1 Voltrra Krnl Estimation for Nonlinar Communication Channls Using Dtrministic Squncs Endr M. Ekşioğlu and Ahmt H. Kayran Dpartmnt of Elctrical and Elctronics Enginring, Istanbul Tchnical Univrsity, Istanbul, 34469, Turky. E-mail: {ndr, kayran}@hb.itu.du.tr Abstract: W prsnt a nw xact mthod for th idntification of communication channls with nonlinaritis. Th channl is modlld as a third-ordr discrt Voltrra filtr and th Voltrra krnls ar masurd using dtrministic input squncs and th corrsponding channl outputs. Th solutions ar in closd form, xact and krnls ar stimatd in a nonrcursiv mannr, thus liminating rror propagation. Complx inputs and complx krnls ar allowd, which prmits th us of PSK or QAM modulatd signals for th idntification of th bandpass Voltrra channl. Simulation xampls and comparison with othr mthods in th litratur ar providd to vrify th mthod. Ky words: Nonlinar channl idntification, Voltrra systms, nonlinar systms
2 I. INTRODUCTION Nonlinar channl idntification is important in mitigating th ffcts of nonlinar distortions and for th qualization of th nonlinar communication channls. Th prformanc of th compnsation fforts for th channl nonlinaritis ar highly dpndnt on th accuracy of th nonlinar channl stimat. Hnc, nonlinar systm and channl idntification has bn a subjct of significanc [1], [2]. Th Voltrra sris rprsntation has bn widly utilizd to dscrib th input-output rlationship of nonlinar systms. Th truncatd (or doubly finit ) Voltrra filtr rprsntation can provid an approximation for a larg class of nonlinar systms, and it is appropriat for modlling th nonlinaritis ncountrd in communication systms. Hnc, th charactrization of nonlinar communication channls as finit ordr discrt Voltrra filtrs has bn studid in th litratur. Th dtrmination of th Voltrra krnls of th nonlinar communication systm modl has bn largly basd on th us of input-output rlations for random inputs. In th cas of zro-man Gaussian input, th closd form stimats for th Voltrra krnls can b formd using th cross cumulants btwn th input and th output [3]. Howvr, th Gaussian assumption is far ftchd for most communication applications. Morovr, th stimats ar found in a squntial mannr, which causs rror propagation btwn krnl stimats. In [4] PSK (phas shift kying) modulatd input signals, which hav vanishing highr ordr momnts, hav bn studid as probing signals for nonlinar channls. A similar mthod has bn givn in [5] for mor gnral input squncs. Anothr Voltrra krnl stimation algorithm basd on cyclostationarity of communication signals is prsntd in [6] whr th krnls for a bandpass nonlinar systm ar idntifid undr i.i.d. M-QAM (quadratur amplitud modulation) and M-PSK inputs. All of th mthods prsntd dpnd on highrordr momnts and cross corrlations btwn input and output squncs. In this work, w focus on dtrministic xcitation squncs for th idntification of nonlinar channls modlld using th third-ordr truncatd Voltrra sris rprsntation. Th algorithm dvlopd in this papr is basd on th novl Voltrra systm idntification mthod prsntd in [7]. Howvr, in [7] only ral input data was considrd. Hr, w will allow for complx inputs and complx Voltrra krnls. Our algorithm provids xact, closd form solutions for th Voltrra krnls. Th rsults do not dpnd on statistics of random squncs; rathr th algorithm utilizs spcially dsignd dtrministic squncs. Thrfor, th algorithm shows bttr prformanc than th abov random-input basd mthods vn
3 for input squncs of shortr lngth. Th krnls ar stimatd sparatly, thus liminating rror propagation amongst th stimats. Complx inputs ar allowd, hnc PSK or QAM modulatd communication signals can b utilizd as th dtrministic input squnc. In th contxt of a narrowband communication channl, th nonlinar systm gts modld by a bandpass Voltrra sris [6], [8]-[1]. W will modify our algorithm for th idntification of bandpass Voltrra channls and prsnt th corrsponding simulation rsults. II. NOVEL VOLTERRA FILTER REPRESENTATION Th third-ordr nonlinar channl modl can b givn as: N N N y(n) = N [x(n)] = b 1 (i 1 ) x(n i 1 ) + b 2 (i 1, i 2 ) x(n i 1 )x(n i 2 )+ i 1 = i 1 = i 2 =i 1 N N N b 3 (i 1, i 2, i 3 ) x(n i 1 )x(n i 2 )x(n i 3 ) i 3 =i 2 i 1 = i 2 =i 1 whr N is th mmory lngth of th nonlinar systm and b k (i 1, i 2,..., i k ) is th Voltrra krnl of dgr k [1]. W introduc a nw rprsntation for th Voltrra systm by rarranging th Voltrra krnls. W rformulat th third-ordr discrt Voltrra systm in trms of th multivariat cross-trm complxity. Th output y(n) can b considrd as th sum of th outputs of 3 diffrnt multivariat cross-trm nonlinar subsystms, H (l) ; that is 3 3 y(n) = y (l) (n) = H (l) [x(n)] (2a) l=1 l=1 whr N H (1) [x(n)] = h (1)T (i) x (1) (n i) (2b) i= N q N 1 H (2) [x(n)] = h (2)T (q 1 ; i) x (2) (q 1 ; n i) (2c) N 1N q 1 H (3) [x(n)] = q 1 =1 i= N q 1 q 2 q 1 =1 q 2 =1 i= (1) h (3)T (q 1, q 2 ; i) x (3) (q 1, q 2 ; n i) (2d) H (l) [ ] is calld as an l-d cross-trm Voltrra oprator, h (l) is calld as an l-d krnl vctor and x (l) is calld as an l-d input vctor. Hr l-d dnots th cross-trm complxity of th input vctor corrsponding to th krnl vctor, rathr than th litral dimnsion of th vctors. It is possibl to giv th l-d krnl vctors and th corrsponding input vctors in trms of th Voltrra krnls and th input signal x(n), rspctivly: h (1) (i) = [b 1 (i) b 2 (i, i) b 3 (i, i, i)] T (3a)
4 x (1) (n) = [ x(n) x 2 (n) x 3 (n) ] T b 2 (i, i + q 1 ) h (2) (q 1 ; i) = b 3 (i, i, i + q 1 ) b 3 (i, i + q 1, i + q 1 ) x(n i)x ( n (i + q 1 ) ) x (2) (q 1 ; n) = x ( n i ) 2 ( x n (i + q1 ) ) x ( n i ) x ( n (i + q 1 ) ) 2 [ h (3) ( ) ] (q 1, q 2 ; i) = b 3 i, i + q2, i + q 1 + q 2 [ x (3) (q 1, q 2 ; n) = x(n i)x ( n (i + q 2 ) ) x ( n (i + q 1 + q 2 ) ) ] (3b) (4a) (4b) (5a) (5b) whr th limits for q 1, q 2 and i ar as givn in (2). W can s that th l-d krnls corrspond to an input vctor, which can b factord into powrs of l distinct input trms with l-distinct dlays. Hnc by cross-trm complxity w considr th numbr of distinct input powrs utilizd in forming th trms of th input vctors and group th krnls accordingly. In th rgular Voltrra rprsntation (1) on th othr hand, th Voltrra krnls ar groupd togthr using th dgr of th nonlinarity of th corrsponding input trms. Th introducd novl rprsntation nabls us to dvis an xact closd form algorithm for idntifying th Voltrra krnls of a third-ordr Voltrra systm utilizing dtrministic input squncs. III. IDENTIFICATION OF THE VOLTERRA KERNELS USING DETERMINISTIC INPUT SIGNALS In this sction, w driv an fficint algorithm to idntify th krnl vctors h (l), l = 1, 2, 3 dfind in (2) by using dtrministic input squncs with 3 distinct lvls (othr than zro). All krnl vctors can b dtrmind by using only th output of th systm. A. Idntification of th 1-D Krnl Vctors W will prov that an nsmbl of thr input squncs composd of singl impulss with distinct valus, x (1) (j; n) = a j δ(n), for j = 1, 2, 3 is adquat to obtain th 1-D krnl vctors in (3a). Looking at th cross-trm rprsntation in (2), w can s that for ths singl impulss th outputs of th 2-D and 3-D subsytms ar zro, i.., H (2) [a j δ(n)] =
5 and H (3) [a j δ(n)] =. Hnc, th output of th ovrall nonlinar systm whn th input is x (1) (j; n) is givn by whr N [ x (1) (j; n) ] = H (1)[ x (1) (j; n) ] = v (1,1) (j; n) = N h (1)T (i)u (1) (j; n i) i= (6a) u (1) (j; n) = [ a j a 2 j a 3 j] T δ(n) (6b) W can writ th thr output squncs togthr in th matrix form as follows: y (1) (n) = H (1) [x (1) ] = N i= U (1) (n i) h (1) (i) (7) whr y (1) (n), x (1) (n) and U (1) (n) dnot th nsmbl output vctor, nsmbl input vctor and th input matrix, rspctivly: v (1,1) (1; n) y (1) (n) = v (1,1) (n) = v (1,1) (2; n) ; v (1,1) (3; n) x(1) (n) = a 1 δ(n) a 2 δ(n) ; a 3 δ(n) U(1) (n) = u (1)T (1; n) u (1)T (2; n) u (1)T (3; n) (8) Hr u (1) (j; n) is as dfind in (6b). W can rplac U (1) (n i) in (7) with U (1) δ(n i), whr th matrix U (1) W can rwrit (7) as is givn as y (1) (n) = N i= U (1) = It follows that if th invrs of th matrix U (1) vctors as Th input matrix U (1) a 1 a 2 1 a 3 1 a 2 a 2 2 a 3 2 a 3 a 2 3 a 3 3 U (1) h (1) (i) δ(n i) = U (1) h (1) (n) (1) (9) xists, w can dtrmin th 1-D krnl h (1) (n) = [ ] 1y U (1) (1) (n) for n =, 1,..., N (11) in (9) is invrtibl if th lvls of th singl impuls input squncs ar distinct and nonzro, i.., a i and a i a j, i j.
6 B. Idntification of th 2-D Krnl Vctors W form th following nsmbl of input squncs which consist of two impulss with distinct amplituds. Th impulss ar sparatd by q 1. x (2)( (1, 2), q 1 ; n ) x (2) (q 1 ; n) = x (2)( (1, 3), q 1 ; n ) a 1 δ(n) + a 2 δ(n q 1 ) x (2)( (2, 3), q 1 ; n ) = a 1 δ(n) + a 3 δ(n q 1 ) a 2 δ(n) + a 3 δ(n q 1 ) Ths squncs will only xcit th 2-D subsystm H (2) and th 1-D subsystm H (1). Th output nsmbl of th ovrall nonlinar systm can b writtn as a sum of th outputs for H (2) and H (1). y (2) (q 1 ; n) = H (1)[ x (2) ( q1 ; n )] + H (2)[ x (2) ( q1 ; n )] = v (2,1) (12) ( q1 ; n ) ( + v (2,2) q1 ; n ) (13) Using (2b) and (12), it is not difficult to show that th rspons of th 1-D systm to th 2-D input nsmbl can b dcomposd in trms of th 1-D rsponss as a 1 δ(n) a 2 δ(n q 1 ) v (2,1) (q 1 ; n) = H (1) a 1 δ(n) + H(1) a 3 δ(n q 1 ) a 2 δ(n) a 3 δ(n q 1 ) v (1,1) (1; n) v (1,1) (2; n q 1 ) = v (1; n) + v (3; n q 1 ) v (1,1) (2; n) v (1,1) (3; n q 1 ) Not that all th trms in (14) ar output squncs which wr found in sction III.A in (6). W can obtain th rspons of th 2-D subsystm alon by subtracting th rspons of th 1-D subsystm from th nonlinar systm output vctor. v (2,2)( (1, 2), q 1 ; n ) v (2,2) (q 1 ; n) = v (2,2)( (1, 3), q 1 ; n ) v (2,2)( (2, 3), q 1 ; n ) = y(2) (q 1 ; n) v (2,1) (q 1 ; n) (15) On th othr hand using (2c) w can writ th 2-D subsystm output for our nsmbl input as, (14) v (2,2) (q 1 ; n) = U (2) h (2) (q 1 ; n q 1 ) (16) U (2) is a matrix which is givn as U (2) = a 1 a 2 a 1 a 2 2 a 2 1a 2 a 1 a 3 a 1 a 2 3 a 2 1a 3 a 2 a 3 a 2 a 2 3 a 2 2a 3 (17)
7 W can idntify th 2-D Voltrra krnl vctors by using (15) and (16), for q 1 = 1,..., N and n = q 1, q 1 + 1,..., N. h (2) (q 1 ; n q 1 ) = [ ] 1v U (2) (2,2) (q 1 ; n) (18) C. Idntification of th 3-D Krnl Vctors Now w apply an input squnc which includs thr distinct impulss to th nonlinar systm. x (3) (q 1, q 2 ; n) = a 1 δ(n) + a 2 δ(n q 2 ) + a 3 δ(n q 1 q 2 ) (19) Th output of th ovrall nonlinar systm can b writtn as th sum of th outputs of th individual subsystms, H (1), H (2), and H (3), y (3) (q 1, q 2 ; n) = 3 v (3,i) (q 1, q 2 ; n) (2) i=1 Th output of th 1-D subsystm for th thr impuls input x (3) (q 1, q 2 ; n) can b writtn as v (3,1) (q 1, q 2 ; n) = v (1,1) (1; n) + v (1,1) (2; n q 2 ) + v (1,1) (3; n q 1 q 2 ) (21) Th output of th 2-D subsystm for th thr impuls input x (3) (q 1, q 2 ; n) can b writtn as v (3,2) (q 1, q 2 ; n) = v (2,2)( (1, 2), q 2 ; n ) +v (2,2)( (1, 3), q 1 +q 2 ; n ) +v (2,2)( (2, 3), q 1 ; n q 2 ) (22) Not that all th trms for th abov subsytm output squncs ar prviously obsrvd and calculatd in th idntification of th 1-D and 2-D krnl vctors. Th output for th 3-D subsystm can b lft alon by subtracting th rsponss of th 1-D and 2-D subsytms from th ovrall systm output. v (3,3) (q 1, q 2 ; n) = y (3) (q 1, q 2 ; n) v (3,1) (q 1, q 2 ; n) v (3,3) (q 1, q 2 ; n) (23) On th othr hand, in a similar fashion to th quations for th 1-D and 2-D subsystms ( ) (3,3) (1), (16), th output of th 3-D subsystm v (q 1, q 2 ; n) can b writtn as, whr U (3) = a 1 a 2 a 3. Hnc, w gt v (3,3) (q 1, q 2 ; n) = U (3) h (3) (q 1, q 2 ; n q 1 q 2 ) (24) h (3) (q 1, q 2 ; n q 1 q 2 ) = [ ] 1v U (3) (3,3) (q 1, q 2 ; n) (25) for q 1 = 1,..., N 1, q 2 = 1,..., N q 1 and n = q 1 + q 2, q 1 + q 2 + 1,..., N.
8 Fig. 1 dpicts th idntification of th Voltrra krnls using th proposd algorithm. Th matrics T and S shown in this figur ar utilizd to form th input nsmbls and to choos th past output nsmbls which gt subtractd as in (15) and (23). Th T matrics ar givn as, 1 1 T 2,1 = 1 ; T 2,2 = 1 ; 1 1 [ ] [ ] T 3,1 = 1 ; T 3,2 = 1 ; T 3,3 = Th S matrics ar givn as 1 1 S 2,1 = 1 1 ; S 3,1 = 1 1 [ ] 1 [ ] [ ] 1 1 1 ; S 3,2 = 1 1 1 Gnral closd form rcursiv algorithms to calculat th matrics T and S from scratch can b found in [7]. Using th approach in [7], th idntification algorithm dvlopd hr can b xtndd to th idntification of systms with nonlinaritis highr than third ordr. Exampl: W considr th third ordr nonlinar systm with N = 2 givn by y(n) = x(n) + 2x(n 1) x 2 (n) 4x(n)x(n 1) 2x 2 (n 1) + 3x 3 (n)+ 4x 2 (n)x(n 1) + 5x(n)x 2 (n 1) 3x 3 (n 1) 5x(n)x(n 2) + 6x(n)x(n 1)x(n 2) W want to find th Voltrra krnls using th mthod outlind in this sction. First w ar intrstd in calculating th 1-D krnl vctors h (1) () = [1 1 3] T and h (1) (1) = [2 2 3] T from th input and th output. W choos th impuls lvls a 1, a 2, a 3 from th QPSK signal st j(2πk/4), k =, 1, 2, 3. Hnc, w dfin th 1-D input nsmbl to th systm as (26) 1 x (1) (n) = 1 δ(n) (27) j Th 1-D output nsmbl from (8) bcoms 3 3 y (1) (n) = 5 δ(n) + 1 δ(n 1) (28) 1 2j 2 + 5j
9 From (9) and (27), th matrix U (1) U (1) = is writtn as, 1 1 1 1 1 1 j 1 j (29) Applying (28) and (29) to (11) yilds, h (1) () = [ ] 3 1 U (1) 1 5 = 1 ; and h(1) (1) = [ U (1) 1 2j 3 ] 1 3 2 1 = 2 2 + 5j 3 (3) Nxt, w want to dtrmin th 2-D krnl vctors h (2) (1; ) = [ 4 4 5] T and h (2) (2; ) = [ 5 ] T of th nonlinar systm givn in (26). Using th 1-D input nsmbl vctor x (1) (n) in (27), th 2-D input nsmbls in (12) can b writtn as 1 1 x (2) (1; n) = 1 δ(n) + j δ(n 1) (31) 1 j 1 1 x (2) (2; n) = 1 δ(n) + j δ(n 2) (32) 1 j For th 2-D input nsmbl in (31), th output of th nonlinar systm in (26) is calculatd as y (2) (1; n) = 3 5 1 3 δ(n) + 6 j δ(n 1) + 2 + 5j δ(n 2) (33) 5 4 + 7j 2 + 5j It is possibl to dtrmin th rspons of th 1-D subsystm to th 2-D input nsmbl in (31). To accomplish this w us th 1-D output nsmbl givn in (28) and (14). 3 8 1 v (2,1) (1; n) = 3 δ(n) + 2 2j δ(n 1) + 2 + 5j δ(n 2) (34) 5 2j 2 + 5j
1 Using this rsult, th rspons of th 2-D subsystm can b obtaind by subtracting v (2,1) (1; n) from th nonlinar systm output y (2) (1; n) in (33). v (2,2) (1; n) = y (2) (1; n) v (2,1) (1; n) 3 (35) = δ(n) + 4 + j δ(n 1) + δ(n 2) 4 + 9j Th dsird 2-D Voltrra krnl h (2) (1; ) can b calculatd by substituting (35) into (18) = h (2) (1; ) = [ ] 1v U (2) (2,2) (1; 1) 1 1 1 1 3 j 1 j 4 + j = j 1 j 4 + 9j 4 4 5 (36) whr th constant matrix U (2) is obtaind from x (1) (n) and (17). For th 2-D input nsmbl in (32), th output of th nonlinar systm is calculatd as 3 3 1 y (2) (2; n) = 3 δ(n) + 3 δ(n 1) + 1 7j δ(n 2) + 2 + 5j δ(n 3) (37) 5 1 1 + 3j 2 + 5j It is possibl to dtrmin th rspons of th 1-D subsystm to th 2-D input nsmbl in (32). To accomplish this w us th 1-D output nsmbl givn in (28) and (14). 3 3 5 1 v (2,1) (2; n) = 3 δ(n) + 3 δ(n 1) + 1 2j δ(n 2) + 2 + 5j δ(n 3) (38) 5 1 1 2j 2 + 5j Using this rsult, th rspons of th 2-D subsystm can b obtaind by subtracting v (2,1) (2; n) from th nonlinar systm output y (2) (2; n) in (37). v (2,2) (2; n) = y (2) (2; n) v (2,1) (2; n) 5 (39) = δ(n) + δ(n 1) + 5j δ(n 2) + δ(n 3) 5j Th dsird 2-D Voltrra krnl h (2) (2; ) can b calculatd by substituting (39) into (18) h (2) (2; ) = [ ] 1v U (2) (2,2) (2; 2)
11 = 1 1 1 j 1 j j 1 j 1 5 5j = 5j 5 (4) Finally, w want to dtrmin th 3-D krnl h (3) (1, 1; ) = 6 of th nonlinar systm. Th 3-D input can b writtn as x (3) (1, 1; n) = 1 δ(n) + ( 1) δ(n 1) + j δ(n 2) (41) For th 3-D input nsmbl in (41), th output of th nonlinar systm in (26) is calculatd as y (3) (1, 1; n) = 3δ(n) 5δ(n 1) + (4 4j)δ(n 2) + (2 + 5j)δ(n 3) (42) It is possibl to dtrmin th rspons of th 1-D subsystm to th 3-D input nsmbl in (41). Th output of th 1-D subsystm is found by utilizing (21) and th 1-D output nsmbl givn in (28). v (3,1) (1, 1; n) = 3δ(n) 8δ(n 1) 2jδ(n 2) + (2 + 5j)δ(n 3) (43) It is also possibl to dtrmin th rspons of th 2-D subsystm to th 3-D input in (41). W find th rspons of th 2-D subsystm by using (22) and th 1-D output nsmbl givn in (28). v (3,2) (1, 1; n) = 3δ(n) 8δ(n 1) + 2jδ(n 2) + (2 + 5j)δ(n 3) (44) Using ths rsults, th rspons of th 3-D subsystm alon can b obtaind by subtracting v (3,1) (1, 1; n) and v (3,2) (1, 1; n) from th nonlinar systm output y (3) (1; n) in (42). v (3,3) (1, 1; n) = y (3) (1, 1; n) v (3,1) (1; n) v (3,2) (1; n) = 6jδ(n 2) (45) Th dsird 3-D Voltrra krnl h (3) (1, 1; ) can b calculatd by substituting (45) into (25): h (3) (1, 1; ) = [ ] 1v U (3) (3,3) (1, 1; 2) = ( j) 1 ( 6j) = 6 Hr, th constant matrix U (3) is calculatd as U (3) = a 1 a 2 a 3.
12 D. Lngth of th Rquird Probing Signal W will giv an uppr-bound for th lngth of th ovrall input squnc, which should b applid to idntify th Voltrra krnls of th nonlinar channl. W considr th third-ordr Voltrra filtr with mmory lngth N as th channl modl. W assum th 1-D, 2-D and 3-D input nsmbls constituting th input signal ar applid srially to a singl nonlinar systm box. W put a guarding intrval of lngth N with all zro lvls at th nd of ach of th individual input nsmbls, x (1) (j; n), x (2)( (i, j), q 1 ; n ), and x (3) (q 1, q 2 ; n). This guarding intrval nsurs to flush out that input nsmbl from th mmory of th nonlinar systm and prpars th nonlinar systm for th nxt nsmbl by claring th mmory. Undr ths assumptions, w calculat th total lngth of th input squnc by summing th lngth of all th ncssary 1-D, 2-D and 3-D input signals and adding N to ach of thm. Hnc, th total input lngth is givn by L = 3(N + 1) + 3 N (q 1,2 + N + 1) + q 1,2 =1 N 1 N q 1,3 q 1,3 =1 q 2,3 =1 q 1,3 + q 2,3 + N + 1 (46) Hr, q 1,2 taks valus btwn 1 and N as suggstd by (2c). Similarly, th sum q 1,3 + q 2,3 taks valus btwn 2 and N, as suggstd by (2d). Hnc, both of ths trms ar upprboundd by N, and w can writ from (46) L 3(N + 1) + 3N(2N + 1) + (N 2 N) (2N + 1) ( ) 2 N + 3 L (2N + 1) 3N 2 Thrfor, th rquird lngth of our total input squnc is uppr-boundd by (2N + 1) ( ) N+3 2 3N. IV. IDENTIFICATION OF BANDPASS VOLTERRA CHANNELS Th bandpass Voltarra sris is mployd in th basband rprsntation of narrow-band communication systms. For bandpass communication signals, whr th carrir frquncy is much largr than th modulatd channl bandwidth, th complx nvlop of th nonlinar channl output signal gts dscribd by a bandpass Voltrra sris rathr than th rgular Voltrra filtr rprsntation as in (1). Th vn-ordr trms in th rgular rprsntation diappar, sinc thy gnrat spctral componnts which fall outsid th channl bandwidth and hnc can b filtrd by bandpass filtr [1]. Th bandpass Voltrra filtr including (47)
13 nonlinaritis up to third ordr is givn as: N N N N y(n) = b 1 (i 1 )x(n i 1 ) + b 3 (i 1, i 2, i 3 )x (n i 1 )x(n i 2 )x(n i 3 ) (48) i 3= i 1 = i 1= i 2= Hr, () dnots complx conjugation. N is th mmory lngth of th bandpass nonlinar systm. b 1 (i 1 ) and b 3 (i 1, i 2, i 3 ) ar th complx-valud linar and cubic bandpass Voltrra krnls, rspctivly [6]. W can asily modify th idntification mthod w dvlopd for th rgular Voltrra filtr to th bandpass Voltrra channl cas. W will first rformulat th input-output rlationship for th bandpass Voltrra filtr. Th nw rprsntation will b similar to (2). Th output y(n) for th bandpass Voltrra filtr in (49) can b considrd as th sum of th outputs of thr diffrnt nonlinar subsystms, H (l) ; that is 3 3 y(n) = y (l) (n) = H (l) [x(n)] (49a) l=1 l=1 whr N H (1) [x(n)] = h (1)T (i) x (1) (n i) (49b) i= N q N 1 H (2) [x(n)] = h (2)T (q 1 ; i) x (2) (q 1 ; n i) (49c) N 1N q 1 H (3) [x(n)] = q 1 =1 i= N q 1 q 2 q 1 =1 q 2 =1 i= h (3)T (q 1, q 2 ; i) x (3) (q 1, q 2 ; n i) (49d) Th krnl vctors and th input vctors utilizd in this rprsntation can b givn in trms of th bandpass Voltrra systm krnls (49) and th input signal x(n), rspctivly. h (1) (i) = [b 1 (i) b 3 (i, i, i)] T (5) x (1) (n) = [ x(n) x(n) 2 x(n) ] T b 3 (i, i, i + q 1 ) h (2) b 3 (i + q 1, i + q 1, i) (q 1 ; i) = b 3 (i, i + q 1, i + q 1 ) b 3 (i + q 1, i, i) x(n i) 2 x ( n (i + q 1 ) ) x (2) x ( n (i + q 1 ) ) 2 x(n i) (q 1 ; n) = x(n i) x 2( n (i + q 1 ) ) x ( n (i + q 1 ) ) x 2 (n i) (51) (52) (53)
[ ] h (3) (q 1, q 2 ; i) = b 3 (i, i + q 2, i + q 1 + q 2 ) (54) [ x (3) (q 1, q 2 ; n) = x(n i) x ( n (i + q 2 ) ) x ( n (i + q 1 + q 2 ) ) ] (55) Hr, th limits for q 1, q 2 and i ar as givn in (49). This rprsntation nabls us to form an algorithm for idntifying th bandpass Voltrra krnls using dtrministic squncs. Th algorithm as w dtaild in Sction III can b usd again for idntification, howvr this tim for th bandpass Voltrra systm krnls. Th sol diffrnc will b in th matrics U (1), U (2) and U (3). For th idntification of th bandpass Voltrra channl, ths matrics will b givn as U (1) = a 1 a 1 2 a 1 a 2 a 2 2 a 2 a 1 2 a 2 a 2 2 a 1 a 1a 2 2 a 2a 2 1 U (2) a 1 2 a 3 a 3 2 a 1 a 1a 2 3 a 3a 2 1 = a 1 2 a 4 a 4 2 a 1 a 1a 2 4 a 4a 2 1 14 (56) (57) a 2 2 a 3 a 3 2 a 2 a 2a 2 3 a 3a 2 2 a 1a 2 a 3 a 2a 1 a 3 a 3a 1 a 2 U (3) = a 1a 2 a 4 a 2a 1 a 4 a 4a 1 a 2 a 2a 3 a 4 a 3a 2 a 4 a 4a 2 a 3 (58) Othr than ths changs in th utilizd matrics, th algorithm as dtaild in Sction III works also for th idntification of th bandpass Voltrra channl. V. SIMULATIONS W prsnt two numrical xampls to illustrat th prformanc of our novl idntification procdur. Exampl 1: W simulat a linar-quadratic-cubic Voltrra filtr with mmory lngth N = 2. W us QPSK modulatd signals as th input, whr th dtrministic input lvls ar chosn from th st 4 j(2πk/4+π/4), k =, 1, 2, 3 Additiv indpndnt GWN obsrvation nois with unit varianc is prsnt. Our rquird dtrministic squnc is of lngth 41. In ordr to compar th prformanc of our algorithm, w also simulat th mthod givn in [4] for this stup. Th PSK data lngth usd for th mthod givn in [4] is 496. Tabl 1 shows th tru valus for th non-rdundant krnls and th man and th standard dviations of th stimats from our algorithm and th PSK input mthod of [4]. Not that th non-rdundant
15 krnls givn in Tabl 1 ar th triangular krnls. In th simulations in [4], symmtric krnl valus ar usd [1, pp.34]. Howvr, w utilizd triangular krnl rprsntation in our simulations to concord with our notation in (1). Thr ar 11 nonzro Voltrra krnls. Th rsults for both mthods ar calculatd ovr 4 indpndnt trials. Th rsults for our algorithm ar bttr than thos for th mthod of [4] vn though our mthod uss an input squnc of lngth almost 1 tims smallr (41 vs. 496). Our stimats ar vry accurat dspit th prsnc of nois and th short lngth of input utilizd. Th mthod in [4] uss highr-ordr momnts. In Exampl 1 of [4], a third ordr filtr with N = 4 is simulatd and th man and dviations of th stimats ar xamind in th absnc of nois for an input lngth of 496. Sinc our algorithm is an xact algorithm, for this filtr our input squnc givs th xact krnl valus for an input lngth as short as 155 in th absnc of nois. Exampl 2: W simulat a linar-cubic bandpass Voltrra filtr, whr th input-output rlationship for th bandpass Voltrra filtr is givn in (48). Th channl modl w simulat has a mmory lngth of N = 2. W us QPSK modulatd signals as th input, whr w choos th input lvls for our dtrministic squnc from th st 2 j(2πk/4+π/4), k =, 1, 2, 3. Additiv indpndnt GWN obsrvation nois with varianc.5 is prsnt. Th lngth of th rquird dtrministic input squnc for our mthod is 32. W rsnd this input squnc 125 tims through th nonlinar channl and calculat krnl stimats for ach turn. Thn, w tak th man ovr ths krnl stimats and form our final stimat. Hnc, for this xampl, th total lngth of th utilizd input squnc is 32x125=4. W also ralizd th mthod for bandpass Voltrra krnl idntification as givn in [6] for th simulation stup givn abov. Th PSK data lngth usd for th mthod of [6] is 496. Tabl 2 shows th tru valus for th non-rdundant krnls and th man and th standard dviations of th stimats from our algorithm and th mthod dtaild in [6]. Th non-rdundant krnls ar th krnls givn as b 3 (i, j, k), b 3 (i, k, k) and b 1 (i) [6]. Thr ar a total of 1 Voltrra krnls. Th rsults for both mthods ar calculatd ovr 4 indpndnt trials. Th rsults for our algorithm ar bttr than thos for th mthod of [6] vn though our mthod mployd an input squnc of shortr lngth. VI. CONCLUDING REMARKS W prsntd a novl mthod for input-output idntification of th Voltrra krnls of a nonlinar channl modlld as a third-ordr Voltrra filtr. Our mthod utilizs carfully
16 dsignd dtrministic squncs as th probing signal and avoids th shortcomings of th us of random signals and corrlation mthods. Th algorithm works also for complx inputs, allowing th us of complx basband communication signals. W giv simulation xampls dmonstrating th prformanc of th algorithm compard to mthods availabl in th litratur. Ths mthods can b asily xtndd to th idntification of nonlinar channls with highr-ordr nonlinaritis following th rsults givn in [7]. REFERENCES [1] V. J. Mathws and G. L Sicuranza, Polynomial Signal Procssing, John Wily&Sons, 2. [2] M. T. Özdn, A. H. Kayran and E. Panayırcı, Adaptiv Voltrra channl qualization with lattic ortogonalisation, IEE Procdings - Communications, vol. 145, no. 2, pp. 19 115, Apr. 1998. [3] P. Koukoulas and N. Kalouptsidis, Nonlinar systm idntification using Gaussian inputs, IEEE Trans. Signal Procssing, vol. 43, no. 8, pp. 1831-1841, Aug. 1995. [4] G. T. Zhou and G. B. Giannakis, Nonlinar channl idntification and prformanc analysis with PSK inputs, Proc. 1st IEEE Sig. Proc. Workshop on Wirlss Comm., Paris, Franc, Apr. 1997, pp. 337 34. [5] N. Ptrochilos and P. Comon, Nonlinar channl idntification and prformanc analysis,, IEEE Intrnational Confrnc on Acoustics, Spch, and Signal Procssing, Istanbul, Turky, Jun 2, vol. 1, pp. 29 212. [6] C. Chng and E.J. Powrs, Optimal Voltrra krnl stimation algorithms for a nonlinar communication systm for PSK and QAM inputs, IEEE Trans. Signal Procssing, vol. 49, no. 12, pp.147 163, Jan. 21. [7] E. M. Ekşioğlu and A. H. Kayran, Nonlinar Systm Idntification Using Dtrministic Multilvl Squncs, Proc. 14th Intrnational Confrnc on Digital Signal Procssing, Santorini, Grc, July. 22, vol. 2, pp. 947 95 [8] C.-H. Tsng and E.J. Powrs, Idntification of cubic systms using highr ordr momnts of I.I.D. signals, IEEE Trans. Signal Procssing, vol. 43, no. 7, pp.1733 1735, July 1995. [9] E. Bigliri, S. Barbris, and M. Catna, Analysis and compnsation of nonlinaritis in digital transmission systms, IEEE Journal on Slctd Aras in Communications, vol. 6, no. 1, pp. 42-51, January 1988. [1] S. Bndtto and E. Bigliri, Nonlinar qualization of digital satllit channls, IEEE Journal on Slctd Aras in Communications, vol. sac-1. no. 1, pp. 42-51, January 1983.
17 d(n) a 3 x (1) (n) T 2,1 T 2,2 z -q 1 + x (1) (n) x (2) (q ;n) 1 N N y (1) (n) S 2,1 (2,1) v (q ;n) (2,2) 1 v (q ;n) 1 y (2) (q ;n) 1 + + - v (1,1) (n) (1) -1 [ U ] (2) -1 [ U ] h (1) (n) h (2) (q ;n-q ) 1 1 T 3,1 T 3,2 T 3,3 z -q 1 z -q 1 -q 2 + x (3) (q,q ;n) 1 2 N y (3) (q,q ;n) 1 2 v (3,1) (q,q ;n) 1 2 S 3,1 S 3,2 - + + + + v (3,2) (q,q ;n) 1 2 - v (3,3) (q,q ;n) 1 2 (3) -1 [ U ] h (3) (q,q ;n-q -q ) 1 2 1 2 Fig. 1. Proposd third-ordr nonlinar channl idntification mthod using dtrministic squncs as inputs. Tabl 1. Rsults for Exampl 1 (i 1 ) () (1) (2) tru b 1 (i 1 ) 1..5 -.8 man of ˆb 1 (i 1 ) for [4].9955.4886 -.815 man of ˆb 1 (i 1 ) for our mthod 1.45.518 -.8164 std of ˆb 1 (i 1 ) for [4].5195.3758.3326 std of ˆb 1 (i 1 ) for our mthod.1219.1266.1237 (i 1, i 2 ) (, ) (, 1) (1, 1) tru b 2 (i 1, i 2 ) 1..6 -.3 man of ˆb 2 (i 1, i 2 ) for [4] 1.35.626 -.2958 man of ˆb 2 (i 1, i 2 ) for our mthod 1.9.5984 -.335 std of ˆb 2 (i 1, i 2 ) for [4].1148.1335.788 std of ˆb 2 (i 1, i 2 ) for our mthod.14.764.5 (i 1, i 2, i 3 ) (,, ) (,, 1) (, 1, 1) (1, 1, 1) (, 1, 2) tru b 3 (i 1, i 2, i 3 ) 1. 1.2.8 -.5.6 man of ˆb 3 (i 1, i 2, i 3 ) for [4].99995 1.25.7993 -.59.5997 man of ˆb 3 (i 1, i 2, i 3 ) for our mthod 1.2 1.26.819 -.4997.636 std of ˆb 3 (i 1, i 2, i 3 ) for [4].164.235.215.281.24 std of ˆb 3 (i 1, i 2, i 3 ) for our mthod.77.196.185.82.285
18 Tabl 2. Rsults for Exampl 2 (i 1 ) () (1) (2) tru b 1 (i 1 ) 1..5+.5j -.6 man of ˆb 1 (i 1 ) for [6].9995+.65j.591+.4971j.5967+.76j man of ˆb 1 (i 1 ) for our mthod.9999 +.5j.53 +.4994j -.5999+.6j std of ˆb 1 (i 1 ) for [6].1131.186.15 std of ˆb 1 (i 1 ) for our mthod.183.193.184 (i 1, i 2, i 3 ) (, 1, 1) (1,, ) (1, 2, 2) (2, 1, 1) tru b 3 (i 1, i 2, i 3 ) 1..4+.4j -.4.6 man of ˆb 3 (i 1, i 2, i 3 ) for [6] man of ˆb 3 (i 1, i 2, i 3 ) for our mthod std of ˆb 3 (i 1, i 2, i 3 ) for [6] std of ˆb 3 (i 1, i 2, i 3 ) for our mthod.9995+.24j.417+.3998j -.3993+.16j.5987-.4j 1.-.7j.3995+.3999j -.4+.4j.62+.1j.27.32.297.259.166.144.16.139 (i 1, i 2, i 3 ) (2,, ) (, 2, 2) (, 1, 2) tru b 3 (i 1, i 2, i 3 ).6+.7j.5.3+.4j man of ˆb 3 (i 1, i 2, i 3 ) for [6].5995 +.6995j.4996 -.1j.32 +.43j man of ˆb 3 (i 1, i 2, i 3 ) for our mthod.65 +.72j.5 +.7j.2993 +.3993j std of ˆb 3 (i 1, i 2, i 3 ) for [6].243.263.26 std of ˆb 3 (i 1, i 2, i 3 ) for our mthod.138.16.217