Giovanni Betta, Domenico Capriglione, Luigi Ferrigno, and Gianfranco Miele DAEIMI, University of...

Giovanni Betta, Domenico Capriglione, Luigi Ferrigno, and Gianfranco Miele

DAEIMI, University of Cassino, Via G. Di Biasio 43,

03043 Cassino (FR), Italye-mail:

betta,capriglione,ferrigno,[email protected]

New Algorithms for the Optimal Selection of the Bandpass Sampling

Rate in Measurement Instrumentation

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

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.

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.

fc

B

fs>2 fu

f[Hz]

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.


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

fc

B

fu=4B

Let us divide the frequency axis in fs/2 wide intervals.

To avoid aliasing the replica has to be included in a fs/2 wide interval.

fs=5B

5B

5B

Let us change the center frequency of the signal.


Bandpass sampling theory

The previous consideration implies that S(f) has to be totally included in a fs/2 wide interval

12

2

Sl

Su

ff n

ff n

Condition for uniform bandpass sampling

2 2

1

1

u lS

ug

f ff

n nf

n IB

Forbidden areas (aliasing)

Alias free areas

Nonadmissible frequencies

Admissible frequencies

Lower guard band BGL

Upper guard band BGU

Upper admissible values DfSuLower admissible values DfSl


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.

B fSf*

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 fS 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 I Algorithm II

Algorithm I

BGL,n,max

BGU,n,max

BGU,n,max

BGL,n,max

DfS

BGU

DfSl

, ,max

GUSl S

GU n

Bf f

B

BGL,x


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

22

ADC ADCS g

f ff m I

m B

2 2 2 21 1

u ADC l u lADC

f f f f fm f m

n m n n n


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.


Measurement station


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

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.


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/s33.33 MS/s

16.158 MS/s

Nyquist rate

Algorithm I

Algorithm II

• B=8 MHz• BGU=BGL=0 MHz

• B=8 MHz• fADC=100 MS/s


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;


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

How we have estimated the PSD.

The PSD estimate was evaluated using two different PSD estimators

Modified periodogra

mBurg

Nonparametric PSD estimator

• Hamming window

Parametric PSD estimator

• model order m=3000 (for Nyquist sampled signal) and m=300 (for bandpass sampled signal)


Measurement results – Channel power

PSD estimators

Figures of merit

Sampling rate [MS/s]

2500 33.33 16.158

Modified periodogram

PC [mW] 85.65 84.56 87.88

sC [mW] 0.53 0.66 0.60

DC [mW] -1.09 2.61

Burg

PC [mW] 85.64 84.68 87.92

sC [mW] 0.46 0.37 0.45

DC [mW] -0.96 2.28DC 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

Measurement results – Occupied bandwidth

PSD estimators

Figures of merit


2500 33.33 16.158

Modified periodogram

B [MHz] 7.5275 7.5353 7.8213

sB [MHz] 0.0041 0.0044 0.0087

DB [MHz] 0.0079 0.2938The 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.




SN

R=

40

dB

fS=2.5 GS/sfS=33.33 MS/s

SN

R=

20

dB

fS=16.158 MS/s

SN

R=

10

dB


PSD estimators

Figures of merit


2500 33.33 16.158

Burg

B [MHz] 7.5435 7.5388 7.8179

sB [MHz] 0.0034 0.0029 0.0059

DB [MHz] -0.0045 0.2744Similar 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.



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.

Thank you for your attention.




fS=2.5 GS/sm=300

Model order too low

fS=2.5 GS/sm=3000

SN

R=

22

.5

dBfS=33.33

MS/sm=300

SN

R=

20

dB

fS=16.158 MS/sm=300

SN

R=

10

dB

Giovanni Betta, Domenico Capriglione, Luigi Ferrigno, and Gianfranco Miele DAEIMI, University of...

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