A Highly Accurate Algorithm for the Estimation of the Frequency of a Complex Exponential in Additive Gaussian Noise

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154 PROCEEDINGS 5TH AUSTRALIAN CO~lMlINlCAT10N THEORY W~':">H0? :'/>1 A Highly Accurate Algorithm for the Estimation of the Frequency of a Complex Exponential in Additive Gaussian Noise Sam Reisenfeld Abstract-A new algorithm for the precise estimation of the frequency of a complex exponential signal ill additive, complex, white Gaussian noise Is presented. The algorithm has low computational complexity is well suited for numerous real time applications. The DIT based algorithm performs an initial coarse frequency estimation using the peak search of an N point complex Fast Fourier Transform. The algorithm forms a frequency estimate using a functional mapping from two modified DIT coefficients which are one half DIT frequency bin below above largest magnitude FFT coefficient. Recursion is used to provide frequencies of the modified DFT coefficients which minimize the variance of the frequency estimation error. For large N large signal to noise ratio, the frequency estimation error variance obtained is 0.063 db above the Cramer-Rae Bound. This excellent performance is achieved with low computational complexity. The algorithm provides exact frequency determination in the noiseless case. Index Terms-s-- Fast Fourier Transform, Discrete Fourier Transform, frequency estimation, frequency tracking, exact frequency determination. I. I:'-iTRODUCTION There has been substantial prior work in the use of the Fast Fourier Transform (FFT) to estimate the frequency of a time sampled complex exponential signal in additive white Gaussian noise [1-4]. A recursive technique for frequency estimation which uses two OF!' coefficients was described in [9]. This paper introduces a new FFT based method to obtain a very accurate frequency estimate [5-6]. In particular, the coarse estimate is the frequency corresponding to the maximum amplitude FFT coefficient. Two modified OFT coefficients are then defined which are functionally related to the index of the maximum FFT coefficient. The frequency estimate is a function of the two modified OFT coefficients. The functional form is provided in this paper. The analysis of the frequency estimation performance of this algorithm the comparison to the Cramer-Rao Bound are also included, Simulation results for the performance of the algorithm are also provided. The algorithm provides exact frequency determination in the noiseless case. II. COARSE FREQUENCY ESTIMATION The received signal plus noise, r[nj, is given by r[nl = s[nl + 1][n), for n = 0,1,2,..., N - 1, (1) This work was supported in part by the Commonwealth Government of Australia tbrough the Cooperative Research Centers Program. The author is with the University of Technology, Sydney, Faculty ofengineeriog, Broadway, NSW 2007, Australia. where sin] is a complex exponential signal with f given by sin] = Ae j21r InT. I][n] is a sequence independent, identically distributed complex Gaussian variables with zero mean variance a 2. It i;; ccsired s process t[n], n = 0, 1,..., N -I, to obtain an estimate off where f is a fixed unknown parameter, f E /ij,) II:: sampling frequency is Is, T, c= }., The sif:i12.1to nol:;,] ratio (SNR) is defined as A 2 SNR = -". a- Rife Boorstyn [1] described a technique for freql1f' estimation using the Fast Fourier Transform (FFJ). The quency corresponding to the maximum amplitude FFT c cient is chosen as a frequency quantized approximation to maximum likelihood estimate, Define r= rio] r[l] 7'[N -1] Y = Y[U Y[l Y[N-l] where Y= FFT(r) FFT( ) is the N point compl operator. Following Rife Boorstyn [l], a coarse fre estimate, fa, may be obtained from kmax = max -1 [jy[kll : 0 s; k S; N -;) f ' - kmaxj 0- N s- Assuming that the SNR is sufficiently high, probable that J E Ifo -!:Fr, J~ +!:FrI. This is all threshold condition [1]. A fine interpolation may be to improve the frequency accuracy. Ill. MODIFIED DFT COEFFICIE:-ITS Define the modified discrete Fourier transform IT";"1'\.,~ ficients c< B as (3 = Y(kmaxl ~) = 'tlr[n]e-j21rnk"~1 I

A EIGHLY ACCURATE ALGORITHM FOR THE ESTl!v1ATJO"i OF THE FREQUDiCY OF A COMPLEX EXPO;-.lENTlAL 1>1 ADDITIVE GAUSSJ,\N NOISE 155 "I--;--~. 0-4\-... "." o'r. 01[--.,,1-1 IT 1m'll = 1m+ - tan [Dm tan(-)] Is (12) IT 2N Recursion is done in which thc index m is incremented. The estimate of the frequency is 1~. The new frequency discriminant, 6.F m (r) is shown in (12), where, 1 I IT ) 6Fm(r) = ;;:tan [Dmtan(2N)1 I, (13) Then putting (13) into (12) yields, fm+l = fm + 6Fm(r) (14) The frequency discriminant 6F',.,,(r) is used in the recursion. FIg. 1: Normalized frequency discnrninaut for "I -~ 1 N = 16 as a function of normalized frequency. A number of functional forms, which map 0',3 to an estimate of the error in 10, are possible. Functional forms contained in a particular class [5 J may be characterized by 1 Im"l - 10'11' 6/0(r) '" 2N 11/31"1 + 10'11' Is> fori> O. (6) The parameter 'Y defines the shape,. of the frequency discnrninant function as a function of f-7~f,. The frequency discriminant fundion shown in (6) with ') I may be used for frequency estimation. This function is 1 1,31 -!0:1 61o(r) = 2N 1;31+ led Is (7) Fig. I shows D.(~~(;) as a function of m- k max for ~/ = 1 N = 16. From Fig. I, it is obvious that the frequency discriminant function described in (7) provides an almost linear discriminant which may be used to obtain the estimate of the frequency of the complex exponential signal [5-6]. This paper introduces an improved frequency discriminant function, denoted 6F m (r ) [5-6], relative to the almost linear discriminant given in (7). IV. The algorithm is defined as For rn. = 0, 1,2,..., define RECURSIVE ALGORITHM, k rnax 10= TIs 11'-1 - _ "". [ ] - j27rn ( f.p;. - ~\. ) am - L...J r 11 e., i31t," - 11.=0,y -l _ c: rn,c,-" n~o ~ l 1 -;27Tn(7'n>.+.,"') (8) (9) (10) ( 11) V. TRUNCATION OF THE RECURSION ALGORITHM TO A FINITE NUMBER OF ITERATIONS feo is the explicit solution of Deo c-: 0, i; is the mth recursive estimate of frequency, m = 1, 2, :~,... In practice, II is sufficiently close to leo to be used as the frequency estimate. The recursion may be truncated to the evaluation of i2. Therefore, the algorithm is rapidly converging the frequency estimate may be obtained with only the computation of the FFT four additional modified DFT coefficients. VI. THE MAPPING FROM THE ERROR DISCRIMINANT TO THE FREQUENCY IN THE NOISELESS CASE In the noiseless case, the exact frequency may be obtained from the identification of the FFT coefficient with the maximum absolute value then forming 0' ;3 as shown in (4) (5), respectively. 1) may be computed from D = 1,61 - I<AI //31 + I<AI It is proven in Appendix A that for the noiseless case, - 1 _\ IT I= JiJ +;;: tan [D tan(2n)] Js (15) kmax 1. 1 IT = {-V' +-tan- \1)tan(:;v)J}I, (16) J IT 2J'i Therefore, (16) is an exact functional mapping from the FFT output two modified DFT coefficients to the frequency of a complex exponential signal in the noiseless case. VII. REDUCTION OF THE RMS ERROR USING RECURSION The rms frequency estimation error due to additive noise, using the proposed frequency discriminant, decreases sharply for ({~~) :::::O. Since the iterative solution adaptively produces 'a frequency estimate which is close to this condition, the recursive use of the discriminant produces a frequency estimate with small rms error. By means of the recursion, the discriminant converges to a minimum rrns error condition. Fig. 2, which was obtained b simulation, shows the normalized rms frequency error, E{I(]1 - J)TsF}, as a function of the normalized frequency (i) - krnr;x.

156 PROCEEDINGS 5TH AUSTRALIAN COMMUNICATION THEORY WORKSHOP 2004,!!! "~H-1 ;- -'-- f- -+ --i -+-1--,] 1110'4 6 r-------r--------,--- I i ----,-----,-----r, I I 5 :. - - ~ - - -:- - - ~ - - -:- - - r- - - -: - - - ~ - - ~ - - - - ~ -1- '; : I I I I I J I I I 1: I I I I I I r J I : 4 511 ': J 1_ _..1 1 L ~ 1 L.J L _, I I ; I I I I I I [ non;;:~zed " -.. :... ~ Tl_...:,._ ~._:.~ ~ ~ ~ _ ; frequency i: '1 I estimation I I J I I I I I I' error )~.: ~"4~', : ~ : ~ _..._: ~ ~.J'~'~! ", J 1 I I I I! " I I. I I I I I I I /' I..! ~.1..~. _ L 1 L I L..._.A!'~ 1 _ I 1... I I J I 1,./ I I I I '~J I I l," I I 2.5 - - ~ _. - -:- - - ~"~-"":.>..~ "::.-'-: ~_~.;::.~::"p:~>j ';.' ~:.~~ ~ - ~ I I I I I I I r I I [ 1 I I I 15-1)4-0:).{12 -Of 0.1 0-; 03 O..t 05 f I~- l:;n~..., 1'1 ~ N I Fig. 2: Normalized rms frequency estimation error as a function of the normalized signal frequency relative to the FFT bin center frequency for SNR ==!OdB, N =: 64. As can be seen from Fig. 2, the rrns frequency error in the noisy case sharply decreases as the signal frequency approaches the mid-point between the frequencies of the two modified DFT coefficients. This justifies the use of the recursion to obtain an estimate that is close to this minimum rms frequency error condition. For the recursion described in (8), (9), (10), (II) (12), the final estimate approaches the minimum rms error condition because the modified DrT coefficient frequencies approach one half DFT frequency bin width above below the actual frequency. The initial estimate is f ' kmaxf o=n which is obtained from the FFT peak search. The second estimate is,, 1 1 [ 11" 1 fl=f01 :;tan- Dotan(2N)f. where, Do = l~~l~l:~" will be in a region of low rms estimation error. The third estimate is,, 1 I.. 11" h = It +:; tan- [DI tan(2n)j!. h D Illll-I"',I btai d f f' were, 1 = IIl,I+I"',I' IS 0 tame rom 1. estimate, J2' is characterized by extremely low error. The recursion essentially converges for 12. s The VIII. PERFORMANCE OF THE ALGORITHM third estimation As shown in Appendix B, using the Taylor Series [7] at high signal to noise ratios, the performance of the algorithm is n N sin2(2!...) tan2(2!...), 2N 2N (Jj = 4 SNR 11"2 ( 17)... 0.Q7--~r-----r----r---r---r------, 005\ : f [ ;,.. '!" j.--. -----_ o.os.. _.. _.--~-_._. -~.""!".--...--.~..._ -0" -- -!- : ; : : : t----- -. r.. : :.. ;.... i...... i. -0 01 ~ zn 400 600 OCO 10c0 2 Fig. 3: _,ul- in db as a function of N. O'CRLB where, a} is the mean square value of (100 - nt. a f is a uniformly distributed rom variable in the [0,(,). The Cramer-Rae lower bound on the unbiased f estimation mean square error [I] is 6 (Jb RL B = -;----:-O:-::-:'77::;;--~=-=- (211")2N(N2-1) SNR Therefore, the performance of the frequency algorithm compared to the Cramer-Rae Lower Bo (J} N2(N2-1) sin 2 (fjv) tan2czil (JbRLB 6 For large SNR large N, (J2 4 lim 10 IOgIO(-2- J -) = 10 loglo(1i"96)=0 N --t00 (J C RI,B This limiting value was also obtained by a di in [8J, Fig.3 shows a plot of (19) in which the co asymptotic limit with increasing N is apparen values of N, the degradation of the estima to the Cramer-Rao Bound is less than 0.06 N, less information is discarded by not coefficients, the performance degradati large N. IX. SIMULATION OF THE PERFO Figs. 4 5 show the rms estimation e of the SNR in db for the cases of a point FFT, respectively. The simulation r analysis for the asymptotic value of the d Cramer-Rae Lower Bound of 0.0633 db. N X. CONCLUSION An algorithm for frequency estimation h For the noiseless case, the algorithm p relationship from the FFT output t ficients to the exact frequency of a comp

"HIGHLY ACCURATE ALGORITHM FOR THE ESTIMATION OF THE FREQUENCY OF A CO:.lPLEX EXPONENTIAL IN ADDITIVE GAUSSIAN NOISE 157 10' :1)!::!11~~.::m1!:m!:m:m::~!:::::~:::1Hm:!~:m:i:m1~11m~::1:1~111:!I!:!1!::!W 10' :~ ~; :;t~~~ffi~;i ~ ;:~ :,J::;;,;: ::I;;;;:: ;~:~ 1 :t ;:;: :I ;;:: :;;;~ ::::::::::::::::::::::::j::::::::::\::::::::::j:::::::::::;:::::::::::'f:::::::::::1:::::::::::...\.. Ifsf.....::::::::::~::::::::::;:l:::::::::::~:::: ::::::::j::::: ::::::i:::::~::::::~:::::::::::j::::: ::::::... ~. ~. (..-,. }... t.. :-. 1.... T.... ~.. ~.. f....r.. ~ r.. 10~.15.10 5 10 1, SNRindB Fig. 4: The rms frequency estimation error as a function of the SNR in db. Resultsare shown for one two iterations. The results are compared to the Cramer-Rae Lower Bound. The FFT length is N = 1024. 10' ;: ~~:!:~~:!~:! ~!~! ~!~~!:!:~!~!:::~!~~: ~!~~:!!:::!;:::!::~!:::;::!!:~:!!:::!!!!: g=!::::!!!:.:::::::::: ~:::::::::::~;::::;: ::::~:::::::::::~:::::::::::~:::::: ::::: :~:::::::::::~::::::::::: 10' 1:1i~iliii11ii;i[ii~,1 ~:~i;1!!1,1,ii:[,ii!1:~1!i1::li!i:iiii~:1 i!i! 11i;!i:!11i:i1i1;i".. i i.. ~ ~..... i.. i......!.. +.. '0' ~~m~~~~~~l ~~~1m ~~[[ ~[~~[11m [[ [: ~[m [[!1[!!![ 1[![~l[~ ~11[1~[mI111! 11~:H1~11~m 11: lan-lido Then, Proal tan(2;"))} f. where, D _ 1.Bo1-laol o - 11101+ laol' lv-i. ao = L r[n]e -j271'n( ~ - i-};r), lv-i "" 110 = 6 rlnle~j '2 (fa. J 1fnr,"'271 ) For the noiseless case, 111m I = ~Icos(7rN ~)I Ism[7r(e - 2N)11 laml = AI COS(lTNe)I Isinl7r( e + 2~ li1ml-laml II1ml+ laml tan( 7re) tan( 2;" ) )JJ 1 IT e ~ -tan-- 1 [Dmtan(2f\')] 1i, for the first iteration in the recursion,,1 1 IT f = fo +;; tan- [Do tan(2n)lf. { kmax 1 I[ (1i) = "IV" +;; tan- Do tan 2N) } f. (21) (22) (23) '5-10 5 o!o 15 SNRindB Fig. 5: The rms frequency estimation error as a function of the SNR in db. Resultsare shown for one two iterations. The results are compared to the Cramer-RaeLower Bound. The FFT length is N = 64. 20 VARIANCE OF THE NORMALIZED FREQUENCY ESTIMATION ERROR Theorem 2: For sufficiently high signal to noise ratio, such that the probability of error in the FFT peak search is negligible, for sufficiently large m such that i;~i: For the case of signal plus Gaussian noise, one iteration of the recursion provides sufficient frequency estimation accuracy f,0rmany applications. For applications requiring performance dose to the theoretical limit, two iterations of the recursion orcvide frequency estimation performance which is only 0.063 ill) above the Cramer-Rae Lower Bound on frequency estimason error. The algorithm has low computational complexity iilld is suitable for real time digital signal processing appli-.e31ionsincluding communications, radar, sonar, spectral I ~na:ysis. ApPENDIX EXACT FREQUENCY DETERMINATION FOR THE NOISELESS CASE Theorem I: In the noiseless case, E:,= (J - fm)t. = ~ tan-i\d m tan( 2~ )), in particular, f = {~+ ~ 2 = V [(f _ f' )T. J = N sin 2 (i]v) tan 2 (i7v) (7f or m s 4 SNR lt2 Proof For high SNR t; ~t. J.1-a J.1-{3 = EliamlJ r-v A --:-'7"::--7 sine 2;" ) = ElII1mlJ A "-'---- sin(2;" ) Also, lam I 111m I arc uncorrelated rom variables. Furthermore,

158 1 a~ = Varll/3mll = 'ina2 Using the stard technique of finding the variance at the output of a nonlinear function of two variables by exping the nonlinearity in a two dimensional Taylor Series Expansion [7], Then, 1 1 [ 71" E=(J-!m)T.=;tan- Dmtan(2N)}, E[Dml = 0 therefore, E[E] = O. For small D m, which is the case for f ~i; Therefore, 1 71" E ~ -D tan(-) 71" m 2N 1 2 71" VadE] ~ 71"2 Var[Dml tan (2N) N'2(1I')t 2(11') _ 1 8m 'in an 2Fi 4 SNR 1f2 PROCEEDINGS 5TH AUSTRALIAN COMMUNICATION THEORY WORKSHOP 2004 REFERENCES [IJ D.C. Rife R.R. Boorstyn, "Single tone parameter estimation from discrete-time observations;' IEEE Trans. Inform. Theory, IT-20, No.5, pp 591-598, September, 1974. [2J D.C. Rife, "Digital tone parameter estimation in the Presence of GsUJo sian noise;' Ph.D. dissertation, Polytechnic Institute of Brooklyn, Juoe,. 1973. [3J B.G. Quinn, "Estimating frequency by interpolation using Fourier c" ficients," IEEE Trans. Signal Processing, vol 42, pp 1264-1268, MIll 1994. [4J B.G. Quinn, "Estimating frequency, amplitude phase from the DFT of a time series," IEEE Trans. Signal Processing, vol 45, pp 814-811, March, 1997. [5J The University of Technology, Sydney, "Frequency Estimation," Australian Provisional Patent 2002950296, 19 July 2002. [6J The University of Technology, Sydney, "Frequency Estimation," International Patent Application PCT!AU03100862, International Filing Dale, 4 July 2003. [7J A.A. Sveshnikov, Ed., Problems in Probability Theory, MtI/!lemalici Statistics Theory of Rom Functions. New York: Dover, 1968. (8) E. Aboutanios, "Frequency Estimation for Low Earth Orbit Saldhlc&" Ph.D. dissertation, University of Technology, Sydney, 2002. (9J S. Reisenfeld E. Aboutanios, "A new algorithm for the eslimaliill of the frequency of a complex exponential in additive Gaussian noilt,' IEEE Communication Letters. Vol 7, Issue II, pp 549-551, Novembe; 2003.

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