49 no Ar.. [9] Tsimbions J. White L. B.: Error roagation and recover in decision-feedback eualizers for nonlinear channels. IEEE Transactions on Communications vol. 49 no Feb.. [] Willink T. Wittke P. H. Cambell L. L.: Valuation of the effects of intersmbol interference in decision-feedback eualizers. IEEE Transactions on Communications vol. 48 no Ar..