New Algorithms for the Optimal Selection of the Bandpass Sampling Rate in Measurement Instrumentation

1. New Algorithms for the Optimal Selection of the Bandpass Sampling Rate in Measurement Instrumentation Giovanni Betta, Domenico Capriglione, Luigi Ferrigno, and Gianfranco Miele DAEIMI, Universityof Cassino, Via G. Di Biasio 43, 03043 Cassino (FR), Italy e-mail: betta,capriglione,ferrigno,g.miele@unicas.it

2. Introduction The modern measurement instruments, involved in telecommunication systems, are generally based on suitable digital signal processing methods. XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

3. Introduction • To avoid aliasing and to obtain a good performance in terms of accuracy and repeatability they are designed to respect the Nyquist-Shannon theorem. • very fast sampling rates • wide acquisition intervals • huge memory installed on board • very fast measurement algorithms • RF telecommunication signals • improving resolution • real-time operations Very expensive instruments. XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

4. Is it necessary to meet the Nyquist-Shannon condition? To avoid the superimposition of the replicas, phenomenon known as aliasing, the Nyquist–Shannon theorem imposes a lower limit to fS. fs>2 fu fc f[Hz] B Passband signals have a null spectrum in the bandwidth [ 0, fc – B/2]. It is possible to sample alias-free these signals considering suitable conditions. XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

5. Bandpass sampling theory The spectrum of the sampled version of a band limited analogue signal s(t) is composed by an infinite set of replicas of the original spectrum centered at fu=4B fs=5B fc 5B 5B B Let us divide the frequency axis in fs/2 wide intervals. Let us change the center frequency of the signal. Toavoidaliasing the replica has to be included in a fs/2 wide interval. XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

6. Bandpass sampling theory The previous consideration impliesthatS(f) hastobetotallyincluded in a fs/2 wide interval Lower guard band BGL Upper admissible values DfSu Upper guard band BGU Lower admissible values DfSl Forbidden areas (aliasing) Nonadmissible frequencies Conditionforuniformbandpasssampling Admissible frequencies Alias free areas XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

7. The optimal choice of the bandpass sampling rate In literature it is possible to find methods for automatic selection of the sample rate for bandpass signals that meet a very common requirement in electronic measurements. Angrisani et al., “Optimal sampling strategies for band-pass measurement signals,” IMEKO-TC4, pp. 343-348, Athens, Greece, Oct. 2004. f* fS B XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

8. Our proposal • In the development of measurement instrumentation two other conditions may be very helpful: to estimate the minimum allowable sampling rate to calculate the minimum admissible fSsubmultiple of the fixed sampling rate of a given analog-to-digital converter (ADC). Useful for the instrument designer that at the design stage can minimize the hardware resources required (in terms of ADC rate, memory buffer, processing unit performance). Useful in such cases where, for an existing ADC stage, the user can select the sampling rate only by using a simple prescaler factor. Algorithm I Algorithm II XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

9. Algorithm I BGU,n,max DfS BGU,n,max BGL,x BGU BGL,n,max DfSl BGL,n,max XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

10. Algorithm II The latter algorithm makes the selection of the minimum sampling frequency that is a submultiple of a given operating one fADC. The maximum value of m has been fixed to avoid the selection of sample rate lower than 2B and m=1 has been neglected because it coincides with the obvious case fS=fADC. The following relation can be obtained by substituting the previous consideration in the condition for uniform bandpass sampling XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

11. Algorithms Assessment We have designed and executed several tests in a suitable emulation environment, to assess the efficiency of the proposed algorithms. To these aims a measurement station was setup. XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

12. Measurement station • Data Acquisition System (DAS): • DC-3 GHz input frequency range • 20 GS/s maximum sampling rate • sampling circuit that could be driven by an external reference clock All the instruments are controlled by a LabVIEW driver that runs on the control unit • RF signalgenerator: • +13 dBmmax output power • 9 kHz-3 GHzfrequencyrange • Vector RF signal generator: • 17 dBm max output power • 250 kHz-6 GHz frequency range • equipped with DVB-T personality XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

13. DVB-T test signals We have imposed the following transmission settings: 8k transmission mode (k = 6817 and Tu = 896 ms); center frequency equal to 610 MHz; nominal bandwidth 7.61 MHz; nominal total power of -10 dBm (100 mW); 64–QAM modulation scheme; code rate equal to 1/2; two guard interval values, 1/4 (D=224 ms) and 1/32 (D=28 ms) respectively. XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

14. Test settings The input signal has been sampled at three different rates: For each transmission setting and sampling rate, 50 tests have been executed considering an observation period equivalent to a time interval associated with 1/4 of DVB–T symbol. 2.5 GS/s 16.158 MS/s 33.33 MS/s Nyquist rate Algorithm I Algorithm II • B=8 MHz • BGU=BGL=0 MHz • B=8 MHz • fADC=100 MS/s XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

15. Test settings The following figures of merit have been analyzed: the mean value of the channel power measurement results (PC) computed by integrating the power spectrum density (PSD) in the nominal channel bandwidth; sC, defined as the experimental standard deviation of PC measurement results; the difference (DC) between the PC measured on the Nyquist sampled signal and that measured on the bandpass sampled signal; XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

16. Test settings The following figures of merit have been analyzed: • the mean value of the occupied bandwidth measurement results (B) computed as the frequency range in which is collocated the 99% of the input signal power; • sB, defined as the experimental standard deviation of B measurement results; • the difference (DB) between B measured on the Nyquist sampled signal and that measured on the bandpass sampled signal; XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

17. How we have estimated the PSD. The PSD estimate was evaluated using two different PSD estimators Modifiedperiodogram Burg • Nonparametric PSD estimator • Hammingwindow • Parametric PSD estimator • modelorderm=3000 (forNyquistsampledsignal) and m=300 (forbandpasssampledsignal) XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

18. Measurement results – Channel power • DC is ever lower than 2.66%. Such value converted in decibel is approximately equal to 0.11 dB, showing a very little bias. • The repeatability of the channel power measurement results does not seem to be influenced by the three sampling rates XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

19. Measurement results – Occupied bandwidth • The occupied bandwidth obtained by using the modified periodogram estimator show a little bias (3.90% in the worst case). • This experienced bias might be dependent by the hypothesis made on the signal bandwidth. We imposed a null guard band. • Even though this condition theoretically assures that the replicas are not overlapped, practically it does not warrant an adequate gap between two adjacent replicas, thus affecting the measurement results. XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

20. Measurement results – Occupied bandwidth SNR=40 dB SNR=20 dB SNR=10 dB fS=2.5 GS/s fS=33.33 MS/s fS=16.158 MS/s XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

21. Measurement results – Occupied bandwidth • Similar consideration to the previous case can be done. • The reference case (sampling rate equal to 2.5 GS/s) has been evaluated by using a model order equal to 3000, instead in the other cases m=300 has been adopted. • Passband sampling allows to reduce the number of samples stored in the memory. In this way it is possible to use a lower model order of the estimator, increasing the processing speed, without worsening the repeatability. XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

22. In conclusion • Two algorithms for the optimal bandpass sampling rate selection in RF measurement instrumentation have been developed and applied on channel power measurement and occupied bandwidth measurements on DVB-T signals. • A tiny bias on channel power measurement results has been experienced. • The occupied bandwidth measurement results seems to be influenced by the bandpass sampling rates, especially if the imposed guard band is null. XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

23. Thank you for your attention. XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

24. Measurement results – Occupied bandwidth Modelordertoo low SNR=22.5 dB SNR=20 dB SNR=10 dB fS=33.33 MS/s m=300 fS=2.5 GS/s m=3000 fS=16.158 MS/s m=300 fS=2.5 GS/s m=300 XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal