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Optimum filters for charge measurements in the presence of 1f current noise
__ __ l!iB
Nuclear Instruments
&
and Methods in Physics Research A 361 (1995) 277-289
NUCLEAR INSTRUMENTS a METNODS IN PHYSICS RESEARCH Secnon A
ELSRVIER
Optimum filters for charge measurements in the presence of l/f current noise Angelo Geraci, Emilio Gatti
*
Politecnico di Milano, Dipartimento di Elettronica, Piazza Leonardo da Vinci 32, 20133 Milano, Italy Received 24 November
1994; revised form received 17 February
1995
Abstract The weighting functions of optimum filters for charge measurements have been identified for infinite and finite width, in the presence of l/f current noise in addition to the normally considered white voltage and current noises and the l/f voltage noise contribution. The corresponding equivalent noise charges have been determined.
1. Introduction The great interest in the effects of l/f voltage noise in charge measurements has been widely discussed [ 11. The shape of the optimum weighting function and the corresponding values of the equivalent noise charge have been calculated: it has been pointed out that the weighting functions are cusp-like with a shape narrower in the middle and larger on the tails in comparison with the cusp of the traditional optimum filters for white noises only. However, these results take into account only a l/f voltage noise, disregarding the l/f current component, associated for example with the gate current in HEMT or with the leakage current of GaAs radiation detectors. The aim of the paper is to investigate the optimum shape of the weighting function and the corresponding values of the equivalent noise charge when the l/f current noise is present in addition to the white voltage and current noises and to the l/f voltage noise contribution. The work deals with optimum filters considering both the weighting functions of infinite time length and the weighting functions with the constraint of a finite time width. The new filters investigated are of zero area due to the fact that the new noise is a l/f noise current which requires a cut off of the continuous component in order that the output noise should not diverge. For the sake of simplicity, the l/f noise has been considered uncorrelated with respect to the other present noises.
2. Model of the physical system The detector-amplifier system is modelled as in Fig. 1. The input signal is a delta current pulse of charge Q. The capacitance’s sum of the detector and the input device is represented by C. The amplifier is assumed to be noiseless and is modelled by its transfer function H(w) or its delta pulse response h(t) [l]. The noise sources of the input stage of the amplifier are taken into account by 1) a voltage generator ei for the white voltage noise of mathematical spectral density “a” [V’/Hz]; 2) a voltage generator e; for the l/f voltage noise of mathematical spectral density a,/] f 1’ (a,[V’]); 3) a current generator i, for the white current noise of mathematical spectral density “b” [AZ/Hz]; 4) a current generator i; for the l/f current noise of mathematical spectral density bf/ ( f 1( bf[A2]). For the sake of simplicity, we consider the dielectric noise included in the voltage l/f noise [l].
* Corresponding author. Tel. + 39 2 23996102, fax + 39 2 2367604. 1In the literature the power spectral density has been denoted by 1) ar/ unilateral
in Ref. 131.
0168-9002/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0168-9002(95)00214-6
1f 1bilateral in Ref. [l]; 2) c/ 1o 1bilateral in Ref. [2]; 3)
Af/f
A. Geraci, E. Gatti/Nucl.
278
hstr.
and Meth. in Phys. Res. A 361 (1995) 277-289
Fig. 1. Noise figure of the detector-amplifier system with all the noise sources referred to the input of the noiseless amplifier of weighting function h(t).
If we transform all the voltage noise generators into the equivalent current generators resulting in parallel and if we represent all the generators as a single noise current generator (Fig. 2) its noise spectral density if is given by N(w)=C2aw2+b+C2af27F(wI+bf2a/lwl.
(1)
As a reference, we assume a time scale defined by the well known noise corner time constant for voltage and current white noises TV= CJa/b. Moreover, we define a corner angular frequency both for the voltage and current noises in order to indicate the spectral densities of the l/f components as a function of the spectral densities of the respective white noises. We define, in this way, the noise corner angular frequency wt at which the white voltage noise spectrum crosses the l/f voltage noise spectrum. By Eq. (1) this statement means that
also wJ275
= a/a.
In the same way, we define the noise corner angular frequency o2 at which the white current noise spectrum crosses the l/f current noise spectrum 2nb,/wz
= b,
so that 02/2 rr = b/b The actual range of w, and w2 variability is between a few Hz and 1 MHz, according to the devices and the operating conditions. By inserting the noise comer time constants and the noise corner angular frequencies into Eq. (l), the noise spectral density can be rewritten as
I 031Tc2+ w,02Tc2+ I WI +o, N(w)=6
Fig. 2. Reduction N(o).
(2)
I WI
of the noise sources of the model in Fig. 1 to a single equivalent
noise current generator
of mathematical
spectral density
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3. Weighting function with no time constraints
3.1. Calculation of ENC’ Let us suppose to have an input signal Qg( t), with G(w) the Fourier transform of g(t), in the presence of a current noise of spectral density N( 0). It is well known [2] that the best possible signal-to-noise ratio is
Q2 =%
optimum
By definition, one
+4G(w)l' dw.
/ --5c N(w)
the equivalent noise charge squared (ENC’)
is the value of Q2 which makes the signal-to-noise
ratio equal to
1 ENC’ = 27~
’
+~IG(w)~’ I -x
N(w)
d”
In the actual case with a delta function as input signal (i.e. G(w) = 1) and the noise density that aC*n
Tr
ENC’=
+= /o
The normalisation wTc=x;
=
1
+m
N(w)
dw
and the statement
I a
IWI 1 Iw31+ w,0’+~IwI+--;0* 7c
of the angular frequencies W,Tc = k,;
ii given by Eq. (2), we have
(3) 1
do
%-
with respect to the noise time comer
w2rc = k, =
3
ENCO, = C& lead Eq. (3) to the form ENC’ -=ENCO&
1
7~ 2 /0
If y = y(k,, ENC’ ___=_
(4)
X
+= x3 +
k,x* +x + k,
dx
kz) is the real root of the cubic equation
71
x3 + k,x2 + x + k, = 0, Eq. (4) becomes
1 ,
X
dx x*+(YX+p)(x-y) where CI= k, + y and p = 1 + (k, + y)y.
-relation between k, and the parameter K of Ref. [I] is k, = 2K. ’ The ENCO, is the equivalent noise charge squared for voltage and current white noises only, processed function.
by the optimum
weighting
A. Geraci, E. Gatti / Nucl. Instr. and Meth. in Phys. Res. A 361 (1995) 277-289
280
20’
b
8.5 16.
7 7.5
12.’
6
6.5
k2
8. 4.5 4-
5
2.5 2
3
0-l 0
4
5.5
3.5 8
4 12
16
20
k2
1 kt
Fig. 3. (a) Contour plot of ENC*/ENCOi in the range 0 + 20 both for k, and k2 in linear scale. (b) Expanded view in the range in logarithmic scale, with iso-curve distance equal to 0.2.
Performing
0.1 + 10
the integration, we get
ENC’
1
n
ENCO,
ENC*
rr
ENCO:
2
-=_
1 for
o*/4>
/3,
(5)
where A= - r/(r2 + cry + p) and i = p/(7&. The contour plot of ENC’/ENCOz is shown in Fig. 3a in the range 0 + 36 both for k, and k,. The figure shows the increase of ENC*/ENCOz due to the voltage and current l/f noises. Fig. 3b shows an expanded view in the range 0.1 + 10 both for k, and k?. 3.2. Calculation of the optimum weighting function h(t) Because of the non-analytical form of N(w) in Eq. (2) due to the presence of the absolute value of w, it is easier to approach the evaluation of the optimum weighting function (WF) in the frequency domain. With the criterion to maximise the signal-to-noise ratio and applying Schwartz inequality, it can be demonstrated [4] that the optimum WF is given by H(jo) Performing
1 = N(o)
(6)
’
the inversion of the Fourier transform 1
H(o),
the WF h(t) is
-+m COS(d)
h(t) = zj_, N(o)
do
(7)
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a
-.
11
Fig. 4. (a) Shapes of the WF with no time constraints just with different values of the current noise component being the voltage noise k, fixed at 1. (b) Expanded view of negative tails of the WFs in Fig. 4a.
and normalising
frequencies +m
h ;
i
from 0 to 40,
to noise comer time constant as in Section 3.1, the WF becomes x COS(Xf/TC) x3+k$+x+k2
The integration
k, varying
gives the following
dx.
(8)
result
I
=h(i)=hi(?)+hj(?)+h,(?),
(9)
where
[ -ci( xi;) W) = (x;-xj;;xixk)
W)
= cxi -xj;;xi
-Xk)
cos( xii)
- sin( xi’)(si(
[ci( xi?) cos( x,S) + sin( xii)si(
xii)]
x,7) + 7~)]
arg(xi)
Iarg(xi)
= 7~,
1 < n,
(10)
and hj(a,la,(?) are similar but for indices rotation, being xi,xj,xk the three roots of the equation x3 + k, x2 fx + k, = 0. Because of the different notations used in literature, we point out that in Eq. (10) the integral sine (si) and cosine (ci) are defined as in Refs. [7] and [8]. Fig. 4a shows the effect of the new noise component on the shape of the WF: it causes the WF to increase slope at small t/q and to develop a negative tail (Fig. 4b) more and more evident when k, rises. It should be noted that the optimum WF is a zero area function, even with no time constraints, and for any small amount of current noise (k,). This is not always
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Fig. 5. Plot of WFs for different values of k, and kz.
evident from the figure but we must remember that, with no time constraints, contribute to a finite area. In fact the area of the WF is x
jo+?r(i)
di=j+rdij+Xx+k 0 Ir
even an infinitesimal
small negative tail can
cos( x?) xl+x+k 1
dx=j+=M(i) 7
dF=m(O)=O,
0
being M(t) the Fourier transform of the function m(x) = x/(x3 + k, x1 + x + k,). The WFs are shown in Fig. 5 for some values of k, and k, given as an example.
4. Weighting function with imposed finite width As just mentioned at the end of Section 3, the WF must be of zero area due to the physical fact that the new noise is a true l/f spectrum current, which requires a cut off of the continuous component in order not to diverge. First let us find a basis. We are looking for a WF h(t) which should be zero at t = 0 and t = 2TP and be symmetrical with respect to the peaking time t = Tp. A convenient representation in the interval 0 -+ 2T_, would be the conventional Fourier series N+
I
h(t) = c K, sin n=O
(2n + 1)n 2T
t,
P
in which case h(t) would have N + 2 degrees of freedom. But if we would like to impose on h(t) a zero area, when it is integrated between 0 and 2Tp, we should add the linear constraint EfziK,(2n + 1) = 0, which would reduce the degrees of freedom to N + 1. However, we prefer the expression (2n + 3)n 2T
(11)
P
which implies by itself the zero area constraint in the interval 0 + 2TP, as each term satisfies this condition. This series with N + 1 terms is constituted by N + 2 different sinusoidal components, but it has only N + 1 degrees of freedom, just as happened by the conventional Fourier series. We can conclude that the set t2n + 3)ll x,(t)
=sin
’
’
2Tp
2n+l
t-
-sin
(2n + 11-n I
2n+3
t
2TP
is not orthogonal but complete with respect to the ensemble of h(t) with the imposed constraints. So the WF constrained in the time domain to have area equal to zero and symmetry with respect to t = Tp, to be identically zero outside the time interval 0 + 2TP and continuous everywhere, can be expressed as n=0,1,2,...,
h(t)=l(t)zA,x,(t)+l(t-2T,)xA,x,(t-2Tp)
n
n
(12)
283
A. Geraci, E. Gatti/Nucl. Instr. and Meth. in Phys. Res. A 361 (1995) 277-289 where the second term forces h(t) to zero for t > 2TP. The sought WF is now represented A,,. Expanding Eq. (12) we obtain (2n + 3)lT 1 2T
2n+Lsin(2n+1)71 2nf3
P
sin (2f12i3)rr
+ l(t--2T,)xA, n
2Tp
r
by the numerable set of variables
1
(t - 2T,) - gsin
(2n2;1)m
P
(t - 2Tp)
P
l-
(13)
In order to calculate the signal-to-noise ratio, we begin to evaluate the output noise. The Laplace transform of Eq. (13) is
r
2nf3
--
H(s) = (1 + eUs2rp) CA,
(14)
”
With straightforward
p=
c&Z
but laborious calculation (see Appendix
A), we can write for the output noise
C CA,AJ,,, m
(15)
n
where A, and A, are the unknown amplitudes of the WF components and Z,, are the constant factors defined in Appendix A. Let’s find out now the squared expression for the amplitude of the signal at the output, that is the value of h’(t) for t = Tp multiplied by Q2 P(2fr + 3) S’=
Q’x
xA,A, m n
sin 2 n(2n sin
rr(2m + 3) sin
2
sin
that is, naming IV,, the term in brackets, s’=
2 x(2m
+ 3)
+ 1) 2
lr(2n sin
1’
+ 1) 2
742/n + 1) sin
2
(16)
Q”C CA,A,W,,. m n
(17)
Finally we can write the signal-to-noise
ratio as a function of the unknown amplitudes
Indicating with p the term in brackets, the squared equivalent
A,,
noise charge ENC’ looks like
ENC’ = ENCO&.
(19)
F
where the term l/w > 1 is the worsening’s factor due 1) to the constraint of the finite pulse duration and 2) to the presence of the voltage and current l/f noise and the consequent zero area of the WF. Let us introduce the ENCO:,, achievable with the optimum unipolar cusp filter of finite width 2Tp, when only voltage and current white noises are present [S] ENCO&
= ENCO, coth
5
(20)
( 7c i This will be used as a reference ENC’ = ENCO;rP f
to discriminate
between the two mentioned
causes of worsening.
The squared ENC results (21)
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A. Geraci, E. Gatti /Mud.
Instr. and Meth. in Phys. Rex A 361 (1995) 277-289
with
(22) In this way, the term l/CL) represents the worsening’s factor due to the cause 2) only. It is also useful to remember [S] that ENCO$rp.bip, the equivalent noise charge achievable with the optimum bipolar cusp filter of finite time duration 2TP for voltage and current white noises, results [5]
(23)
This represents the worsening’s noises only. Writing ENC’ = ENCOzrr.,A
factor due to the forced imposition
of zero area to the optimum finite time WF for white
(24)
3 P
with
(25)
the factor l/g’ is just the worsening’s factor due to the addition of the two components of the l/f noise. The ENC* and the WF are calculated looking for the A, which make stationary the signal-to-noise ratio (S/N)&
Considering obtained
a set of n A,,
~[Wmn-pZmn]A,=O m
a system of n + 1 homogeneous
m=O,l;..,n.
equations
in the 12+ 1 unknowns
A,, A,, . . . , A,, can be
(27)
The proper condition of homogeneous systems’ solution, that is the determinant of coefficients equal to zero, determines the eigenvalue p which substituted in Eq. (19) gives the ENC’ [6]. It is well known that the substitution of the eigenvalue in the system implies that the last equation becomes a linear combination of all the others. So that it can be replaced by the amplitude expression forced to be one in Tp. The resultant system is no more homogeneous and has a determinant different from zero and its solution gives the sought A, coefficients. This makes possible to calculate the WF. Iso-ENC’/ENCO&, curves are shown from Figs. 6a to 6c for decreasing pulse widths 2T,/r,: it can be realised the ENC2 increases when the width of the WF is reduced with a constant amount of voltage and current noise (k, and k,). The only due to the current noise (k,) becomes more and more less important increase of the (ENCZ-ENCOg, )/ENCO& decreasing the processing time d:ration 2Tp,is the input spectrum of the current noise is proportional to l/o. On the other side, the (ENS* - ENco&)/ENCO,~, only due to voltage noise (k,) tends to saturate to a constant value (of about ten times the ENCOzrD) when &e width of \he WF decreases, because this kind of noise depends on o.
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A. Geraci, E. Gatti / Nucl. Instr. and Meth. in Phys. Res. A 361 (1995) 277-289
-__
20
16
11.5 II
10.5
10
b
a i9.5
9
12
12
8.5
8
kl
k) 8
7.5
I
6.5
5.5
5 3.5
8
4.5
4 4 8
k2
I2
I6
k2 I
20’
16.
C
._.
5.4 5.3
5.2 5.1 12.
5
~ -4.9 _
4.8
kl 8.
4.7
.4.6 ~ p---4.5
4-
~-- 4.4 4.3
O0
4
I 8
12
16
20
k2 Fig. 6. [so-ENC2/ENC0& for different 2Tt, /rc = 1; (c) 2Tp /T, = 0:25.
finite pulse processing
times and for values of the parameters
k, and k,: (a) 2Tp /rc = 4; (b)
Fig. 7. Plot of two WFs for different noise conditions: (a) k, = k, = 0 (only white noises); (b) k, = k, = 20. The shapes of the WFs are almost the same for different temporal pulse duration. The undulations are due to the finite number of terms (30) of the series development.
286
A. Geraci, E. Gatti/Nucl.
Instr. and Meth. in Phys. Rex A 361 (1995) 277-289
II-I
I,
2T,lr,=co
0
4
a
8
12
16
U
20
4
8
kl
12
16
20
kr 2T, = 0.25 ~~
2T, = 5,
1
t0
4
8
12
16
20
0
_i 4
kl Fig. 8. Plot of ENC’/ENCO&
12
8 kl
as function of k, and k, = 0, i.e. unipolar WF, (continuous
lines) and k, + 0, i.e. zero area WF, (dashed
lines), for different values of thi pulse width 2TP/TV =m (a); 4 (b); 1 (cl; 0.25 cd).
The l/f current noise component is quantitatively less important than the l/f voltage one decreasing the pulse width: this is clear in Figs. 6a, 6b and 6c, where the iso-ENC’ curves (nearly straight lines) have low tending to zero slopes, showing a minor sensitivity of the ENC’ to k, with respect to k,.
Fig. 9. Plot of WFs for different values of the parameter number of terms (30) of the series development.
2Tp /TV with k, and k, fixed to 10. The present undulations
are due to the finite
A. Geraci, E. Gatti/Nucl.
Instr. and Meth. in Phys. Res. A 361 (1995) 277-289
287
Fig. 7 shows an example of the shape of the WF for two different noise situations. Let us point out the role of the zero area imposition in terms of the ENC’. In order to do that, the ENC’ has been calculated varying k,, both with k, tending to zero and with k, equal precisely to zero. For infinite time duration (Fig. S), the curves obtained are indistinguishable because for k, # 0 and k, = 0 they merge themselves with continuity when k, + 0 (for instance, when k, = 0 the WF becomes the conventional unipolar cusp which can be also considered as bipolar with zero area when k, + 0 is present). related to zero area WF and those related to On the contrary, there is a jump between the values of ENC’/ENCO& unipolar WF; this jump increases and saturates to 4 when the width of the tfme window is reduced, in agreement with Eq. (23). From Fig. 9 it can be realised that the three lobes WF departs in shape more and more from the unipolar cusp function when the total pulse width decreases. The WFs tend to saturate to a fiied three lobes shape for pulses shorter than a few 7c because, at these short processing times, their shape is mainly determined by the prevalent voltage noise contribution.
5. Conclusions This work gives the optimum filter in the presence of four noise components, white voltage and current and l/f voltage and current contributions. The shape of the optimum WF for different processing times and the corresponding values of the equivalent noise charge have been calculated: the worsening in resolution becomes relevant for processing times shorter than the noise time constant. In particular, the effect of the l/f current noise has been studied. Its real presence determines a jump in the ENC’ for finite processing times due to the zero area constraint necessary for convergence. The resulting WFs are cusp-like with a slope causing practical problems of ballistic deficit. The increase in the ENC*, due to a smoothing of the cusp, is easily evaluated by using the h(t) series development limited to a number of terms small enough: the resulting worsening is of about 4% in the ENC with six terms. For small values of k, and k,, the calculated WFs are those introduced by Deighton [5] in the case of white noises and constraint artificially imposed by zero area. For the sake of simplicity, only the three lobes symmetrical WF has been introduced in this paper dealing with finite response filters. Certainly, for continuity, two lobes asymmetrical filters would give lower ENC2 even in the presence of small l/f noise contribution, as it happens for Deighton’s two lobes filters. This will be dealt in a forthcoming paper.
Acknowledgements We thank Ing. G. Bertuccio for unpublished interesting for interesting suggestions. This work was supported by INFN and by MURST.
experimental
data on l/f
current noise and Prof. M. Sampietro
Appendix A In this case of finite width, we refer frequency and comer angular frequencies of T= to be consistent with notations of Ref. [l] 2T,o=x;
2Tpw,=K1;
to the total width of the pulse 2Z”, instead
2Tpw,=K2.4
However, in the main text we continue with the normalisation used for infinite pulse, which makes all the discussion and more understandable. In fact, K, and K, change when the pulse width is decreased. The input noise spectrum density becomes
a&
N(x)
= (2Tp$
I x3I +K,
(A.11 easier
x*+(++(T)~K~ 1x1
4 The relation between k, of the main text and the parameter K, of this Appendix is K, = k,(2T, /TJ
(A.21
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The output power spectrum (s = jw) results IH(x)12=2[1
+cos(x)]
x
CA,A,
~~,,U,,[(2n+i)***-xZ][(2m+1)~~“-x*]
[ m.n
(
+ ~[(2n+3)*a*-xi][(2~+3)~~i-X2] 3n
3m
-~[(2n+~)2n’-*‘][(2~+3~~~‘-r’])
(2Q2 [ (2n + 3)*n* -X2]
x [ (2n + 1)2P* -x2]
[ (2m + l)*lra
-X’]
[ (2m + 3)*lr2 -X’]
where u,,
= (2m + l)n,
U,,, = (2n + l)n,
U,,,, = (2m + 3)n,
1 ’
(A.3)
U,, = (2n + 3)1r.
The total noise at the output is
@=
f~~+m~(x)lH(x)12 dx P
By inserting Eqs. (A.2) and (A.31 in Eq. (A.4) and performing be obtained
the integration,
the final expression
for the output noise can
where
/?‘(ln I,(a,
P)
a-ci
a)-a*(ln
p-ci
spy
In a-l 1a((Y, P)=(y4+3-4
/3*
Si ff
-
a’)
ci ff a=P,
a
ln(P/a)+ci LT(a, P) =
p) a+B,
=
a*-p2
a-ci
p afP,
The integral functions Si( a) and ci( a) are the notations stated in Ref. [7] and [8] and 6(
a, p)
is Kronecker symbol.
A. Geraci, E. Gatti/Nucl. Instr. and Meth. in Phys. Res. A 361 (1995) 277-289
Defining the Z,,,, factors
we get the final expression (15) of the main text for the output noise p.
References [l] [2] [3] [4] [5] [6] [7] [8]
E. Gatti, M. Sampietro and P.F. Manfredi, Nucl. Instr. and Meth. A 287 (1990) 515. E. Gatti, P. Manfredi, Nuovo Cimento 9 (1986) 38. V. Radeka, Annu. Rev. Nucl. Part. Sci. 38 (1988) 217. Ref. [2], p. 96. M.O. Deighton, Proc. Ispra Nuclear Electronics Symposium, 1969, EURATOM CID Brussels (1969) p. 47. W. Press et al., Numerical Recipes: The Art of Scientific Computing (Cambridge, 1986) Chap. 9. I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals Series and Products (Academic Press, 1992) p. 406, Sections 3.722-3.723 M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (Dover Publications, 1970) p. 232.
289
Nuclear Instruments
&
and Methods in Physics Research A 361 (1995) 277-289
NUCLEAR INSTRUMENTS a METNODS IN PHYSICS RESEARCH Secnon A
ELSRVIER
Optimum filters for charge measurements in the presence of l/f current noise Angelo Geraci, Emilio Gatti
*
Politecnico di Milano, Dipartimento di Elettronica, Piazza Leonardo da Vinci 32, 20133 Milano, Italy Received 24 November
1994; revised form received 17 February
1995
Abstract The weighting functions of optimum filters for charge measurements have been identified for infinite and finite width, in the presence of l/f current noise in addition to the normally considered white voltage and current noises and the l/f voltage noise contribution. The corresponding equivalent noise charges have been determined.
1. Introduction The great interest in the effects of l/f voltage noise in charge measurements has been widely discussed [ 11. The shape of the optimum weighting function and the corresponding values of the equivalent noise charge have been calculated: it has been pointed out that the weighting functions are cusp-like with a shape narrower in the middle and larger on the tails in comparison with the cusp of the traditional optimum filters for white noises only. However, these results take into account only a l/f voltage noise, disregarding the l/f current component, associated for example with the gate current in HEMT or with the leakage current of GaAs radiation detectors. The aim of the paper is to investigate the optimum shape of the weighting function and the corresponding values of the equivalent noise charge when the l/f current noise is present in addition to the white voltage and current noises and to the l/f voltage noise contribution. The work deals with optimum filters considering both the weighting functions of infinite time length and the weighting functions with the constraint of a finite time width. The new filters investigated are of zero area due to the fact that the new noise is a l/f noise current which requires a cut off of the continuous component in order that the output noise should not diverge. For the sake of simplicity, the l/f noise has been considered uncorrelated with respect to the other present noises.
2. Model of the physical system The detector-amplifier system is modelled as in Fig. 1. The input signal is a delta current pulse of charge Q. The capacitance’s sum of the detector and the input device is represented by C. The amplifier is assumed to be noiseless and is modelled by its transfer function H(w) or its delta pulse response h(t) [l]. The noise sources of the input stage of the amplifier are taken into account by 1) a voltage generator ei for the white voltage noise of mathematical spectral density “a” [V’/Hz]; 2) a voltage generator e; for the l/f voltage noise of mathematical spectral density a,/] f 1’ (a,[V’]); 3) a current generator i, for the white current noise of mathematical spectral density “b” [AZ/Hz]; 4) a current generator i; for the l/f current noise of mathematical spectral density bf/ ( f 1( bf[A2]). For the sake of simplicity, we consider the dielectric noise included in the voltage l/f noise [l].
* Corresponding author. Tel. + 39 2 23996102, fax + 39 2 2367604. 1In the literature the power spectral density has been denoted by 1) ar/ unilateral
in Ref. 131.
0168-9002/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0168-9002(95)00214-6
1f 1bilateral in Ref. [l]; 2) c/ 1o 1bilateral in Ref. [2]; 3)
Af/f
A. Geraci, E. Gatti/Nucl.
278
hstr.
and Meth. in Phys. Res. A 361 (1995) 277-289
Fig. 1. Noise figure of the detector-amplifier system with all the noise sources referred to the input of the noiseless amplifier of weighting function h(t).
If we transform all the voltage noise generators into the equivalent current generators resulting in parallel and if we represent all the generators as a single noise current generator (Fig. 2) its noise spectral density if is given by N(w)=C2aw2+b+C2af27F(wI+bf2a/lwl.
(1)
As a reference, we assume a time scale defined by the well known noise corner time constant for voltage and current white noises TV= CJa/b. Moreover, we define a corner angular frequency both for the voltage and current noises in order to indicate the spectral densities of the l/f components as a function of the spectral densities of the respective white noises. We define, in this way, the noise corner angular frequency wt at which the white voltage noise spectrum crosses the l/f voltage noise spectrum. By Eq. (1) this statement means that
also wJ275
= a/a.
In the same way, we define the noise corner angular frequency o2 at which the white current noise spectrum crosses the l/f current noise spectrum 2nb,/wz
= b,
so that 02/2 rr = b/b The actual range of w, and w2 variability is between a few Hz and 1 MHz, according to the devices and the operating conditions. By inserting the noise comer time constants and the noise corner angular frequencies into Eq. (l), the noise spectral density can be rewritten as
I 031Tc2+ w,02Tc2+ I WI +o, N(w)=6
Fig. 2. Reduction N(o).
(2)
I WI
of the noise sources of the model in Fig. 1 to a single equivalent
noise current generator
of mathematical
spectral density
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3. Weighting function with no time constraints
3.1. Calculation of ENC’ Let us suppose to have an input signal Qg( t), with G(w) the Fourier transform of g(t), in the presence of a current noise of spectral density N( 0). It is well known [2] that the best possible signal-to-noise ratio is
Q2 =%
optimum
By definition, one
+4G(w)l' dw.
/ --5c N(w)
the equivalent noise charge squared (ENC’)
is the value of Q2 which makes the signal-to-noise
ratio equal to
1 ENC’ = 27~
’
+~IG(w)~’ I -x
N(w)
d”
In the actual case with a delta function as input signal (i.e. G(w) = 1) and the noise density that aC*n
Tr
ENC’=
+= /o
The normalisation wTc=x;
=
1
+m
N(w)
dw
and the statement
I a
IWI 1 Iw31+ w,0’+~IwI+--;0* 7c
of the angular frequencies W,Tc = k,;
ii given by Eq. (2), we have
(3) 1
do
%-
with respect to the noise time comer
w2rc = k, =
3
ENCO, = C& lead Eq. (3) to the form ENC’ -=ENCO&
1
7~ 2 /0
If y = y(k,, ENC’ ___=_
(4)
X
+= x3 +
k,x* +x + k,
dx
kz) is the real root of the cubic equation
71
x3 + k,x2 + x + k, = 0, Eq. (4) becomes
1 ,
X
dx x*+(YX+p)(x-y) where CI= k, + y and p = 1 + (k, + y)y.
-relation between k, and the parameter K of Ref. [I] is k, = 2K. ’ The ENCO, is the equivalent noise charge squared for voltage and current white noises only, processed function.
by the optimum
weighting
A. Geraci, E. Gatti / Nucl. Instr. and Meth. in Phys. Res. A 361 (1995) 277-289
280
20’
b
8.5 16.
7 7.5
12.’
6
6.5
k2
8. 4.5 4-
5
2.5 2
3
0-l 0
4
5.5
3.5 8
4 12
16
20
k2
1 kt
Fig. 3. (a) Contour plot of ENC*/ENCOi in the range 0 + 20 both for k, and k2 in linear scale. (b) Expanded view in the range in logarithmic scale, with iso-curve distance equal to 0.2.
Performing
0.1 + 10
the integration, we get
ENC’
1
n
ENCO,
ENC*
rr
ENCO:
2
-=_
1 for
o*/4>
/3,
(5)
where A= - r/(r2 + cry + p) and i = p/(7&. The contour plot of ENC’/ENCOz is shown in Fig. 3a in the range 0 + 36 both for k, and k,. The figure shows the increase of ENC*/ENCOz due to the voltage and current l/f noises. Fig. 3b shows an expanded view in the range 0.1 + 10 both for k, and k?. 3.2. Calculation of the optimum weighting function h(t) Because of the non-analytical form of N(w) in Eq. (2) due to the presence of the absolute value of w, it is easier to approach the evaluation of the optimum weighting function (WF) in the frequency domain. With the criterion to maximise the signal-to-noise ratio and applying Schwartz inequality, it can be demonstrated [4] that the optimum WF is given by H(jo) Performing
1 = N(o)
(6)
’
the inversion of the Fourier transform 1
H(o),
the WF h(t) is
-+m COS(d)
h(t) = zj_, N(o)
do
(7)
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a
-.
11
Fig. 4. (a) Shapes of the WF with no time constraints just with different values of the current noise component being the voltage noise k, fixed at 1. (b) Expanded view of negative tails of the WFs in Fig. 4a.
and normalising
frequencies +m
h ;
i
from 0 to 40,
to noise comer time constant as in Section 3.1, the WF becomes x COS(Xf/TC) x3+k$+x+k2
The integration
k, varying
gives the following
dx.
(8)
result
I
=h(i)=hi(?)+hj(?)+h,(?),
(9)
where
[ -ci( xi;) W) = (x;-xj;;xixk)
W)
= cxi -xj;;xi
-Xk)
cos( xii)
- sin( xi’)(si(
[ci( xi?) cos( x,S) + sin( xii)si(
xii)]
x,7) + 7~)]
arg(xi)
Iarg(xi)
= 7~,
1 < n,
(10)
and hj(a,la,(?) are similar but for indices rotation, being xi,xj,xk the three roots of the equation x3 + k, x2 fx + k, = 0. Because of the different notations used in literature, we point out that in Eq. (10) the integral sine (si) and cosine (ci) are defined as in Refs. [7] and [8]. Fig. 4a shows the effect of the new noise component on the shape of the WF: it causes the WF to increase slope at small t/q and to develop a negative tail (Fig. 4b) more and more evident when k, rises. It should be noted that the optimum WF is a zero area function, even with no time constraints, and for any small amount of current noise (k,). This is not always
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Fig. 5. Plot of WFs for different values of k, and kz.
evident from the figure but we must remember that, with no time constraints, contribute to a finite area. In fact the area of the WF is x
jo+?r(i)
di=j+rdij+Xx+k 0 Ir
even an infinitesimal
small negative tail can
cos( x?) xl+x+k 1
dx=j+=M(i) 7
dF=m(O)=O,
0
being M(t) the Fourier transform of the function m(x) = x/(x3 + k, x1 + x + k,). The WFs are shown in Fig. 5 for some values of k, and k, given as an example.
4. Weighting function with imposed finite width As just mentioned at the end of Section 3, the WF must be of zero area due to the physical fact that the new noise is a true l/f spectrum current, which requires a cut off of the continuous component in order not to diverge. First let us find a basis. We are looking for a WF h(t) which should be zero at t = 0 and t = 2TP and be symmetrical with respect to the peaking time t = Tp. A convenient representation in the interval 0 -+ 2T_, would be the conventional Fourier series N+
I
h(t) = c K, sin n=O
(2n + 1)n 2T
t,
P
in which case h(t) would have N + 2 degrees of freedom. But if we would like to impose on h(t) a zero area, when it is integrated between 0 and 2Tp, we should add the linear constraint EfziK,(2n + 1) = 0, which would reduce the degrees of freedom to N + 1. However, we prefer the expression (2n + 3)n 2T
(11)
P
which implies by itself the zero area constraint in the interval 0 + 2TP, as each term satisfies this condition. This series with N + 1 terms is constituted by N + 2 different sinusoidal components, but it has only N + 1 degrees of freedom, just as happened by the conventional Fourier series. We can conclude that the set t2n + 3)ll x,(t)
=sin
’
’
2Tp
2n+l
t-
-sin
(2n + 11-n I
2n+3
t
2TP
is not orthogonal but complete with respect to the ensemble of h(t) with the imposed constraints. So the WF constrained in the time domain to have area equal to zero and symmetry with respect to t = Tp, to be identically zero outside the time interval 0 + 2TP and continuous everywhere, can be expressed as n=0,1,2,...,
h(t)=l(t)zA,x,(t)+l(t-2T,)xA,x,(t-2Tp)
n
n
(12)
283
A. Geraci, E. Gatti/Nucl. Instr. and Meth. in Phys. Res. A 361 (1995) 277-289 where the second term forces h(t) to zero for t > 2TP. The sought WF is now represented A,,. Expanding Eq. (12) we obtain (2n + 3)lT 1 2T
2n+Lsin(2n+1)71 2nf3
P
sin (2f12i3)rr
+ l(t--2T,)xA, n
2Tp
r
by the numerable set of variables
1
(t - 2T,) - gsin
(2n2;1)m
P
(t - 2Tp)
P
l-
(13)
In order to calculate the signal-to-noise ratio, we begin to evaluate the output noise. The Laplace transform of Eq. (13) is
r
2nf3
--
H(s) = (1 + eUs2rp) CA,
(14)
”
With straightforward
p=
c&Z
but laborious calculation (see Appendix
A), we can write for the output noise
C CA,AJ,,, m
(15)
n
where A, and A, are the unknown amplitudes of the WF components and Z,, are the constant factors defined in Appendix A. Let’s find out now the squared expression for the amplitude of the signal at the output, that is the value of h’(t) for t = Tp multiplied by Q2 P(2fr + 3) S’=
Q’x
xA,A, m n
sin 2 n(2n sin
rr(2m + 3) sin
2
sin
that is, naming IV,, the term in brackets, s’=
2 x(2m
+ 3)
+ 1) 2
lr(2n sin
1’
+ 1) 2
742/n + 1) sin
2
(16)
Q”C CA,A,W,,. m n
(17)
Finally we can write the signal-to-noise
ratio as a function of the unknown amplitudes
Indicating with p the term in brackets, the squared equivalent
A,,
noise charge ENC’ looks like
ENC’ = ENCO&.
(19)
F
where the term l/w > 1 is the worsening’s factor due 1) to the constraint of the finite pulse duration and 2) to the presence of the voltage and current l/f noise and the consequent zero area of the WF. Let us introduce the ENCO:,, achievable with the optimum unipolar cusp filter of finite width 2Tp, when only voltage and current white noises are present [S] ENCO&
= ENCO, coth
5
(20)
( 7c i This will be used as a reference ENC’ = ENCO;rP f
to discriminate
between the two mentioned
causes of worsening.
The squared ENC results (21)
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A. Geraci, E. Gatti /Mud.
Instr. and Meth. in Phys. Rex A 361 (1995) 277-289
with
(22) In this way, the term l/CL) represents the worsening’s factor due to the cause 2) only. It is also useful to remember [S] that ENCO$rp.bip, the equivalent noise charge achievable with the optimum bipolar cusp filter of finite time duration 2TP for voltage and current white noises, results [5]
(23)
This represents the worsening’s noises only. Writing ENC’ = ENCOzrr.,A
factor due to the forced imposition
of zero area to the optimum finite time WF for white
(24)
3 P
with
(25)
the factor l/g’ is just the worsening’s factor due to the addition of the two components of the l/f noise. The ENC* and the WF are calculated looking for the A, which make stationary the signal-to-noise ratio (S/N)&
Considering obtained
a set of n A,,
~[Wmn-pZmn]A,=O m
a system of n + 1 homogeneous
m=O,l;..,n.
equations
in the 12+ 1 unknowns
A,, A,, . . . , A,, can be
(27)
The proper condition of homogeneous systems’ solution, that is the determinant of coefficients equal to zero, determines the eigenvalue p which substituted in Eq. (19) gives the ENC’ [6]. It is well known that the substitution of the eigenvalue in the system implies that the last equation becomes a linear combination of all the others. So that it can be replaced by the amplitude expression forced to be one in Tp. The resultant system is no more homogeneous and has a determinant different from zero and its solution gives the sought A, coefficients. This makes possible to calculate the WF. Iso-ENC’/ENCO&, curves are shown from Figs. 6a to 6c for decreasing pulse widths 2T,/r,: it can be realised the ENC2 increases when the width of the WF is reduced with a constant amount of voltage and current noise (k, and k,). The only due to the current noise (k,) becomes more and more less important increase of the (ENCZ-ENCOg, )/ENCO& decreasing the processing time d:ration 2Tp,is the input spectrum of the current noise is proportional to l/o. On the other side, the (ENS* - ENco&)/ENCO,~, only due to voltage noise (k,) tends to saturate to a constant value (of about ten times the ENCOzrD) when &e width of \he WF decreases, because this kind of noise depends on o.
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-__
20
16
11.5 II
10.5
10
b
a i9.5
9
12
12
8.5
8
kl
k) 8
7.5
I
6.5
5.5
5 3.5
8
4.5
4 4 8
k2
I2
I6
k2 I
20’
16.
C
._.
5.4 5.3
5.2 5.1 12.
5
~ -4.9 _
4.8
kl 8.
4.7
.4.6 ~ p---4.5
4-
~-- 4.4 4.3
O0
4
I 8
12
16
20
k2 Fig. 6. [so-ENC2/ENC0& for different 2Tt, /rc = 1; (c) 2Tp /T, = 0:25.
finite pulse processing
times and for values of the parameters
k, and k,: (a) 2Tp /rc = 4; (b)
Fig. 7. Plot of two WFs for different noise conditions: (a) k, = k, = 0 (only white noises); (b) k, = k, = 20. The shapes of the WFs are almost the same for different temporal pulse duration. The undulations are due to the finite number of terms (30) of the series development.
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A. Geraci, E. Gatti/Nucl.
Instr. and Meth. in Phys. Rex A 361 (1995) 277-289
II-I
I,
2T,lr,=co
0
4
a
8
12
16
U
20
4
8
kl
12
16
20
kr 2T, = 0.25 ~~
2T, = 5,
1
t0
4
8
12
16
20
0
_i 4
kl Fig. 8. Plot of ENC’/ENCO&
12
8 kl
as function of k, and k, = 0, i.e. unipolar WF, (continuous
lines) and k, + 0, i.e. zero area WF, (dashed
lines), for different values of thi pulse width 2TP/TV =m (a); 4 (b); 1 (cl; 0.25 cd).
The l/f current noise component is quantitatively less important than the l/f voltage one decreasing the pulse width: this is clear in Figs. 6a, 6b and 6c, where the iso-ENC’ curves (nearly straight lines) have low tending to zero slopes, showing a minor sensitivity of the ENC’ to k, with respect to k,.
Fig. 9. Plot of WFs for different values of the parameter number of terms (30) of the series development.
2Tp /TV with k, and k, fixed to 10. The present undulations
are due to the finite
A. Geraci, E. Gatti/Nucl.
Instr. and Meth. in Phys. Res. A 361 (1995) 277-289
287
Fig. 7 shows an example of the shape of the WF for two different noise situations. Let us point out the role of the zero area imposition in terms of the ENC’. In order to do that, the ENC’ has been calculated varying k,, both with k, tending to zero and with k, equal precisely to zero. For infinite time duration (Fig. S), the curves obtained are indistinguishable because for k, # 0 and k, = 0 they merge themselves with continuity when k, + 0 (for instance, when k, = 0 the WF becomes the conventional unipolar cusp which can be also considered as bipolar with zero area when k, + 0 is present). related to zero area WF and those related to On the contrary, there is a jump between the values of ENC’/ENCO& unipolar WF; this jump increases and saturates to 4 when the width of the tfme window is reduced, in agreement with Eq. (23). From Fig. 9 it can be realised that the three lobes WF departs in shape more and more from the unipolar cusp function when the total pulse width decreases. The WFs tend to saturate to a fiied three lobes shape for pulses shorter than a few 7c because, at these short processing times, their shape is mainly determined by the prevalent voltage noise contribution.
5. Conclusions This work gives the optimum filter in the presence of four noise components, white voltage and current and l/f voltage and current contributions. The shape of the optimum WF for different processing times and the corresponding values of the equivalent noise charge have been calculated: the worsening in resolution becomes relevant for processing times shorter than the noise time constant. In particular, the effect of the l/f current noise has been studied. Its real presence determines a jump in the ENC’ for finite processing times due to the zero area constraint necessary for convergence. The resulting WFs are cusp-like with a slope causing practical problems of ballistic deficit. The increase in the ENC*, due to a smoothing of the cusp, is easily evaluated by using the h(t) series development limited to a number of terms small enough: the resulting worsening is of about 4% in the ENC with six terms. For small values of k, and k,, the calculated WFs are those introduced by Deighton [5] in the case of white noises and constraint artificially imposed by zero area. For the sake of simplicity, only the three lobes symmetrical WF has been introduced in this paper dealing with finite response filters. Certainly, for continuity, two lobes asymmetrical filters would give lower ENC2 even in the presence of small l/f noise contribution, as it happens for Deighton’s two lobes filters. This will be dealt in a forthcoming paper.
Acknowledgements We thank Ing. G. Bertuccio for unpublished interesting for interesting suggestions. This work was supported by INFN and by MURST.
experimental
data on l/f
current noise and Prof. M. Sampietro
Appendix A In this case of finite width, we refer frequency and comer angular frequencies of T= to be consistent with notations of Ref. [l] 2T,o=x;
2Tpw,=K1;
to the total width of the pulse 2Z”, instead
2Tpw,=K2.4
However, in the main text we continue with the normalisation used for infinite pulse, which makes all the discussion and more understandable. In fact, K, and K, change when the pulse width is decreased. The input noise spectrum density becomes
a&
N(x)
= (2Tp$
I x3I +K,
(A.11 easier
x*+(++(T)~K~ 1x1
4 The relation between k, of the main text and the parameter K, of this Appendix is K, = k,(2T, /TJ
(A.21
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in Phys. Res. A 361 (1995) 277-289
The output power spectrum (s = jw) results IH(x)12=2[1
+cos(x)]
x
CA,A,
~~,,U,,[(2n+i)***-xZ][(2m+1)~~“-x*]
[ m.n
(
+ ~[(2n+3)*a*-xi][(2~+3)~~i-X2] 3n
3m
-~[(2n+~)2n’-*‘][(2~+3~~~‘-r’])
(2Q2 [ (2n + 3)*n* -X2]
x [ (2n + 1)2P* -x2]
[ (2m + l)*lra
-X’]
[ (2m + 3)*lr2 -X’]
where u,,
= (2m + l)n,
U,,, = (2n + l)n,
U,,,, = (2m + 3)n,
1 ’
(A.3)
U,, = (2n + 3)1r.
The total noise at the output is
@=
f~~+m~(x)lH(x)12 dx P
By inserting Eqs. (A.2) and (A.31 in Eq. (A.4) and performing be obtained
the integration,
the final expression
for the output noise can
where
/?‘(ln I,(a,
P)
a-ci
a)-a*(ln
p-ci
spy
In a-l 1a((Y, P)=(y4+3-4
/3*
Si ff
-
a’)
ci ff a=P,
a
ln(P/a)+ci LT(a, P) =
p) a+B,
=
a*-p2
a-ci
p afP,
The integral functions Si( a) and ci( a) are the notations stated in Ref. [7] and [8] and 6(
a, p)
is Kronecker symbol.
A. Geraci, E. Gatti/Nucl. Instr. and Meth. in Phys. Res. A 361 (1995) 277-289
Defining the Z,,,, factors
we get the final expression (15) of the main text for the output noise p.
References [l] [2] [3] [4] [5] [6] [7] [8]
E. Gatti, M. Sampietro and P.F. Manfredi, Nucl. Instr. and Meth. A 287 (1990) 515. E. Gatti, P. Manfredi, Nuovo Cimento 9 (1986) 38. V. Radeka, Annu. Rev. Nucl. Part. Sci. 38 (1988) 217. Ref. [2], p. 96. M.O. Deighton, Proc. Ispra Nuclear Electronics Symposium, 1969, EURATOM CID Brussels (1969) p. 47. W. Press et al., Numerical Recipes: The Art of Scientific Computing (Cambridge, 1986) Chap. 9. I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals Series and Products (Academic Press, 1992) p. 406, Sections 3.722-3.723 M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (Dover Publications, 1970) p. 232.
289