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

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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

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Introduction XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal 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.

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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 f S. fcfc B f s >2 f u f[Hz] Passband signals have a null spectrum in the bandwidth [ 0, f c – B/2]. It is possible to sample alias-free these signals considering suitable conditions. XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

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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 fcfc B f u =4B Let us divide the frequency axis in f s /2 wide intervals. To avoid aliasing the replica has to be included in a f s /2 wide interval. f s =5B 5B Let us change the center frequency of the signal. XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

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Bandpass sampling theory The previous consideration implies that S(f) has to be totally included in a f s /2 wide interval Condition for uniform bandpass sampling Forbidden areas (aliasing) Alias free areas Nonadmissible frequencies Admissible frequencies Lower guard band B GL Upper guard band B GU Upper admissible values f Su Lower admissible values f Sl XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

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The optimal choice of the bandpass sampling rate XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal 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. B fSfS f*f*

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Our proposal XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal 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 f S submultiple 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 IAlgorithm II

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Algorithm I B GL,n,max B GU,n,max B GL,n,max fSfS B GU f Sl B GL,x XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

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Algorithm II The latter algorithm makes the selection of the minimum sampling frequency that is a submultiple of a given operating one f ADC. 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 f S =f ADC. 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

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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

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Measurement station XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal Vector RF signal generator: 17 dBm max output power 250 kHz-6 GHz frequency range equipped with DVB-T personality 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 RF signal generator: +13 dBm max output power 9 kHz-3 GHz frequency range All the instruments are controlled by a LabVIEW driver that runs on the control unit

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

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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 33.33 MS/s 16.158 MS/s Nyquist rateAlgorithm IAlgorithm II B=8 MHz B GU =B GL =0 MHz B=8 MHz f ADC =100 MS/s XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

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

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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; B, defined as the experimental standard deviation of B measurement results; the difference ( B ) 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

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How we have estimated the PSD. The PSD estimate was evaluated using two different PSD estimators Modified periodogram Burg Nonparametric PSD estimator Hamming window Parametric PSD estimator model order m=3000 (for Nyquist sampled signal) and m=300 (for bandpass sampled signal) XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal

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Measurement results – Channel power PSD estimators Figures of merit Sampling rate [MS/s] 250033.3316.158 Modified periodogram P C [ W] 85.6584.5687.88 C [ W] 0.530.660.60 C [ W] -1.092.61 Burg P C [ W] 85.6484.6887.92 C [ W] 0.460.370.45 C [ W] -0.962.28 C is ever lower than 2.66%. Such value converted in decibel is approximately equal to 0.11 dB, showing a very little bias. XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal The repeatability of the channel power measurement results does not seem to be influenced by the three sampling rates

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Measurement results – Occupied bandwidth PSD estimators Figures of merit Sampling rate [MS/s] 250033.3316.158 Modified periodogram B [MHz]7.52757.53537.8213 B [MHz] 0.00410.00440.0087 B [MHz] 0.00790.2938 he 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

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

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Measurement results – Occupied bandwidth PSD estimators Figures of merit Sampling rate [MS/s] 250033.3316.158 Burg B [MHz]7.54357.53887.8179 B [MHz] 0.00340.00290.0059 B [MHz] -0.00450.2744 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

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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.

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

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Measurement results – Occupied bandwidth XIX IMEKO World Congress, September 6-11, 2009, Lisbon, Portugal f S =2.5 GS/s m=300 Model order too low f S =2.5 GS/s m=3000 SNR=22.5 dB f S =33.33 MS/s m=300 SNR=20 dB f S =16.158 MS/s m=300 SNR=10 dB

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