ADSP ADSP ADSP ADSP. Advanced Digital Signal Processing (18-792) Spring Fall Semester, Department of Electrical and Computer Engineering

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1 NOTES ON LINEAR REDICTION AND LATTICE FILTERS 1. Introducton ADS ADS ADS ADS Advanced Dgtal Sgnal rocessng (18-792) Sprng Fall Semester, In ths note we revew aspects of our class dscusson on lnear predcton and lattce flters. The lnear predcton approach s one of several ways to accomplsh flter desgn by modelng, where we try to come up wth a parametrc representaton that most closely matches the power spectral densty functon of an unknown random process. In general, there are three types of models that can be consdered. The movng average (MA) model has zeros but not poles: Hz Bz The autoregressve (AR) model has poles but not zeros: Hz G Az Department of Electrcal and Computer Engneerng The thrd type of model has both poles and zeros and s called (unsurprsngly) the autoregressve movngaverage (ARMA) model: Hz Bz Az Of the three types of flter desgn by modelng, the all-pole AR model s the most commonly used, largely because the desgn equatons used to obtan the best-ft AR model are smpler than those used for MA or ARMA modellng. Serendptously, the all-pole model also has the ablty to descrbe most types of speech sounds qute well, and for that reason t has become wdely used n speech processng. In these notes, we wll begn wth some general comments about lnear predcton, whch drves us to consder an all-pole model of the power spectrum of a one-dmensonal sgnal. We wll then consder n varyng degrees of depth the three major ways n obtanng the parameters of the model, the autocorrelaton method, the covarance method and the partal correlaton (ARCOR) method. Ths revew also wll nclude a short ntroducton to lattce flters.

2 Notes on Lnear redcton -2- Fall, Lnear predcton of the current sample of a random process In our ntal consderaton of lnear predcton, let us magne that we are observng a random process xn. We would lke to determne how to obtan a best predcton of the current sample of xn from the prevous samples of the random process: xˆ n k xn k k 1 (1) We can defne the error of the approxmaton to be en xn xˆ n xn k xn k k 1 Later we wll consder n some detal n some detal the z-transform of the error functon: Ez Xz 1 k z k k 1 Xz Az It s convenent to defne a short-term verson of the error functon: 2 Ee 2 n Exn xˆ n 2 2 E xn k xn k k 1 (2) (3) (4) Our mmedate goal s to determne the set of k that mnmze E. For a partcular coeffcent ths s typcally accomplshed by dfferentatng 2 wth respect to, settng the dervatve equal to zero, and solvng for, or 2 E 2xn k xn kxn k 1 0 Exn xn ˆ kexn xn k k 1 (5) (6) where 1 and ˆ represents the estmated value of. Defnng k Exn xn k (7) Copyrght 2018, Rchard M. Stern

3 Notes on Lnear redcton -3- Fall, 2018 we obtan k k 0 k 1 (8) It can also be shown that 2 00 k 0 k k 1 (9) You may recognze Eqs. (8) and (9) as beng a form of the Yule-Walker equatons, whch you have already been exposed to (usng a slghtly dfferent set of notatonal conventons) n the context of our dscusson of MEM spectral estmaton. We wll solve these equatons makng use of two specfc assumptons about the defnton of k n Sec Relatonshp of lnear predcton to the all-pole flter (autoregressve) model u[n] h[n] x[n] Although the dscusson of the prevous secton was framed strctly n the context of determnng the coeffcents k that produce the best lnear predcton of the current sample of xn from the prevous samples, t s also often useful to consder xn to be the output of an all-pole flter as n the fgure above. the Assume that the unt sample response of the flter hn has a z-transform of the form Hz Xz Uz G G Az k z k k 1 Note that ths notatonal conventon (whch partly follows that of RS) s unusual: un represents the determnstc or stochastc nput (rather than the unt step functon), whle xn represents the system output (rather than ts nput). In spectral estmaton and system dentfcaton problems, the nput functon un s typcally assumed to be whte nose, and the squared magntude of the frequency response of the flter s a parametrc estmate of the power spectral densty functon of the output random process. In speech processng, the nput un represents the exctaton of the vocal tract, whch could be a quas-perodc functon correspondng to glottal pulses for voced speech segments, and a broadband nose source for unvoced speech segments. By takng the nverse z-transform of both sdes of Eq. (10) we obtan the expresson (10) Copyrght 2018, Rchard M. Stern

4 Notes on Lnear redcton -4- Fall, 2018 en xn xˆ n xn k xn k Gun k 1 (11) It s typcally assumed that the predctor coeffcents k that mnmze the average square of en provde a good model of the vocal tract confguraton that s used to produce a partcular segment of speech. The gan parameter G s used n syntheszng an output waveform from the nput exctaton that has a power that best matches that of the observed output. 2. Soluton of the LC equatons 2.1. General soluton of the LC equaton We wll frst consder the soluton of the LC equatons for the general expresson of k as defned above, and then consder the two specal cases that lead to the so-called autocorrelaton and covarance solutons. Let us assume for the sake of example that 1 k as These equatons can be wrtten n matrx-vector form as. The system of equatons n Eq. (8) can be wrtten for (12) (13) Note that Eq. (13) s of the form R (14) where R s a matrx of autocorrelaton coeffcents, s a 1 vector of the k, and s a 1 vector of autocorrelaton coeffcents. Ths equaton s known as the Wener-Hopf equaton, whch s encountered frequently n optmal sgnal processng. In general, a drect soluton to the Wener Hopf equaton can be obtaned by pre-multplyng both sdes of Eq. (14) by the nverse of R: Copyrght 2018, Rchard M. Stern

5 Notes on Lnear redcton -5- Fall, 2018 R 1 (15) The nverson of the R matrx can be accomplshed by Gaussan elmnaton and other smlar technques, whch are O 3 n computatonal complexty. In the next secton we dscuss a partcular formulaton of the LC problem that produces a soluton that s far more computatonally effcent The autocorrelaton method and Levnson-Durbn recurson As you have noted, we stll have not specfed how the autocorrelaton coeffcents k are obtaned from the orgnal waveform xn. One reasonable approach s to obtan the correlaton coeffcents by frst extractng a fnte-duraton segment of xn by multplyng by a wndow functon and then computng the frst autocorrelaton coeffcents of the fnte length segment. In other words, we assume that xn xn wn (16) where wn s nonzero only for 0 n N 1. In ths case we can defne the autocorrelaton functon as N 1 + Mn k k xm xm k m Max k N 1 + Mn k m + Max k xm xm + k (17) for 1 and 0 k. Note that despte the nelegant notaton, there wll always be N k nonzero terms n the sum. It can easly be seen that Eq. (17) represents the short-term autocorrelaton functon of a segment of xn and that k k (18) Because the number of nonzero elements of the argument of the summaton decreases as t s common to use a tapered wndow such as the Hammng wndow for wn. k ncreases, Usng ths defnton of the autocorrelaton functon and agan droppng the subscrpts n, Eq. (13), the Wener-Hopf equaton, reduces to (19) Ths method of soluton s known as the autocorrelaton soluton of the LC equatons. Because the resultng correlaton matrx R s Toepltz (.e. the elements of each dagonal of the matrx, major and mnor, are dentcal), a smpler soluton known as Levnson-Durbn recurson s possble. As we wll see n a moment, Levnson-Durbn recurson s O 2 n complexty. Copyrght 2018, Rchard M. Stern

6 Notes on Lnear redcton -6- Fall, 2018 The equatons of the Levnson-Durbn recurson, whch are used to compute the correspondng reflecton coeffcents and LC parameters are E 0 0 (20) k 1 1 j j j , calculated for (21) E 1 1 k (22) j 1 1 j k j, for 1 j 1 (23) E 2 1 k E 1 (24) Equatons (21) through (24) are solved recursvely for 12 and the fnal soluton s gven by j j for 1 j (25) The coeffcents k for 1 are referred to as the reflecton coeffcents. They consttute an alternate specfcaton of the random process xn that s as unque and complete as the LC predctor coeff- cents k. The reflecton coeffcents are actually far more robust to coeffcent quantzaton than the predctor coeffcents, so they are frequently the representaton of choce n applcatons such as speech codng or speech compresson. If the magntude of the reflecton coeffcents k s less than 1 for 1, all of the roots of the polyno- k mal Az 1 k z wll le nsde the unt crcle. Ths means that f k 1, the resultng flter Hz k 1 wll be stable. It can be shown that dervng the Durbn recurson guarantees that k 1. k We wll make extensve use of the reflecton coeffcents 2.3. The covarance soluton of the LC equatons n the fashon descrbed above usng Levnson- k n our dscusson of lattce flters. An alternate way of dervng the autocorrelaton functons used to solve the LC equatons would be to always compute all lags of the autocorrelaton functon over the same tme ndces. In a sense, ths means that we are correlatng and then wndowng, whle wth the autocorrelaton method we wndow and then Copyrght 2018, Rchard M. Stern

7 Notes on Lnear redcton -7- Fall, 2018 correlate. Under these crcumstances the autocorrelaton computaton becomes N 1 k xm xm k m 0 N+ 1 m xm xm + k (26) whch as before s evaluated for 1 and 0 k. Although ths computaton s smlar to that of Eq. (17), t s not dentcal to t because the lmts of the summatons are dfferent. Specfcally, t s very mportant to note that k k k (27) Because the number of samples used n the autocorrelaton computaton s always N regardless of the magntude of the correlaton lag, t s not consdered necessary to multply the argument of the summaton by a tapered wndow before summng. Under these crcumstances, Eq. (13), the Wener-Hopf equaton reduces to (28) Ths soluton to the LC equatons s called the covarance soluton because the autocorrelaton matrx on the left has the Hermtan symmetry that s characterstc of an arbtrary covarance matrx. We note that the Levnson-Durbn recurson cannot be used to solve ths equaton because the autocorrelaton matrx s Hermtan symmetrc but not Toepltz. Ths equaton s typcally solved usng the Cholesky decomposton method whch was dscussed superfcally n class and s treated n more detal n RS. In general, the covarance soluton provdes a lnear predcton of the current sample of a random process wth somewhat less mean-squared predcton error than the autocorrelaton soluton. Nevertheless, the autocorrelaton soluton s far more commonly used n applcatons such as speech processng because t s so much more computatonally effcent. The pole locatons wll be nsde the unt crcle provded that the reflecton coeffcents k are all less than one n magntude, but ths s not guaranteed when the covarance method s used. In practce, f t s observed that at least one reflecton coeffcent s greater than one n magntude, t s qute straghtforward to calculate the actual pole locatons that correspond to that partcular analyss segment, and then reflect the poles that le outsde the unt crcle to mrror-mage locatons nsde the unt crcle. Copyrght 2018, Rchard M. Stern

8 Notes on Lnear redcton -8- Fall, Recursve relatonshps between the LC coeffcents and reflecton coeffcents In Secs. 2.2 and 2.3 above, we dscussed how the LC coeffcents can be obtaned from the autocorrelaton coeffcents of an observed random process. We also noted n that secton that the reflecton coeffcents k completely specfy the LC characterzaton of a random process just as the LC coeffcents do. In fact, gven ether set of coeffcents, we can always obtan the other by a smple lnear recurson. Specfcally, to convert from the reflecton coeffcents k to the LC coeffcents, we use the recurson Let l k startng at l k l for 1 l 1 (29) Repeat for 12 Smlarly, we can convert from the LC coeffcents to the reflecton coeffcents k f we have all of the l for 122 : Let k Startng wth, let k 1 l + k l l for 1 21 (30) 2 1 k 2.5. Computaton of the LC gan parameter The LC gan parameter G can be computed n ether of two ways. From the zero th -order Yule Walker equaton, 00 2 k 0 k + x k 0 k+ G 2 k 1 k 1 (31) we easly obtan G 2 00 k 0 k k 1 (32) More prosacally, we can also estmate the gan parameter f both the nput and output of the flter are avalable. Combnng Eqs. (36) and (37) we obtan Az G GUz Ez Hz Xz Xz (33) Copyrght 2018, Rchard M. Stern

9 Notes on Lnear redcton -9- Fall, 2018 mplyng, of course, that Ez G or en Gun (34) Uz Snce both en and un are typcally tme-varyng stochastc sgnals, we typcally estmate G from the short-term rato of ther energes: G 2 N 1 m 0 e 2 m m N 1 u 2 m (35) 2.6. The LC predcton error n tme and frequency u[n] h[n] x[n] Consder agan the all-pole flter representaton dscussed n Sec. 2, wth the all-pole transfer functon Hz Xz Uz As noted above n Eq. (11), en G G Az k z k k 1 xn xˆ n xn k xn k Gun k 1 (36) These relatons also mply that Ez Xz Az (37) as was noted n our dscusson of lattce flters (wth slghtly dfferent notaton). The error sgnal en s peaker than the orgnal tme functon xn and ts spectrum Ee j s flatter than that of Xe j. Because of ths peakness, the error sgnal s frequently used as the bass for estmatng the nstantaneous fundamental frequency of a sgnal. It s also useful to consder the error sgnal n the frequency doman n more detal. If we are usng the auto- Copyrght 2018, Rchard M. Stern

10 Notes on Lnear redcton -10- Fall, 2018 correlaton method of the LC soluton, we can wrte by applyng arseval s theorem E N 1 e 2 1 m Ee j 2 d 2 m Xe j 2 2 Ae j 2 d (38) where Ae j jk 1 k e k 1 (39) Snce He j G Ae j we obtan E G 2 Xe j He j 2 d (40) As you may recall from our dscusson of maxmum entropy spectral estmaton, the autocorrelaton coeffcents of the flter wth mpulse response hn are dentcal to those of the random process xn for lags of magntude less than or equal to. In other words, xx m hh m for m (41) Hence, lm hh m xx m (42) lm He j Xe j and (43) lm E n G 2 (44) We note from Eq. (40) that the energy of the error sgnal s the ntegrated rato of the squared magntude of Xe j 2 and He j 2. Ths causes the match between these two quanttes to be closer for frequences where these functons are of greater magntude than those n whch the functons have lesser magntudes. In other words, Xe j 2 and He j 2 wll match each other more closely at ther peaks than at ther valleys. Ths s also especally good for the applcaton of LC analyss to speech processng, as perceptual studes ndcate that the peaks of the frequency response are much more mportant n determnng percepton than the correspondng valleys. Copyrght 2018, Rchard M. Stern

11 Notes on Lnear redcton -11- Fall, The FIR lattce flter x[n] e 0 [n] e 1 [n] e 2 [n] e N 1 [n] e N [n] y[n] k! k 2 k N k! k 2 z 1 z 1 z 1 z 1 b 0 [n] b 1 [n] b 2 [n] b N 1 [n] k N b N [n] Consder the basc lattce flter structure n the fgure above. It should be obvous that ths s an FIR flter structure, as t contans no feedback loops. In addton, f we set the nput xn to be equal to n, we can observe easly by nspecton that h0 1 and hn k N. The value of hn for other values of n s obtaned by observng all the dfferent ways of passng a sgnal through the lattce whle ncurrng exactly n delays, and addng all of the correspondng branch transmttances. It can be seen that the sample response wll be a lnear combnaton of the k Tme-doman and frequency-doman characterzaton of the lattce flter Although t may not be totally obvous from the fgure above, the FIR lattce flter s defned by the followng recursve relatons: xn e 0 n b 0 n (45) e n e 1 n k b 1 n 1 b n k e 1 n + b n 1 1 yn e N n (46) (47) (48) Because the structure s FIR, we can make use of the followng general characterzaton of ts transfer functon for the entre flter: We wll also make use of the transfer functon from the nput to the ths, let N Yz N l Xz l z Az l 1 A z E z l E 0 z l z l 1 The correspondng transfer functon from the nput to the b n e n (49) at a gven stage of the lattce. For (50) at a gven stage of the lattce s smlarly Copyrght 2018, Rchard M. Stern

12 Notes on Lnear redcton -12- Fall, 2018 à z B z B 0 z (51) We note that A 0 z à 0 z 1 and A N z Yz Xz. Usng ths notaton, we can wrte the z-transforms of the equatons that defne the lattce as E 0 z B 0 z Xz E z E 1 z k z 1 B 1 z B z k E 1 z+ z 1 B 1 z (52) (53) (54) Yz E N z (55) It s shown n Appendx A that f the l and k are related by the Levnson-Durbn equaton (and specfcally Eq. (23) above), then A z A 1 z k z A 1 z 1 (56) and à z z A z 1 (57) These equatons are mportant because they enable us to develop a recursve characterzaton of the transfer functon of the lattce flter stage by stage. Substtutng Eq. (50) nto Eq. (56) we obtan: 1 1 l 1 l 1 l z 1 l z k z 1 l 1 l z l 1 l 1 l 1 (58) Matchng the coeffcents of an arbtrary term of power z r we obtan r z r 1 r r z 1 + k r r z or, of course r 1 r 1 k r (59) as specfed by the Levnson-Durbn recurson. Hence, just as the standard FIR flter mplements the unt sample response of a system, wth the sample response values as the coeffcents or parameters of the flter, the lattce flter mplements the Levnson-Durbn recurson, wth ts reflecton coeffcents k as the parameters of the flter! The all-zero transfer functon of ths lattce flter s the recprocal of the all-pole model used to descrbe the orgnal random process, or n other words the flter Az s the nverse of Hz n Eq. (10) f we set the gan parameter G equal to 1. Copyrght 2018, Rchard M. Stern

13 Notes on Lnear redcton -13- Fall, hyscal nterpretaton of the functons e n and b n Up untl now we have been thnkng of the functons e n and b n as arbtrary nternal functons. Nevertheless, they each do have a physcal meanng relatng to lnear predcton error. Consder frst the transfer functon to the functons n the upper ral of the lattce: A z E z E z E 0 z Xz 1 l z l l 1 (60) Takng the nverse z-transform we obtan e n xn xn l l xn l 1 xˆ n (61) whch s dentcal (to wthn a sgn) to the lnear predcton error defned n Eq. (2). Agan, ths expresson descrbes the dfference between the current sample xn and the best lnear predcton of xn usng the prevous samples. Hence the expresson e s referred to as the th n -order forward predcton error. Let us now consder the functons obtan à z b n n the lower ral of the lattce. Combnng Eqs. (51) and (57) we z A z 1 B z B z and (62) B 0 z Xz B z z Xz 1 l z l l 1 z l l z l 1 (63) Agan, takng the nverse z-transform we obtan b n xn l xn + l l 1 (64) Comparng Eqs. (61) and (64) we observe that b n represents the dfference between xn, the value of the nput functon samples ago, and some lnear combnaton of the followng samples of the nput, runnng from xn 1 rght up to xn. In fact, the same lnear predcton coeffcents are used, but they are appled backward. One way of thnkng about ths s that b n s what we would have obtaned f we calculated e n but wth the nput functon presented n tme-reversed order. Because of all ths, s referred to as the th -order backward predcton error Dervng the reflecton coeffcents from the forward and backward predcton error: b n Copyrght 2018, Rchard M. Stern

14 Notes on Lnear redcton -14- Fall, 2018 the ARCOR method of lnear predcton In Sec. 2.2 we derved the LC coeffcents and the reflecton coeffcents k of the best-ft all-pole model to the samples of a random process by mplementng the Levnson-Durbn recurson to solve the autocorrelaton equatons. As you wll recall, equatons were developed by startng wth the dfference equaton relatng the nput and output, xn m 1 and fndng the values of the xn m m + m Gun that mnmze the expected value of the square of the forward error. Specfcally, startng wth E 2 xm k xm k m k 1 we computed the partal dervatve of E wth respect to each of the m we obtaned the equatons m 1 m m where Exn xn + Wth knowledge of the values of the autocorrelaton coeffcents n for 012 we can use the Levnson-Durbn recurson to obtan all the LC coeffcents correspondng reflecton coeffcents k. for model orders 1 through and the We can also obtan estmates of the reflecton coeffcents k (and subsequently the LC coeffcents m ) usng expressons for forward and backward error developed n the prevous secton. Specfcally, f we let E f N 1 n 0 e n 2 we can compute the dervatve of E f wth respect to k usng the expresson for e n n Eq. (61). Settng that dervatve equal to zero provdes a value of the reflecton coeffcent k of a gven stage of the lattce m Copyrght 2018, Rchard M. Stern

15 Notes on Lnear redcton -15- Fall, 2018 flter k f N 1 e 1 nb 1 n 1 n N 1 n 0 2 b 1 n 1 (65) where the superscrpt f n the symbol derved usng the mean square forward error k f remnds us that ths verson of the reflecton coeffcent was e n Note that ths estmate for the reflecton coeffcent s expressed n terms of the expected values of the products of the forward and backward errors of the prevous stage n the numerator, and the expected value of the square of the backward predcton error n the denomnator. The expresson n the numerator s actually the cross-correlaton of the forward and backward error functons of the prevous stage, and the expresson n the denomnator s the energy of the backward error of the prevous stage. Because of these physcal nterpretatons, ths method of obtanng the estmate of the reflecton coeffcents s referred to as the partal correlaton or ARCOR method. In the approach of Sec. 2 we began by calculatng the autocorrelaton functons of the nput drectly; n ths approach the nput autocorrelaton s calculated ndrectly, through a recursve computaton of cross-correlaton of predcton error functons. Ths approach has some very attractve statstcal propertes and s wdely used. Of course, there s nothng magc about the forward predcton error. We can just as easly perform a smlar calculaton wth the backward predcton error b n. erformng a smlar set of operatons on the backward predcton error produces the very smlar estmate for the reflecton coeffcent k b N 1 e 1 nb 1 n 1 n N 1 2 e 1 n n 0 Varous methods have been proposed for combnng the two estmates of the reflecton coeffcents f b obtaned usng the ARCOR method, k and k. For example, the Itakura estmate of the reflecton coeffcents s obtaned by combnng these two results accordng to the equaton (66) I k l k l f kl b (67) The Burg estmate of the reflecton coeffcents produced by combnng these two results accordng to the equaton Copyrght 2018, Rchard M. Stern

16 Notes on Lnear redcton -16- Fall, 2018 k l B f b 2k lkl f fb k l + k l (68) 4. IIR lattce flters As noted above, we developed n Sec. 3 an all-zero lattce flter wth the transfer functon Az N N l E N z 1 l z E 0 z l 1 Referrng to the fgure at the begnnng of Sec. 3, we note that the nput s yn would obtan the transfer functon and the output s whch clearly s an all-pole transfer functon, and n fact s exactly the transfer functon of the orgnal flter consdered, Hz, wth the gan factor G set equal to 1. xn e 0 n e N n. If we could mantan the same flter structure but nterchange the nput and output, we E 0 z E N z N N l 1 l z (69) Recall that the orgnal defntons of the stages of the FIR lattce flter were e n e 1 n k b 1 n 1 b n k e 1 n + b n 1 1 Wth a trval amount of algebra, Eq. (70) can be rewrtten as e 1 n e n+ k b 1 n 1 (70) (71) (72) Eqs. (71) and (72) suggest the followng lattce structure for a sngle stage: e [n] e 1 [n] +k b [n] k z 1 b 1 [n] Copyrght 2018, Rchard M. Stern

17 Notes on Lnear redcton -17- Fall, 2018 Combnng nto multple stages, we obtan the followng IIR lattce structure: x[n] e N [n] e N 1 [n] e 2 [n] e 1 [n] e 0 [n] y[n] +k N k N b N [n] +k 2 k 2 +k 1 k 1 z 1 z 1 z 1 z 1 z 1 b N 1 [n] b 2 [n] b 1 [n] b 0 [n] Note that e N n s now the nput and that e 0 n s the output. Ths flter wll have the transfer functon Hz N N l 1 l z l 1 (73) where the LC parameters are related to the reflecton coeffcents accordng to the usual Levnson-Durbn relatonshp. Snce the flter s IIR wth feedback loops, t does have the potental to be unstable. However, t s guaranteed to reman stable f k 1 5. Addtonal readng for all. (74) The dscussons on lnear predcton were based on my class notes, whch n turn are largely derved from the text Dgtal rocessng of Speech Sgnals by L. R. Rabner and R. W. Schafer (rentce-hall, 1978), whch was passed out n class. A update of ths text was wrtten by the same authors under the ttle Theory and Applcatons of Dgtal Speech rocessng, publshed by earson n Addtonal materal on lattce flters was derved from Secton 6.6 of Dscrete-Tme Sgnal rocessng by A. V. Oppenhem and R. W. Schafer (earson, 2010). The text Dscrete-Tme Speech Sgnal rocessng by T. F. Quater (rentce- Hall, 2002) s also hghly recommended and goes nto deeper detal than Rabner and Schafer n some aspects of the topcs consdered. Copyrght 2018, Rchard M. Stern

18 Notes on Lnear redcton -18- Fall, 2018 AENDIX A: ROOF OF THE RECURSIVE LATTICE FILTER RELATIONSHI To prove: A z A 1 z k z 1 A 1 z 1 From the defnton of A z 1 A j z 1 j z j and the Levnson-Durbn relaton we have j j k j, 1 j 1 k Substtutng for, we have Substtutng for j, 1 A j z 1 j z j 1 k z 1 A 1 j 1 j z 1 j z k j z k z j j 1 j 1 j z + k j z j 1 j 1 k z In the second term, let j j' +, then 1 1 A 1 j 1 j z 1 j z + k j' z j 1 j' 1 k z j 1 j z k z 1 1 j' z j' j 1 j' 1 A 1 z k z A 1 z 1 Copyrght 2018, Rchard M. Stern