AN EVALUATION OF DIGILTAL ANTI-ALIASING FILTER FOR SPACE TELEMETRY SYSTEMS

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1 AN EVALUATION OF DIGILTAL ANTI-ALIASING FILTER FOR SPACE TELEMETRY SYSTEMS Alion de Oliveira Morae (1), Joé Antonio Azevedo Duarte (1), Sergio Fugivara (1) (1) Comando-Geral de Tecnologia Aeroepacial, Intituto de Aeronáutica e Epaço, Praça Mal. Eduardo Gome, 50 Vila da Acácia, São Joe do Campo São Paulo Brazil aom@iae.cta.br; duarte@iae.cta.br; and fugivara@iae.cta. ABSTRACT The neceity of accelerometer enor on launching vehicle and ounding rocket generally preent a concern for deigning the telemetry ytem, due to the amount of bandwidth required to repreent thi kind of data. Commercially available enor and conditioning amplifier are not band-limited to frequencie of interet; and the uually analog anti-aliaing filtering technique fail to perform an effective band limiting in PCM data acquiition ytem. Thee factor make even more difficult the managing of high dynamic enor in traditional telemetry ytem. Thi work report an invetigation about uing digital anti-aliaing filter for data from accelerometer enor in order to optimize the bandwidth of rocket telemetry ytem. 1. INTRODUCTION The ue of accelerometer enor on launching vehicle and ounding rocket i neceary to decribe a complete dynamic performance of a flight. However, thi kind of data generally require a large bandwidth of the telemetry ytem. Beide, the commercially available enor and conditioning amplifier are not band-limited to frequencie of interet; and the uually analog anti-aliaing filtering technique in a PCM data acquiition ytem fail to perform an effective band limiting. To reduce the effect of the aliaing due to the limitation decribed above, the PCM encoder uually overample the data from thi kind of enor. Thi proce reult in an increaing of the bit rate. In thi work, an evaluation of the ue of digital anti-aliaing filter i made and applied to accelerometer data before the PCM encoder ha acquired it. Uing a finite impule repone (FIR) filter, with a frequency repone almot ideal, the ampling rate i reduced, with mall increae in ize, weight, and power of the onboard hardware. Reult from tet with digital anti-aliaing filter, are analyzed and compared to traditional accelerometer data filtered by an analog filter following by an overampling in the PCM encoder. Extending thi analyi, thee reult are alo compared to an accelerometer data acquired with ground ytem intrumentation where it ample rate and quantization reolution are much higher than any pace telemetry ytem. The analye howed that digital anti-aliaing filter preent good reult, making poible the replacement of the traditional meaurement of accelerometer data for the propoed one. With thi propoed filter in an onboard telemetry ytem i poible to enhance the performance of a typical ampling ytem o that it approache Nyquit rate, and a conequence effectively reducing the amount of bit required in the telemetry ytem to repreent thi information. Therefore, the reduction in the amount of bit to repreent thi ort of information can be relevant in telemetry link with bandwidth limitation. 2. DATA SAMPLING According to Nyquit Shannon ampling theorem, an analog ignal, (t), can be perfectly recontructed baed on ample of thi analog ignal. The condition for thi perfect recontruction i that the ample frequency, f, mut be higher than twice the maximum frequency, B, of (t). So the analog ignal mut be bandlimted with S(f)=0 for B>f /2. The block diagram of a ampling ytem i how in Figure 1. Figure 1. The proce of continuou to dicrete time converion. The Fourier tranform of (t) and d [n] are how repectively in Figure 2 (a) and (b). The difference between both pectrum i the fact that S d (f) i periodic in the frequency domain. Baed on the dahed line in Figure 2 (b), it i poible to recover (t), applying an ideal lowpa filter on d [n]. Thi ideal lowpa filter i a pule in the frequency domain with width of 2B. In the time domain the impule repone of thi ideal filter i given by: ( πt /(1/ f )) πt /(1/ f ) h( t) = in (1) Proc. 19th ESA Sympoium on European Rocket and Balloon Programme and Related Reearch, Bad Reichenhall, Germany, 7 11 June 2009 (ESA SP-671, September 2009)

2 Then, the proce of recovering the original ignal (t) baed on the equence of ample of d [n] i given by: ( t) r = in [ n] d ( π ( t n(1/ f )) /(1/ f )) ( π ( t n(1/ f )) /(1/ f )) Figure 2. Fourier tranform howing the effect of ampling. (2) The proce of recontruction of (t) uing d [n] uing Eq. (2) i called interpolation. The ideal interpolation proce i how in Figure 3. 2 (c). In practice, thi effect i alway preent, and it ha to be managed to guaranty the minimum ditortion. The tak of minimizing the effect of aliaing on electronic ytem i done by anti-aliaing filter. 3. ANTI-ALIASING FILTER In mot meaurement ytem, the anti-aliaing filter i implemented with a paive analog filter. More ophiticated ytem work with active analog filter that are more effective if compared to the paive one. To be effective, an analog active filter need to have a high order that reult in a large number of component and higher price. Beide, even a high order analog filter would fail to eliminate element cloe to the maximum frequency B. Thi happen becaue the analog filter doe not preent a harp repone cloe from the cut-off frequency. In thi work, a digital FIR (Finite Impule Repone) filter i ued to perform the tak of anti-aliaing filter. The FIR filter are choen by the fact that they are alway table becaue all their pole are in the origin. Beide thi kind of filter preent a linear phae repone which i important to analyze data in the time domain. The ue of a digital FIR filter allow to et the cut frequency cloe to the value B, with a minimum ripple and no attenuation cloe to thi bound like in analog filter. But the main advantage of thi kind of filter i the harpne of the tranition between the cut frequency and the rejected band. Figure 4 how an example of the frequency repone of thi filter. Figure 3. Signal recontruction baed on ideal interpolation [2]. Unfortunately, device deigned for analog to digital converion are limited and it i not poible to implement the ideal lowpa filter for recontruction proce. Thee factor avoid the perfect recontruction of (t) baed on the ample of d [n]. Depite thee factor it i poible to achieve good approximation for (t). A it ha been hown, all the theory that allow decompoing the continuou ignal and recontruct that baed on it ample, conider that the ignal (t) i bandlimited. However, in the real world, there are no bandlimited ignal. When the ignal i not bandlimited, the periodic copie on the pectrum of S d (f) overlap each other and the ignal i not recoverable any more. Thi proce i called aliaing and it i hown in Figure Figure 4.Frequency repone of the digital anti-aliaing filter of thi work.

3 The trategy in thi work i to take the advantage of the frequency repone of the digital filter to conider the meaured ignal a bandlimited. The value of B mut be adequate to the region of the beginning of the rejection band. With thi approach, the telemetry ytem can acquire the data with a mall ample frequency. Beide the data reulted from thi proce i expected to be the bet approximation for the original meaured ignal. The baic architecture of the digital anti-aliaing filter ued in thi work i preented in Figure 5. The ignal from one monoaxial accelerometer, A(t), i firt amplified and filtered by a traditional 6 pole analog anti-aliaing filter. Thi amplified and filtered verion of the accelerometer ignal, X(t), i than acquired by the PCM encoder with a ample frequency 10, 5 and 2.5 time the maximum frequency of A(t). In parallel, the ignal X(t), i applied to the FIR filter, generating the ignal W(t). The ignal W(t) i alo applied to the PCM encoder to be acquired with a ample frequency of 2.5 time. Figure 7. Block diagram of the meauring ytem. Figure 5. Architecture of the digital anti-aliaing filter. 4. METHODOLOGY OF THE MEASUREMENTS The accelerometer ued in thi tet wa placed on the motor adapter of the VSB-30 ounding rocket during the dynamic acceptance teting of thi module. Figure 6 illutrate the motor adapter and how the poition where the accelerometer ha been fixed. Figure 6. Motor Adapter. In order to analyze the effectivene of the digital antialiaing filter for accelerometer data, it ha been conidered the diagram of Figure 7 for the tet. The reaon for thee ample frequencie can be jutified in the following way. In a traditional rocket telemetry ytem, the accelerometer data i ampled between 4 and 5 time the maximum frequency, which the enor can repond. Thi i why one channel wa et to the ample frequency of 5 time, which i conidered the tandard ratio for thi kind of data. Another PCM channel ha been acquired with a ample frequency of 10 time. Thi channel ha been conidered a an onboard reference for the telemetry meaurement. Two PCM channel were et for the ample rate of 2.5 maximum frequency. Thee particular channel i the focu of thi work, once that the intention i to compare the quality of the data with and without the digital anti-aliaing filter in a ample rate that i cloe to Nyquit bound. Signal acquired by the PCM encoder were digitalized with 10 bit of quantization. The erial output data from the PCM encoder wa proceed by a telemetry ground tation, which conit of PC baed bit ynchronizer and PCM decommutator. The reult of thi proceing are four file, with the tochatic proce X 1 [n], X 2 [n], X 3 [n] and W[n]. The file, with X 1 [n], X 2 [n], X 3 [n] contain the data related to X(t), ampled in 10, 5 and 2.5 time the maximum frequency of A(t). The proce W[n] repreent the dicrete verion of X(t) filtered by the digital filter. Figure 8 how an example of W[n]. 5. RESULTS In thi ection we preent the reult related to the accelerometer data during the dynamic acceptance tet. Figure 8 how an example of the acquired data during the tet in the time domain.

4 The power pectrum denity of the ground baed intrumentation ytem i preented in Figure 9, a an example. Figure 8. Signal x 2 [n]. During acceptance tet, an intrumentation ground ytem i the ource of data, ued to analyze the dynamical performance of the module. The data from thi intrumentation ground ytem i alo preented in thi ection a an auxiliary reference. Thi data i conidered a ground reference becaue it ha been acquired with a 32 bit of quantization and analog filter with 24 pole and a ample frequency that i 50 time higher than the maximum frequency of A(t). Beide that, thi intrumentation ytem i completely iolated from the onboard telemetry ytem. Thu, the ignal might be conidered the cloet approximation of A(t). The reult preented in thi ection are divided in two cae. The firt cae preent reult related to a tet where there i no interference or noie outide from the bandwidth of interet. The econd cae preent an example where ome kind of interference i preent. Thi interference wa not initially predicted in thi reearch. It wa from an unknown ource and it wa dicovered during the analye of the data for thi work. Before preenting the reult, it i preented ome aumption for the analyze. The random vibration that the VSB-30 motor adapter module i ubmitted i generated by a haker. The ignal obtained from thi tet i conidered a wide ene tationary. In thi cae, the power pectrum denity, S W (f), of the tochatic proce W[n], i the Fourier tranform of the autocorrelation function, R W [n 1 -n 2 ], of the data enor, defined by: S W 2 [ Wm ( ) ] 1 ( f ) = lim Ε f m 2m where Ε[ W m ( f ) ] i given by: (3) Figure 9. Power pectrum denity of ground baed intrumentation ytem for the econd cae. Firt Cae The power pectrum denity of X 1 [n], X 2 [n], X 3 [n] and W[n] for the cae where there i no interference in the meaurement ytem, during the tet, are how repectively in Figure 10, 11, 12 and 13. Figure 10. Power pectrum denity of X 1 [n]. Ε m m 2 j 2πf ( n1 n2 ) [ Wm( f ) ] = RW [ n1 n2] e n m n 2= 1= m (4) and the autocorrelation function i defined by: R W * [ n n ] = Ε[ W n ] W [ ]] (5) 1 2 [ 1 n2

5 Second Cae The power pectrum denity of X 1 [n], X 2 [n], X 3 [n] and W[n] for the cae with interference are how repectively in the Figure 14, 15, 16 and 17. Figure 11. Power pectrum denity of X 2 [n]. Figure 14. Power pectrum denity of X 1 [n]. Figure 12. Power pectrum denity of X 3 [n]. Figure 15. Power pectrum denity of X 2 [n]. Figure 13. Power pectrum denity of W[n]. Analyzing thee reult, it i poible to conclude that all the power pectrum denitie are alike. The reult from thi cae would ugget that if all the reult are very imilar, the bet implementation for onboard telemetry ytem would be uing only the accelerometer with it own amplifier and filter and a ample frequency cloe from 2.5B. Figure 16. Power pectrum denity of X 3 [n].

6 3. Krihnan, V. Probability and Random Procee, 1. ed. New Jerey: John Wiley & Son, Figure 17. Power pectrum denity of W[n]. In thi cae, it i poible to ee an interfering ignal together with the teted ignal. In Figure 14, it i poible to ee thi ignal preent around 6kHz. Probably, thi interfering ignal i due to ome kind of witching, maybe provided by other circuit preent in the module or around, in the laboratory. Baed on Figure 14, 15 and 16 it i poible to conclude that interference can damage the quality of the data. In a flight, interference like that might difficult the analye of the flight introducing to a wrong concluion baed on thi kind of corrupted data. 6. CONCLUSION Baed on the firt cae, the olution of uing only the accelerometer with it own amplifier and filter with a ampling rate cloe to Nyquit bound, eem to be perfect by the fact that there i no extra circuit onboard and the telemetry ytem i aving bit if compared with the traditional ytem. However, when there i ome kind of interference, a in the econd cae, thi ytem will fail. Baed on the econd cae, it wa hown that ome kind of interference, depending on the ituation, might reult in a wrong interpretation of the dynamical performance of the flight. The reult from the econd cae allow the concluion that the digital anti-aliaing filter preent good olution to guaranty the integrity of the data provided by the telemetry ytem during the flight. Thi reult i more relevant if taking into account the apect of ave bit for the telemetry ytem. A a future work, thi filter will be teted in a ounding rocket to evaluate it performance in a real flight. 7. REFERENCES 1. Oppenheim AV, Shafer RW. Dicrete time ignal proceing. Prentice Hall; Englewood Cliff, NJ: Proaki, J. G. Digital communication, 3. ed. New York: McGraw-Hill Inc., 1995.