An Improved Square-Roo Nyquis Shaping Filer fred harris San Diego Sae Universiy fred.harris@sdsu.edu Sridhar Seshagiri San Diego Sae Universiy Seshigar.@engineering.sdsu.edu Chris Dick Xilinx Corp. chris.dick@xilinx.com Karl Moerder Broadband Innovaions kmoerder@broadbandinnovaions.com Absrac A Nyquis filer can have any odd symmeric aper as a ransiion beween is pass band and sop band. The mos well known aper is he raised cosine (RC) aper specified in mos communicaion sandards. The square roo (SR- RC) Nyquis filer formed from his prooype exhibis high specral side lobe levels ha fail o mee severe specral mask requiremens. This paper presens a SR Nyquis filer wih an alernae aper ha, for he same lengh filer, achieves side lobe levels beween one and wo orders of magniude below hose obained from a cosine apered square roo filer. This alernae filer also offers up o an order of magniude reducion in residual ISI levels.. Inroducion The radiional RC apered Nyquis specrum is obained by convolving he ideal recangle specrum of widh /T wih a half cycle cosine shaped specrum of widh α/t. The equivalen process seen in he ime domain is he produc of he wo ime series represening he inverse ransforms of he recangle and he half cycle cosine [], [] as is shown in (). h RC α sin( π ) cos( π ) () = ( ) T α ( π ) [ ( ) ] The (SR-RC) Nyquis specrum [], he specral desired response for he shaping filer a he ransmier and of he mached filer a he receiver has he impulse response shown in (). The filers presened in () and () are symmeric abou he ime origin and have infinie exen. The realizable version of hese filers is obained by symmerically runcaing he ime exen and hen ranslaing along he ime axis ill he response resides in he posiive ime inerval. The runcaion process is anoher ime domain window ha leads o a second specral convoluion which is he cause of high specral side lobes exhibied by he SR-RC () Nyquis filer. We should know beer han o simply runcae he impulse response of a filer! h SR RC (4 α )cos( π( + α) ) () = ( ) T T T ( π )[ (4 α ) ] sin( π( α) ) + ( ) T T ( π )[ (4 α ) ] Furher when we convolve he runcaed versions of he SR-RC Nyquis filers we do no obain an RC-filer bu raher an approximaion ha exhibis non-equally spaced zero crossings. The non-zero sample values aken a equally spaced sample posiions are collecively he cause of non-channel induced iner-symbol inerference (ISI).. Iniial Design Procedure Our ask is o replace he sandard SR-RC filer wih one ha does no suffer he defecs relaed o he runcaion or window applied o he impulse response of he prooype SR-RC filer. One successful approach o many filer designs involves varians of he Parks-McClellan (PM or Remez) algorihm ha ieraively forms a weighed Chebyschev frequency domain approximaion o he desired specral response for a filer of specified lengh. The filer design requires a vecor of frequencies idenifying inerval boundaries, a vecor of gains a hese frequency boundaries, and a penaly (or weigh erm) vecor o conrol he olerance bands abou he desired response profile in each inerval. Since we are considering an alernae ransiion in his design we are free o selec any arbirary odd symmeric ransiion shape and use he square roo of ha shape as he desired ransiion band response. Raher han resric he ransiion shape, we ake a minimalis approach and specify hree poins in he ransiion band ha he SR filer has o saisfy and permi he algorihm o acquire hese poins wih he hope ha he ransiion beween hese poins will be well behaved. Wha ever ()
well behaved means! We know ha he Nyquis filer gain a he half symbol rae is 0.5 (or -6 db) hence he SR Nyquis filer gain a he half symbol rae is (0.5) / (or -3 db). As shown in figure, we require he specral ransiion conour o pass hrough (0.5) / a he bandwidh normalized frequency of 0.5. f f sym (- α ) 0 (+ α) Figure. Frequency Response Specificaion for SR Nyquis Filer We can accommodae his requiremen a number of ways and here we employ an opion of he PM algorihm [3], [4] ha permis frequency inervals of zero widh. The poins we assign o he algorihm are he boundaries defining he pass band, he sop band, and he zero widh inerval in he ransiion band locaed beween hese wo bands. These lead o he bandwidh normalized frequencies, gains, and weighs shown in (3) for a filer designed wih raio of sample rae o symbol rae of 4. We will commen on he firs weigh, of value.4535, shorly. Figure. Specrum and Pass Band Deail of SR-RC Nyquis and hm- Filers. ha he SR-RC and hm- filers exhibi RMS ISI levels of 0.4% and of 0.46% (-47.65 and -46.77 db respecively) which are nearly comparable ISI levels for filers wih very differen specral side lobe characerisics. Wha is ineresing here is he ISI shown in he hird subplo which presens he mismach performance Freq v: [0 (- α )/ / / (+ α)/ ]/ Gain v: [ / / 0 0 ] W v: [.4535 ] (3) Figure presens he log-magniude specral response of a 97-ap, α = 0., SR-RC Nyquis filer and of a SR filer designed by he approach presened here. We refer o his opion as he harris-moerder- (hm-) filer. Also seen here is a zoom o he in-band ripple deail. We firs noe and compare he -40 db and -70 db side lobe levels respecively of he SR-RC and of he hm- filers. This addiional ou-of-band specral response is he goal of his design. We also noe he 0-o- reducion of in-band ripple of he hm- filer relaive o he SR-RC filer. The anomaly in he hm- filer is he 0.045 db (or 0.5 %) specral bump a he sar of he ransiion band. This is he behavior we alluded o earlier when we hoped ha he unresriced specral ransiion would be well behaved. We improve his behavior in he nex secion. Figure 3 presens deails of he iner-symbol inerference (ISI) exhibied by a cascade of shaping and mached filers implemened as wo SR-RC filers, as wo hm- filers and as an hm- and an SR-RC filer. Here we noe Figure 3. ISI levels of Mached SR-RC, hm- and of Mismached SR-RC and hm- Filers beween filers a he modulaor and he demodulaor. This corresponds o he case in which we have seleced he hm- filer a he ransmier o access is reduced specral side lobe levels which faces a SQ-RC mached filer in a legacy receiver. We see here he same ISI levels for he hm- o hm- mached filer as for he mismached hm- o SR-RC filers. This was no a remarkable coincidence bu raher he resul of adjusing he firs weigh in he PM algorihm weigh vecor shown in (3) o achieve his mach.
3. Improved Design Procedure The anomalous specral bump a he edge of he ransiion band can be removed by redefining he pass band edge. The modified algorihm slides, in small incremens, he pass band edge from (-α)/ o β(-α)/. Afer each slide, he ISI obained wih he new filer weighs is compared o he ISI obained wih he previous weighs and he slide is coninued ill he ISI no longer decreases. Minimum ISI is obained by a search refinemen ha ieraively reduces incremen size and slide direcion. The frequency vecor used in he PM-algorihm is modified as shown in (4). Here β modifies he pass band edge o form a second se of weighs hm- while β 3 modifies he sop band edge o form a hird se of weighs hm-3. Freq v: [0 β (- α)/ / / β (+ α)/ ]/ 3 The ISI levels exhibied by mached filers hm- o hm- and by mismached filers hm- o SR-RC are 0.06% and 0.6% (-64. and -44. db respecively) while he ISI levels exhibied by mached filers hm-3 o hm-3 and by mismached filers hm-3 o SR-RC are 0.037% and 0.5% (-68.6 and -45.7 db respecively). For comparison, he ISI exhibied by he mached filer SR-RC o SR-RC is 0.4% (-47.65 db). Figure 4 presens he log-magniude specral response for filer hm-3. Here we see he frequencies o which he wo band edges have shifed o minimize is associaed ISI. Figure 5 presens deails of he iner-symbol inerference (ISI) exhibied by he cascade of shaping and mached filers implemened as wo SR-RC filers, as wo hm-3 filers and as an hm-3 and an SR-RC filer. As a maer of ineres, figure 6 presens and compares he ransiion band edges of he specra obained a he oupu of he (4) Figure 5. ISI levels of Mached SR-RC, hm-3 and of Mismached SR-RC and hm-3 Filers mached filer processing for he hree filers we have examined, SR-RC, hm-, and hm-3. Also shown are he specral slopes for he hree specra. These are approximaely he funcions ha are convolved wih he ideal recangle specrum o form he finie duraion, excess bandwidh SR Nyquis filer. The conour for he SR-RC is supposed o be an even symmeric half cosine. We clearly see he specral disorion of he half cosine caused by ime domain runcaion. A pleasan observaion is ha he corresponding conour for he hm-3 filer is even symmeric. Finally we see he specra formed by folding he ou-of band specrum back ino band, an even ha occurs when he oupu of he mached filer is sampled a symbol rae, one-sample per symbol. Ideally, hese would all be consan uniy level specra and i he deviaion from a consan ha is responsible for he filer ISI. Figure 4. Specrum and Pass Band Deail of SR-RC Nyquis and hm-3 Filers. Figure 6. Transiion Band Edge, Specral Derivaive, and Folded Specra of SR-RC, hm-, and hm-3 Filers
4. Improving Mismach Performance The design echnique we have been examining was formulaed as a mechanism o obain SR Nyquis filers wih significanly reduced ou of band specral side lobes. This is accomplished by embedding a design wih an unconsrained ransiion conour in he PM filer design algorihm. As expeced, a filer wih significanly improved ou-of-band aenuaion levels is characerized by a smooher ransiion beween he pass band and sop band. We can see he smooher ransiion of he SR filers in figure 7 where he specral derivaive shows ha he ransiion likely has many coninuous derivaives. We noe, by examining he specral derivaives, ha he hm filers have a narrower ransiion bandwidh han does he SR- RC. This, of course, is due o he zero derivaives a he boundaries of he ransiion bandwidh. We know ha his difference in ransiion bandwidhs will cause a mismach beween he shaped filer and he mached filer. As indicaed earlier, his is a likely occurrence if an hm-3 filer is employed as a shaping filer a a new generaion modulaor which is processed by a SR-RC mached filer in a legacy receiver. illusrae he benefis of his opion, we have designed hm filers wih excess bandwidh of 0. for use as a shaping filer operaing in conjuncion wih a legacy SR-RC filer wih excess bandwidh 0.0. Figure 8 presens deails of he iner-symbol inerference (ISI) exhibied by he cascade of hese shaping and mached filers implemened as wo SR-RC filers, as wo hm-3 filers and as mismached α hm-3 and SR-RC filers. By adjusing he firs weigh erm of (3) we were able o obain he same RMS ISI level for all 3 filer pairings. Hence by permiing a minor α mismach beween he hm shaping filer and he legacy SR-RC mached filer we incur no performance penaly while accessing he improved ou-of band specral response of he hm filers. Figure 9 shows he specra of he α-mismached filer pair. The anomalous 0.04 db bump in he ransiion band is removed in he hm- and hm-3 versions of his α When we do he numbers, we find he mismach loss is quie small. The inner produc beween he wo mismached signals, each normalized for uni energy, is 0.9978 or -0.0096 db. We conclude ha, in spie of he differen ransiion conours, here is very lile energy difference in he ransiion band. This suggess ha we migh be able o widen he ransiion bandwidh of he hm-3 shaping filer when we know ha i is likely o be Figure 8. ISI levels of Mached SR-RC, hm-3 and of Mismached α SR-RC and hm- Filers Figure 7. Deail of Transiion Band Edge and Specral Derivaive of Band Edge of SR-RC and hm-3 Filers used wih a legacy SR-RC mached filer. We can do his wih very lile performance loss in colleced energy bu perhaps wih he promise of improving ISI performance of he mismached filers. In fac his proves o be rue. To Figure 9. Specrum and Pass Band Deail of α Mismached SR-RC Nyquis and hm- Filers.
mismached filer. The bump free hm-3 version of his filer exhibis an improved mached filer RMS ISI level of 0.0007 (-6.9 db) bu has a reduced α and shape mismached filer RMS ISI level of 0.0054 (-45.4 db). This compares o he SR-RC Nyquis RMS ISI level of 0.004 (-47.7 db). Figure 0 presens he eye diagrams formed from he ime series a he oupu of he hm-3 and SR-RC Nyquis mached filers. These are seen o be quie similar bu he hm filer pair has slighly narrower widh eye-opening and slighly larger peak ransiions as expeced for he is narrower effecive ransiion bandwidh. We expec ha his minor variaion will no impac iming recovery loops when an hm shaping filer is subsiued for he SR-RC filer. design, hm-, exhibis a small specral bump a he saring edge of he ransiion band. The ampliude of his bump is approximaely he magniude of he in-band ripple exhibied by he sandard SR-RC Nyquis filer. By adjusing he weighing erm in he PM filer design algorihm we were able o obain he same ISI level in he mached hm- o hm- filer pair as in he mismached hm- o SR-RC Nyquis filer pair. The specral bump in he ransiion band is suppressed by shifing he pass band edge owards he ransiion band while monioring he ISI performance of he corresponding ime domain filer. This varian, denoed hm- and is shifed righ band edge cousin, denoed hm-3 exhibi considerably reduced ISI levels. While he hm- and hm-3 mached filer pairs exhibi improved mached filer performance hey also exhibi slighly reduced performance levels when mismached wih SR-RC Nyquis filers. We suggesed and demonsraed he use of a small inenional excess bandwidh (α value) mismach beween he hm filer and legacy SR-RC Nyquis filers. This opion offer he abiliy o use he hm filer, wih is improved specral characerisics, a a modulaor wihou incurring a mismach performance loss when used wih SR-RC Nyquis filers residing in legacy receivers. 6. References []. John Proakis, Digial Communicaions, Fourh Ediion, 00, McGraw-Hill, Ch. 9., pp. 554-56. Figure 0. Eye Diagrams for SR-RC and hm-3 Shaping and Mached Filer Pairs 5. Conclusions We have examined a design echnique based on use of he PM-algorihm o form SR Nyquis filers wih unconsrained ransiion conours ha exhibi significanly reduced ou-of-band side-lobe levels. One varian of his []. fred harris, Mulirae Signal processing for Communicaion Sysems, 004, Prenice-Hall, Ch. 4.- 4.4, pp. 8-97 [3]. Sanji Mira and James Kaiser, Handbook for Digial Signal processing, 993, John Wiley & Sons, Ch. 4.8, pp. 98-4 [4]. MATLAB, FIRPM.m scrip file help response.