Performance analysis of a RLS-based MLP-DFE in time-invariant and time-varying channels

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1 Dgtal Sgnal Processng 18 (2008) Performance analyss of a RLS-based MLP-DFE n tme-nvarant and tme-varyng channels Kashf Mahmood, Abdelmalek Zdour, Azzedne Zergune Electrcal Engneerng Department, Kng Fahd Unversty of Petroleum and Mnerals, Dhahran 31261, Saud Araba Avalable onlne 27 Aprl 2007 Abstract In ths work, a recently derved recursve least-square (RLS) algorthm to tran mult layer perceptron (MLP) s used n an MLP-based decson feedback equalzer (DFE) nstead of the back propagaton (BP) algorthm. Its performance s nvestgated and compared to those of MLP-DFE based on the BP algorthm and the smple DFE based on the least-mean square (LMS) algorthm. The results show mproved performance obtaned by the new structure n both tme-nvarant and tme-varyng channels. As wll be detaled n ths work, the newly proposed structure s a compromse between complexty and performance Elsever Inc. All rghts reserved. Keywords: Mult layer perceptron (MLP); Decson feedback equalzer (DFE); Least-mean square (LMS); Recursve least-square (RLS) 1. Introducton A serous lmtaton n attemptng to acheve a hgh transmsson rate through a partcular band-lmted channel s the tme dsperson suffered by the sgnal at the recevng end of ths channel [1]. In data transmsson, the tme dsperson mparted on the transmtted sgnal results n a tme overlap between successve symbols, known as ntersymbol nterference (ISI). Equalzers have been used to descrbe flters used to compensate for such dstortons n the ampltude and delay characterstcs of the channel. Nonlnear equalzers [1,2] are superor to lnear ones n applcatons where the channel dstorton s too severe for a lnear equalzer to handle. In partcular, a lnear equalzer does not perform well on channels wth deep spectral nulls n ther ampltude characterstcs or wth nonlnear dstorton. A decson feedback equalzer (DFE) s a nonlnear equalzer that s wdely used n stuatons where the ISI s very large. It has been proved theoretcally and expermentally that the DFE performs sgnfcantly better than a lnear equalzer of equvalent complexty [1]. The basc dea of DFE s that f the values of the symbols already detected are assumed to be correct, then the ISI contrbuted by these symbols can be canceled exactly by subtractng past symbol values wth approprate weghtng from the equalzer output [2]. To further enhance the performance of the DFE, the multlayer perceptron (MLP) has been ncorporated to the DFE. It s shown that the MLP-based DFE (MLP DFE) [3] and the MLP DFE wth lattce structure [4], usng the * Correspondng author. E-mal addresses: kashf_mahmood@hotmal.com (K. Mahmood), malek@kfupm.edu.sa (A. Zdour), azzedne@kfupm.edu.sa (A. Zergune) /$ see front matter 2007 Elsever Inc. All rghts reserved. do: /.dsp

2 308 K. Mahmood et al. / Dgtal Sgnal Processng 18 (2008) back propagaton algorthm (BP) [5], gve a sgnfcantly mproved performance over the smple DFE [2]. Moreover, the MLP DFE wth lattce structure [4] gves better performance over the MLP DFE, however, ts hgh computatonal complexty stll greatly lmts the applcatons. Snce the back propagaton algorthm s no more than a generalzed least-mean squares (LMS) algorthm [6], t then suffers from the same problems that the LMS algorthm exhbts, partcularly the relatvely slow rate of convergence when appled to channels wth spectral nulls n ther frequency responses. These channels are known to yeld a large egenvalue spread (ES) of the autocorrelaton matrx of the sgnal at ther outputs. To extend the applcablty of the MLP DFE to areas nvolvng fast tme-varyng channels, e.g., moble communcaton channels, the recursve least squares (RLS) algorthm [6] rather than the back propagaton algorthm s a necessary tool for the MLP DFE to be able to track varatons n these envronments. Moreover the effect of the egenvalue spread, encountered n the BP algorthm, wll be reduced substantally n both tme-nvarant and tme-varyng channels as the RLS algorthm s unaffected by ths factor. Also, t s known n adaptve flterng that the RLS algorthm s typcally an order of magntude faster than the LMS algorthm [6]. Eventually ths wll have a great mpact on the convergence behavor of the MLP DFE. A couple of RLS adaptve algorthms [7 9] desgned for the learnng of MLP were proposed. These two algorthms were later nvestgated by Peng [10] n equalzaton where she reported the drawbacks of these algorthms. Among them s, e.g., the RLS adaptve algorthm proposed by Scalero [9] for the learnng of MLP requres fndng the nverse of actvaton functon for each neuron whch s a real heavy load on the algorthm. Recently, a new RLS algorthm [11] s proposed n the learnng of the MLP but so far t has only been appled for XOR functon approxmaton problems. Ths algorthm, however, does not requre fndng the nverse of actvaton functon and moreover approxmate the values of true symbols. Consequently, the RLS algorthm proposed by [11], rather than the BP algorthm, s used here to update the MLP DFE even though the latter has a lower complexty than the former one. One of the most powerful algorthms for the tranng of feedforward networks s undoubtedly the Levenberg Marquardt (LM) method [12] whch combnes the excellent local convergence propertes of Gauss Newton method near a mnmum wth the consstent error decrease provded by (a sutably scaled) gradent descent far away from a soluton. The Levenberg Marquardt optmsaton algorthm [13] apples, smlarly as the conugate gradent method [13], numercally estmated nformaton from the second dervatve of the cost functon. A dsadvantage of the LM method, however, s ts ncreased memory requrements arsng from the demand to calculate the Jacoban matrx of the error functon and the need to nvert matrces wth dmensons equal to the number of the weghts of the neural network. Another dsadvantage orgnates from the fact that, snce LM s an unconstraned optmzaton method, t s not guaranteed to converge to the global mnmum of the cost functon, but t s globally convergent n the sense that t s guaranteed to converge to a mnmzer (local or global) of the cost functon where the necessary and suffcent condtons for optmalty hold. The ncreased memory requrements of the LM algorthm however render such a practce clearly unacceptable. Other newly derved technques to enhance further the performance of the back propagaton algorthm are found n [14,15]. In ths work, the performance of the MLP DFE usng the RLS algorthm [11] s evaluated and t wll be called MLP(RLS)-DFE algorthm. It s shown that a great mprovement n performance s obtaned through the use of ths technque over both the smple DFE based on the LMS algorthm and the MLP DFE n both tme-nvarant and tme-varyng channels. The rest of the paper s organzed as follows. In Secton 2, a bref revew of artfcal neural networks s gven. Secton 3 reports the RLS algorthm [11] along wth the proposed algorthm and the computatonal complexty of the proposed algorthm, whle ts performance s demonstrated by the smulaton results of Secton 4. Secton 5 s concerned wth the dscusson of the results and conclusons. 2. Artfcal neural networks Because of the capabltes of artfcal neural networks n effcently modelng arbtrary nonlneartes, there has been recent nterest n employng them n adaptve equalzaton for data communcaton channels [3,16,17]. In ths case, the lnear adaptve flter s replaced by a neural network. Dfferent artfcal neural network archtectures such as multlayer perceptron, radal bass functons, and recurrent neural networks have all been proposed n the lterature for channel equalzaton [18]. Among all these structures, the most commonly and wdely-used s the MLP structure. The

3 K. Mahmood et al. / Dgtal Sgnal Processng 18 (2008) popularty of MLP-based equalzers s due n part to ther computatonal smplcty, fnte parameterzaton, stablty, and smaller structure sze for a partcular problem as compared to other structures. A multlayer perceptron conssts of several hdden layers of neurons that are capable of performng complex, nonlnear mappngs between the nput and output layer. The hdden layers provde the capablty to use the nonlnear sgmod functon to create ntrcately-curved parttons of space wth complex nonlnear decson boundares [19]. Furthermore, t has been shown that only three layers are needed by the MLP to generate these boundares [20]. The basc element of the multlayer perceptron s the neuron. Each neuron n the layer has prmary local connectons and s characterzed by a set of real weghts [w 1,w 2,...,w N ] appled to the prevous layer to whch t s connected and a real threshold level I.Theth neuron n the pth layer accepts real nputs v (p 1) h (p 1)th layer and produces an output v (p) ( N ) v (p) = f h=1 w h v (p 1) h + I (p). (h = 1, 2,...,N) from the, whch s also a real scalar, expressed n the followng way: Ths output value v (p) serves also as nput to the (p + 1)th layer (next layer) to whch the neuron s connected. In the above expresson, f ( ) represents the nonlnearty functon. The most commonly used one n the perceptron s of the sgmod type, defned as [20] f (x) = 1 e x 1 + e x, (2) where f (x) s always n the range [ 1, 1], x R (the set of real numbers). The weghts {w h } and thresholds levels {I } are updated durng tranng [3]. The MLP dd not receve much attenton n applcatons untl the ntroducton of the BP algorthm [12]. The BP algorthm was used n both lnear equalzers [21] and nonlnear equalzers (DFE) [3], and t was found that n both cases, the neural network-based confguraton outperformed ts nonneural network-based counterpart. In ths work, however, only the MLP DFE [3] wll be consdered as t s more advantageous than ts lnear counterpart Learnng phase In the BP algorthm, the output value s compared wth the desred output, resultng n an error sgnal. Ths error sgnal s fed back through the network whose weghts are adusted to mnmze ths error. The ncrements used n updatng the weghts, w h, and threshold levels, I,ofthepth (p [1, 2,...,P]) layer are updated, respectvely, accordng to the followng relatons: ( + 1) = ηδ(p) and I (p) ( + 1) = βδ (p), where η s the learnng gan, α s the momentum parameter, and β s the threshold level adaptaton gan. The error sgnal δ (p) for layer p s calculated startng from the output layer P,as w (p) h v (p 1) + α w (p) h () (3) δ (P ) = (z v (P ) )(1 v 2(P ) ) 2 and s then recursvely back propagated to lower layers (p [1, 2,...,P 1]) accordng to δ (p) = (1 v 2(p) ) l δ (p+1) l w (p+1) l 2, where l s over all neurons above the neuron n the (p + 1)st layer and z s the desred output. To allow for a rapd learnng, a momentum term, w (p) h (), scaled by α, s used to flter out hgh frequency varatons of the weght vector. Consequently, the convergence rate s much faster and the fast weght changes are smoothed out. (1) (4) (5) (6)

4 310 K. Mahmood et al. / Dgtal Sgnal Processng 18 (2008) Fg. 1. Model of the th neuron n the kth layer. 3. RLS algorthm In the ensung, a detaled descrpton of the RLS algorthm developed n [11] used for the learnng of the MLP n the context of nonlnear equalzaton (DFE) s reported n ths work. For more detals about ths algorthm, the reader may refer to [11]. The error at the output layer of the neural network s calculated as ε L (n) = dl (n) yl (n), where y L(n) s the output of the output layer L, dl (n) and s the desred output or the target for the partcular output. For the rest of the layers, 1 k L 1, the error s calculated n the followng way: N ε (k) k+1 (n) = =1 f ( s (k+1) (n) ) w (k+1) (n)ε (k+1) (n), (8) where N k s the number of neurons n the kth layer, f s the dervatve of the actvaton functon already defned n (2) and reproduced here for more clarfcaton: f(x)= 1 e x 1 + e x, (9) where s (k) (n) s the lnear output of the th neuron n the kth layer, w (k) (n) s the weght of th neuron of the kth layer connectng ths neuron wth the th nput. Fgure 1 depcts the model of the th neuron n the kth layer. The weght vector of the th neuron, w (k) (n), s updated accordng to the followng recurson: w (k) (n) = w (k) (n 1) + g (k) (n)ε k (n), where g (k) (n) s the Kalman gan and s calculated n the followng way: g (k) (n) = f (s (k) (n))p (k) (n 1)x (k) (n) λ + f 2 (s (k) (n))x (k)t (n)p (k) (n 1)x (k) (n). (11) Fnally, x (k) (n) and λ are the vector of nput sgnals for the kth layer and the forgettng factor n the RLS algorthm, respectvely, and P (k) (n) s the nverse of the correlaton matrx defned by P (k) (n) = λ 1[ I f ( s (k) (n) ) g (k) (n)x (k)t (n) ] P (k) (n 1). (12) (7) (10)

5 K. Mahmood et al. / Dgtal Sgnal Processng 18 (2008) Proposed scheme Because the BP algorthm s no more than a generalzed least-mean squares (LMS) algorthm [6], t then suffers from the same problems as the LMS algorthm, partcularly the slow rate of convergence when appled to channels wth spectral nulls n ther frequency responses. These channels are known to yeld a large egenvalue spread of the autocorrelaton matrx of the sgnal at ther outputs. Ths knd of channel characterstcs s often encountered n tmevarant channels. Moreover, as shown n [4], the BP-based MLP DFE has an equal performance to that of the smple DFE n tme varyng channels. The MLP DFE equalzer based on the BP algorthm can then hardly be ustfed for the equalzaton of such channels. Eventually, a moderate confguraton n complexty and performance must be proposed to mprove the performance of the MLP DFE based on the BP algorthm and as well havng less complexty than that of MLP-DFE wth lattce structure. Therefore, a compromse between complexty and performance must be reached n order to solve ths conflct. The RLS algorthm proposed by Blsk and Rutkowsk [11] has not been used so far n the feld of equalzaton. In ths work, the RLS algorthm [11] s used to update the MLP nstead of the BP algorthm. As assessed n the smulaton results secton, great mprovement s obtaned through the use of ths technque n tme-nvarant and tmevaryng channels. Fgure 2 detals the MLP-based DFE where the RLS nstead of the BP s used to tran the MLP Computatonal complexty of the algorthms Most common operatons used n adaptve flterng algorthms are addtons, multplcatons and dvsons. When dgtally mplemented, the last two operatons are known to be more computatonally costly than the frst one and are very prone to causng nstablty specally n fxed-pont computatons. Even n systems equpped wth floatng-pont arthmetc, these two operatons can sometme lead to overflow, hence causng the algorthm to dverge. To avod ths problem of overflow, floatng-pont arthmetc s preferred. From a computatonal vewpont and whenever possble, algorthms nvolvng fewer multplcatons and dvsons should always be sought after, especally n applcatons nvolvng trackng n fast-changng envronments, e.g., wreless communcaton. Table 1 gves the computatonal complexty of the three algorthms used n ths work, namely, the smple DFE, the MLP DFE, and the MLP(RLS)-DFE. The notatons used n Table 1 are defned as follows: N 1 s the number of feedforward taps, N 2 s the number of feedback taps, (N 1 + N 2 ) s the number of nputs to the MLP, L 1 and L 2 are the numbers of neurons n the frst and second hdden layers of the MLP, respectvely, and fnally L 3 = 1 s the number of neurons n the output layer of the MLP. Table 1 Computatonal load for dfferent adaptve DFE algorthms used n ths work Total number of addtons and multplcatons DFE MLP DFE MLP(RLS)-DFE 2(N 1 + N 2 ) + 1 N 1 + N L L 2 16N N L L 2 + 5(N 1 + N 2 ) L 1 + 6L 1 L (N 1 + N 2 ) L 1 + 6L 1 L 2 18 Dvsons 0 2L 1 + 2L (N 1 + N 2 ) L 1 + 2L 1 L 2 Fg. 2. Block dagram of MLP(RLS)-DFE.

6 312 K. Mahmood et al. / Dgtal Sgnal Processng 18 (2008) From Table 1, we see that the LMS algorthm s the smplest of all and by usng the RLS algorthm nstead of the BP algorthm for the MLP DFE, a load of (15N N 2 27) computatons s added. 4. Smulaton results The performance of the new structure MLP(RLS)-DFE s compared to those of the LMS DFE (usng LMS algorthm) and MLP DFE. Both the LMS DFE and the MLP DFE structures use four samples n the feedforward secton and one sample n the feedback secton. For the latter structure, ths results n fve nput samples n ts nput layer. For the MLP DFE the number of neurons n the frst and second hdden layer, and the output layer are 9, 3, and 1, respectvely, whereas for the MLP-RLS DFE the number of neurons n the frst and the output layers are 9 and 1, respectvely. Ths s made so that both structures have comparable complexty. The back propagaton algorthm s used to update the MLP DFE where the learnng gan parameter η, momentum parameter α, and threshold level adaptaton gan β, have been chosen as n [3], namely 0.07, 0.3, and 0.05, respectvely. The step sze for the LMS algorthm used n the LMS DFE s The forgettng factor n the RLS algorthm, λ, s chosen to be The dgtal message appled to the channel s made of unformly dstrbuted bpolar random numbers ( 1, 1). The channel nose s taken to be Fg. 3. Sgnal constellaton: (a) at the nput of the equalzer for channel H 1 (z) under SNR = 10 db; (b) after equalzaton for channel H 1 (z); (c) at the nput of the equalzer for channel H 2 (z) under SNR = 10 db; (d) after equalzaton for channel H 2 (z).

7 K. Mahmood et al. / Dgtal Sgnal Processng 18 (2008) addtve whte Gaussan nose. The system performance wll be evaluated usng both tme-nvarant and tme-varyng channels, as detaled next System performance n tme-nvarant channels To assess the performance of the proposed structure, two tme-nvarant channel models are used n the smulaton and are descrbed by ther transfer functons H 1 (z) = z z 2 and H 2 (z) = z z 2 wth egenvalue spreads of 25 and 81, respectvely. Also, t should be ponted out here that the frst channel model matches the one used n [3], whle the second channel matches one of the tme-nvarant channel models used n [22]. Durng ths part of the smulatons, the performance measure s obtaned through the use of sgnal constellatons, eye dagrams and learnng curves. Sgnal constellatons and eye dagrams are plotted for channel H 1 (z) and H 2 (z). Fgures 3 and 4 show these dagrams for the MLP(RLS)-DFE equalzer for frst channel and second channel, respectvely, and wth a sgnal to nose rato (SNR) of 10 db. In Fg. 3, parts (a) and (c) show the unequalzed data for channels H 1 (z) and H 2 (z), respectvely, whle parts (b) and (d) depct ther respectve equalzed versons. A huge dfference n performance, as can be observed from these dagrams, s obtaned through the use of the proposed algorthm. Fgure 4 depcts the performance of the MLP (RLS)-DFE as far as the eye dagram s concerned. In summary, the equalzer s performance Fg. 4. Eye dagram: (a) at the nput of the equalzer for channel H 1 (z) under SNR = 10 db; (b) after equalzaton for channel H 1 (z); (c) at the nput of the equalzer for channel H 2 (z) under SNR = 10 db; (d) after equalzaton for channel H 2 (z).

8 314 K. Mahmood et al. / Dgtal Sgnal Processng 18 (2008) Fg. 5. Learnng curves of dfferent types of equalzers wth channel H 1 (z) = z z 2. s much mproved and the symbols are seen to converge closer to ther orgnal postons when the MLP (RLS)-DFE s used. The postve effect of the RLS algorthm s clear. In the case of the learnng curves, these are obtaned by averagng 600 ndependent runs. Each run has a dfferent random sequence and random startng weghts for the perceptron structure, and an SNR of 20 db s used. Fgure 5 depcts the convergence behavor of the three algorthms for the frst channel. Ths fgure shows a clear mprovement n both the convergence tme and the steady-state MSE when the MLP (RLS)-DFE algorthm s deployed. Ths result llustrates also that even though the MLP DFE confguraton converges more slowly than that of the smple DFE, t nevertheless results n a lower steady-state MSE value than that of the latter. It should also be clear from ths fgure that the steady-state MSE of both MLP DFE confguratons s below the nose level. Ths results from the nonlnear nature of the equalzer transfer functon [3]. Furthermore, the MLP DFE equalzer s capable of generatng hghly nonlnear decson regons, n contrast to the LMS DFE equalzer whch only forms a hyperplane decson boundary [17]. Also, t s generally understood n lnear sgnal processng that the l 2 norm error crteron produces a parabolc error surface wth no local mnma and has a contnuously dfferental nature. However, the l 2 norm error crteron for perceptrons wll not generally produce a parabolc error surface owng to ts nonlnear nature [23]. Smlarly here, usng the same reasonng one can reach the concluson that the least squares error crteron for perceptrons, whch the RLS algorthm s a member, s unlkely to produce a parabolc error surface. A smlar mprovement s also obtaned for the second channel despte ts larger egenvalue spread, as shown n Fg. 6. The dfference n convergence tme between the MLP DFE and the MLP DFE usng the RLS algorthm s now more pronounced. The nsenstvty of the RLS algorthm to the egenvalue spread s very clear. To further nvestgate the consstency n performance of the MLP(RLS)-DFE n the past scenaro, ts computatonal complexty s reduced almost by 50%, the number of neurons n the frst layer s now 5 nstead of 9. Agan as depcted n Fgs. 7 and 8, the performance of MLP(RLS)-DFE s stll better than those of the LMS DFE and the MLP DFE. Ths s ndcated n the fgures by the label MLP(RLS)-DFE (5) to dfferentate t from the MLP(RLS)-DFE (9) (the orgnal confguraton shown n Fgs. 5 and 6) System performance n tme-varyng channels In ths second part of the smulatons, a tme-varant channel s used to evaluate the capablty of the equalzer to track the changes n a tme-varyng dspersve channel. The dscrete-tme channel model for tme-varyng channel s

9 K. Mahmood et al. / Dgtal Sgnal Processng 18 (2008) Fg. 6. Learnng curves of dfferent types of equalzers wth channel H 2 (z) = z z 2. Fg. 7. Learnng curves of dfferent types of equalzers wth channel H 1 (z) = z z 2. descrbed by the followng transfer functon H(z)= a 0 (t) + a 1 (t)z 1 + a 2 (t)z 2, where a 0 (t), a 1 (t), and a 2 (t) are the tme-varyng coeffcents of the channel mpulse response. These are generated by passng whte Gaussan nose through a low-pass flter of a specfed bandwdth [24]. If we assume that we have a nomnal 3 khz HF channel, the sgnalng rate s 2400 symbols/s, and the low-pass flter s a two-pole Butterworth flter, then the 3-dB bandwdth of

10 316 K. Mahmood et al. / Dgtal Sgnal Processng 18 (2008) Fg. 8. Learnng curves of dfferent types of equalzers wth channel H 2 (z) = z z 2. Fg. 9. Tap coeffcents of tme-varant channel wth lowpass flter bandwdth of 0.5 Hz. the low-pass flter can be used as a parameter to control the rate of varaton of the channel mpulse response. The curves representng the tme varaton of the coeffcents are depcted n Fg. 9 for bandwdth of 0.5 Hz. Fgures 10 and 11 show the BER performance of the three equalzers for the tme varatons of the coeffcents for bandwdths of 0.1 and 0.5 Hz, respectvely. The results llustrate the superorty of the MLP(RLS)-DFE. In both

11 K. Mahmood et al. / Dgtal Sgnal Processng 18 (2008) Fg. 10. BER performance of dfferent types of equalzers for tme varyng channel wth lowpass flter bandwdth of 0.1 Hz. Fg. 11. BER performance of dfferent types of equalzers for tme varyng channel wth lowpass flter bandwdth of 0.5 Hz. fgures, the other two equalzers performances are not attractve. Moreover, as can be notced from these fgures that comparable performance s obtaned for both MLP DFE and LMS DFE confguratons. To further nvestgate the consstency n performance of the MLP(RLS)-DFE, the rate of varaton of the channel mpulse response s ncreased. Ths s done by ncreasng the bandwdth of the low-pass flter, e.g., bandwdth of 1.0 Hz. There s a deteroraton n the performance of both the LMS DFE and the MLP DFE and the dfference

12 318 K. Mahmood et al. / Dgtal Sgnal Processng 18 (2008) Fg. 12. BER performance of dfferent types of equalzers for tme varyng channel wth lowpass flter bandwdth of 1.0 Hz. between them and the MLP(RLS)-DFE s ncreased as clearly shown n Fg. 12. The MLP(RLS)-DFE attans lower error floor than both of the LMS DFE and the MLP DFE. The LMS DFE and the MLP DFE saturate after the SNR of 13 db to approxmately the same BER. Agan n ths case the MLP DFE and the LMS DFE have comparable performance. The saturaton effect (error-floor) for both LMS DFE and MLP DFE n Fgs s due manly to when SNR s larger than 15 db, nose has lttle nfluence on the BER, because at that tme, errors are manly caused by tap-gan lag due to the varaton of channels, ths s depcted n Fgs Fnally, as was detaled n Secton 3.2, the computatonal load of the MLP(RLS)-DFE s larger by far than those of the LMS DFE and MLP DFE. Therefore, t was made sure that both MLP confguratons have smlar computatonal load such that a far comparson s made. Moreover, even wth almost 50% decrease n computatonal complexty, the performance of the MLP (RLS)-DFE s much better than those of the LMS DFE and MLP DFE. 5. Concluson Ths work has presented the mprovements brought about by the RLS algorthm used for the frst tme n equalzaton to tran the MLP nstead of the BP algorthm. The computer smulatons have llustrated the better performance of the MLP(RLS)-DFE over both the MLP DFE and the LMS DFE n tme-nvarant and tme-varyng channels. The results of our study can be summarzed brefly as follows: 1. The use of the RLS algorthm nstead of the back propagaton algorthm for the MLP DFE results n substantal mprovements n terms of convergence rate, steady-state MSE and BER. 2. The convergence rate of the MLP(RLS)-DFE s nsenstve of the egenvalue spread of the channel correlaton matrx for tme-nvarant channels. 3. The proposed MLP(RLS)-DFE has better BER performance than both the LMS DFE and MLP DFE n tmevaryng channels. 4. The smulaton results ndcated that the MLP (RLS)-DFE s stable and no sgn of nstablty was shown for both tme-nvarant and tme-varyng channels. 5. Further work amed at applyng the proposed scheme wth respect to nonlneartes s currently beng pursued.

13 K. Mahmood et al. / Dgtal Sgnal Processng 18 (2008) Fnally, the mproved performance of the proposed structure s obtaned wthout any addtonal ncrease n complexty. Acknowledgment The authors would lke to acknowledge the support of KFUPM. References [1] S. Quresh, Adaptve equalzaton, Proc. IEEE 73 (9) (1985) [2] J.G. Proaks, Dgtal Communcatons, McGraw Hll, New York, [3] S. Su, G.J. Gbson, C.F.N. Cowan, Decson feedback equalzaton usng neural network structures and performance comparson wth standard archtecture, IEE Proc. 137 (1990) [4] A. Zergune, A. Shaf, M. Bettayeb, Multlayer perceptron-based DFE wth lattce structure, IEEE Trans. Neural Networks 12 (6) (2001) [5] R.P. Lppmann, An ntroducton to computng wth neural nets, IEEE ASSP Mag. 4 (1987). [6] S. Haykn, Adaptve Flter Theory, Prentce Hall, Englewood Clffs, NJ, [7] S. Azm, R.J. Lou, Fast learnng process of multlayer neural networks usng recursve least squares method, IEEE Trans. Sgnal Process. 40 (2) (1992) [8] J. Coloma, R.A. Carrasco, MLP equalzer for frequency selectve tme-varyng channels, Electron. Lett. 30 (6) (1994) [9] R.S. Scalero, A fast new algorthm for tranng feedforward neural networks, IEEE Trans. Sgnal Process. 40 (1992) [10] M. Peng, Neural networks applcatons n lnear and nonlnear channel equalzaton, Ph.D. dssertaton, Northeastern Unversty, Boston, MA, June [11] J. Blsk, L. Rutkowsk, A fast tranng algorthm for neural networks, IEEE Trans. Crcuts Syst. II 45 (6) (1998) [12] D.E. Rumelhart, J.L. McClelland, Parallel Dstrbuton Processng: Exploratons n the Mcrostructure of Cognton, vol. 1, MIT Press, Cambrdge, MA, [13] W.H. Press, S.A. Teukolsky, W.T. Vetterlng, B.P. Flannery, Numercal Recpes, second ed., Cambrdge Unv. Press, Cambrdge, [14] S. Halgamuge, L. Wang (Eds.), Computatonal Intellgence for Modelng and Predctons, Sprnger-Verlag, Berln, [15] L. Wang (Ed.), Soft Computng n Communcatons, Sprnger-Verlag, Berln, [16] S. Chen, G.J. Gbson, C.F.N. Cowan, P.M. Grant, Adaptve equalzaton of fnte nonlnear channels usng multlayer perceptrons, Sgnal Process. 20 (1990) [17] G.J. Gbson, S. Su, C. Cowan, The applcaton of nonlnear structures to the reconstructon of bnary sgnals, IEEE Trans. Sgnal Process. 39 (1991) [18] B. Mulgrew, Applyng radal bass functons, IEEE ASSP Mag. 13 (1996) [19] A. Weland, R. Leghton, Geometrc analyss of neural network capabltes, n: 1st Internatonal Conference on Neural Networks, June 1987, pp [20] S. Haykn, Neural Networks: A Comprehensve Foundaton, Macmllan Co., New York, [21] G.J. Gbson, S. Su, C. Cowan, Multlayer perceptron structures appled to adaptve equalzers for data communcatons, IEEE Proc. ICASSP, Glasgow, May 1989, pp [22] F. Lng, J.G. Proaks, Adaptve lattce decson-feedback equalzers: Ther performance and applcaton to tme-varant multpath channels, IEEE Trans. Commun. COM 33 (1985) [23] S. Su, C.F.N. Cowan, Performance analyss of the l p norm back propagaton algorthm for adaptve equalsaton, IEE Proc. F 140 (1) (1993) [24] F. Lng, J.G. Proaks, Lattce decson feedback equalzers and ther applcaton to fadng dspersve channels, n: Proc. Int. Conf. Comm., Boston, MA, June 1983, pp. C8.2.1 C Kashf Mahmood receved the B.Sc. degree from Pakstan n 1997, and the M.Sc. degree from Kng Fahd Unversty of Petroleum & Mnerals (KFUPM), Dhahran, Saud Araba, n 2000, n electrcal engneerng. Snce 2000, he s a Lecturer at Hafr Al-Batn Communty College. Mr. Mahmood research nterests nclude sgnal processng for communcatons, neural networks, and local area networks. Dr. Abdelmalek Zdour s a Senor IEEE member and an Assstant Professor n the Department of Electrcal Engneerng at Kng Fahd Unversty of Petroleum and Mnerals, Dhahran, Saud Araba. He holds a Master of Scence n control engneerng from Bradford Unversty, UK, n 1984, and a Doctor of Engneerng n appled electroncs from Tokyo Insttute of Technology Japan, n Hs research nterests are n the feld of sgnal processng and pattern recognton. In partcular character recognton and document mage analyss. Dr. Zdour has publshed many refereed ournal and conference papers. He has supervsed many senor proects and M.Sc. theses. He s currently the head of the Dgtal Sgnal Processng Group at the EE Department. He s a member of Engneerng Educaton Socety, Sgnal Processng Socety, and Communcatons Socety.

14 320 K. Mahmood et al. / Dgtal Sgnal Processng 18 (2008) Azzedne Zergune receved the B.Sc. degree from Case Western Reserve Unversty, Cleveland, OH, n 1981, the M.Sc. degree from Kng Fahd Unversty of Petroleum & Mnerals (KFUPM), Dhahran, Saud Araba, n 1990, and the Ph.D. degree from Loughborough Unversty, Loughborough, UK, n 1996, all n electrcal engneerng. From 1981 to 1987, he was workng wth dfferent Algeran state owned companes. Durng the perod from 1987 to 1990, he was a research and teachng assstant, Electrcal Engneerng Department, KFUPM. Dr. Zergune s presently an Assocate Professor, Electrcal Engneerng Department, KFUPM, workng n the areas of sgnal processng and communcatons. Dr. Zergune s a Senor Member of IEEE and currently he s servng as an assocate edtor wth the EURASIP Journal on Advances n Sgnal Processng. Dr. Zergune research nterests nclude sgnal processng for communcatons, adaptve flterng, neural networks, multuser detecton, and nterference cancellaton.