Improving FFT Frequency Measurement Resolution by Parabolic and Gaussian Spectrum Interpolation

Transcription

1 CERN-AB-4-3 BDI Iproving FFT Frequency Measureent Resolution by Parabolic an Gaussian Spectru Interpolation M. Gasior, J.L. Gonzalez CERN, CH-,Geneva 3, Switzerlan Abstract Discrete spectra can be use to easure frequencies of sinusoial signal coponents. Such a easureent consists in igitizing a copoun signal, perforing winowing of the signal saples an coputing their iscrete agnitue spectru, usually by eans of the Fast Fourier Transfor algorith. Frequencies of iniviual coponents can be evaluate fro their locations in the iscrete spectru with a resolution epening on the nuber of saples. However, the frequency of a sinusoial coponent can be eterine with iprove resolution by fitting an interpolating parabola through the three largest consecutive spectru bins corresponing to the coponent. The abscissa of its axiu constitutes a better frequency approxiation. Such a etho has been use for tune easureent systes in circular accelerators. This paper escribes the efficiency of the etho, epening on the winowing function applie to the signal saples. A typical interpolation gain is one orer of agnitue. Better results are obtaine with Gaussian interpolation, offering frequency resolution iproveent by ore than two orers of agnitue when use with winows having fast sielobe ecay. An iproveent beyon three orers of agnitue is possible with steep Gaussian winows. These results are confire by laboratory easureents. Both ethos assue the easure frequency to be constant uring acquisition an the spectral pea corresponing to the easure coponent to constitute a local axiu in a given ban of the input signal iscrete spectru. Presente at BIW May 4 Knoxville TE - USA Geneva, Switzerlan May, 4

2 FFT FREQUENCY MEASUREMENT Assue that a banliite copoun signal s(t) has been uniforly saple with frequency f s an contains a sinusoial coponent s in (t), whose frequency f in is to be easure. The Discrete Fourier Transfor (DFT) agnitue spectru of the signal saple sequence s[n]=s(nt s ), usually copute using the Fast Fourier Transfor (FFT) algorith, is given by N πn S [ ] = s[ n]exp j a) n= N where T s = f s is the sapling perio an N is the total nuber of saples. The iscrete spectru () is calculate at frequencies that are integer ultiples of f =f s /N. If S[] has an observable local axiu corresponing to s in (t) at the spectru bin, f in can be approxiate as f s fin = = = b) N N Ts L where L = N T s is the sapling uration of the input signal. The largest approxiation error ε = ax( f f in ) occurs for a frequency locate exactly between two bins. This error can be consiere as the resolution of the FFT frequency easureent an f s ε = f = = = c) N NTs L The resolution can be increase consierably by iscrete spectru interpolation, which has been use in tune easureent systes []. The etho principle is setche in Fig.. When the frequency coponent f in is locate exactly on a local axiu of the iscrete agnitue spectru at bin, f in can be calculate accoring to () with no error. This is the ieal case shown in Fig. a. When f in increases, the aplitue of bin gets saller an of bin + bigger, as presente in Fig. b. The spectru value S[ ] reains the biggest, until f in is equiistant between bins an +, see Fig. c. In this case, eterining f in with () leas to the largest error (3). When f in is increase further, as shown in Fig., bin + becoes the biggest. For f in saller than f the analysis is siilar resulting in syetrical cases. S [] S [] S [] S [] (a) f - + (b) f in - + (c) - + () + + FIGURE. The principle of iscrete spectru interpolation. This exaple illustrates the fact that continuous frequency f in can be estiate, even if it is locate between two bins, by calculating the axiu abscissa of an interpolation curve of the iscrete spectru pea. This axiu can be locate between / an +/, where is the inex of the biggest bin within the range of interest. If one nees to resolve the cases presente in Fig. an syetrical ones, Geneva, Switzerlan May, 4

3 it is necessary for the interpolation to have at least three noe points. As seen in Fig. c, for efficient interpolation the inial with of the spectral pea is 3 bins. If in the continuous-tie Fourier Transfor (FT) S( f ) = s( t) exp( jπ f t) t ) one assues tie oain sapling t = nt s an frequency oain sapling f = f, then the integral can be replace by a su with N eleents an FT (4) becoes DFT as in () when taing T s an f as units in the iscrete tie an frequency oains. For this reason iscrete spectra in the paper are consiere as continuous spectru saples taen at ultiplies of f, allowing to exaine iscrete spectra between the bins by eans of the FT (4). Replacing the orinary frequency f in (4) by f N ϕ = = f = f NTs = f L e) f f s referre to as the noralize frequency, ajusts continuous spectra to the scale of the iscrete spectru bin inexes. It allows expressing iscrete an continuous spectra in one convenient scale (e.g. ϕ =.5 for frequency lying exactly between bin an ). Interpolate spectral peas have shapes corresponing to the spectru of the winow function, since winowing can be consiere as a oulation applie to each sinusoial coponent of the analyze signal. Consequently, the goal of the interpolation ethos iscusse hereafter is to fin the axiu abscissa of such a shape while nowing only its three iscrete spectru saples. Results of the ethos are presente for weighte cosine winows with fast ecaying sall sielobes, escribe by Nuttall [] 3 πi w c ( t) = wr ( t) c cos t f) i i= L where w r (t) is the rectangular winow with uration fro L/ to L/. Basic properties an coefficients of the weighte cosine winows are liste in Table. In the winow naes of the for atb, a stans for the nuber of ters on which the ain lobe with epens, an b for the orer of the highest continuous erivative, eterining the sielobe asyptotic fall-off. As an exaple, 4T is a four-ter winow with the ain lobe full with of 8 bins (a) an continuous first erivative, resulting in the sielobe ecay of 8 B/octave (6b +). Three Gaussian winows are also consiere. They are efine as = t w ( t) w ( t) exp r g) g r L where r is the ratio between the winow length L an its stanar eviation σ, i.e. the Gaussian shape is truncate at ±σ r/. Geneva, Switzerlan May, 4

4 TABLE h). Paraeters of winows use in the paper. Winow Main lobe -6 B with [bin] Highest sielobe [B] Sielobe Winow coefficients fall-off [B/oct] c c c c 3 Hanning / / Blacan T T /8 / /8 4T T T /3 5/3 6/3 /3 Gaussian r = Gaussian r = Gaussian r = The winow choice is a coproise between a few paraeters, three of which are liste in Table. The ain lobe with eterines the inial istance between two spectral peas, which still can be resolve. The highest sielobe an sielobe fall-off characterize the spectral leaage to the coponent of interest fro respectively close an far interference. As seen in Table an Fig., showing an exaple of agnitue spectra of four winows, spectral properties of Gaussian winows are inferior to those of the weighte cosine ones. This paper eals only with interpolation on iealize (i.e. not perturbe by noise nor interference) spectra an in this case only the shape of the ain lobe is iportant, exclusively within the interpolation range of ±.5 ϕ units. This range of Fig. is agnifie in Fig. 3. Nor. agnitue spectru [B] Hanning 3T 4T Gaussian r = Noralize frequency FIGURE. Winow agnitue spectra. Main lobe noralize aplitue Hanning 3T 4T Gaussian r = Noralize frequency FIGURE 3. Magnifie ain lobes of Fig.. PARABOLIC INTERPOLATION As explaine before, the goal of the interpolation process is to fin the abscissa of the spectral pea axiu, nowing only three pea saples in the for of three bins of the iscrete agnitue spectru. The pea continuous shape, i.e. the winow spectru ain lobe, oes not nee to be accurately reprouce as long as the axiu abscissa of the interpolation shape follows the easure frequency. This is why a siple parabolic interpolation (PI) can iprove the iscrete spectru frequency resolution by an iportant factor even when, as shown in Fig. 3, the shape of the winow spectru ain lobe is quite far fro a parabola. To quantify this eviation, a Geneva, Switzerlan May, 4

5 winow spectru agnitue W(ϕ) can be expane into a (noralize to the coefficient upon ϕ ) Maclaurin series of the for 4 W ) a i) + ϕ + a4ϕ Coefficient a 4, liste in Table, can be use as a easure of this eviation. Let S[] be the iscrete agnitue spectru of N saples of a signal s(t) containing a sinusoial coponent of frequency ϕ in = f in L, an be the inex of the biggest bin of the corresponing iscrete spectru pea. Inex can be foun if the bin constitutes a local axiu within a given range. Fitting a parabola S ) = a( ϕ ϕ ) h j) p + through interpolation noes S[ ], S[ ], S[ +] an fining the abscissa of the interpolation axiu ϕ, gives S[ + ] S[ ] ϕ in ϕ = + = + ) ( S[ ] S[ + ] S[ ] ) uner conition S [ ] S[ + ] + S[ ]. > The quantity in () is the abscissa correction of the iscrete spectru axiu. It is a real nuber lining both, the iscrete an continuous spectra, ranging fro / when S[ ] = S[ ], to / for S[ +] = S[ ]. The shape of the interpolate agnitue spectru pea correspons to the spectru of the winowing function applie to the signal saples. If winow w(t) with agnitue spectru W(ϕ) is use, then the interpolation error E(ϕ ) = ϕ ϕ in is [] W + ) W ) E ) = ϕ l) ( W ) W + ) W ) ) where ϕ = ϕ in. The interpolation error corresponing to four winows is shown in Fig. 4 an is given in units of f. For non-perturbe spectra the error is the sae aroun each iscrete spectru bin, i.e. is perioic with perio of f. The interpolation errors for other winows have siilar shapes an can be characterize by the error axiu E ax = ax( E(ϕ ) ) an its abscissa. They are liste in Table. Perforance of an interpolation etho can be characterize by the interpolation gain, efine as the ratio of the FFT frequency resolution (3) an the etho axiu error G = ε = ) Eax Eax As liste in Table, the PI can increase the resolution of iscrete spectra by ore than one orer of agnitue. GAUSSIAN INTERPOLATION The interpolation gain can be significantly iprove by fitting a Gaussian shape to fin the abscissa of the spectral pea axiu locate between two iscrete spectru bins. Since a Gaussian curve S ) = exp a ϕ ) + h n) ( ) g Geneva, Switzerlan May, 4

6 is a parabola in the logarithic scale, the Gaussian interpolation (GI) reuces to the PI on the natural logarith of the agnitue spectru. Analogically to PI, the winow spectru ain lobe eviation fro a Gaussian shape can be quantifie by coefficient b 4 of a noralize Maclaurin series expansion 4 lnw ) b o) + ϕ + b4ϕ The coefficient is liste in Table. Its value is uch saller than the corresponing a 4 of (8), especially for Gaussian winows. The GI can be erive fro () using logarithic spectru values, i.e. the interpolation noes S[ ], S[ ], S[ +] are replace by natural logariths ln(s[ ]), ln(s[ ]) an ln(s[ +]). Thus, after logarith grouping, () becoes S[ + ] ln S[ ] ϕ = + = + in ϕ p) S[ ] ln S[ + ] S[ ] Siilarly, the GI error can be erive irectly fro () lnw + ) lnw ) E ) = ϕ q) ( lnw ) lnw + ) lnw ) ) It is plotte in Fig. 5 for four winows. The errors for other winows have siilar shapes an are characterize by the error axiu an its abscissa, as liste in Table. The interpolation gains are about two orers of agnitue for cosine weighte winows an well beyon three orers of agnitue for the Gaussian winow of r = 8. PI error [% of ] Hanning 3T 4T Gaussian r = 8 GI error [% of ] - Hanning 3T 4T Gaussian r = ϕin - FIGURE 4. Parabolic interpolation errors ϕin - FIGURE 5. Gaussian interpolation errors. TABLE r). Parabolic an Gaussian interpolation perforance. Parabolic interpolation Gaussian interpolation Winow a 4 E ax ϕ in - b Gain 4 E ax ϕ in - in (8) [% of f E ax in (4) [% of f E ax Gain Hanning Blacan T T T T T Gaussian r = Gaussian r = Gaussian r = Geneva, Switzerlan May, 4

7 MEASUREMENTS The parabolic an Gaussian interpolation ethos were exaine with a laboratory setup shown scheatically in Fig. 6, base on a tune easureent evelopent syste []. It consiste of a channel with a 4-bit analog to igital converter (ADC) precee by an antialias low-pass filter (LPF), a eory for fast ADC saple storage, an a boar with a floating-point igital signal processor (DSP). A PC was use to prepare processing software an to ownloa it to the DSP boar. To achieve input an cloc frequencies of sufficient phase stability, the frequencies were generate by two Direct Digital Synthesizers (DDS), riven by the reference source of a frequency eter. During each acquisition, 48 ADC saples of the sine wave input signal were store in the eory. Then, the DSP successively perfore winowing, the FFT an the power spectru calculation. Next, the spectru bin with the biggest aplitue was foun an finally the input frequency f in was calculate accoring to () or (5). The easureents were one aroun bin =8, with the ADC cloc frequency f s =.5 MHz an f in about 78 Hz (i-range of the frequency span of the setup). Measureent results are shown in Fig. 7. Crosses ar extree values fro consecutive easureents with the sae setup frequencies an ashe lines show theoretical errors as in Fig. 4 an 5. The easureent results were sprea ue to aplitue noise present in the analyze spectra, which was converte uring the interpolation process into a frequency jitter. This uncertainty was cause ostly by noise present in the input signal, originating in the DDS output -bit igital to analog converters. The quantization noise of the ADC ight have contribute as well. The observe noise becae ore visible as the interpolation gain increase. Nevertheless, as seen in Fig. 7, the total of the systeatic an "noise" error was 7 pp of f, that is soe 65 Hz. The interpolation gain was close to 47 an relative easureent error of about.8 pp (for f in close to only f s /6). The interpolation gain of 47, obtaine at the expense of perforing the Gaussian interpolation (5) within soe icrosecons, is equivalent to the frequency resolution of an FFT easureent without interpolation with N saples an the sapling tie L ultiplie by this factor. For the presente easureent it correspons to increasing N fro 48 to alost 7 an L fro.6 s to 7.5 s. Such an aount of ata woul increase the FFT calculation tie by a factor of 4, fro about s to s. DDS DDS f ref input frequency freq. eter cloc frequency LPF evelopent syste ADC logic eory PC DSP internal bus FIGURE 6. Measureent setup. Geneva, Switzerlan May, 4

8 Interpolation error [% of ] (a) Noralize input frequency ϕ in calculation extree values fro easureents Parabolic interpolation with 4T winow ϕ in - Interpolation error [% of ] (b) Noralize input frequency ϕ in calculation extree values fro easureents Gaussian interpolation with 4T winow ϕ in - Interpolation error [pp of ] (c) Noralize input frequency ϕ in calculation extree values fro easureents Gaussian interpolation with Gaussian winow of r = ϕ in - Interpolation error [pp of ] () Noralize input frequency ϕ in calculation extree values fro easureents Gaussian interpolation with Gaussian winow of r= ϕ in - FIGURE 7. Measureent results of the parabolic an Gaussian interpolation ethos. CONCLUSIONS Theoretical an experiental stuies have been unertaen to enhance FFT frequency easureent resolution, using parabolic or Gaussian interpolations on the iscrete agnitue spectru to fin abscissa of spectral peas axia locate between iscrete spectru bins. The interpolation yiel strongly epens upon the winowing function use an, to be significant, the winow spectru ain lobe shoul be at least 3 bins wie. This paper shows that the parabolic interpolation can iprove the frequency resolution by ore than one orer of agnitue. The etho coputing cost is one ivision an one ultiplication. The frequency resolution iproveent can still be better with Gaussian interpolation. A gain larger than two orers of agnitue is achievable with winows having very goo spectral properties an well beyon three orers of agnitue when using steep Gaussian winows. The cost of the Gaussian interpolation is three ivisions, three ultiplications an calculation of two natural logariths. Both interpolation ethos assue the easure frequency to be constant uring acquisition an the spectral pea corresponing to the easure coponent to constitute a local axiu in a given ban of the input signal spectru. These ethos o not help to resolve nearby spectral peas an assue bin spacing to be sall enough to avoi pea erging. This paper escribes systeatic errors of the interpolation ethos, assuing ieal iscrete spectra. Soon, results will be publishe, concerning the behavior of the Geneva, Switzerlan May, 4

9 ethos when spectra are perturbe by noise, interference fro strong coponents an the exponential ecay of the input signal. A irect application of these ethos are FFT-base tune easureent systes. The Gaussian etho with 4T winow is use in such a syste for the CERN PS Booster accelerator. In the future, siilar systes will be ae for the PS an LEIR achines. Further etails concerning the ethos an winowing can be foun in [], containing also a list of suppleental references. ACKNOWLEDGMENTS The authors are inebte to all those involve in the evelopent of the tune easureent syste for the PSB accelerator. A special ention is given to E.T. Aico an A. Chapan- Hatchett. The authors also acnowlege the help of J. Bellean, J. Bosser, J.P. Potier an U. Raich. REFERENCES. M. Gasior, J.L. Gonzalez, "Iproving FFT Frequency Measureent Resolution by Parabolic an Gaussian Interpolation", AB-Note-4- BDI, CERN, February 4, with all references therein. Geneva, Switzerlan May, 4