dynamic maximization of filter length

8/13/2019 Dynamic Maximization of Filter Length

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IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 59, NO. 5, OCTOBER 2012 2451

Dynamic Maximization of Filter Lengthin Digital Spectroscopy

Andrea Abba and Angelo Geraci

AbstractIn many modern applications based on detected

radiation measurements, high rate and high resolution are moreand more features of primary importance. In this scenario, it is

well known that high resolution of measurements and high rateof pulses are conflicting issues, which implies a trade-off betweenlive time of the system and precision of measurements. We pro-pose a digital processing technique based on adaptive filters for

maximizing both achievable resolution and sustainable rate.

Index TermsAdaptive filters, digital Spectroscopy, live time,rate, resolution.

I. INTRODUCTION

I N terms of energy measurement resolution, it is well knownthat the best approximation of the optimum filter corre-sponds to weighting functions (WFs) of time duration as long

as possible in case of white noise [1]. Of course, depending on

the statistical occurrence of the events, every fixed choice of the

WF temporal duration results in a trade-off between resolution

and efficiency of processing. This is true in most modern ap-

plications of radiation detectors where the achievement of both

high rate and high resolution is critical.

In [2] we have proposed a concept for maximizing both the

resolution and the rate, by using for each pulse the WF of thelongest temporal duration compliant with the distance of the

pulse from the adjacent ones. In this paper we carry out the

experimental setting of this technique with particular attention

to both implementation and calibration.

The considered reference setup consists of an analog

(pre)filter followed by an analog-to-digital converter (ADC)

and a digital processing stage (Fig. 1). In particular, the pream-

plifier output feeds the analog section that consists of a poletion,

necessary to cancel the long exponential decay of the pream-

plifier, and of a single pole signal filter. The quasi-exponential

pulse seen at the analog output is sampled by the ADC and the

sampled data stream enters the digital stage. Here, the signalfirst undergoes a last digital fine pole-zero compensatin [3] and

then the samples go through digital FIRfilters that extract the

required information (e.g., energy, occurrence time, etc.). In

this regard, we are interested in measuring the pulse energy by

using the best WF.

Manuscript received February 21,2012; revised May18, 2012; accepted June20, 2012. Date of publication August 10, 2012; date of current version October09, 2012. This work was supported by the Italian MIUR.

The authors are with the Department of Electronics-DEI, Politecnico di Mi-lano, Milan, Italy (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TNS.2012.2206827

The baseline restoration is performed in digital form as well,

by using zero-area WFs. In particular, the class of asymmetric

Deightons filters [4] is implemented, whose shape can be

thought as the superposition of a trapezoid and two parabolas

[5]. In order to reduce pile-up phenomena, strictly finite-width

weight functions are enforced. Furthermore, the flexibility in

implementing the filter shape allows the use of a sufficiently

large flat top, which prevents inaccuracies due to ballistic

deficit and jitter.

In this contribution we present a processing technique with its

full implementation for both maximizing the temporal length of

the filters, i.e., the resolution, and minimizing the probabilityof pile-up and the rejection of too close pulses, i.e., dead-time.

In practice, for each pulse, the processor uses the longest filter

that is consistent with the position of the two adjacent pulses.

It should be considered that the proposed technique is suited

for any kind of detector, providing the de-convolution process

adapted to the signal released by the detection stage.

II. PROPOSEDTECHNIQUE

Let us consider three consecutive pulses after the de-con-

volution stage [3], which realizes the equivalent of an analog

pole/zero transfer function in the discrete time domain that re-

turns the pulse with a shape delta-like (Fig. 2). ConsideringDeightonsfilters with flattop, the longestWF available for

the energy measurement of the central pulse can be expressed

analytically as

(1)

where is the flat top length that is set a-priori, and are the

distances of the flat top edges from the correspondent adjacent

pulses. As (1) shows, the WF is uniquely determined by ,

and , which means that a bench of different WFs can be stored

in form of different sets of values of these parameters. In partic-

ular, the flat top prevents inaccuracies due to ballistic deficit,

especially in case of bulky detectors, and to an approximated

de-convolution of the digitized pulses that returns quasi-delta

shapes. In practice, once established the experimental setup, the

0018-9499/$31.00 2012 IEEE


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2452 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 59, NO. 5, OCTOBER 2012

Fig. 1. Schematic representation of the considered experimental setup. The

proposed technique is implemented into the digital section where the energymeasurement of the pulses is performed. The graphic representation of the en-ergy measurement procedure in the box Digital Section is only symbolic anddoes not represents other elaboration items, such as pulse triggering, de-convo-lution process and baseline restoration.

Fig. 2. Sequence of three delta-like pulses after the de-convolution process.Regarding to the energy measurement of the central pulse, the longest zero-areaWF in the class of asymmetrical Deightons filters is plotted. Flat top lengthand temporal distances , of the flat top extremes from adjacent pulses aremarked.

contribution of these effects is stationary and consequently the

value can befixed once and for all.

For each incoming and de-convoluted pulse, the technique

consists of measuring its distance from the adjacent ones, and

choosing the maximum values compatible among those

stored, in order to synthesize the WF of maximum length. The

procedure requires that at least samples among three consecu-

tive pulses are stored in a buffer into the used computing device

(typically a Field Programmable Gate ArrayFPGA or a Dig-

ital Signal ProcessorDSP). To build the most versatile tool,

the size of this buffer should be set to the maximum allowableby the available hardware resources. Theoretically, the granu-

larity of and values should correspond to the number of the

buffer taps.

The use of WFs of different length is cause of deterioration

of the resolution of the energy spectrum. In fact, although the

operation of de-convolution is accurate, there are always sec-

ondary poles introduced by the analog stage that are not taken

into account and compensated. This determines that the signal

has a quasi-delta shape showing a tail that exits the boundary

of the flat top. If the filter had constant length, the samples of

this tail would always be weighted by the same coefficients.

Conversely, in the case of variable length of the filter, even for

mono-energetic pulses, the tails are multiplied by different co-

efficients thus giving a different estimate of energy. This results

Fig. 3. One of the carried out experiments measured the energy of a peri-odic monochromatic signal from a pulser changing the length of the WF onlythrough the parameter . The fixed flat top duration was equal to 10 samplesand contained the de-convoluted pulse shape. The plot highlights an absolutespread of the measured energy values of almost one hundred percent. The set ofcorrection factors that would return the energy values to be equal regardless ofthe fi lter length is plotted.

Fig. 4. Pulse shape at the output of the de-convolution stage for different preci-sions of the pole-zero compensation process, spanning in a range of 0.5% below(under-compensation) and 10% above (over-compensation) the exact compen-sation. It shouldbe considered that also ringing trendscan occur in thetail shape.

in a broadening of the energy spectrum and its shift in case of

significantly different lengths of the filters. It has been experi-

mentally verified that this detrimental effect can influence the

accuracy of measurement up to some tens of percent, as for in-

stance the Fig. 3 shows.

The distribution of the measured energies depending on the

WF length is also a function of the de-convolution process, i.e.,of the value of the de-convolution time-constant since sampled

pulses have ideally exponential shape. By the way, this also ac-

counts for different shapes of energy distributions in Fig. 3. The

sensitivity depends on the different shapes of the residual tails

when the pulse is de-convoluted with different time-constant

values, as Fig. 4 shows.

For a realistic experimental condition, Fig. 5 gives a synoptic

view of the variation of the pulse energy estimate as function of

the duration of the WF and of the time-constant value used in

the de-convolution operation.

The problem of the dependence of the energy estimate on

the length of the WF requires the introduction of a calibration

procedure of the system at startup and an operation of correction

during the measurement process.


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ABBA AND GERACI: DYNAMIC MAXIMIZATION OF FILTER LENGTH IN DIGITAL SPECTROSCOPY 2453

Fig. 5. Error of the energy estimate of the reference curve as a function of theextension of the symmetric deightons WF and of the time-constant used inthe de-convolution process. Considered time-constants are normalized to thecorrect value.

Fig. 6. Calculated reference shapes of the sampled and de-convoluted inputpulse. The quotation of the x-axis in continuous time is not relevant.

III. CALIBRATION

We assume that the shape of the ideally exponential sampled

pulse, normalized to unit area and called reference curve in the

following, is stationary during an experiment.

At system startup, the calibration procedure begins with the

construction of the reference curve by averaging a statistically

significant set of sampled signals, which can be easily caught

by suitable experiments. In addition to calibration, the refer-

ence curve is used to estimate the time constant with which we

initialize the de-convolution phase. This step returns the nor-

malized reference shapes of input and de-convoluted pulses, asshown in Fig. 6.

Once the maximum length of the WF and the extension of

the flat top are set, the calibration process proceeds to measure

the energy of the reference as a function of and values. For

ease of explanation, a symmetric Deightons WF, i.e., equal

to , is used to illustrate the technique in the following. The

variation of the energy estimation of the reference pulse as a

function of WF extension is represented in Fig. 7, which also

shows the sequence of corrective factors that multiplied by the

correspondent energy estimates would return a constant value

of energy regardless of the filter length.

For each value of , the operation is repeated for every value

of and the surface of corrective coef ficients,

which are functions of the variables , , is synthetized.

Fig. 7. Variation of the energy estimate of the reference curve as a function ofthe extension of the WF, according to a fixed value of the flat-top duration andto the maximum interval tolerated between adjacent pulses. The plot also showsthe curve of the corrective coefficientsthat multiplied by the estimates of energycorrespondent to the same values return the correct normalized energy of thereference curve.

Fig. 8. Error of the energy estimate of the reference curve as a function of theextension of the symmetric deightons WF and of the time-constant used inthe de-convolution process. Time-constants are normalized to the correct value.Arbitrary units of amplitude values are the same of Fig. 5 and, consequently, thecompensation has reduced the error offive orders of magnitude.

Fig. 9. Possible scenario of the circular buffer content. It is evident from thetimeline of the processing stages CONV 1 and CONV 2 that the same samplesmayhave to be used twice forthe calculation of energies of differentpulses. Thisrequires the implementation of two parallel processing branches (convolutionstage). Note that graphics representing the WFs is only symbolic and does notrepresents the deightons fi lters that are actually used.

The operation of correction during the measurement process

consists of normalizing the calculated energy value of the

incoming pulse by the correction factor corre-

sponding to the parameters , of the WF used.

After the application of the correction, the energy estimate

does not depend significantly on both the WF length and the

de-convolution time-constant value, as Fig. 8 demonstrates.

IV. IMPLEMENTATION

Since the processor operates in real time synthesizing for each

pulse a different WF, the necessary computing power is con-


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Fig. 10. Data-path andflow of processing for energy calculation with WF as long as possible. Note that graphics representing the WFs is only symbolic and doesnot represents the deightons filters with flat top that are actually used. The length of the flat-top is omitted in the scheme.

siderable and requires a quite sophisticated computing architec-

ture. In support of this assertion, consider that in case of fil-ters positioned as close as possible to each other and maximum

rate,roughly 900 Moperations/s are required.

In the presented realization, the technique has been imple-

mented in a digital spectrometer based on a Field Programmable

Gate Array (FPGA). In order to minimize the error due to ma-

chine finite precision, the entire architecture has been developed

in single-precision 32-bit floating-point data format (8-bit ex-

ponent and 24-bit mantissa). We have used standard Intellectual

Propertyblocks in the FPGA logicwhile no resources of theem-

bedded processor of the device have been used. The time cost

of this architectural choice has been verified be compliant with

the maximum data rate fixed by the ADC sampling frequency.According to the experimental setup topology of Fig. 1, the

ideally exponential signal at the output of the analog section is

sampled at the rate of 100 Msamples/s (MSPS) with 16-bit res-

olution and the samples are converted in floating-point format.

The sampled data pass through the de-convolution stage that

processes the pulses in the stream to having delta-like shape

(Fig. 6).

The stream enters a circular buffer that is a fixed-size memory

in which the last tap coincides with the first one. The buffer has

4096 taps, which is the maximum interval containing three con-

secutive pulses that can be observed in the built implementation

of the technique.The buffer content is scanned by a threshold-based trigger

procedure that detects the presence and the position of pulses.

For each pulse, the values of and parameters are calculated.

As the Fig. 9 graphically puts in evidence, the energy estima-

tion of pulses in general requires the presence of two parallel

processing branches called convolution stages. This is due to

the fact that in the time interval between two adjacent pulses

one rising edge and one falling edge of WF may have to be al-

located.

The whole data-path and flow of processing described are

shown in Fig. 10.

Since a customized WF has to be synthesized for each pulse,

an efficient generation architecture is mandatory. For this pur-

pose, (1) that represents the WF shape as function of the free

parameters , , , is implemented by grouping the numeric

coefficients in the terms , , ,

(2)

As Fig. 11 shows, the terms , , , can be properly com-

bined by means of multiplexing stages in order to synthetize the

characteristic parts of the WF, i.e., ascent, descent, and flat-top.

The global parameters , , , are tabulated and stored

in memory resources of the processing device with double-pre-

cision floating-point resolution. In the present implementation

into a FPGA device, theused memory data structure is a look-uptable(LUT)[6], where , values address the correspondent ,

, , memory locations.

The number of different shapes offilters that are feasible in

real time is limited by the memory available for storing the dif-

ferent sets . For instance, 4 memory blocks of

2 k-words x 9 bit are required to generate 256 different asym-

metric WFs that are more than enough to cover a wide range of

experimental conditions.

The architecture has been tested both on a Xilinx Virtex-5

FX-100T device and on a Xilinx Spartan-6 LX-25 device, which

are very far in terms of performance and cost [7]. The former is

able of operate over 300 MHz (i.e., 300 MSPS sampling rate)with less than 5% of resource occupancy, the latter uses 60%

of available resources operating at 100 MHz (i.e., 100 MSPS).

Considering a minimum filter length of 30 samples, the pro-

cessor into the Spartan-6 device is able to operate with live time

of 50% at 2 Mcounts/s, with respect to a live time of 80% of the

Virtex-5 solution.

A screenshot of a sequence of de-convoluted pulses and re-

lated WFs synthetized by the presented technique is depicted in

Fig. 12.

V. EXPERIMENTS

The reference processing setup is a digital pulse processor

based on a Xilinx Spartan 6 LX-25 FPGA [8]. For complete-

ness of discussion, we recall that the experiments reported in [2]


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ABBA AND GERACI: DYNAMIC MAXIMIZATION OF FILTER LENGTH IN DIGITAL SPECTROSCOPY 2455

Fig. 11. (a) Diagram of thearchitecturethat implementing (1)generatesthe WFs. Starting from thetop of thepicture,the discrete valuesand thesquared ones of thetime variable multiply the outputs of thefirst two multiplexers that correspond to the coefficients of the regarded construction phase of the WF, i.e., 00 ascent,01 flat-top, 10 descent, 11 reset. The multiplication coefficient of the output of the third multiplexer is in turn the output of a selection procedure.The samples of the synthetized WF and the input data are multiplied and accumulated to perform the calculation of the central value of their convolution, i.e., ofthe area and so of the energy of the pulse. The signal energy strobe checks out the end of the convolution process. (b) The and values address the memorylocations of the correspondent , , , . (c) Example of synthetized WF with the codes corresponding to the different construction phases.

Fig. 12. Example offiltering process of a sequence of de-convoluted pulses.As already underlined, the technique generates deightons WFs with flat-top,both symmetrical and asymmetr ical.

confi

rmed that shortfi

lters have poor resolution and are suitedfor processing very close pulses, while long filters guarantee

better resolution but can be used only in case of slow rate, i.e.,

distant pulses. In both cases, a live time around 90% has been

demonstrated to be achievable. These results have been veri-

fied in several different experimental conditions. For instance,

Fig. 13 shows the spectra of two X-ray sources at different

activity obtained by using WFs of different length.

Particular attention has been reserved to the validation of the

calibration procedure. For this purpose, an experiment has been

carried out with artificial pulses at constant different rates that

correspond to different values of the parameter . Specifically,

four different rates of pulses at two energies have been gener-

ated and the spectra have been measured. First, the spectra have

been grown without correction of measured values [Fig. 14(a)]

and then activating the calibration procedure [Fig. 14(b)]. The

shift in the peaks in the first case is evident. This means that with

no calibration the spectra remain separated even if they physi-

cally belong to the same spectrum. With calibration procedure

activated, the measured spectra grow up superposed and can be

summed together.

From experiments the efficiency of the algorithm is evident.For instance, using a X-ray source of emitting at a rate

of 750 kcounts/s and a silicon drift detector (SDD), we mea-

sured an energy spectrum with resolution of 230 eV and effi-

ciency of 88% with a filter temporal length equal to 200 ns. In

the same source conditions but using a set of 64 filters with tem-

poral lengths equally distributed between 200 ns and 1600 ns,

we built a spectrum with resolution of 184 eV and efficiency

maintained at 88%.

The same experiment has been repeated changing the source

rate to 160 kcounts/s. With a single filter s long we mea-

sured a spectrum with resolution of 142 eV and efficiency of

85%. The resolution has been lowered to 131 eV just using aset of 64 filters with temporal lengths equally spaced between

s and 6.8 ns with the same efficiency of the former mea-

surement.

VI. CONCLUSION

A processing technique based on filters of different length is

proposed and has been investigated and implemented.

The user sets WFs with different lengths, from the longest

that is optimum for energy resolution down to the shortest that

complies the rate specification of the application.

The system automatically and in real time synthetizes the best

filter for each single event measuring the time distance between

the current event and the two adjacent ones.


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Fig. 13. (a) Spectrum of a source of low activity (20 kcps) with live time of 88% and resolution of 5.8 electrons, and (b) spectrum of a source of highactivity (800 kcps) with live time of 90% and resolution of 20.9 electrons, that is signi ficantly worse than the previous case as expected.

Fig. 14. Spectra of artificial pulses of two energies at constant different ratesthat correspond to different values of the parameter , i.e., equal to 1, 2, 5,14 normalized to the minimum considered value (a) Spectra corresponding tothe different values with correction procedure de-activated (b) Spectra calcu-lated with correction procedure activated. The vertical reference lines betweenthe subplots put in evidence that the correction procedure aligns the generated

spectra, as is consistent with the reality.

This procedure allows to recover in real time the maximum

number of piled up events maintaining best performance in reso-

lution with the constraints preset by the user. In particular, if two

pulses are closer than the temporal length of the shortest filter,they are discarded. If two pulses are closer than the temporal

duration of the de-convolutor pulse response, they are undistin-

guishable and cause pile-up.

ACKNOWLEDGMENT

The authors would like to thank L. Bombelli, R. Alberti, and

T. Frizzi for their support in the experimental activity.

REFERENCES

[1] A. Geraci and E. Gatti, Optimum filters for charge measurements in

presence of 1/f current noise, Nucl. Instrum. Meth. A, vol. 361, pp.277289, 1995.

[2] A. Abba, A. Geraci, and G. Ripamonti, Digital adaptive fi ltering forresolution and live-time maximization, presented at the IEEE NuclearScience Symp., Valencia, Spain, Oct. 2329, 2011.

[3] A. Geraci, A. Pullia, and G. Ripamonti, Automatic pole-zero/zero-pole digital compensator for high-resolution spectroscopy: Design andexperiments,IEEE Trans. Nucl. Sci., vol. 46, no. 4,pp. 817821,Aug.1999.

[4] M. O. Deighton, Minimum-noisefilters with good low-frequency re-jection,IEEE Trans. Nucl. Sci., vol. 16, no. 5, pp. 6875, Oct. 1969.

[5] A. Geraci, I. Rech, E. Gatti, and G. Ripamonti, Shared baselinerestoration at minimum noise for high resolution spectroscopy, Nucl.

Instrum. Meth. A, vol. 482, pp. 441448, 2002.[6] R. J. Francis, A tutorial on logic synthesis for lookup-table based

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dynamic maximization of filter length

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