A Faster and Accurate Method for Spectral Testing Applicable to Noncoherent Data

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1 A Fater and Accurate ethod for Spectral Teting Applicable to Noncoherent Data inhun Wu 1,2, Degang Chen 2, Guican Chen 1 1 School of Electronic and Information Engineering Xi an Jiaotong Univerity, Xi an, P. R. China wuminhun@tu.xjtu.edu.cn 2 Department of Electrical and Computer Engineering Iowa State Univerity, Ame, IA, USA Abtract The Fat Fourier Tranform i the tandard approach for pectral teting. However, it correct application to inuoidal ignal require either trict coherent ampling, or careful windowing, or other method that are not computationally efficient. Thi paper introduce an improved method for achieving accurate and robut pectral teting for inuoidal ignal without the need for coherent ampling or windowing. Theoretical analyi, extenive imulation reult, and experimental reult how that the propoed method i alway fater than the original method and robut when the ignal frequency i cloe to Nyquit frequency. Statitical analyi and comparative tudie demontrate that the propoed algorithm achieve almot the ame pectral teting accuracie a thoe obtained under perfect coherent ampling. I. INTRODUCTION Spectral performance of an integrated circuit i of critical concern in many important application area uch a ignal proceing and communication. It i well nown that DFT (Dicrete Fourier Tranform) or FFT (Fat Fourier Tranform) i the mot prevalent method for pectral teting. However, when performing FFT for pectral teting of inuoidal ignal, one mut mae ure that the data record being ued in the FFT algorithm repreent exactly an integer number of period of the ignal. In other word, the ignal frequency and the ampling cloc frequency of the data acquiition ytem mut atify coherent condition. In FFT algorithm, even the lightet mimatche between the two frequencie will caue the frequency leaage phenomenon in which energy from the fundamental pectral line i pread into neighboring frequencie cauing the appearance of a irt around the pectral line. In order to combat the frequency leaage, the IEEE tandard [1] a well a indutry bet practice i to require coherent ampling, meaning that the ampling cloc ignal hould be perfectly ynchronized with the ignal under tet o that an integer multiple of ignal period are captured in a data record. When thi i guaranteed, direct ue of FFT i permitted and the data analyi i computationally very efficient. Unfortunately, trict coherent ampling i difficult to maintain, epecially for on chip implementation. Another alternative method i to ue the windowing technique [2, 3] while allowing noncoherent ampling. Thi technique doe not remove the irting due to noncoherency, rather it merely uppree the irting level at frequencie far away from the fundamental frequency. By doing o it alter the height of the original pectral line. Care mut be taen in order to correctly recover the pectral line. Another limitation i due to the fact the amount of irt uppreion i limited and hence it i not ufficiently accurate for many high reolution application. Other method to combat pectral leaage include ingular value decompoition [4], 2-D FFT [5] and filter ban [6]. Thee method are accurate but they are computationally very inefficient. In order to overcome the hortcoming of the above method, the concept of fundamental identification and replacement wa firt introduced in [7]. In thi method, the amplitude, frequency and phae of the fundamental harmonic component are etimated firtly. Then the noncoherent fundamental harmonic component wa replaced by a ine component that ha the ame amplitude and phae but a lightly modified frequency o that it become coherent with the ampling cloc. The method did not require coherent ampling and windowing. However, the method i robut only when the ignal frequency i not very high. It i vulnerable when the ignal frequency i cloe or above ADC Nyquit frequency. Furthermore, the method i occaionally not computationally efficient if the data record length ha larger prime factor, epecially i a prime number. An improved fundamental identification and replacement technique i propoed in thi paper. The method can achieve accurate and robut pectral teting for inuoidal ignal without the need for coherent ampling or windowing. The improved method wor well when the ignal frequency i cloe to Nyquit frequency. Furthermore, the method i computational more efficient than the original one. In Section II, an improved fundamental identification and replacement technique i propoed. The imulation and experimental reult are reported in Section III and IV repectively. Section V preent tatitical analyi of extenive imulation reult howing that the propoed method achieve pectral teting accuracie comparable to thoe obtained with perfect coherent ampling in an ideal noie-free environment. Section VI conclude thi paper /1/$ IEEE 7

2 II. THE IPROVED FUNDAENTAL IDENTIFICATION AND REPLACEENT TECHNIQUE Let f be the ampling frequency, T =1/f the ampling interval, f i the unnown input ignal frequency, and the nominal data record length. Then J= f i /f =J + will be the number of period input ignal in the data record, where J i the integer part of J, i the fraction part of J. J and are aumed to be co-prime. i unnown, o i J (J could be nown). Let the input ignal be x( t) = Ain(2 π fit + θ) + ( bn in 2π nfit + an co 2 π nfit) (1) Where A 1, θ [,2 π ), a n, b n for are all unnown, but together they atify 2 2 ( bn + an ) 1 (2) ( bn in 2π nfit + an co 2 π nfit) 1 (3) The ample of x(t) at ampling rate f are given by: x[ ] = Ain(2 π fi + θ ) f Since + ( b in 2π f + a co 2 π f ) n i n i f f f J J + f i = =, J x[ ] = Ain(2 π + θ ) + ( b in 2π f + a co 2 π f ) f f (4) n i n i J + = A π + θ + b π f + a π f in(2 ) ( n in 2 i n co 2 i ) f f = x [ ] [ ] 1 + xh (5) In (5), x 1 [] i the fundamental harmonic component of x[], x h [] i the um of the 2nd and higher harmonic component of x[]. From [7], we now that a long a i non-zero, which mean the data record length i not exactly an integer number of ignal period, the DFT algorithm introduce an error term (irt term) in the Fourier tranform of the fundamental component. Thi leaage term can be o large that it completely inundate the harmonic ditortion component, maing it impoible to correctly tet the true pectrum of the ignal. In order to etimate and remove the irt term from the DFT pectrum, the concept of fundamental identification and replacement wa firt introduced in [7]. In thi method, the amplitude, frequency and phae of the fundamental harmonic component are etimated firtly. Then the noncoherent fundamental harmonic component wa replaced by a ine component that ha the ame amplitude and phae but a lightly modified frequency o that it become coherent with the ampling cloc. After that the tandard FFT pectral analyi i done a uual. Unfortunately, the method in [7] ha two hortcoming. The firt one i that it i occaionally computationally unefficient. The method in [7] chooe the bet data record length by earching ample from to 2. A we all now, the algorithm of FFT i efficient and require O(log) operation if ha mall factor, the algorithm of FFT will require O( 2 ) operation if i a prime number. In order to avoiding the unliely cae that i a prime number, we chooe the bet data record length having only prime factor of 2 or 3, the two mallet prime number. Furthermore, the pair far away from zero, uch a the pair whoe abolute value are larger than A / 2, are ued to elect the bet data record length o that the effect of noie can be reduced greatly. For example, earch through 1, 2,, i, ( i factor comprie only 2 or 3) point, to find the (1+ i )th data point that mot cloely matche the 1t point in the data equence. That i, x[1] through x[ i +1] mot cloely match an integer number of ignal period. Then i i the bet data record length. Therefore, the propoed method would be computationally more efficient than the method in [7]. After chooing the bet data record length, we till ue the poitive zero-croing point a the tarting point of the data record. For intance, in the data record from x[1] to x[], x[1] i the poitive zero-croing point. By doing o, θ will be approximately and the error in etimating θ will have le effect. The econd one i that it i vulnerable when the ignal frequency i near ADC Nyquit frequency. The reaon may be that the method in [7] doen t count the integer cycle of input ignal correctly when the ignal frequency i high. Therefore, a new accurate method for counting the integer cycle of input ignal i applied in the improved fundamental identification and replacement technique. The new method for counting the integer cycle J int in the data equence from x[1] to x[] i introduced a follow (here we call method I). a) Let y[]=1 if x[ ] A / 2, y[]=-1 if x[ ] A / 2, y[]= if A / 2 < x[ ] < A / 2 then get a new equence y[] ( by doing thi, the effect of noie can be reduced greatly ). b) In the equence y[], if the adjacent element are the ame, chooe only one element, then get a new equence w[]. c) Define the variable c If for all, we have w[]=w[+4], then let c= indicating f in f /4. If there i at leat one with w[] w[+4], then let c=1 indicating f in >f /4. d) Let z[ ] = y[ ], then get the new equence z[]. 71

3 e) In the equence z[], if the adjacent element are the ame, chooe only one element, then get a new equence zz[]. f) Count the um um(zz) of the non-zreo element in equence zz[], g) Compute the integer cycle J int J int =floor(um(zz)/2) if c= J int =/2-floor(um(zz)/2) if c=1. The algorithm (in tep c) determining whether the ignal frequency i larger than f / 4 or not can be eaily verified by pigeonhole principle. Becaue of page limitation, we won t dicu it in thi paper. It hould be pointed out that thi method for counting the integer cycle i not robut when f in i cloe to f /4. In thi cae, the tep ize of x[] phae i cloe to π/2. Becaue of the noie, y[] value i not accurate when x[] i cloe to A / 2 or A / 2. Therefore, when f in i cloe to f /4, we ue another method to count the integer cycle of input ignal (Here we call it method II ). After method I i executed, we get J int, if Jint 2% < < 3%, we ue method II to recount J int. Becaue of page limitation, we cannot dicu method II in detail. ethod II ue the nature of ine wave, riing and falling, to count the integer cycle of input ignal. ethod II i extremely robut if f in i cloe to f /4. The procedure of the improved fundamental identification and replacement technique can be outlined in following 11 tep. 1) Capture a ufficient number of ample, 2) Find the firt point x[1] whoe abolute value i larger than A / 2, 3) chooe the bet data record length. compare x[ 1 ] with x[ ], x[ ], x[ ],,x[ i + 1 ], x[ N + 1 ] ( i ha only prime factor of 2 or 3, N i cloe to the length of original data record ), to find the point x[ i + 1 ] that mot matche x[ 1 ] in the data equence. Then i i jut the bet data record length. 4) count the integer cycle J int of the equence from x[ 1 ] to x[ 1 +-1] uing method I, Jint 5) If 2% < < 3%, recount J int uing method II, 6) Find the poitive zero-croing point x[ 2 ] near x[ 1 ], ue all the data point between x[ 2 ] and x[ 2 +-1] a the data record. 7) normalize the data record uing the power-baed normalization, and get the firt etimate of the fundamental harmonic magnitude A. 8) Compute the fractional cycle 1 = [arcin( x [ ] / ) arcin( [ ] / )] 2 A x 2 + A 2π Then the input ignal frequency i f J (6) int in = f (7) + 9) At a ubet of data point write x[ ] = A in(2 π f t + θ ) in = A co( θ)in(2 π f t ) + A in( θ)co(2 π f t ) in And ue leat quare method to identify A co(θ) and A in(θ). in (8) 1) Perform the fundamental component replacement x[ ] = x[ ] A in(2 π fint + θ ) (9) + A in(2 π f J / + θ ) int The new data x[ ] i generated by replacing the fundamental component from the original data (which i ampled noncoherently and caue poibly large irt) with one that i coherent with the ampling cloc. Thi i done by imply ubtracting a ine component with the identified parameter and adding a ine component with the ame A and θ but with being rounded to zero. 11) Perform FFT analyi on x[ ] a uual. The improved method ue the data record length that ha only prime factor of 2 or 3, the two mallet prime number. Thi guarantee that the new method i alway computational efficient. Furthermore, the new algorithm for counting the integer cycle of input ignal mae the improved method more robut and more immune to noie. It wor well when the ignal frequency i cloe to Nyquit frequency. Therefore, the improved method i robut and immune to noie. III. SIUALTION RESULT Extenive imulation tudy and experimental tudy have been conducted in order to verify the performance of the propoed algorithm. During the imulation, ADC i modeled a a et of tranition level. It nonlinearity error i choen to be a Gauian random variable with zero mean and tandard deviation σ DNL. In thi ection we preent the pectral teting example of 12-bit ADC with σ DNL of.2 LSB. Additive noie of input ignal i alo choen to be a Gauian random variable with zero mean and tandard deviation of.25 LSB. For comparion, three different pectral teting method are imulated. They are: 1) traightforward application of DFT auming periodic ampled equence, 2) the propoed method, 3) perfect coherent ampling. 72

4 1.5 Coherent (o) and noncoherent (*)ample -2 After coherency correction ADC reolution: 12 Additive noie of input ignal:.25 LSB DNL igma:.2 LSB SFDR=84.3 db THD= db Figure 1 Data ample from coherent and noncoherent ampling Normalized power pectrum in db Normalized power pectrum in db Frequency bin (16384= cloc frequency) Figure 2 Straightforward application DFT to the noncoherent data ample A time domain illutration of the coherent and noncoherent data i hown in Fig.1. Fig.2 how the pectrum of traightforward application DFT to the noncoherent data ample. From Fig.2, we can ee that there are large irt around the pectral line. So traightforward application of DFT uffered from large error due to noncoherency. The pectrum of the noncoherent dada ample uing the propoed method and the pectrum from perfect coherent ampling data are hown in Fig.3 and Fig.4 repectively. From Fig.3 and Fig.4 we ee that both pectrum how zero or minimal irt. Therefore, the imulation reult how that the propoed method can achieve pectral teting accuracie imilar to thoe obtained with perfect coherent ampling. IV. Before any coherency correction ADC reolution: 12 Additive noie of input ignal:.25 LSB DNL igma:.2 LSB EXPERIENTAL RESULT SFDR=24.8 db THD= db Since the propoed method exhibited excellent pectral performance, we want to validate the algorithm with experimental data. Fig.5 how the egment of captured noncoherent data in time domain. The data i collected in an indutry etting and the original data record length i To analyze the pectral content of the ignal, one can traight forwardly apply DFT to the raw data. The reultant pectrum i hown in Fig.6. A mall zoomed-in piece i hown on Fig.7. From Fig.7 we can ee there are a large irt around the pectral line. So the pectral leaage i eriou. Fig. 8 how the pectrum of the noncoherent data ample uing the propoed method. A mall zoomed-in Normalized power pectrum in db Frequency bin (5832= cloc frequency) Figure 3 Spectrum of the noncoherent data ample uing the propoed method Frequency bin (496=cloc frequency) TABLE I. With perfectly coherent ampling ADC reolution: 12 Additive noie of input ignal:.25 LSB DNL igma:.2 LSB SFDR=83.51 db THD= db Figure 4 Spectrum from perfect coherent ampling ENOB, TIE AND USING DIFFERENT ETHODS ethod ENOB Time () Original method =3*337*379 Propoed method =2 7 *3 7 piece i hown on Fig.9. From Fig.9, we can ee that there are no irt in the pectrum. In order to tet the performance of the propoed method, we alo conduct the pectral analyi uing the original method in [7]. Table I ummarize the ENOB, run time, and the bet data record length of both method. From Table I we can ee that the pectral teting accuracie of both method are imilar. However, the propoed method i ignificantly fater than the original method. The reaon i that the bet data record length of original method ha large prime factor, 337 and 339. Therefore, the experimental reult how that the propoed method i fater and robut. 73

5 Normalized power pectrum in db Segment of raw data Sampling intant index Figure 5 Raw data of noncoherent ample Normalized power pectrum in db SNR=14.16 db SNDR=14.16 db SFDR=2.97 db THD= db ENOB=2.6 Before any coherency correction Frequency bin (524288= cloc frequency) x 1 5 Figure 6 Spectrum by tandard application of DFT Before any coherency correction Frequency bin (524288= cloc frequency) Figure 7 A zoomed-in piece howing the irting V. STATISTICAL PERFORANCE STUDY Statitical performance tudy i conducted in order to verify the performance of the propoed method. The imulation environment in atlab i et up o that many parameter are randomly generated. The input ignal frequency i generated by electing a random ratio of f in to f. The ditorted ine wave ignal i generated by adding random amount of harmonic ditortion component to a pure ine wave. Additive meaurement noie which i choen to be Gauian random variable i introduced at the input node of ADC with a tandard deviation of around.25 LSB. Normalized power pectrum in db Normalized power pectrum in db SNR=51.2 db SNDR=5.82 db SFDR=65.58 db THD= db ENOB=8.15 After coherency correction Frequency bin (279936=cloc frequency) x 1 5 Figure 8 Spectrum of the noncoherent data ample uing the propoed method After coherency correction Frequency bin (279936=cloc frequency) Figure 9 A zoomed-in piece howing no irting During the imulation, we avoid uing the certain ignal and cloc frequency combination if the ratio fin / f can be reduced to a rational number with mall integer. Furthermore, we alo avoid uing extremely low frequencie. The reaon ha been explained in [7]. Simulation of 1 cae wa conducted. Fig.1 and Fig.11 illutrate the ignal SFDR teting error and THD teting error uing the propoed method repectively. Notice that in all 1 run, the SFDR teting error and THD teting error are within ± 3.4 db and ±.6 db repectively. We alo notice that SFDR teting error and THD teting error are mall when the ignal frequency i cloe to Nyquit frequency. Therefore the propoed method i demontrated to be robut and immune to noie. Table II ummarize the tatitic of thee comparative tudy reult. Fig.12 how the time in 1 run uing the propoed method and the original method in [7] repectively. From Fig.12 we can ee that the propoed method i alway fater and ometime ignificantly fater than the original method. 74

6 Etimation error of SFDR in db Time () Propoed method Original method Ratio of ignal frequency to cloc frequency in % Figure 1 SFDR teting error in 1 run uing the propoed method, v f in/f * Bet data record length x 1 4 Figure 12 Time in 1 run uing the propoed method and original method repectively, v the bet data record Etimation error of THD in db Ratio of ignal frequency to cloc frequency in % TABLE II. Figure 11 THD teting error in 1 run uing the propoed method, v f in/f *1 SFDR AND THD TESTING ERRORS IN 1 RUNS USING PROPOSED ETHOD SFDR error (db) THD error (db) max min mean td more immune to noie. Simulation and experimental reult, tatitical analyi validate thi propoed method. REFERENCES [1] IEEE Standard for Digitizing Waveform Recorder, IEEE Std. 157T-27, April 28. [2] P. Carbone, E. Nunzi, and D. Petri, "Window for ADC Dynamic Teting via Frequency-Domain Analyi", IEEE Tran. Intr. & ea., vol.5, No.6, pp , 21. [3] F.J.Harri, "On the Ue of Window for Harmonic Analyi with the Dicrete Fourier Tranform", Proc. IEEE, vol.66, No.1, pp.51-83, [4] J. Zhang and S.J Ovaa, " ADC Characterization baed on ingular value decompoition", IEEE Tran. Intr. & ea., vol.51, No.1, pp , 22. [5] X. Gao, S.J Ovaa and S. Shenghe,et al, " Analyi of econd-order harmonic ditortion of ADC uing bipectrum", IEEE Tran. Intr. & ea., vol.45, No.1, pp.5-55, [6] C. Rebai, D. Dallet and P archegay, " Non-coherent Spectral Analyi of ADC uing Filter Ban", IEEE Tran. Intr. & ea., Tech. Conf., pp , 22. [7] Z. Yu, D. Chen, and R. L. Geiger, "A Computationally Efficient ethod for Accurate Spectral Teting without Requiring Coherent Sampling ", Proceeding IEEE Int. Tet Conference, pp , 24. [8] S..Kay, Fundamental of Statitical Signal Proceing: Etimation Theory, Prentice-Hall, VI. CONCLUSIONS An improved method for fater and accurate pectral teting i propoed that doe not require coherent ampling or the uing of windowing. The new method ue the bet data record length that ha only prime factor of 2 or 3, the mallet prime number. Thi guarantee that the new method i alway computational efficient. Becaue of thi, the new method i alway fater and ometime it can be ignificantly fater than the original method. The paper alo introduce a new algorithm for counting the ignal period in the data which mae the propoed method i more robut and 75