Asymmetrical optimum filters for charge measurements in presence of 1f current noise

Nuclear Instruments

and Methods

in Physics

Research

A 386 (1997) 487-491

NUCLEAR

INSTRUMENTS LMElNooS IN PHVSICS R%%Tn

EL‘33’lEK

Asymmetrical

optimum filters for charge measurements of 1 lf current noise

in presence

A. Geraci Pofitecnico di Milana.

Dipartimenfo

di Elettronica

Received

e Informazione,

Piaxu

ZOI.?.?. 1~1~

L. da Vinci 32, Milan

30 July 1996

Abstract The optimum filter for 1 (f current noise with no constraints on the position of the peaking time is presented in this work. In particular, the two lobes asymmetrical weighting functions of optimum filters for charge measurements have been identified, also in the presence of voltage and current 1lf noises in addition to normally considered noise contributions. The corresponding equivalent noise charges have been investigated.

1. Introduction

The interest in the effects in charge measurements of the associated for example with the gate current in HEMT or with the leakage current of GaAs radiation detectors, has recently grown up. Symmetrical optimum weighting functions (WFs). of both finite and infinite time duration, designed to take into account a l/f current noise component in addition to the other classical noises, have been studied in previous papers [ 1.21. Here, as announced in Ref. [2], the analysis is completed by removing the symmetry constraint for the WFs. This may reduce in same cases the equivalent noise charge (ENC) similarly to what found by Deighton, for the case of zero area asymmetrical WFs in presence of white voltage and current noises [3]. Other additional useful constraints, e.g. a flat top in the WF, could be introduced by this method but are not taken into account here for the sake of simplicity. It is well known [2] that the optimum WFs designed to take into account a I lf current noise component in addition to the other classical noises, should have zero area in order to avoid a divergent noise power at the output. Due to this area constraint, the resolution obtainable with these filters, if run with classical noises, is worse than that of corresponding unipolar filters. Despite this worsening, zero area WFs are also useful in filtering slow disturbing signals as baseline fluctuations due to C-R couplings, excess noise, microphonics, a.c. hum, etc. In fact, their power spectral density increases slowly from zero in the low frequency range, precisely as w4 for symmetrical filters and as w2 for asymmetrical ones, while unipolar filters have a power spectral density substantially constant over the same region.

Ilfcurrent noise component,

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The paper is so organized: Section 2 introduces the adopted model for the considered noises: Section 3.1 outlines the mathematical method of optimum WF calculation; Section 3.2 discusses the trend of optimum resolution and optimum WF shape as a function of the relative intensity of the noise contributions and of the total measurement time intervals.

2. Model of the noise Consider the detector-amplifier

system modeled in Fig.

I. Noise is assumed to arise from four uncorrelated sources at the input: c, and i_ generate white voltage and current

e,,]a f

I

I

I

I

I

i

at

“‘=m

n WI

nl LJ

I

II

I 1 I

(a)

I

I

Fig. 1. Plot of the detector-amplitier system: (a) with all the noise sources referred to the input of the noiseless amplifier h(t) and (b) with a global equivalent current noise generator

0 1997 Elsevier Science B.V. All rights resewed

488

A. Geraci I Nucl. Instr. and Meth. in Phys. Res. A 386 (1997) 487-491

noise of spectral densities a and b respectively; e, and i, generate 1 lfvoltage and current noises of spectral density urllfl and b,llfl respectively’ (Fig. la). Therefore, the input of the noiseless amplifier H(w) for a charge Q from the detector can be modeled (Fig. lb) as a Q&r) pulse signal plus a global equivalent input current noise of spectral density N(w) = Paw’

+ 6 + C’awJwl + b,llw( ,

(1)

obtained by setting 2ra, = a_ and 2n-b, = b_. Referring the time scale to the noise comer time constant r, (7, = I lo, = C&), we introduce two other characteristic comer angular frequencies (see Fig. 2): o, the angular frequency at which white and 1 lf voltage noise have equal spectral densities and w, the angular frequency at which white and l/f current noise have equal spectral densities C2awf = C’a_o,

* 13, = awla,

b = b,Jw, + is2 = bulb.

(2)

In terms of w, , w2 and w,, the noise spectra1 density (I ) at the input of the amplifier can be written:

(3) or, equivalently, ponents

in terms

of voltage

and current

N(o) = aC’(w’

+ w,loj) + bf I +

w,ljwJ) .

3. Calculation

of the optimum

weighting

com-

(4)

function

3. I. Computation We are looking for the optimum WF for a 8(t) signal in presence of a noise whose spectral density N(o) is given by Eq. (4). The constraints imposed are a finite time duration T and zero area. This is obtained by maximizing the signal-to-noise ratio at the output of the filter having pulse response h(t) (which is, disregarding a time shift, specular to the sought WF for time invariant filters, as it is well known). In Ref. [4] it has been shown that a convenient general representation of the sought waveform is a truncated Fourier sine series composed by symmetrical and antisymmetrical harmonics [4]

h(t)=TA,Zsin(n$r) h(t) = 0

elsewhere

O
.

\-I

The zero area constraint is fulfilled with a proper choice of odd A,, parameters. In fact, each function associated to even A+, coefficients has inherently zero area. The constraint is

T++=o. In order to calculate the optimum WF, i.e. the coefficients A,!, we require the signal-to-noise ratio (S/N) at the output of the filter to be maximum at the peaking time T,. The full procedure of calculation of the sets of A,$ which determine the optimum WF and the corresponding EN? is dealt with in Ref. [4]: there, a general method for WF calculations with assigned time domain constraints (finite width, flat top, peaking time, etc.) in presence of general type of uncorrelated noises is developed and applied to typical examples. 3.2. Discussion of results The waveform optimisation is achieved by calculating the optimum shape and the corresponding ENC’ for several given values of the peaking time 7’,. The dependence of ENC’ on T, allows to determine the ENC’ versus TP function f(T,). In several cases, a central maximum of f(7’,) and two symmetrical lateral minima, one on each half side, are found. The best pair of values of T, define two specular solutions for the absolute optimum filter shape, corresponding to a common absolute minimum of the N/S ratio. In general, EN? can be regarded as a function f(T,, T, w,. w,, b) of the peaking time T,, of the total measurement width T and of the amount of noises expressed by w,, w, , wz, b. In order to investigate the dependence of ENC’ on the two variables 7’,, T, in addition to that from the relative weights of the noise terms, let us consider three different values of the WF duration T referred to the noise corner time constant 7,: (a) T/Q-~= IO, (b) T/T~ = 2 and (c) T/T~ = 0.1. For each one of these conditions, we present in Figs. 3, 4, 5 the optimum peaking time and the associated optimum ENC’, varying the relative contribution of voltage and current I lf noises for several couples of w, and %. 3.2.1. First case: (a) T/Q-~= 10

’ Dielectric tern.

noise is included

in the l/f voltage

noise spectral

One can see from Fig. 2 that for 1IT =f;./lO (frequencies corresponding to the duration of the filter low compared with that corresponding to the noise time comer)

A. Geraci

I Nucl. Instr. and Meth. in Phys. Res. A 386 (1997) 487-491

489

1.7 or=

os=

1.2 0,

1.6.

Fig. 2. Components of the power spectral density N(o) of the noise at the input of the noiseless amplifier: white current spectrum (dashdot line), white voltage spectrum (dotted line), 1lf current spectrum (continuous line). 1 /fvoltage spectrum (dashed line) and global spectrum (continuous marked line). The corner angular frequencies w,, w,. w2 referred in the text are highlighted. The difference between two horizontal ticks corresponds to I decades and the difference between vertical ticks corresponds to 10 dB.

the most important 1lf spectral contribution is the 1 lf current noise. We define ENCO, the equivalent noise charge for the same white noises with WF of infinite duration. Fig. 3a shows ENC/ENCO, curves versus peaking time T, for different values of 1 lfcurrent and voltage noise contributions o, and 02, starting from the limit case of only white voltage and current noises (w, = w, = 0) to extremely dominant 1 lf spectral noise contribution (w, = wZ = 1.2q). The intermediate cases of the two ENC’ curves (w, = 0, o, = 1.20,) and (0, = 1.2w,, o2 = 0), emphasize the dominance of just one of the I/fnoise contributions. These two curves cross. Let us first consider the second curve. When T, is very low, the large slope of the corresponding asymmetrical WF (see waveform in Fig. 6) emphasizes the high frequency noise with spectral density w (i.e. the 1 lf voltage noise), which is responsible of the resolution worsening. But, as TP increases, the worsening for the large slope decreases and a minimum of ENC’ is reached: going to the right of the minimum, the w noise begins to increase again because of the global increasing of the slopes of the WF. This determines that in presence of l!f voltage noise, the favorite WF is the asymmetrical one. On the contrary observing the first curve, the presence of 1 lfcurrent noise is dominant over all other noise contributions at low frequencies and the consequently optimum T, corresponds to a symmetrical

0.2

0.4

0.6

0.8

1

1.2

Oz/% Fig. 3. Case of a measurement time window T ten times larger than the noise corner time constant for white noises 7,. (a) Plot of ENC/ENCO, curves versus peaking time T, for voltage (w, ) and current (wz) l/f noise contributions varying from 0 to 1.2~~. i.e. from only white voltage and current noises case to dominant 1 !f noise contributions. (b) Plot of iso-ENCIENCO, curves at optimum peaking times T, for different 1 lfnoise contributions. The time T, is normalized to half the total measurement time 7’. Dashed curve represents Deighton’s analytical curve calculated just for white noises.

WF. This WF is characterized by having zero first moment, I.e., a power spectrum wJ at low frequency, which determine a large attenuation of the low frequency components of the noise. The dominance of the 1 lfcurrent noise determines the requirement of symmetrical three lobes WF also in the case of contemporary presence of 1lfvoltage and current noises (see w, = o, = 1.2~~ in Fig. 3a). In case of just white noises (w, = wZ = 0), Fig. 3a also

490

A. Geraci

I Nucl. Instr. and Meth. in Ph.vs. Res. A 386 (1997) 487-491

shows the excellent fitting of the corresponding ENC* curve to the analytical Deighton’s one [3]. Fig. 3b shows iso-ENC/ENCO, curves for different values of I/fnoise contributions w, and w, at optimum peaking times. The plot highlights the ENC’ stronger dependence on l/f current noise rather than on voltage one. Figs. 4b and 5b show, as expected, progressive reduction and an inversion of this trend, as the measurement time window becomes narrower. 3.2.2. Second case: (b) TlrC = 2 In this case, the shorter duration of the WF corresponds to a bandwidth centered around w,/2: Fig. 2 qualitatively shows a dominance of the 1/f voltage over the I/fcurrent noise contribution. In fact, the ENC/ENCO_ curves of Fig. 4a strongly depend on this noise term. There is a strong separation between the two couples of curves (w, = 0, wz = 6q), determined by the presence of I/fvoltage noise; while the presence of the 1/fcurrent noise accounts for the small separation between curves inside the same couple, having the same amount of l/fvoltage noise. The dominance of 1/fvoltage noise makes the asymmetrical optimum WF favorite, with peaking times T,,, nearly constant as shown in Fig. 4a for different amounts of I/f voltage noise. Fig. 4b plots iso-ENCIENCO, curves at optimum peaking times for different values of 1/fnoises contribution, w, = 0-60~~ and wz = 0-6w,. This plot highlights again the greater sensitivity of ENC’ (parameter of the curves in Fig. 4b) to voltage noise, with respect to the amount of l/f current noise. 3.2.3. Third case: (c) T/rC = 0. I As it is well known, short measurement times give prominence to high frequencies, i.e. to voltage noise. In the present case, the bandwidth is centered at low, and voltage l/f noise is dominant with respect to I/fcurrent noise. This is clear from Fig. 5a, where the sensitivity of ENC/ ENCO, curves on the 1/fcurrent noise is invisible at all. From Fig. 2 is apparent that both white (w’) and 1/f(w) voltage noises dominate. Fig. 5a shows that the optimum WF is asymmetrical: this was to be expected since the voltage noises are dominant. Peaking time positions are slightly dependent on the amount of 1 /fvoltage noise. In particular, for very little T,,, the noise with higher spectral power (w* in this case) is dominant and ENC/ENCO, curves differ less and less in percentage. This is also confirmed by Fig. 5b which shows iso-ENC/ENCO, curves as straight lines parallel with the direction of variation of the current noise parameter w,. In Fig. 6 we plot three optimum WFs for different noise spectral densities to show the shape of the optimum symmetrical and asymmetrical filters, at various relative intensities of the noise contributions and at various measurement times. The curves correspond to the case of a large measurement time window T/r, = 10, so that the peak positions of the optimum synthesised filters corre-

(4

3.441

I

I

I

0.1

0.2

0.3

0.5

0.4

5

T

(b) :: ‘-25

1

I I

2

3

4

5

6

Fig. 4. Case of a measurement time window T two times larger than the noise corner time constant for white noises 7,. (a) Plot of ENCIENCO, curves versus peaking time T, for voltage (w,) and current (w,) l/f noise contributions varying from 0 to 6y, i.e. from only white voltage and current noises case to dominant Ilf noise contributions. (b) Plot of iso-ENCIENCO, curves at optimum peaking times T, for different 1lf noise contributions.

spond to the minimum same noise conditions.

ENC* values of Fig. 3a for the

4. Conclusions In this paper we have calculated the optimum timelimited filter in presence of l/f current noise, with a peaking time position placed at an arbitrary position inside the measurement interval. In presence of a small amount of l/f current noise, the two lobes asymmetrical filters generally give lower ENC’: this was to be expected, since, mathematically, this small

A. Geraci

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I Nucl. Ins?r. and Meth.

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in Phys. Res. A 386 (1997)

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487-491

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T

5 7

Fig. 6. Plot of three optimum WFs for the same measurement

(W

time window T= 120.

,11.5

w, = I .20

10~~: (a) w, = w, = 0. “Deighton’s

and wz = 0; (c) w, = 0 and w, = I .?y

filter”:

(b)

Peak positions

correspond to those of minimum ENC“ in Fig. 3a.

,ll.l looI 10.8 80

.

3’ 3’

circuit gives overall (filter plus baseline restorer) zero area WFs 151. Consequently, unipolar shaping of pulses buried in a noise comprising parallel I/f will not exhibit a divergent noise contribution. due to the presence of baseline restorers. In many cases, it could even be advantageous to use a baseline-restorer and an unipolar filter. because the overall filtering time will be higher. However, this is surely not true when the I lf current noise contribution is dominant in a frequency range comparable with the inverse of the filter duration.

,10.4 .lO.l

607 ,9.x .9.38 40 L

,9.04 .8.89

20.

,834

20

40

60

80

100

120

@z/O, Acknowledgements Fig. 5. Case of a measurement time window T ten times shorter

noises 7,. (a) Plot of curves versus peaking time T, for voltage (CO,)and

than the noise comer time constant for white ENCIENCO,

current (w?) I lf noise contributions varying from 0 to 1200~. i.e.

from only white voltage and current noises case to dominant 1 !f noise

contributions.

(b)

Plot

of

iso-ENCIENCO,

optimum peaking times ‘F, for different

curves

at

The author is indebted with E. Gatti for the critical discussions and with G. Ripamonti for his constant interest and stimulating support. This work was supported by INFN and by MURST.

1 lf noise contributions.

amount of 1lf noise only forces the WF to be areabalanced. For long shaping times T > 10~~, the three lobes zero area symmetrical WFs are the optimum filters and they asymptotically approach the ENC’ of the optimum infinite unipolar cusp for T + m. In many cases, however, the three lobes symmetrical filter could be preferred, even at the expense of a slightly larger ENC’, for its better rejection of low frequency noise, since its squared transfer function is of the type a“ at low frequency. For small amounts of I lf parallel noise contribution, the synthesised WFs tend to be those introduced by Deighton for the case of white noises. It is to be noted that the presence of a baseline-restorer

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