f noise

NUCLEAR INSTRUMENTS AND METHODS 130 (I975) 565--570;

© NORTH-HOLLAND PUBLISHING CO.

O P T I M U M FILTERING IN T H E P R E S E N C E OF D O M I N A N T 1/f NOISE* JORGE LLACER Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720, U.S.A. Received 16 September 1975 Optimum step response waveshapes for filtering in high resolution opto-feedback X-ray spectrometer systems in which 1If noise is dominant are calculated. The resulting cusps are shown to reduce the electronic noise line width of very good spectrometer systems by approximately 10% at long peaking times, lmplemen~Lationof such filters by transversal or by time variant methods is

discussed and, as an example, it is shown theoretically that for a system with 90.5 eV fwhm electronic noise as filtered with a 7th order pseudo-Gaussian waveshape at v0 = 35/~s, the corresponding noise line widths for the optimal cusp and for a particular time variant filter would be 82.5 and 75 eV fwhm respectively.

1. Introduction

o f o p t i m a l filter shaping for the case o f d o m i n a n t l / f noise, to show calculated quantitative c o m p a r i s o n s with the case o f p s e u d o - G a u s s i a n filters a n d to suggest ways o f realizing such o p t i m u m filtering.

I n recent years, it has b e c o m e evident t h a t highresolution pulse-light f e e d b a c k X - r a y s p e c t r o m e t e r systems t h a t generally o p e r a t e at long m e a s u r e m e n t times are limited in electronic noise p e r f o r m a n c e by 1 I f noise13). The ideal solution to the p r o b l e m o f better resolution w o u l d be to reduce the generation o f this type o f noise at the source, which resides, without d o u b t , in the F E T itself1). This pursuit m a y prove to be very hard, so it is worthwhile to see whether better filtering could be devised to reduce the d o m i n a n t l [ f noise. I n t e g r a t i o n o f noise whose p o w e r varies as l / f over a frequency b a n d ft to fh p r o d u c e s a total noise p o w e r p r o p o r t i o n a l to ln(fh/ft ) 4). This l o g a r i t h m i c dependence m a k e s the filtering o f l [ f noise quite insensitive to the shape o f the filter response. In fact, if one represents the c o n t r i b u t i o n to noise line width due to c u s t o m a r y sources t) as 2.35g NLW(fwhm) = - {qI~ ( N ~ ) + q

+ 2kTrtC~.+A~

(NI~>}~, (1) 2

it can be shown that the factor ( N ~ / y ) , which determines the p e r f o r m a n c e o f a filter for series lff" noise, changes only f r o m 7.54 to 6.43 in going from a simple R C - C R filter to a 7th o r d e r pseudo-Gaussian~). K o e m a n has calculated the variation in l / f noise contributions for two types o f idealized filters with different p a r a m e t e r s a n d his results also show only a small effect on l / f noise f r o m changes in filtering2). The p u r p o s e o f this p a p e r is to present calculations * This work was performed under the auspices of the United States Energy Research and Development Administration. 565

2. Optimum filter shapes A n o p t i m a l filter is defined as the one that maximizes the ratio o f signal p o w e r to noise p o w e r at the o u t p u t o f the filter at an a p p r o p r i a t e m e a s u r e m e n t time z0 on the signal. R a d e k a ' s description o f the ' m a t c h e d filter' concept is very clearS). It is shown that the signalto-noise ratio is m a x i m u m when the transfer function o f the filter H(o)) is given by H(,o) = k{S*(eg)/Wn(o3)}e

3.... ,

(2)

where k is a constant, S*(o3) is the complex conjugate o f the F o u r i e r t r a n s f o r m o f the signal waveshape and W,(og) is the p o w e r spectral density o f the noise. F o r the simple case o f only series and parallel noise c o m p o n e n t s , evaluation o f H(og) leads to a filter with a step response given by an exponential cusp. I t is well k n o w n t h a t p s e u d o - G a u s s i a n , t r i a n g u l a r or trapezoidal step responses a p p r o x i m a t e the o p t i m u m noise p e r f o r m a n c e o f a cusp within a few percent a n d are much easier to implement. In the presence o f strong l [ f noise, evaluation o f eq. (2) can best be carried out by numerical m e t h o d s using a F a s t F o u r i e r T r a n s f o r m ( F F T ) algorithm. F o r a signal o f the f o r m s ( t ) = e -~t for t > 0 , in the limit o f very small et ( a p p r o a c h i n g a pure step function), S(~o) a p p r o a c h e s 1/rio. The p o w e r spectral density o f the noise is given 1) b y Wn(f) - - qI'L + A-,- + 2 k T r ; C ~ , , ( 2 ~ f ) 2 lfl

--oo < f <

0%

(3)

566

J. LLACER

and the transfer function of the optimum filter becomes, from eq. (2)

J _ _ L {q l'L

H(oa) =

__A,+zkTr,C~2 ,

r

)'

e -~"'°. (4)

2~J'\(2=J') 2 + Ifl The inverse Fourier transform of H(~o) can be obtained from eq. (4) by F F T without difficulty provided that enough points are used (2048 for the results reported here) extending high enough in frequency. The impulse response function thus obtained is found to contain a residual imaginary part which is 102-103 times smaller than the real part at any point in the calculation. The integral of the impulse response function always yields a cusp-like step response function the exact shape depending on the relative magnitude of the three noise components. Fig. la shows the step response of a filter calculated to match the noise i

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a) STEP RESPONSE OF FILTER

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Fig. 1. (a) Step response o f an optimal matched filter for the noise parameters o f an excellent X-ray spectrometer front end, c o m p a r e d to a p s e u d o - G a u s s i a n o f the 7th order. M e a s u r e m e n t time ~0 chosen arbitrarily to be 64/~s. i

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The evaluation of the factors ( N if) and

1

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3. Noise performance

FILTER FOR

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PSEUO0 GAUSS"~,.:,

spectrum of a X-ray spectrometer system with noise source parameters as low as we have seen in our laboratory'), along with the step response of a pseudo-Gaussian filter of order n = 7 for which f ( t ) = (t/%)" exp [n (1 - t/to)]. A measurement (i.e. peaking) time ro = 64 its was chosen for both waveforms. Fig. l b shows the corresponding impulse responses. Fig. 1 and all subsequent graphs are normalized to unity step response at t = %. The effect of varying the 1/'fnoise component on the shape of the optimum cusp is shown in fig. 2. The I J ' s o u r c e parameter A~ ranges from 1 x 10 -4° (negligible l/f) to 2 x 10 -37 (dominant at % = 64 ps) in these curves.

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MATCHED FILTER STEP RESPONSES FOR

1.0

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(5)

where H(e)) is the transfer function of the filter. The evaluation of eq. (5) for cusp-like step responses presents no serious problems but care must be exercised at low frequencies where the integral changes rapidly. Enough points must be taken and df should be small enough to ensure good accuracy in the integration. Fig. 3a shows IH(~o)[ 2 for a pseudo-Gaussian, n = 7, and for the matched filter for the conditions of fig. 1, r0 = 64/~s. The conditions of the calculation were A t = 2.5x 10 -6 s, 2048 points, and a resulting A f =

b) IMPULSE RESPONSE OF FILTER 5×10 '

If I-' iH(~)12d/;

Aa=2~ 10 37

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t(psec)

t(psecl

Fig.

I.

(b) Corresponding impulse responses.

Fig. 2, Matched filter step responses showing effect of I / f n o i s e on o p t i m u m waveshapes.

OPTIMUM FILTERING

1.563 × 10 3 Hz. The complete integrand of eq. (5) is plotted in fig. 3b and the areas obtained by integration using a four-point routine are also shown. These areas are the corresponding values of (N2t/s), from

eq. (5). It is quite evident from fig. 3 that the matched cusp is a superior shape for filtering l / f dominant noise, compared to a pseudo-Gaussian. This is due to the fact that the cusp has a bandpass that extends to higher frequency at the expense of low frequency response, where, of course, there is more l / f noise. The expected improvement in noise line width by the use of a matched cusp, taking into consideration the other two noise sources, can be investigated by evaluating eq. (1) for a pseudo-Gaussian, n = 7, at different peaking times ~o under a significant set of noise conditions, and comparing it to the cusp optimized for those noise conditions and peaking time. Fig. 4 shows noise line width (NLW) vs measurement time ~o for the parameters o f the opto-feedback system already used in fig. 1. The noise filter factors of I

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567

eq. (1) are also shown in the figure. It is interesting to note that as the measurement time ~o is increased the optimum cusp adjusts itself to a shape such that the parallel noise factor ( N ~ ) and the 1If factor (N2//) decrease in magnitude, while the series factor ( N ~ ) increases. For the conditions described, it becomes clear that at long ro the percentage improvement in N L W by using a matched cusp is quite significant. At r o = 64/as, for example, more than a 10% benefit is obtained. 4. Suggested realizations The implementation of a cusp as a filter response has not been very appealing in the past because of the difficulty encountered in the filter design. Recently Koeman2), using digital techniques, has succeeded in realizing a transversal filter with response quite close to a cusp which he calculates should give him a substantial improvement in l If performance. The published tests of his system were unfortunately carried out using a detector-preamplifier combination of rather poor quality. His results are therefore not as convincing as might be hoped. More recently, Miller and Robinson 7) have discussed the realization of a transversal filter using either capacitively tapped delay lines or charge transfer devices and have demonstrated the use of the former to create a trapezoidal response although the base width was limited to a few microseconds by the characteristics of commercially available delay lines. Transversal filter methods can be expected to provide a basic tool for the implementation of analog filtering with arbitrary response waveforms, although research will be needed

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64

128

q'o (psec)

Fig. 4. Calculated noise line widths vs measurement time r0 for noise source parameters o f fig. 1 for optimal cusps and pseudoGaussian n = 7. N o i s e filter factors are also shown.

568

J. LLACER significantly affect filter noise characteristics at long

TABLE 1 R e s u l t s o f c a l c u l a t i o n s to s t u d y the effect o f a flat t o p o n the noise parameters.

r0

Convolution rectangle

(/ts)

width

:N~)

< N a2,>

( N *z I f

NLW (fwhm) (eV)

(/is) 32

0 1.10 2.03 5.15

0.451 0.460 0.475 0.523

2.428 2.361 2.335 2.275

4.519 4.598 4.721 5.086

82.5 82.2 82.4 83.1

64

0 1.25 2.18 5.31

0.379 0.380 0.390 0.412

2.864 2.753 2.723 2.625

4.208 4.212 4.310 4.556

70.7 70.15 70.4 70.9

"C0 .

A third alternative for the implementation of a filter with good l/f characteristics is strongly suggested by observing the step response of the time variant filter used by Kandiah et al. 3) in their pulse processor for X-ray spectrometry. F r o m the point o f view of response to a signal, the filter consists of a simple integrating time constant ~1 followed by a gated integrator with high input impedance which is active for a measurement time %. Switch S, fig. 5, is opened at the beginning o f measurement time, t = 0, and closed again at t = %. The step response o f the filter is then of the form

V(t) =

I - exp(-t'/rl)d¢' o

0 1.25 2.50 5.62

128

0.308 0.302 0.309 0.328

3.671 3.502 3.424 3.285

3.903 3.824 3.910 4.098

63.3 62.4 62.5 63.1

to develop those ideas particularly for long measurement times %. The fact that a cusp has a sharp point makes its practical realization impossible; furthermore, it is desirable to have a somewhat flat top to allow for possible differences in detector charge col lecting times s). In order to study the effect of a flat top on noise parameters, the optimum cusp for the system of fig. 1 has been convolved with rectangles of approximately 1, 2 and 5 [~s and the noise parameters for the resulting combinations have been recalculated. Table 1 shows the results, taking the noise source parameters o f fig. 1 for the N L W calculations. It is apparent that broadening the top by a few microseconds does not i

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= t-r1 [1- exp(-t/rl)],

for 0 < t < r o. (6)

Fig. 5 shows V(t) for r o = 32/~s, ro/rl = 1.33, as used in ref. 3. The optimum cusp for % = 32/~s and for the noise parameters of the system of fig. I is also shown for comparison. The similarity of the two waveforms for t < ro is quite evident. From the point of view of noise performance, with switch S normally closed before the arrival of a recognized signal, capacitor C2 forms a differentiator o f essentially zero time constant. When switch S opens at signal arrival time, t = 0, the input of the high input impedance gated integrator will follow the residual effects of noise disturbances which occurred at t < 0 only to the extent that they change with time after t = 0. This is treated in detail in the appendix. The calculation of noise performance o f such a filter can then be carried out in time domain for ( N ~ ) and ( N ~ ) by the methods described in ref. 6, and for (N~/j.) by following the method indicated by Koeman in ref. 2. The appendix shows the procedures followed to obtain the values of the above noise filter factors. The results for ro = 32 Its, rl = z0/1.33 are shown in fig. 5. Although the noise filter factor for parallel noise, ( sN) ,2 is larger than that of the optimum cusp, that source of noise is usually not a limiting factor in highquality opto-feedback systems for X-ray spectrometry. The improved ~N 2) and (N2/y) over the cusp values are, however, quite important in determining the operation of a system. At % = 32 its, the expected N L W from the noise source parameters of fig. 1, calculated from eq. (1) is 75 eV fwhm, compared with 82.5 for the optimum cusp and with 90.5 for the 7th order Gaussian. The contributions to N L W from 1If noise are 50.3, 53.0 and 63.3 eV fwhm respectively.

569

OPTIMUM FILTERING

5. Conclusion It has been shown theoretically that the electronic noise p e r f o r m a n c e o f excellent o p t o - f e e d b a c k X - r a y fluorescence systems can be further i m p r o v e d by 8-10 eV f w h m by tailoring a cusp to the specific noise p a r a m e t e r s o f the system. Transversal filtering, digital or analog, can provide the tools for filter realization. In addition, it has been shown t h a t suitable gated integration filters can p r o v i d e even better filtering than time invariant cusp for high quality spectrometers. The a u t h o r is i n d e b t e d to F. S. G o u l d i n g and D. A. L a n d i s for their illuminating c o m m e n t s , a n d to V. P a d e k a , G. L. Miller a n d K. K a n d i a h for useful discussions. Appendix

Calculations of noise .filter factors Jot gated integrator case. The calculations o f ( N ~2>, (WE} a n d (N2/y} for the gated integration circuit o f ref. 3 have been carried out in time d o m a i n following the m e t h o d s indicated in refs. 6 a n d 2. The calculations for each p a r a m e t e r are divided into two parts: (1) c o n t r i b u t i o n s f r o m noise disturbances which occurred before the o p e n i n g o f switch S, fig. 5, i.e., for t~ < 0 , and (2) for disturbances which occurred at 0 < t~ < To. In all cases, functions r(t~) are defined which give the residual effect at t = r0 f r o m noise disturbances which occurred at t = t 2. Then, the c o n t r i b u t i o n s to a noise filter factor ( N 2} are o b t a i n e d by (N2> =f

[r(tl)]Zdtl/[f(z°)]

2,

(7)

-~ to 0 results in a noise factor c o n t r i b u t i o n o f 1T 1 • F o r a noise step occurring at 0 < t 1 _< Zo, the filter response at t = To is given by

r(tl) =

dt{l -

tl

exp[-(t-tl)/zl] }

= ( T o - T O - Zl { 1 - exp [ - ( T o -

t t ) / r , ] }.

(10)

S u b s t i t u t i o n into eq. (7) a n d integration between t~ = 0 to To gives a c o n t r i b u t i o n o f S±T3 (3 0

T1,/72+ Zl2 To [1 - 2 e x p ( - z o / Z 0 -

+ ½t~ [1 -- exp ( -- 2Zo/tO]}/[f(to)]

]+

2.

F o r Zo/'rl = 1.333 a n d t 0 = 32 ps, we o b t a i n f r o m the two contributions, respectively, (N2> = 1.2 × 10 -5 +0.728 × 10 -5 = 1.928 × 10 -5 a n d therefore, ( N 2 > / r o = 0.602. Series noise factor. F o r a noise impulse generated at - ~ < t 2 < 0 the response o f the filter at time To is given by

r(t,) = I [to d t { e x p [ - ( t - t , ) / r 2 ] - V o } ,

(11)

272 ,Jo

where Vo is the voltage at which c a p a c i t o r C2 is charged at t = 0 for an impulse which occurred at t = t2, a n d is given by exp (tl/~1). Integrating a n d substituting into eq. (7) gives a c o n t r i b u t i o n o f ( 2 r l ) -a. F o r a noise impulse generated at 0 < tl < To, the filter response at To is

r(tO = 1 t TM d t e x p [ - ( t - t l ) / z l ] "c2 ,/o

with suitable limits o f integration d e p e n d i n g on which range o f t~ is considered. The n o r m a l i z i n g function f ( % ) is o b t a i n a b l e f r o m eq. (6) and c o r r e s p o n d s to the filter response at To for a signal step at t = 0 .

= 1- exp[-

(~'0--tl)/271].

(12)

Substitution into eq. (7) gives a c o n t r i b u t i o n o f z o + r 2 {2 exp ( - Zo/Z 0 - ½ exp ( - 2 To/Z1)- ~z}/[f(Zo)] 2.

Parallel factor. F o r a noise step occurring at -- ~ < t2 < 0 the filter response at t = To is given by

r(t2) =

fo

dt {(1 -

exp[-(t-tl)/Z2] ) - 1%},

(8)

0

where Vo is the voltage at which c a p a c i t o r C2 is charged at t = 0 from the noise step at t l , which is Vo = l - e x p (t~/zj, with t~ < 0 . I n t e g r a t i o n o f eq. (8) leads to r ( t 0 = exp (t I/z2) f(To).

(9)

Substitution into eq. (7) a n d integration between t I =

F o r ro/Z~ = 1.333 a n d To = 32 gs, we o b t a i n f r o m the two c o n t r i b u t i o n s ( N J > = 2.083 x 104+3.81 x 104 = 5.89 x 104 o r (N~> To = 1.88.

I/f parameter. Impulses generated at t l are assumed passed t h r o u g h a filter with frequency response H(f) = (jf)-~, - ~
570

J. LLACER

the o u t p u t o f the gated integrator at t = ro is given by

T1 ,,]0

X [exp(--'c/~cl)(t--ll--r)-~]]-

k o,

(.13) References

where the voltage across C2 at t = r0 is given by -f1

C

1% = (V/2/~t)

drexp(-r/'zl)(-tr-r)

~.

L, 0

F o r an impulse generated at 0 _< t 1 < %, the response is

ritl ) = x/2 ~°dt tl

r = t - t 1 the integral does exist and the calculations can be carried o u t numerically with g o o d convergence. The o b t a i n e d results are, respectively, 0.58 a n d 3.47, with an estimated error o f less t h a n 1% in each figure. We have then (N2/I) = 4.05+0.04.

dt[exp(_z/Zl)(t_tl_r)-,~]. 0

(14) Substitution into eq. (7) yields the two c o n t r i b u t i o n s to (N~/y). A l t h o u g h the integrand has a singularity at

1) j. Llacer, Accurate m e a s u r e m e n t o f noise parameters in ultralow noise opto-feedback spectrometer systems, L B L Report no. 3671, to appear in Proc. 2rid I S P R A Nuclear Electronics Symp., M a y 1975. ~) H. K o e m a n , Nucl. Instr. and Meth. 123 (1975) 161, 169 and 181. ~) K. K a n d i a h , A. J. Smith a n d G. White, A pulse processor for X-ray spectrometry with Si(Li) detectors, to appear in Proc. 2nd I S P R A Nuclear Electronics Syrup., May 1975. 4) V. Radeka, IEEE Trans. Nucl. Sci. NS-16, no. 5 (1969) 17. ,3) V. Radeka, Nucl. Instr. and Meth. 52 (1967) 86. ~) F. S. Goulding, Nucl. Instr. and Meth. 100 (1972) 493. 7) G. L. Miller and D. A. H. Robinson, Transversal filters for pulse spectroscopy, to appear in Proc. 2nd I S P R A Nuclear Electronics Syrup., May 1975. s) V. Radeka, IEEE Trans. Nucl. Sci. NS-19 (1972) 412.