A Low-Cost and High Performance Solution to Frequency Estimation for GSM/EDGE

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1 A Low-Cost and High Prformanc Solution to Frquncy Estimation for GSM/EDGE Wizhong Chn Lo Dhnr Wst Parmr Ln 77 Wst Parmr Ln Austin, TX7879 Austin, TX7879 ABSTRACT Frquncy synchronization in GSM/EDGE includs Frquncy Burst (FB) dtction and FB ton frquncy stimation. This papr focuss on th FB frquncy stimation, which is th frquncy stimation from a singl-ton signal. Diffrnt from convntional suboptimal solutions, th solution dscribd in this papr is basd on optimal Maximum Liklihood (ML) stimation with a squnc of fficint implmntation tchniqus. As compard with typical convntional solutions, th nw frquncy stimation provids about 5dB gain in diffrnt channl conditions whil rquiring significantly lss silicon gats. Th tchniqus prsntd in this papr ar also gnric and can b qually applid to many diffrnt applications in digital communication systm dsigns.. TRODUCTO Frquncy synchronization is th first stp whn powring up a GSM/EDGE mobil handst and is nabld rgularly in standby. Although GSM/EDGE has bn on markt for mor than yars, many companis hav not rvisitd its original dsign in frquncy synchronization. This papr prsnts a nw solution that is significantly mor cost ffctiv whil providing th optimal prformanc. n GSM/EDGE, frquncy synchronization is achivd through th dtction of th prsnc of a Frquncy Burst (FB) and th frquncy stimation from th dtctd FB. This papr is focusd on th scond half of th frquncy synchronization: frquncy stimation from th dtctd frquncy burst, which is a singl ton signal. GSM/EDGE spcification rquirs that th frquncy stimation rror is lss than Hz. A singl ton digital signal can b rprsntd in th form x ( = Axp{ j( ω n + ϕ)} + z(, n=,,- () whr A is th magnitud of th ton, ω is th normalizd frquncy of th ton, ϕ is a random phas of th ton, and z( is th zro-man complx Additiv Whit Gaussian ois (AWG) with varianc σ. Th frquncy stimation from th digital signal x( is a non-linar stimation, which xhibits th thrshold ffct typical to non-linar stimation: blow an SR thrshold, th man-squard stimation rror incrass rapidly. Diffrnt solutions xhibit diffrnt thrsholds. Maximum Liklihood (ML) stimation is considrd th optimal solution. ML solution, howvr, has not bn widly usd in practic du to its complxity. Many sub-optimal solutions hav bn dvlopd to rduc th complxity by compromising th prformanc. Th Maximum Liklihood (ML) stimation [] of th frquncy ω can b rprsntd as follows ˆ = arg max ω n= ω x( xp( Th ML stimation () abov is simply th sarch for th Fourir spctrum pak location. Using th FFT approach to calculat th Fourir spctrum is normally not nough to provid sufficint rsolution. To incras th rsolution, sufficint zro-padding is ndd, which is quivalnt to calculating Fourir spctrum on mor qually spacd frquncy bins. ormally, th frquncy sarch window is much lss than [ π, + π ]. Thrfor, th FFT approach with high frquncy rsolution is normally not fasibl in singl ton frquncy stimation. To avoid th complxity of th ML stimation, a numbr of sub-optimal solutions hav bn dvlopd. On widly usd solution is basd on th signal s autocorrlation r ( = x( x( n l) () n= l () 6 Txas Wirlss Symposium 5

2 Th simplst form was proposd by Lank t al [] as follows ˆ ω = arg r( l) () l For l=, this approach can b intrprtd as a linar prdictor stimator []. Othr forms of autocorrlation basd stimation wr proposd in [] in th form of J l arg[ r( l)] l = = J l l = ˆω (5) Anothr st of sub-optimal algorithms oprat on th signals phas. Kay[5] considrd using th phas diffrncs φ ( = arg( x( x ( n )), n=,,,- (6) Th carrir cntr of th rcivd signal could hav a variation btwn ±KHz du to mobil station s LO uncrtainty in powr-up mod. Whil in standby mod, this variation is within ±KHz. A FB in frquncy domain can, thus, b rprsntd by Figur. FRAME# tail bits FCCH bits Frquncy Burst Multifram (5 frams) tail bits Figur. Frquncy Burst in GSM/EDGE Kay shows that for larg SR, φ( can b xprssd as φ = ω + u( (7) f KHz f KHz f whr u( is a zro-man colord Gaussian procss. This lads to th stimation fs f = n= ˆ ω w( φ( (8) whr w( ar wights drivd from th corrlation of th nois trm u(. Both th autocorrlation basd and phas-diffrnc basd frquncy stimation xhibit significantly highr SR thrshold than ML. n othr words, th sub-optimal solutions prform substantially wors than th ML solution in low SR. What follows dscribs a solution that is basd on th ML stimation () with a squnc of fficint tchniqus. Th rsulting solution is provd to hav ngligibl compromis on optimal prformanc. Manwhil it is significantly lss complx than many suboptimal solutions, spcially in ASC implmntation.. PROBLEM DEFTO n GSM/EDGE, Frquncy Bursts (FB) ar allocatd through Frquncy Control Channl (FCCH), as shown in Figur. A burst, including FB, in GSM/EDGE has a duration of 576.9us, which is quivalnt to 56.5 symbol priods with 7.8KHz symbol rat. A FB is a singl ton signal at 67.7KHz rlativ to th carrir cntr. Figur. Frquncy burst in frquncy domain f th rcivd signal is sampld at th symbol rat 7.8KHz, th nominal frquncy of th FB will b at ¼ of th sampling rat. Thus in th normalizd frquncy domain, th nominal frquncy of th FB ton will b xactly at ½ π, as shown in Figur. Th frquncy uncrtainty of ±KHz in powr-up mod and ±KHz in standby mod ar quivalnt to th uncrtainty of ±.5 and ±.65 in th normalizd frquncy domain. Th frquncy stimation of th FB ton is mad aftr th FB is dtctd. Th timing of th dtctd FB has crtain tim uncrtainty. ormally th signal availabl for th FB ton frquncy stimation is lss than sampls. Th problm of frquncy stimation in GSM/EDGE synchronization can thus b formulatd as th frquncy stimation from about sampls of th rcivd digital signal, whr th frquncy sarch window is cntrd at ½ π with a width of ±.5 and ±.65 in th powr-up mod and standby mod rspctivly and th stimation rror is rquird to b lss than Hz or.665. What follows is to dscrib a solution that is basd on th ML stimation () with a squnc of fficint implmntation tchniqus. 7

3 Figur. Frquncy burst in normalizd frquncy domain.. EFFCET AD OPTMAL FREUECY ESTMATO Th basic ida to achiv complxity rduction for th ML solution () is a four-stp sarch as shown in Figur. Stp- is th coars frquncy sarch usd for powr-up mod only. t calculats th DTFT (Discrt Tim Fourir Transform) from sampls of th signal ovr frquncy bins. Stp- is a fin sarch around th sarchd pak from Stp- with 8 sampls of th rcivd signal, in which th DTFT ovr 9 frquncy bins ar calculatd. Th 9 frquncy bins hav twic th frquncy rsolution than that in Stp-. Stp- is a nd ordr polynomial pak match from th thr highst DTFT magnituds sarchd from Stp-. Th polynomial sarch furthr incrass th rsolution by tims. Th final stp is th convrsion of th sarch rsults from Stp- through Stp- to th frquncy stimat. Stp- Stp- Stp- Stp-V π 7. 8 π Figur. Four-stp solution for frquncy sarch Th cor of th four-stp solution is th DTFT of th rcivd signal, which is usd rpatdly in Stp- and Stp-. Th fficincy of th DTFT calculation, thus, has a dirct impact on th fficincy of th ovrall solution. Th DTFT is calculatd basd on Gortzl Algorithm[6], with som simplifications. Gortzl Algorithm stats that th DTFT X(ω) of a complx signal x( is calculatd rcursivly as follows π - + Convrt th sarchd rsult to Lo-Offst.65 ω + X( ω) = ) ), = cosωy ( n ) n ) + x(, Diffrnt from FFT algorithm which calculats th DTFT on qually spacd frquncis ovr th ntir [-π~+π] rang, th Gortzl Algorithm calculats th DTFT on any frquncy, thus allowing th calculation of th DTFT only on thos frquncis to b sarchd. f th sarch window is small and th rquird rsolution is high, Gortzl Algorithm is a vry dsirabl choic. n addition, Equation (9) shows that th rcursion can b sparatd into th ralpart and th imaginary part for th ral and imaginary parts of th signal rspctivly. Thrfor, on rcursion circuitry is ndd for both, making Gortzl Algorithm a cost fficint solution for ASC implmntation. What follows dscribs simplifications for th Gortzl Algorithm. Lt A = cosω, th rcursion in (9) bcoms y ( = A n ) n ) + x(, n =,,..., () n our problm, cos ω = cos( π / + ) = sin, thus, A = sin, whr = π f / f with s f Hz bing th rcivr LO offst in powr-up mod and f Hz in standby mod, and f s = ( / 8) Hz bing th sampling rat. Convntional practic quantizs ω, which rsults in a squnc of qually spacd ω that rquirs a larg numbr of bits to rprsnt th rsulting paramtr A = sin. Th larg numbr of bits in A will rquir a big multiplir to calculat An-). n addition, th convrsion from ω to A rquirs xtra circuitry or look-up-tabl. Th basic idal to simplify th rcursion (9) is to quantiz th variabl A, instad of th frquncy ω to b sarchd. n Stp- coars sarch, considring sin A π =.5, lt 6 A k da ( / 8) n =,,, (9) =, whr da = and k = [,,..,...,,]. Th multiplication An-) is rducd to th multiplication k n ) sinc th factor of da = rquirs a simpl shift. This way of quantization maks th multiplir for An-) significantly smallr than by quantizing ω. With th granularity of da =, th maximum sarch stp-siz for f is lss than.5khz. 8

4 n Stp-, fin sarch, lt A = da + kda, whr is 5 th coars sarch rsult from Stp-, and da = da / = and k = [,..,,.. + ]. Th allocation of th sarch window in Stp- is in ordr to covr th rquird frquncy sarch rang of f < KHz in standby mod. Th sarch 5 granularity da = in Stp- is quivalnt to a frquncy sarch stp siz lss than 75Hz. Th final stp of th Gortzl Algorithm for th DTFT calculation is th combining of th rsults of th last two rcursion stps, -) and ), by X ( ω ) = ) ) () whr th ky challng is ) = [ y ( )cosω + )sinω] j[ y ( )sinω y ( )cosω] () whr cosω = A/ and sinω = sin = A /. t is obvious that th onlin calculation of sin ω from A is not dsirabl. On altrnativ is to us look-up-tabl to obtain sin ω. Howvr, th accurat rprsntation of sinω rquirs larg numbr of bits, which lads to a larg multiplir to calculat y ( )sinω and y ( )sinω in (). To simplify th calculation, th function f ( A) = A / can b approximatd with diffrnt ordrs. Th th and st ordr approximations ar as follows f ( kda ) f ( kda ) k f (da ) th st ordr ordr Th approximation rrors ar shown in Figur 5. th Ordr Approximation f ( kda ) = ( kda ) / st Ordr Approximation k Figur 5. Approximation of th f(k da ) () t appars that th th ordr approximation for sin ω is out of considration du to its larg approximation rror. Howvr, th goal in our problm is not an accurat DTFT. Th goal is th accurat pak location of th rsulting DTFT X (ω) or X (ω). n othr words, what is important to our problm is th prsrvation of th shap around th pak. Simulation analysis indicats that th th ordr approximation is sufficint in our application. With th ordr approximation, jω ) can thn b approximatd by ) [ y ( ) + [ y ( ) Ay ( ) y ( )] Ay ( ) + y ( )] () whr Ay ( ) and Ay ( ) hav bn calculatd in th last stp of rcursion (9) whn calculating th y () and y (), thus no multiplication is nd for th combining opration (). Figur 6 summarizs th ovrall calculation for th simplifid DTFT. x ( x ( A z - z - - Figur 6. Simplifid DTFT calculation As indicatd arlir, th sarch stp siz in Stp- is th 75Hz, whil th rquird rsolution is much lss than Hz. Stp- is to incras th stimation rsolution by using a scond ordr polynomial pak match from th largst DTFT magnituds sarchd from Stp-. Assum th thr largst DTFT magnituds sarchd from Stp- ar P(-), P(), and P(). Th scond-ordr polynomial pak location rlativ to th sarchd pak location of P() can b calculatd by frac = () b whr a = [ P( ) + P( + )]/ P() and b = [ P() P( + )]/. t is asy to s that frac< implis that th pak is on th lft sid of sarchd pak of P(). Th sign of frac can b asily dtrmind by comparing th P() and P(-). Th division in Equation () can b implmntd fficintly by noting th fact that frac. gnoring th sign, frac = b / a can b rprsntd in s powr by a z - z - X (ω) X (ω) 9

5 + b + + b () frac = b with b k = or to b dtrmind. Th valus of b k can b dtrmind by a simpl logic without multiplications. t is found that =5 is sufficint, which is quivalnt to qually spacing th da by or da =da /= -. Th rsult of th sarch Stp-, Stp- and Stp- is thn combind as follows A = + () da + da da whr da = -, da = -5, and da = -, and =,±,...,±, =,±,...,±, and =,±,...,±. Sinc th DTFT pak sarch is mad on th paramtr A, instad of frquncy f, now coms th issu of convrsion from th sarchd A to th final goal f. f and A ar rlatd by π A f = sig A) sin ( ) () ( f s Th magnitud of A can b rprsntd by A = ˆ da + ˆ da (5) whr ˆ =,,...,, and ˆ =,,..., 6. Th valus of { ˆ, ˆ } can b obtaind dirctly from th s powr rprsntation of A without rquiring any calculation. A look-up-tabl (LTU) is usd for th convrsion from ˆ, ˆ } to f. { [ + δf ( ˆ )] f = f ˆ ) + ˆ (6) ( Equation () rprsnts a sgmnt-wis linar intrpolation, whr [ + δf ( ˆ ) ] is th slop of th ˆ th sgmnt. Thus two LUTs ar ndd for f ( ˆ ) and δ f ( ˆ ) rspctivly, ach with ntris. Th rsulting frquncy stimat will hav a rsolution lss than Hz.. SUMMARY AD GEERALZATO This papr dscribd a singl ton frquncy stimation basd th optimal ML solution () with a squnc of fficint tchniqus. Th rsulting solution is provd to hav ngligibl compromis on th optimal prformanc, whil it rquirs significantly lss complxity than many sub-optimal solutions in practic. Th rsulting solution is a four-stp solution: a) Stp- is a coars sarch with a small portion of th signal, b) Stp- is th fin sarch, c) Stp- furthr incrass th sarch rsolution by using th nd -ordr polynomial pak match from th sarchd rsults of Stp- and Stp-. Stp-V is th convrsion of th sarch rsults of Stp-, Stp-, and Stp- to th final frquncy stimat. n standby mod, th Stp- can b skippd. n Stp- and Stp-, th ky is th DTFT calculation basd Gortzl algorithm with simplifications. Th simplification includs a)quantization of th paramtr A, instad of th quantization of th frquncy to b sarchd in normal practic, and b) focusing on th prsrvation of th DTFT pak shap, instad of th accuracy of th DTFT. Most of th tchniqus dscribd abov can b qually applid to any singl ton frquncy stimation with a fw xcptions. First, if th frquncy sarch window is small, th Stp- can b skippd. Scond, if th nominal frquncy to b sarchd is clos to or ±π, rathr than at +½π or -½π, th Gortzl algorithm cannot b usd dirctly sinc th rcursion (9) will gnrat intrmdiat valus so larg that any softwar or ASC implmntation bcoms unralistic. This problm can b asily avoidd by shifting th input signal in frquncy by ithr + ½π or - ½π. This frquncy shift can b implmntd as follows ± j π n x ( = x( (7) t is shown that th frquncy shift (7) rquirs no calculation.. REFERECES [] D. Rif and R. Boorstyn, Singl-ton Paramtr Estimation from discrt-tim obsrvation, EEE Trans. nformation Thory, Vol. T-, pp59-598, Spt. 97. [] G. Lank,. Rd, and G. Pollon, A Smicohrnt dtction and Dopplr stimation statistics, EEE trans. Arosp, Elctron. Systm., Vol. AES-9, pp. 5-65, Mar. 97. [] L. B. Jackson and D. W. Tuffs, Frquncy Estimation by Linar Prdiction, in Proc. nt. Conf, Acost, Spch, Signal Procssing, Vol. ASSP-6, 978, pp [] M. Filz, Furthr rsults in th fast stimation of a singl frquncy, EEE Trans. Commun., Vol., pp , Fb. 99. [5] S. Kay, A fast and accurat frquncy stimator. n EEE Trans. Acousti. Spch, Sgnap Procssing, vol. 7, Dc. 989, pp [6] Gortzl, G., An Algorithm for th Evaluation of Finit Trigonomtric Sris, Amrican Math. Monthly, Vol. 65, pp. -5, Jan. 958.

COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.

COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH. C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH