TUTORIL ois Powr Ratio (PR) 65-Yar Old Tlphon Systm Spcification Finds w Lif in Modrn Wirlss pplications ITRODUTIO by Walt Kstr Th concpt of ois Powr Ratio (PR) has bn around sinc th arly days of frquncy division multiplxd (FDM) tlphon systms. Th PR is simply a masur of th "quitnss" of an unusd channl in a multi-channl systm whn thr is random activity on th othrs. ois and intrmodulation distortion products fall into th unusd channl causing lss than idal prformanc. Originally usd to chc 4-Hz wid voic channls in FDM lins, th sam concpt is usful today in charactrizing multichannl widband communication systms but thr ar som important diffrncs in th modrn masurmnt tchniqus. HISTORY OF PR ois powr ratio tsting has bn usd sinc th arly days of Frquncy Division Multiplxd (FDM) communication systms. In a typical FDM systm, 4-Hz wid voic channls ar "stacd" in frquncy bins for transmission ovr coaxial, microwav, or satllit quipmnt. Th numbr of channls dpnds on th systm. group is composd of voic channls and occupis a bandwidth of 48 Hz. Similarly, a suprgroup has 60 channls and occupis a bandwidth of 40 Hz, and a mastrgroup has 300 channls and occupis a bandwidth of approximatly.3 MHz. Suprgroups and mastrgroups ar oftn combind to ma up vn highr capacity systms. For instanc, an 800-channl systm occupis a bandwidth of approximatly 8 MHz. t th rciving nd of th transmission lin, th FDM data is dmultiplxd and convrtd bac to 4-Hz individual voicband channls. Th FDM signal is thrfor composd of many individual voic channls and passs through amplifirs, rpatrs, channl bans, tc., which add nois and distortion to th signal. Early studis at Bll Tlphon Labs (Rfrnc ) ld to th conclusion that th composit signal in an FDM systm having mor than approximatly 00 channls can b approximatd by Gaussian nois having a bandwidth qual to th bandwidth of th combind FDM signal. For instanc, a 800-channl FDM signal is approximatd by Gaussian nois with a bandwidth of 8. MHz. Th "quality" of an individual voic channl is thn masurd by first assuming that thr ar random "talrs" on all channls xcpt th spcific 4-Hz channl undr tst. n individual 4- Hz channl can thrfor b masurd for "quitnss" using a narrow-band notch (bandstop) filtr and a spcially tund rcivr which masurs th nois powr insid th 4-Hz notch as shown in Figur. Rv., 0/08, WK Pag of
() GUSSI OISE SOURE LPF OTH FILTER TRSMISSIO SYSTEM RROWBD REEIVER (B) GUSSI OISE SOURE LPF OTH FILTER D BUFFER MEMORY D FFT PROESSOR f s RMS OISE LEVEL (db) LOWER OTH WIDTH UPPER OTH WIDTH PR OTH WIDTH FREQUEY 0.5f s Figur : ois Powr Ratio (PR) Masurmnts ois Powr Ratio (PR) masurmnts ar straightforward in an analog transmission systm (Figur ). With th notch filtr out, th rms nois powr of th signal insid th notch is masurd by th narrowband rcivr. Th notch filtr is thn switchd in, and th rsidual nois insid th notch is masurd. Th ratio of ths two radings xprssd in db is th PR. Svral notch frquncis across th nois bandwidth (low, midband, and high) ar tstd to charactriz th systm adquatly. Dtails of arly PR tst quipmnt and th masurmnts can b found in Rfrnc 4. PR masurmnts on Ds ar mad in a similar mannr, xcpt th analog rcivr is rplacd by a buffr mmory and an FFT procssor which prforms th calculations as shown in Figur B. Thr ar som cass whr th combind FDM signal is convrtd to digital with an D, transmittd, and thn convrtd bac to analog using a D at th rcivr. In this cas, th analog mthod shown in Figur is utilizd in prforming th PR tst. In a 939 articl (Rfrnc ), Holbroo and Dixon prformd an analysis of FDM systms in an ffort to dtrmin th optimum channl "loading" lvls. Thir wor ld to th fundamntal thory of multichannl nois loading. Th goal is to st th signal lvl (or "loading") to a valu which will giv th highst PR. Th PR is plottd as a function of rms nois lvl rfrrd to th pa rang of th systm. For vry low nois loading lvls, th undsird nois (in nondigital systms) is primarily thrmal nois and is indpndnt of th input nois lvl. Ovr this rgion of th curv, a -db incras in nois loading lvl causs a -db incras in PR. s th nois loading lvl is incrasd, th amplifirs and rpatrs in th systm bgin to ovrload, crating intrmodulation products which caus th nois floor of th systm to incras. s th input nois continus to incras, th ffcts of "ovrload" nois prdominat, and th PR is rducd dramatically. FDM systms ar usually opratd at a nois loading lvl a fw db blow th point of maximum PR to allow hadroom for pa busy hours. Pag of
Systm PR rcommndations for FDM systms wr formalizd in 966 by th ITT/IR to masur th transmission charactristics of Frquncy Division Multipl (FDM) communications lins (s Rfrnc 4). In a digital systm containing an D, th nois within th notch is primarily quantization nois whn low lvls of input nois ar applid. Howvr, for vry low amplitud signals (lss than - LSB pa-to-pa), th rsulting nois rvrts to th input-rfrrd nois of th D. For signals that xrcis svral LSBs of th D, th PR curv is linar, and quantization nois prdominats. s th nois lvl incrass, thr is a on-for-on corrspondnc btwn th nois lvl and th PR. t som lvl, howvr, "clipping" nois causd by th hard-limiting action of th D bgins to dominat. Th D hard-limiting "clipping" nois is somwhat diffrnt from th soft-limiting "ovrload" nois of an analog FDM and rsults in a "stpr" downward slop in th clipping rgion. THEORETIL PR FOR DIGITL SYSTEM Svral paprs hav bn writtn ovr th yars driving th thortical PR of an idal n-bit D (for xampl, Rfrncs 5, 6, and 7). Rfrnc 6 is th most complt, and shows th drivation for both uniformly distributd nois and Gaussian nois. Howvr, Gaussian nois is much mor rlvant to PR tsting. Th drivation is not difficult but dos involv som partial intgration. Sinc th "clipping" nois componnt not hav a closd-form solution, numrical mthods must b usd to actually comput th thortical PR numbrs. thortical curv for 0,, 4, and 6-bit Ds is shown in Figur. Undrstanding th dfinitions of th trms V O,,, and th rms loading lvl ( 0log 0 ) ar vry important in ordr to avoid confusion. PR (db) 86 8 76 7 66 6 56 6 BITS 4 BITS BITS 85.40dB 5.47dB 6 BIT 74.0dB 4.79dB 4 BIT 6.7dB BIT 4.00dB D RGE ±V O V O / RMS OISE LEVEL 5 46 4 0 BITS 5.56dB 3.06dB 0 BIT 36 30 5 0 5 0 RMS OISE LODIG LEVEL 0log 0 () db Figur : Thortical PR for 0,, 4, and 6-bit Ds Pag 3 of
It is important to undrstand that ths curvs ar basd on an idal D whr th only nois is th quantization nois and th clipping nois. In practic, th actual lvl of prformanc will b lss than thortical, dpnding upon th particular D undr tst. Th D input rang is bipolar, and is ±V O full-scal (hnc V O pa-to-pa). Th input rms nois lvl is, and th nois-loading factor (also calld th crst factor) is dfind as V O /. Th valu of is thrfor th pa-signal-to-rms-nois-ratio, whr is xprssd as a numrical ratio. gain, it is important to not that a pa signal of V O implis a pa-to-pa full-scal input of V O. This can bcom a point of confusion. nothr way to put it is that a fullscal sinwav givn by v(t) V O sinωt xactly fills th D input rang. This is why V O is rfrrd to as th pa amplitud. Th rciprocal of is th rms-nois-to-pa-signal-ratio, and th rms nois loading lvl is dfind as / xprssd in db: RMS ois Loading Lvl 0log0 0 log(). Eq. Th drivation for thortical PR can b bron into two parts. Th first part drivs th thortical quantization nois powr of an idal n-bit D. Th scond part drivs th "clipping nois" powr du to th limiting action of th D. Th total nois powr is th sum of th two nois powrs. Th complt rror wavform showing th two rgions is shown in Figur 3. Th thory is basd on svral assumptions. First, th quantization rror signal is not corrlatd to th input signal. This is valid providd th signal amplitud is at last svral LSBs in amplitud and th rsolution of th D is at last 6-bits. Scond, th sampling rat is twic th input nois bandwidth. Third, th D acts as an idal limitr for out-of-rang signals. Ths thr assumptions ar valid for most practical systms and lad to a rlativly straightforward solution. (x) LIPPIG REGIO q LSB (x) x V O x LIPPIG REGIO V O 0 V O (x) ERROR (IPUT OUTPUT) Figur 3: Idal D Error Wavform Pag 4 of
Th quantization nois componnt (xprssd as th squar of th actual quantization nois voltag to yild nois powr), has bn shown to b (s Rfrnc, for xampl): q Q, Eq. whr q is th wight of th last significant bit (LSB). It should b notd that this is th quantization nois powr masurd ovr th full yquist bandwidth dc to f s /. If th signal bandwidth is rducd, th nois in th rducd bandwidth is proportionally lss, and a corrction factor must b addd (discussd latr in this papr). ontinuing with th drivation w now that, q V O / n. Thrfor, from Equation : Q n VO / q V O n 3. Eq. 3 Howvr, V O /, thrfor V O, and substituting for V O in Equation 3 yilds: Q n 3. Eq. 4 ow, rfr to Figur 3 for th drivation of th clipping nois powr,. Th clipping nois powr is givn by th following gnral quation: From Figur 3B, (x) (x)p(x) dx Eq. 5 x, for x > V O, and thrfor Eq. 6 V O (x V ) P(x) dx, Eq. 7 whr P(x) is th Gaussian probability dnsity function and is givn by: V O O x / P(x). Eq. 8 Pag 5 of
Substituting V O, and combining Equation 8 with Equation 7 yilds: dx ) (x / x Eq. 9 Th final rsults of th intgration (s ppndix for complt drivation) yilds: [ ] / () ) ( Eq. 0 Whr () is th ormal Distribution Function: dt () / t. Eq. For calculation purposs, th function [ ()] can b approximatd by th following xprssion: 6 4 5 4) () / 0 8 6 4 9 8 6 4 9. Eq. Th total nois, T, can now b calculatd by adding Equation 4 and Equation 0: [ ] / n Q T () ) ( 3, Eq. 3 [ ] / n T () ) ( 3. Eq. 4 T T 0log 0log PR Eq. 5 Pag 6 of
Figur 4 shows th thortical pa valu of PR and th corrsponding valu of for Ds having rsolutions btwn 8 and 0 bits. Th vrtical axis is PR (xprssd in db pr Equation 5). Th horizontal axis is th Gaussian nois loading lvl with rspct to th pa signal lvl, /V O, xprssd in db. BITS 8 9 0 3 4 5 6 8 0 OPTIMUM 3.9 4. 4.50 4.76 5.0 5.6 5.49 5.7 5.94 6.34 6.78 (db) MX PR (db).87 40.60.50 46.05 3.06 5.56 3.55 57. 4.00 6.7 4.4 68.35 4.79 74.0 5.5 79.70 5.47 85.40 6.04 96.88 6.6 08.4 D Rang ±Vo Vo / RMS ois Lvl Figur 4: Thortical Maximum PR for 8 to 0-bit Ds gain it is important to rmmbr that this is th PR obtaind whn th input signal nois occupis th full yquist bandwidth, dc to f s /. For th cas of ovrsampling, whr th signal bandwidth, BW, is lss than f s /, th corrction factor of 0log 0 [f s /( BW)], oftn rfrrd to as procss gain, must b addd to th PR givn in Equation 5: PR 0log T f 0log s. Eq. 6 BW In multi-channl high frquncy communication systms, whr thr is littl or no phas corrlation btwn channls, PR can b usd to masur th distortion and nois causd by a larg numbr of individual channls, similar to an FDM systm. notch filtr is placd btwn th nois sourc and th D, and an FFT output is usd in plac of an analog rcivr. Th width of th notch filtr is st for about 500 Hz to MHz as shown in Figur 5 for th D99 -bit 65-MSPS D. Th sampling rat is 65 MSPS, th notch is cntrd at 8 MHz, and th PR is th "dpth" of th notch. n idal D will only gnrat th thortical valu of quantization nois, howvr a practical on has additional nois componnts du to additional nois and intrmodulation distortion causd by D imprfctions. otic that th PR is about 60.8 db compard to 6.7-dB thortical. Pag 7 of
Maing PR masurmnts digitally rquirs that th FFT hav a sufficint numbr of sampls such that thr ar at last 5 to 50 sampls within th filtr notch. Thr ar obviously tradoffs btwn th width of th notch and th FFT siz. Howvr, th notch width should not b widr than about 0% of th nois bandwidth, or th tst rsults may not b valid. In th xampl shown in Figur 5 for th D99, th FFT siz was 6,384 which givs a frquncy rsolution of 65 MSPS/6,384 3.97 Hz. Sinc th notch filtr width is approximatly MHz at th bottom of th notch, approximatly 50 sampls fall within th notch. Du to th spcific rquirmnts rgarding th cntr frquncy, width, and band-stop rjction, custom-mad notch filtrs ar gnrally rquird in ordr to implmnt PR tsts on Ds. chiving good rsults is difficult using just a simpl filtr and widband nois sourc. Widband Gaussian nois gnrators, such as th oisom DG7500, ar availabl that allow th usr to custom shap th nois according to thir application. Using a combination of a Gaussian nois shaping gnrator and notch filtr mas this tst asir to implmnt. Th rsults of svral FFTs must thn b avragd in ordr to rduc th variation in th PR rsults from run to run sinc thr ar only a limitd numbr of sampls which fall insid th notch itslf. Th data shown in Figur 5 rprsnts th avrag of th PR rsults for 5 individual FFT runs. PR should b masurd at svral diffrnt frquncis across th nois bandwidth, thrby rquiring svral notch filtrs. Som dgradation will occur at th highr frquncis vry similar to th dgradation in othr D ac spcifications such as SR and SFDR. PR 60.8dB OTH ETER 8MHz OTH WIDTH 3MHz PR 60.8dB f s Figur 5: D99 -bit, 65-MSPS D PR Masurs 60.8 db (6.7 db Thortical) Pag 8 of
SUMMRY W hav shown how PR is usd in standard FDM systms to charactriz th nois and intrmodulation distortion of multi-channl systm whr th voic channl width is 4 Hz. It can also b usd to dtrmin th optimum signal lvl to giv maximum dynamic rang. This 65-yar old concpt is still usful today in modrn multichannl wirlss systms. Bandwidths and channl spacings ar highr, but th sam concpts still apply. In many cass, PR is a good approximation to complicatd multi-ton tsting and mbodis th spcific faturs of many applications whn tsting your systm's dynamic rang (Rfrnc 7). lthough th singl ton or two-ton sinwav signal is by far th most popular mthod for tsting Ds for widband applications, PR tsting offrs a rlativly asy mthod using a Gaussian nois input to simulat a broadband multiton signal without th nd for gnrating a larg numbr of singl ton sinwavs. REFEREES. B. D. Holbroo and J. T. Dixon, "Load Rating Thory for Multi-hannl mplifirs," Bll Systm Tchnical Journal, Vol. 8, pp. 64-644, Octobr 939.. W. R. Bnntt, "Spctra of Quantizd Signals," Bll Systm Tchnical Journal, Vol. 7, pp. 446-47, July 948. 3. W. R. Bnntt, H. E. urtis, and S. O. Ric, "Intrchannl Intrfrnc in FM and PM Systms undr oiss Loading onditions," Bll Systm Tchnical Journal, Vol. 34, pp. 60-636, May 955. 4. M.J. Tant, Th Whit ois Boo, Marconi Instrumnts, July 974. 5. G.. Gray and G.W. Zoli, "Quantization and Saturation ois du to /D onvrsion," IEEE Trans. rospac and Elctronic Systms, Jan. 97, pp. -3. 6. Frd H. Irons, "Th ois Powr Ratio Thory and D Tsting," IEEE Transactions on Instrumntation and Masurmnt, Vol. 49, o. 3, Jun 000, pp. 659-665. 7. IEEE Std. 4-000, IEEE Standard for Trminology and Tst Mthods for nalog-to-digital onvrtrs, IEEE, 00, ISB 0-738-74-8. 8. oisom DG7500 Digital ois Gnrator, http://www.noiscom.com Pag 9 of
PPEDIX In this appndix, w will show how to valuat th following intgral from Equation 9. x / (x ) dx. Eq. This intgral is of th form: Bx (x ) dx Eq. whr, B,. Eq. 3 Intgrating Eq. : (x ) Bx dx Bx x dx Bx x dx Bx dx Eq. 4 ow, valuat th first intgral using partial intgration: x Bx dx B Th fundamntal quation for partial intgration is: Bx Lt u x and dv Bx dx. x ( Bx) Bx dx. Eq. 5 udv uv vdu. Eq. 6 Pag 0 of
Thn Bx Bx Bx x dx x dx B Eq. 7 B Evaluating th scond intgral in Equation 4: B B Bx dx Eq. 8 Bx Bx Bx B x dx ( Bx) dx B Eq. 9 B B Substituting Equation 8 and 9 into Equation 4: Bx ( x ) dx B B B Bx dx B B Bx dx B B B B Bx dx B B B Bx dx Eq. 0 ow from Eq. 3, substituting, B, and into Eq. 0 : / x / dx Eq. Dfin x t, x t, dx dt Eq. Pag of
Thn substituting into Equation and rarranging yilds: t / / dt Eq. 3 / ( )[ () ] Eq. 4 Whr () t / dt, th ormal Distribution Function Eq. 5 opyright 009, nalog Dvics, Inc. ll rights rsrvd. nalog Dvics assums no rsponsibility for customr product dsign or th us or application of customrs products or for any infringmnts of patnts or rights of othrs which may rsult from nalog Dvics assistanc. ll tradmars and logos ar proprty of thir rspctiv holdrs. Information furnishd by nalog Dvics applications and dvlopmnt tools nginrs is blivd to b accurat and rliabl, howvr no rsponsibility is assumd by nalog Dvics rgarding tchnical accuracy and topicality of th contnt providd in nalog Dvics Tutorials. Pag of