Multiple read-out of signals in presence of arbitrary noises Optimum filters

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1 Nuclear Instruments and Methods in Physics Research A 417 (1998) Multiple read-out of signals in presence of arbitrary noises Optimum filters Emilio Gatti*, Angelo Geraci, Chiara Guazzoni Politecnico di Milano, Dipartimento di Elettronica e Informazione, Piazza Leonardo da Vinci, 32, Milano 20133, Italy Received 14 May 1998 Abstract We present a new method for optimizing the multiple read-out technique of the signal charge detected by a CCD with floating gate output amplifier. The procedure consists in calculating the optimum filter with assigned time domain constraints (e.g. arbitrary finite duration), and in the presence of any kind of uncorrelated, stationary, additional noises. The method is fully developed and applied in practical designing conditions. It can be easily translated into a computer software, which can be used as a tool for optimizing a digital signal processing set-up in its digital filter section Elsevier Science B.V. All rights reserved. Keywords: Optimum filters, Multiple read-out; CCD 1. Introduction As early as 1972 a Charge-Coupled Device (CCD) was proposed to be used on the Hubble Space Telescope [1]. Only more recently, CCDs have been proposed in the field of X-ray astronomy [2] and the three major X-ray observatories planned for launch in the coming years (XMM, AXAFS and JET-X) will have CCD cameras on board. As in the X-ray detection the CCD is usually operated in the single photon counting mode, in order to detect individual X-rays, the requirement of an accurate read-out of the signal charge is essential to * Corresponding author. Tel.: # ; fax: # ; egatti@elet.polimi.it. be able to detect X-rays even in the low energy scale. To fulfill these requirements it is necessary to improve the energy resolution and the quantum efficiency. The quantum efficiency is limited by the absorption in the electrodes for front illuminated devices, by the dead layer in case of back illumination and, in general, by the extension of the depleted volume. In the case of pn-ccd [3] the wafer full depletion allows back illumination and high photon efficiency. The energy resolution is limited by the charge transfer efficiency, by the charge collection efficiency (for back illuminated devices) and by the read-noise. Let us focus on the readnoise, neglecting the other two limiting factors, well discussed in the literature [4,5]. The read-noise is dominated by the noise of the output preamplifier. The two most common forms of CCD output /98/$ Elsevier Science B.V. All rights reserved. PII: S ( 9 8 )

2 E. Gatti et al./nucl. Instr. and Meth. in Phys. Res. A 417 (1998) circuits are floating diffusion (FDA) [6] and floating gate (FGA) amplifiers [7]. In the case of the FDA the signal charge is transferred through the device and dumped directly on the output node (the floating diffusion). The CCD output node is connected to the gate of a FET operated as a source follower. In order to be read, the information brought by the signal charge adds to the charge of the input FET and is no more available. Moreover, before each pixel is read the voltage of the output node must be reset to a reference voltage. The uncertainty in the reset voltage is an additional source of noise (reset noise) [8]. On the contrary, in the case of the FGA the signal charge can be readout multiple times. The CCD output node is not directly connected to the channel where the signal charge is transferred, hence the signal charge is only capacitively coupled to the output node. By clocking multiple times the signal charge underneath the output node a periodic current signal is induced at the output electrode. After the multiple read-out is completed a real CCD output node, that can be directly connected to the transfer channel, clears the already processed signal charge. With this technique it is possible to overcome the 1/f limit and to eliminate the q» C reset noise, introduced by the reset FET. The read-noise of the FGA will be slightly higher than the FDA for a single read-out, because the output node of the FGA cannot be made arbitrarily small, because it must create a potential well underneath itself large enough to hold the desired maximum charge and, consequently, its capacitance will be higher than the one of the collecting anode of a FDA. For multiple read-outs per charge packet, however, significantly higher resolution can be achieved with the FGA. Kraft et al. [9] reached sub-electron read-noise performance (0.9 electron r.m.s.) by applying a floating gate output amplifier with 16 read-out per pixel to a CCD. In this paper we present a detailed analysis of the effect of the multiple read-out technique on the different noise contributions in order to understand the optimum resolution that can be achieved. The first time implemented method to derive the optimum signal response in the presence of different noise sources in the case of the multiple read-out technique is developed in Section 2. Section 3 addresses the description of the effects of the multiple read-out on the different noise contributions and, in general, on the achievable resolution compared to the optimum processing of a collected delta pulse. Finally, in Section 4 the effects of the real shape of the signal on the achievable resolutions are discussed. 2. Description of the method We have recently introduced a method for synthesizing the optimum time-limited weighting function (WF) in the presence of noise of arbitrary power density and for input signals of arbitrary shape [10,11]. We look for the WF for the optimum measurement of the signal i(t) shaped by the induced charge function in a CCD whose output pulse is induced by a cloud of electrons periodically presented to a sensing output electrode. It is well known that the optimum WF for an i(t) input is obtained by minimizing the noise-to-signal ratio: N S " N(ω)I[WF(t)] dω i(t)wf(t)dt. (1) The only constraints imposed a priori are a finite time duration ¹ of the WF and its zero value at the extremes of this interval and outside it. Other constraints suitable to cope with particular experimental requirements may also be added. The synthesis algorithm represents the sought WF as a truncated Fourier sine series composed of symmetrical and antisymmetrical harmonics. π A sin WF(t)" n t ¹ 0(t(¹, (2) 0 elsewhere. The optimum WF is univocally defined by the coefficients A, whose full calculating procedure is dealt with in Ref. [10] for δ-like input pulses, and in Ref. [11] for input signals of arbitrary shape. Let us assume that the cloud of electrons Q is presented to the sensing output electrode with a law of motion suitable to give an induced charge

3 344 E. Gatti et al./nucl. Instr. and Meth. in Phys. Res. A 417 (1998) function of the type: Q" Q 2 1!cos 2π ¹ t, t"i¹ 2 j¹ (3) and zero elsewhere. The measurement time interval is taken from 0 to ¹ "p¹ (0(i(j(p integers), where ¹ is the time duration of each oscil- lation. The induction is not complete, resulting in a reduction of the amplitude of the signal to be measured. The induction can be however maximized by a careful design of the geometry of the electrodes and by a proper choice of the biasing voltages in order to displace the potential well in which the electrons are stored as close as possible to the sensing output electrode. The corresponding output current of the device is i(t)" dq dt "Q π 2π sin t (4) ¹ ¹. Note that the total induced charge is 0 for t'j¹, when the charge is definitely moved away from the sensing electrode: this is the consequence of never collecting the inducing charge. The denominator of Eq. (1) contains the shape of the input waveform, projected over the WF sinusoidal basis N S " Q A ( N(ω)I[WF(t)] dω π 2π sin t ¹ ¹ sin n π t ¹ dt. By equating to one the expression in brackets in the denominator of Eq. (5), we obtain an expression that can be introduced in the quoted algorithm [10] as a Lagrange constraint, instead of the classical requirement of the WF being unity in correspondence with the peaking time. In order to obtain a parameter equivalent to the classical equivalent noise charge (ENC), we write Eq. (5) as Q N S " N(ω)I[WF(t)] dω, (6) (5) with the associated equation A π 2π sin ¹ ¹ t sin n π t dt!1"0, ¹ to be used as a Lagrange constraint. If we impose (N/S) to be equal to one, Eq. (6) becomes ENC" N(ω)I[WF(t)] dω, (8) and, consequently, (ENC ) equals (Q ) similarly to the case of the δ-like pulse. In fact, minim- izing the functional (8) with the constraint (7) we get the optimum WF so normalized, as to give to Q the significance of indicating by Eq. (3) the amplitude to be given to the waveform i(t) expressed by Eq. (4), in order to get the signal-tonoise ratio equal to 1. The assumption that the signal induced on the read-out electrode by the periodic oscillation of the signal charge is sinusoidal, is just a useful approximation. (In some real devices the signal could be better approximated by a series of two delta pulses of opposite areas (see Section 4).) We will indicate with q"j!i the number of signal oscillations during ¹ and with m "i the number of waiting periods before the q signal oscillations, i.e. the time in which the signal pulse is not present and the shape of the WF is determined only by the actual noise. p is the total number of periods in the measurement time (p"¹ /¹ ), including the empty m "p!j periods following the signal oscillations ( lagging periods ). We can write explicitly the projections of the output current signal over the WF sinusoidal basis. The projection over the iso-frequency base term results in ¹, (9) while the other projections are given by 2 p¹ sin π m p n cos π q p n #cos π m p n sin π q p n!sin πm π(n!4p) (7) n p. (10)

4 3. Effect of the different noise sources on the optimum filtering 3.1. Particular cases E. Gatti et al./nucl. Instr. and Meth. in Phys. Res. A 417 (1998) Let us take into account some particular cases in order to understand the influence of the different noise sources on the shape of the optimum filter and the effect of the multiple read-out technique on the reduction of the different noise contributions. We consider a realistic case for a multiple read-out system that could be applied to a high-resolution X-ray spectroscopy setup. In particular, part of the front-end electronics is assumed on-chip, hence the resulting total input capacitance is only C "0.5pF. Assuming a FET transistor as the first element of the signal preamplifier, the white voltage noise is related to the channel thermal noise of the first transistor of the signal preamplifier, while the white current noise is due to the leakage current of the gate of the transistor. For all the foregoing analysis (if not specified differently) we estimate the following mathematical spectral densities: white voltage noise a "1.510 V/Hz white current noise b "1.610 A/Hz that can be considered as typical values for on-chip electronics and devices operated well below roomtemperature. Other noise contributions, e.g. 1/f voltage noise, will be also considered. The noise corner time constant for the white noises is defined as τ "C a b "1.53 μs. (11) In Fig. 1 the optimum weight functions in the presence of either white current noise or 1/f voltage noise are shown. The optimum processing of the white current noise imposes as weight function a replica of the signal itself. Its spectrum has a dominant component at the frequency corresponding to the repetition rate of the signal. When no waiting period is left before and after the processing of the signal, the shape of the weight function is the same in presence of either white current noise or white voltage noise and the action of the Fig. 1. Optimum weight function in the presence of either white noise or 1/f voltage noise. The time duration of each oscillation is ¹ "1 μs. No periods have been left before and after the processing of the signal oscillations. WF is an averaging. When the 1/f voltage noise is present, the WF weights differently the signal oscillation with weights for the first and the last lobe of the signal oscillations greater than for the others. When a waiting and a lagging period are left, the shape of the optimum WF during the waiting and the lagging time is determined by the noise spectral densities that are present. When only the white current noise is present the WF is a replica of the signal itself, while when only the white voltage noise is present, the optimum WF can be seen as the sum of a saw-tooth wave whose maximum and minimum are centered, respectively, on the maximum of the first signal oscillation and on the minimum of the last signal oscillation and a second component at the frequency of the signal itself (see for instance the inset of Fig. 2). For the voltage noise case, the shape of the filter during the waiting and lagging time is close to the finite straight line (optimum processing of a delta pulse in presence of white voltage noise). The choice of the number of virtual periods before and after the effective signal processing affects the achievable energy resolution. We verified that, as expected, the minimum ENC always occurred

5 346 E. Gatti et al./nucl. Instr. and Meth. in Phys. Res. A 417 (1998) Fig. 2. ENC normalized to the ENC achievable with a delta pulse optimally processed in the same measurement time and in the presence of the same noise spectral densities as a function of the number of periods left before the processing of the signal with the constraint of symmetry on the input signal for different time duration of each oscillation (¹ "1 μs, ¹ "2 μs, ¹ "5 μs) and corresponding total measurement time imposed (¹ "20 μs, ¹ "40 μs, ¹ "100 μs). The upper X-axis reports the number of signal oscillations in correspondence with the number of waiting periods. The inset shows the WF for the case ¹ "1 μs and m "m "1. for equal number of virtual periods to be left before and after the processing time of the signal. For these symmetric cases the study has been carried out to evaluate which is the ratio between waiting and lagging periods and signal oscillations that gives the best resolution. In Fig. 2 the results of the simulations for different time durations ¹ of the signal oscillations are shown. The imposed total number of periods is p"20. As it can be seen, the white current noise contribution increases as the number of signal oscillations decreases because of the reduction in the number of averages. The behavior of the white voltage noise contribution depends on the ratio between the time duration of each oscillation ¹ and the noise corner time constant τ. When ¹ is shorter than τ, adding a virtual oscillation gives a slight improvement in the achievable resolution because the weight function can adjust its shape in order to optimally process the amount of voltage noise. On the other hand, when ¹ is longer than τ the achievable resolution worsens as the number of signal oscillations is reduced. Let us now evaluate the effect of the multiple read-out of the signal charge on the different noise contributions. For sake of simplicity, we disentangle the current white, voltage white and 1/f voltage noise contributions and calculate the optimum waiting function and the ENC as if only one of these noise components were present. We impose the waiting and lagging period (m "m ), in order to obtain the optimum resolution. The time duration of each oscillation is imposed (¹ "1 μs and ¹ "5 μs) and, consequently, the total measurement time ¹ increases as the number of signal oscillations is increased. The effect of the multiple read-out on the white current noise can be easily understood as an averaging; hence the reduction in the ENC is inversely proportional to the number of averaging, independently of the number of virtual periods and on the oscillation time. In Fig. 3a the effective reduction of the noise is shown. The best fit gives exactly the foreseen behavior. When we take into account only the contribution of the voltage noise, the improvement in the resolution is given by the increase in the total measurement time and not by the increase in the number of oscillations (see later for the case with ¹ constant). When no virtual periods are left, the ENC is inversely proportional to q, which means that it is inversely proportional to the total measurement time ¹ as in the case of the conventional processing of a delta pulse. Moreover, the optimum processing of a delta pulse in the same measurement time would result in an ENC lower by a factor two. This fact can be easily understood by comparing the ENC calculated for a delta-like current pulse and for the sinusoidal current pulse (Eq. (4)), in presence of the white voltage noise. When one virtual period is left before and after the processing of the signal the effect of the multiple read-out on the reduction of the noise is less significant. The contribution of the white voltage noise decreases approximately as 1/q, as it is shown in Fig. 3b. For a number of signal oscillations great enough (q'5), the ENC is still inversely proportional to ¹ (equal to (q#2)¹ ). The situation is different when we impose the duration of the measurement time. We now focus

6 E. Gatti et al./nucl. Instr. and Meth. in Phys. Res. A 417 (1998) Fig. 3. ENC as a function of the number of signal oscillations. The time duration of each oscillation is ¹ "1 μs (z) and m "1 and ¹ "5 μs () and m "0. (a) Only white current noise is present and (b) only white voltage noise is present. The resolution achievable with a delta pulse optimally processed in the same measurement time and in the presence of the same white voltage noise spectral density is shown for comparison. Note that ENC for a delta pulse optimally processed is proportional to 1/¹ for both the cases (,). The bending in the curve with circular symbols is due to the plotting versus the number of signal oscillations. on the effects of the multiple read-out technique, neglecting the effects of the increase of the WF time duration. In the case of the optimum processing of a delta pulse, as well known, the longer is the time duration of the WF the lower will be the contribution of the voltage noise and the higher the contri- Fig. 4. ENC as a function of the number of signal oscillations. The total measurement time imposed is equal to ¹ "25 μs. The cases with m "0() and m "1() are shown. (a) Only white current noise is present and (b) only white voltage noise is present. The resolution achievable with a delta pulse optimally processed in the same measurement time and in the presence of the same white noise spectral density is shown for comparison. The insets show examples of the WF for three different cases. bution of the white current noise. When no virtual period is left and in presence of current noise contribution, Fig. 4a shows that the ENC decreases faster and is inversely proportional to the square of the number of signal oscillations. When a waiting and a lagging period is imposed, the decrease in the ENC can be well approximated by 1/q and approaches 1/q for a great number of signal oscillations. In both the cases the decrease is more

7 348 E. Gatti et al./nucl. Instr. and Meth. in Phys. Res. A 417 (1998) significant than in the case of Fig. 3a. By analogy with what happens while processing a delta pulse, we can attribute this decrease to the absence of worsening due to the increase in the measurement time. The behavior of the contribution of the voltage noise is shown in Fig. 4b. When no virtual period is left before and after the processing of the signal, the ENC equals twice the ENC achievable by the optimum processing of a delta pulse (constant because the total measurement time is constant). When a waiting and a lagging period is left, with the first three oscillations there is a great reduction in the voltage noise related to the increase of the time of presence of the signal (equal to q¹ /q#2). With a further increase in the number of oscillations this time becomes proportional to ¹ and the ENC approaches twice the value achievable in the same measurement time by optimally processing a delta pulse. Let us now discuss the effect of the multiple read-out on the 1/f voltage noise. This contribution is very important from an experimental point of view. In fact, when the noise obtained in reading the signal charge is white, it can be reduced also with classical charge-collecting read-out techniques by increasing the measuring time ¹ and/or by lowering the temperature. However, when the white noise is much reduced, then the 1/f voltage noise begins to dominate and its contribution cannot be reduced in the case of the classical chargecollecting read-out techniques. In fact, the contribution of the 1/f noise is independent of the measurement time and is not reduced by cooling down the detection system and the read-out electronics. Concerning the multiple read-out technique, we have verified that the achieved ENC is independent of the oscillation time in the presence of only the 1/f voltage noise and imposed number of oscillations. If the number of signal oscillations and the total measurement time are imposed, the best resolution is achieved by reducing the oscillation time in order to leave a virtual period before and after the processing of the signal. However, if it is possible to reduce the time duration of each signal oscillation, the best resolution is achieved by maximizing the number of signal oscillations and hence for p"q. Fig. 5 shows the very effective reduction of the 1/f noise by the multiple read-out with no Fig. 5. ENC as a function of the number of signal oscillations in the presence of only 1/f voltage noise. No virtual periods have been left before and after the processing of the signal. The time duration of each oscillation is ¹ "1 μs. It is interesting to note that, when the total measuring time is imposed, the same decrease law with the number of signal oscillations is obtained. virtual periods. The achieved ENC is inversely proportional to the number of signal oscillations, q, both in the case of imposed time duration of the signal oscillation and in the case of imposed measuring time. As a conclusion, we can say that, in the case of imposed measuring time: (i) ENC due to the white voltage noise is independent of the number of signal oscillations; (ii) ENC due to the 1/f voltage noise decreases as 1/q; (iii) ENC due to the white current noise decreases as 1/q. In the case of imposed time duration of the signal oscillations, ENC due to all the different noise contributions decreases as 1/q General case Let us consider a more realistic case, in which different noise sources are contemporary present. For the sake of simplicity, we first discuss the situation in which only the two white noise sources are present. In Fig. 6 the ENC is shown as a function of the number of signal oscillations with an oscillation time ¹ "5 μs. In the following, no virtual periods are present before and after the presence of the signal. The contributions to ENC of the

8 E. Gatti et al./nucl. Instr. and Meth. in Phys. Res. A 417 (1998) Fig. 6. ENC as a function of the number of signal oscillations with an oscillation time ¹ "5 μs in the presence of both white current and white voltage noise. The contributions to ENC of the current and the voltage noise are also shown. current and the voltage noise have also been shown separately. Since the weight function is the same for both the white voltage and the white current noise (see Section 3.1), no differences are expected when both the white contributions are present with respect to the situations discussed in Section 3.1. The interpolation of the achievable resolution indicates a decay of the ENC as (1/q). Both the contributions of the current and the voltage noises exhibit the same behavior. With only four signal oscillations a reduction of ENC of more than 60% can be achieved with respect to the conventional optimum processing for a delta pulse for equal measurement time. For comparison in Fig. 7, in the presence of the same noise contributions, the duration of the whole measurement time (¹ "25 μs) was imposed. The achievable resolution saturates to the amount of white voltage noise as explained in Section 3.1. The decrease of the current noise contribution is the same we obtained in Section 3.1 when it was the only noise component taken into account. In practical cases, however, the limiting factor is the presence of the 1/f voltage noise, as discussed in Section 3.1. Fig. 8 shows the results in the most general case for two different values of the oscillation time, ¹ "5 μs and ¹ "1 μs. The power Fig. 7. ENC as a function of the number of signal oscillations in the presence of both white current and white voltage noise. The duration of the measurement time has been imposed equal to ¹ "25 μs. The contributions to ENC of the current and the voltage noise are also shown. No virtual periods have been imposed before and after the presence of the signal. spectral density of the 1/f voltage noise has been assumed equal to a "1.510 V. For comparison, the energy resolution achievable with the optimum processing of a delta pulse in the same measurement time is shown. The contributions of the white voltage, white current and 1/f voltage noise are shown, both in the case of multiple readout and in the case of the delta pulse processing. In the case of multiple read-out, the ENC due to all the noise contributions decreases linearly with the number of signal oscillations (and consequently with the measurement time). For the delta pulse we can recognize the well-known behavior. The voltage 1/f noise contribution is independent of the total measurement time, while the white voltage noise decreases and the white current noise increases as the measurement time is increased. The multiple read-out with q"10 oscillations and a time duration of each oscillation ¹ "5 μs achieves an ENC less than 20% the ENC achievable with a single delta pulse optimally processed. When the time duration of each oscillation is reduced to ¹ "1 μs, the ENC achievable with q"10 oscillations is 32% the ENC achievable with a single delta pulse optimally processed.

9 350 E. Gatti et al./nucl. Instr. and Meth. in Phys. Res. A 417 (1998) Fig. 8. ENC as a function of the number of signal oscillations in the presence of white current and voltage noises and 1/f voltage noise (dark symbols). The contributions to ENC of the white current () and the 1/f () and white voltage noise () are also shown. The resolution achievable with a delta pulse optimally processed in the same measurement time and in the presence of the same noise spectral densities is shown for comparison (light symbols). (a) ¹ "5 μs, (b) ¹ "1 μs. It can be of interest to analyze the effects of different values of the noise components, since this read-noise reduction technique can been applied for measurements in the optical region of the energy spectrum also with different transistors as the first amplifying element. For a device cooled below!100 C, the slowest tolerable frame rate, related to the rate of the detected events, will determine the maximum number of averages, not a theoretical limit. For applications that cannot be cooled suffi- Fig. 9. ENC as a function of the number of signal oscillations for different noise spectral densities of the 1/f voltage noise. (a) The time duration of each oscillation is ¹ "1 μs. (b) The measurement time imposed is equal to ¹ "25 μs. ciently, dark current generation will be the limit [12]. In the present analysis we suppose that the device is always sufficiently cooled to neglect the contribution of the thermally generated charge. The method presented here can be extended to also take into account the effect of the thermally generated charge on the induced signal and consequently on the achievable resolution and on the shape of the optimum weighting function. The assumption of a white current noise b "1.610 A/Hz, equivalent to a leakage current of the gate of the first transistor of 1 pa, is generally verified. Fig. 9a and b shows the ENC normalized to the ENC for a delta pulse optimally-processed in the presence of the same noise spectral densities, as

10 E. Gatti et al./nucl. Instr. and Meth. in Phys. Res. A 417 (1998) a function of the number of signal oscillations for different noise spectral densities of the 1/f voltage noise. In one case the time duration of each oscillation has been imposed (¹ "1 μs) while in the other case the total measurement time has been imposed (¹ "25 μs). The maximum number of oscillations taken into account when the total measurement time is imposed, is limited by the minimum duration of each oscillation (assumed equal to 1 μs [9]). No waiting and lagging periods have been introduced in all the cases. When only one signal oscillation is considered, the achievable resolution is worse than in the case of the optimum processing of a delta pulse in the presence of the same noise spectral densities. However, two signal oscillations are enough to overcome this worsening and with the increase in the number of oscillations the resolution is improved. When the total measurement time is imposed the improvement is less significant with the increase in the number of oscillations when the white voltage noise becomes dominant, because, as shown in the previous section, when the total measurement time is imposed its contribution remains constant with the number of signal oscillations. This is true for low values of the 1/f voltage noise spectral density and for sufficiently high number of signal oscillations. On the other hand when the time duration of each oscillation is imposed the decrease in the ENC is inversely proportional to the number of signal oscillations independently of the amount of the 1/f voltage noise. 4. Effects of the shape of the signal on the achievable resolution In Section 2 we assumed that the shape of the charge induced on the output electrode is given by Eq. (3). This is equivalent to assuming that the cloud of signal electrons is not stored underneath the sensing output electrode but it travels towards and away from that electrode with a periodic motion. As we said in Section 2, this assumption is just a useful approximation. In this section we want to show the effects of considering different shapes of the induced signal. In many real devices [9,13], the shape of the induced signal can be very different from a sinusoidal function. First of all, in order to achieve a high charge transfer efficiency from the potential well underneath the storing electrode (the electrode nearby the sensing output electrode) to the region underneath the sensing output electrode, a very high fringing field is obtained by properly tailoring the geometry of the electrodes and by properly tuning the bias voltages needed to transfer the cloud of electrons. Assuming that the charge transfer is completely caused by the drift motion and neglecting the motion due to the diffusion of the signal charge, the transfer velocity approaches the mobility times the obtained fringing field. Moreover, the required transfer efficiency and the maximum allowed power dissipation impose a limit to the minimum duration of each signal oscillation (¹ ). Hence, for a fraction of ¹ the signal charge is stored in the potential well reproduced underneath the sensing output electrode. The shape of the induced charge function caused by this law of motion is trapezoidal. The leading edge and the trailing edge are determined by the time needed by the signal electrons to cover the distance from the storing electrode to the sensing output electrode (and vice versa), while the time duration of the flat top equals the time interval in which the signal charge is stored underneath the sensing output electrode. The corresponding output current of the device can be obtained by deriving the induced charge function. Since in practical cases the rise and fall times of the trapezoidal function are much shorter than the time duration of the flat top, the induced signal charge can be well approximated by a square wave. The corresponding output current will be given by a doublet of delta pulses of opposite areas placed in correspondence with the leading and the trailing edge of the square function. Assuming that the time during which the cloud of electrons induced on the sensing output electrode equals the time during which the induction is ideally zero and extending the notation used in Section 2 for the sinusoidal signal, the current signal can be written as Q δ t!m ¹ p!δ t!m ¹ p #4i!3 4 #4i!1 4 ¹ p ¹ p. (12)

11 352 E. Gatti et al./nucl. Instr. and Meth. in Phys. Res. A 417 (1998) waiting and of the lagging time have been assumed as multiples of the oscillation time. 5. Conclusions Fig. 10. ENC as a function of the number of signal oscillations for two different shapes of the current signal. The squared symbols () represent the achievable resolution for a current signal as given by Eq. (12). The rounded symbols () represent the achievable resolution for a sinusoidal current signal. The oscillation time is ¹ "1 μs. No virtual periods have been left before and after the processing of the signal. For comparison in Fig. 10 the ENC in the presence of the white voltage noises is shown for the case of a sinusoidal current signal and for the case of a current signal given by Eq. (12) as a function of the number of signal oscillations. As shown in the figure, the effect of the multiple read-out technique on the ENC is approximately the same for both the shapes of the current signal. The only difference is the absolute value of ENC that can be obtained. The current signal given by Eq. (12) gives a resolution factor of two in the ENC better than the sinusoidal current signal. However in real devices the shape of the current signal is not sinusoidal or like the one given in Eq. (12). These two signals can be considered as the extreme cases of the signal shape obtained in real devices, hence the achievable resolution will be in between the values given by the two current signals (the region shaded in Fig. 10). Just for sake of mathematical simplicity, an assumption has been taken in all previous analyses, which does not reduce the general validity of the obtained results. That is, the duration of the We have introduced a method aimed at calculating the optimum filters for the multiple read-out technique of signals, e.g. CCD signals, with arbitrary time domain constraints in presence of arbitrary noise power spectra. The method is easily implemented on a work station, and it is fast since the optimum filter waveform and corresponding ENC are obtained in a few seconds with currently employed personal computers. By using the method, we present a detailed analysis of the effects of the multiple read-out technique on any kind and/or amount of the noise contributions, in order to understand the optimum resolution that can be actually achieved. Cases of experimental interest have been taken into account and fully treated, outlining the most important design characteristics of the read-out mechanism for optimally processing the signal. The discussion is developed both for a sinusoidal signal induced on the read-out electrode by the periodic oscillation of the signal charge and for a signal approximated by a series of delta pulses of opposite areas. These assumptions are useful because they are easy to be dealt with and because they are the boundary cases of the signal shapes in real devices. The main limitations to increase the number of read-out per pixel arises from the increase of the total measurement time. This limits the maximum rate of the processed signals and increases the effects of the generation current, which is neglected in these analyses. As the read-noise would be reduced to a small fraction of an electron, problems such as ADC quantization error and low levels of spurious current can become increasingly significant. Acknowledgements The authors would like to thank A. Castoldi, A. Fazzi, A. Longoni, A. Pullia, P. Rehak, S. Rescia and G. Ripamonti for interesting and stimulating discussions.

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