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Investigations of the dynamic compression principle for fast detector pulses
Nuclear lnstrulnents and Methods in Physics Research A 396 11997) 19X-213
ELSF!VIER
Investigations
NUCLEAR INSTRUMENTS 8 METHODS IN PHYSICS RESEARCH section A
of the dynamic compression principle for fast detector pulses Werner
KurzbauePb,*,
Bo Lofstedt”
Received 14 January 1997 Abstract For detectors at LHC, fast and accurate data acquisition systems are required. The method of analog dynamic range compression in order to overcome the technological limits of quantization circuits is presented and shown for an application at the electromagnetic calorimeter (ECAL) for CMS. A model of the detector pulses and an analysis of the compressed pulses in time and frequency domain is given. As a consequence the demands on the electronics are derived. A method to specify the bandwidth and the slew rate requirements for the very front end electronics is presented. For nonlinear circuits approximated by piecewise-linear circuits a detailed analysis is worked out and applied to the dynamic range compressor. This work is meant to be a guideline to determine bandwidth requirements also for other topologies and should help to optimize overall system performance.
1. Introduction To analyse the suitability of electronic circuits for high energy physics experiments, tests are usually carried out in a testbeam arrangement using test signals with typical shapes of detector pulses. To evaluate the performance of the electronics statistical methods in the time domain are applied on the test data, whilst in order to specify a circuit electrically, unity steps or delta pulses are used and tests are accomplished in an electronics laboratory environment. However, investigations in frequency domain can also be very helpful to optimize circuits for specific applications. One well known example is the bode diagram which offers a relatively simple way of analysing the stability of feedback systems. Especially in analog systems where pulses are to be processed, an analysis in frequency domain can be used to elaborate the bandwidth requirements for the various components. This is important to minimize noise and can help to find the cheapest solution fulfilling the requirements.
* Corresponding author 016%9002/97/$17.00 Copyright PI1 SO1 68-9002(97)00747-X
The restrictions for performing the necessary transformations in an analytical way are driven by the difliculties to set up mathematical models for complex pulses and to calculate the transfer characteristics in frequency domain, especially for nonlinear circuits. One approach is to simplify as required the models for the pulses and the circuit and to calculate the power spectral density (PSD) using Laplace- and/or Fourier transformation. One drawback of this method is the problem of estimating the error due to the simplification of the models. A useful method to circumvent a symbolical analysis is to apply Fast Fourier Transformation (FFT) on which allows to perform the simulation results, transformations fast and with high precision. The precision mainly depends on the computation power available and has to be considered when the results are interpreted. A direct application of this kind of analysis is given by the method of dynamic range compression by means of a nonlinear amplifier as used in the FERMI system [ 11. The goal of this investigation is to analyse the compressed signal in time and frequency domain and to work out a specification method to define the bandwidth requirements of the dynamic compressor circuit.
‘1 1997 Elsevier Science B.V. All rights reserved
CT Kurzhauer.
B. LqfStedt / Nucl. hstr.
2. Principle of analog dynamic compression -3.1. General considerutions To measure signals using a read out system four factors are driving the design and the effort of the system: ~ precision provided by the signal to be measured. - precision needed by the application, _ technological possibilities, ~ cost of the system. To design a read out system for a specific application these factors have to be weighted against each other to achieve an optimized solution. In the field of high energy physics the precision which is needed is theoretically infinity. Thus the system limiting factors are (neglecting the cost factor) the accuracy of the signals to be measured e.g. the detector resolution and. especially for experiments planned at LHC [2], technological constraints. One current constraint is the limited product of speed and resolution of present Analog to Digital Converters (ADC), as both of these factors are essential for high energy physics applications. In order to overcome this problem and nevertheless keep the required performance of the system a method of fitting the properties of the read out system to the error characterisitcs’ of the detector is being used. To accomplish this, an analog signal conditioner in front of the ADC is used to perform the matching. Such a circuit, located in the front part of a data acquisition system, should perform signal manipulation by keeping the precision of the signal as close as possible to the resolution, which is provided by the subsequent stages of the acquisition chain. For example in a system, where the resolution is determined by the number of quantization intervals, it is a waste of performance (and money) to use electronics which provides a much higher precision than for what the digital system is built. In other words the ratio between the error contribution of the electronics and the uncertainty of the signal being processed should not exceed a specified limit. Because the error contributions of all components in a chain sum up, it is up to the system designer to determine a maximum factor of merit for every part in order to meet a given overall performance. This factor of merit defines the error contribution of every component with regard to the precision of the signal it processes. In general, every manipulation of a signal can only decrease its information content. However, by doing this
’ In the following sections the error characteristics signal IS referred to as the fractional energy resolution detector.
of the of the
and Met/l. in Phys. Rex A 396 (IY97) I%‘,713
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manipulation in an appropriate way the information of interest can be separated from the uninteresting part (noise, etc.), thereby improving the signal quality. The goal is to build a read out system which is optimized to extract only the relevant information of the input signal by matching the resolution of the system to the precision of the signal. For example, assume a detector signal having a very simple fractional energy resolution’ which consists only of a calibration term a(E)/E = c. The constant c is supposed to be k 0.5% (In reality a certain lower limit exists for this kind of accuracy, caused by noise, etc., but this is of no relevance in this example). This signal is assumed to be represented by a voltage with a maximum of 1 V,. To measure this voltage a 10 bit read out system is supposed to be used. A common way would be to use linear quantization leading to an LSB of - 1 mV: For a signal level equal to 0.1 V the error of the measured signal equals f 0.5 LSB (0.5% of 0.1 V = 0.5 mV). At this signal level the system can be considered matched. For signals between 0.1 V and 1 V the error of the measured signal will occupy between 1 and 10 LSBs; thus the system is slightly overdesigned. For signals < 0.1 V the resolution of the ADC is too small to get a digital replica with the required precision. By matching the resolution of the read out system to the precision of the signal using a nonlinear amplifier the significance of all bits can be made equal over the whole input voltage range. In other words the LSB size is varied according to the absolute accuracy of the value to be measured. The exact calculation is shown in the appendix. Despite of this improvement it has to be taken into account that any amplifier has a deviation from its ideal model. Error sources like noise, limited bandwidth, distortion, etc. have to be considered when such a input conditioning circuit is put in front of the system. 2.1. Application ECAL
of dwamic
range cotnpTession at CiMS
In the read out system, which is required for the electromagnetic calorimeter in CMS 121, one of the most challenging problems is to handle pulses with a dynamic range of - 100 dB at a sampling rate of 40 Msamples/s. Investigations lead to a system, where the dynamic range of the original signal is being compressed before analog
‘This example should just illustrate the principle. in reality the fractional energy resolution of a detector consists of more terms 121.
to digital conversion and afterwards digitally reconstructed by means of a look-up table. The decisive factors have been [t]: ~ the absolute accuracy of the detector signal is decreasing at higher signal levels. ~ especially the low energy range (small signal amplitudes) should be measured with high resolution. ~ technological limit of analog to digital converters is at the moment far from the demanded performance, _ fast analog electronics devices (ASICs) are available at reasonable costs, _ system should be flexible and easy to be adapted to new technological possibilities. The principle of the dynamic compression is to map the dynamic range of the detector signal to the resolution of the ADC. If the error introduced by this input conditioning is kept below a defined fraction of the finite accuracy of the detector, the overall resolution and the fractional energy resolution of the detector have a well defined relation [3]. This can be achieved by applying the mathematical method shown in the previous section on the fractional energy resolution of the detector [2]. The resulting function is approximated by the input conditioning circuit using a piecewise-linear function. One way to approximate a function using piecewiselinear transfer function fitting is to apply the method of least mean squares. In this application the slopes and
Fig. 1. Basic block diagram
breakpoints of the transfer function have been found empirically using mathematical software tools [4]. This is required, because the energy resolution of the read out system has to emphasize the precision in some energy regions which are of special interest for the experiment
PI. Fig. 1 shows the basic block diagram of such a circuit. The circuit performs a piecewise-linear function fitting [S] with 4 linear regions, which is accomplished by summing the output currents of gain segments with different clamping voltages. The knee voltages and gains are calculated to optimize the error contribution at various signal levels. Fig. 2 shows the SPICE simulation of the (ideal) transfer characteristics and the required values for the gains and breakpoints. For investigations, which lead to the shape of the transfer curve and to the choice of four linear section refer to 131. In Fig. 1 the blocks are assumed to be ideal. However, each cell has a limited bandwidth, clamping stages have a finite response time. etc. These effects have to be kept sufficiently small to render the applicability of the compression technique in the high frequency domain. The following chapters will show how to determine the performance requirements of each stage for the given application.
of a compressing
circuit
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section
I I I
0
0.5
Ylimit
3
1.2v
4
201
I
-
I
1.0 Input Voltage [V]
a.31
1 I
0.085
I
1 J
1.5
2.c
Fig. 2. Transfer curve of the nonlinear circuit.
3. Analysis of the signals
In CMS ECAL the detector signals are short charge pulses coming from an avalanche photo diode (APD), converted into voltage pulses by means of a charge amplifier [I]. The signal then passes a CRC shaper which reduces the bandwidth and improves the signal-tonoise ratio [6]. The pre-amplifier and the shaper comprise a unit which is referred to as the very front end electronics (VFE) of the detector. The compressor is located together with the VFE close to the APD and gets its input signal from this amplifier/shaper part of the chain. As it is the main task of the VFE section to amplify and filter the signal, the transfer functions of these sections can be assumed to be well controlled. If these functions are chosen appropriately a reconstruction of the original signal shape from the filtered signal is possible. As the compressor section performs a nonlinear operation insufficient bandwidth causes signal distortion in such a way, that a reconstruction is not possible by means of a first order system. One of the goals of this investigation is to specify the bandwidth requirements for the compressor such, that the distortion due to bandwidth limitation is smaller than the required resolution and it can be regarded as being ideal within the read out system. This approximation can only be valid with some constraints and it is
therefore important to model the boundary given by the application.
conditions
A typical shaped pulse of a scintillating calorimeter can be modelled by a double exponential pulse according to (I). The shape of such a pulse is shown in Fig. 3. This pulse will be used to derive a specification for the compressor electronics and it will also be utilized to compare the analytical method which the discrete Fourier Transformation method (DFT) and to estimate the error due to the time discrete sampling method. .u(t) = k((1 - e-“7l) - (1 - e-“‘2)) where
(I)
t = time in sec. T, = rise time constant in set, constant in sec. k = amplitude factor in
T- = fall time
volts. 3.2. Power spectral densi@ @shaped
detector pulses
The power spectral density (PSD) of a pulse describes its energy characteristics in frequency domain. It provides a powerful method to estimate bandwidth requirements and to yield valuable information about the energy involved in a process.
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!
Time [ns] Fig. 3. Model of a shaped detector pulse.
One technique for the derivation of the PSD is to use direct Fourier transform respectively to use the direct FFT method [7]. To transform the exponential pulse into frequency domain Laplace transformation on (1) can be applied which leads to
s co
L{x(t))
=
.x(t)e-“‘dt
0
= k ~
1
1 - ~
s + (l/t21
s + (l/Zl)
.
approximation
The total energy of the pulse in time domain must equal the total energy in frequency domain (Parseval’s theorem (5)) [7]. This relation can be used to estimate the error introduced by bandwidth limitation of the electronics, when the energy of the pulse is of importance:
(2) Substituting s in (2) by jw and integrating from - x to + CC results in the Fourier integral (3) which can be directly used to obtain the PSD of the pulse:
F{.w(t)} =
s
cc
x(t)e-’ mot
2k(r2 - tJ
Jw’df =(1 + jwrr)(l
+ jwz,)
(3)
Note that the factor 2k(zz - TV) represents a peak value and has to be multiplied by l/G in later calculations in order to obtain the RMS value. The two poles of the polynom in (3) create two break points in the loglog diagram of the PSD. After the first pole the curve drops with 20 dB/decade, after the second pole with 40 dB/decade. In the case when the time constants or and x2 are close to each other the - 3 dB point of the spectral density can be estimated using the
EL 2n
;
IX(w)l’dw ,u
s
= j”‘(t)dt. 0
To calculate the needed bandwidth for a given percentage of the total energy of the pulse the integral in frequency domain (6) has to be solved. It yields the energy in the power spectrum within the bandwidth limit w1 and CII~.It should be noted, that w1 and w2 are treated as being ideal bandwidth limits. Usually the bandwidth characteristics of a circuit has a rolloff rate, so the result of the integral represents a lower limit for the energy.
E’=;
s
w2/X(w)12dw
WC
2k2(T2 - zl)’ 1 + oJ”(T: + T:) + w”#
do.
(6)
W. Kurzhaurr.
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-8O-----
Frequency [MHz] Fig. 4. PSD of the double exponential pulse in Fig. 3
The solution E,
=
of this integral
k2(r2~ ~1)(TVatan(cut,) n(z: +
resistor can be calculated which leads to
is given by
- 7L atan(oJ~,))i:;;.
(7)
Tl)
As an example the calculation of the PSD of a typical physical pulse with a peaking time of 50 ns and a peak amplitude of 1 V, is used. The constants for the double exponential pulse are: Z, = 49.75 ns, ?Z = 50.25 ns, k = 271.8 V. The pulse shape is according to Fig. 3 and the PSD has been calculated using an FFT with 215 (32768) points. The sampling frequency was 3.28 GHz with a time window of 10 ps. This leads to a frequency resolution of 100 kHz. Fig. 4 shows the normalized PSD in dB. The time constants 7, and 52 differ very little, therefore (4) can be used to calculate the - 3dB points of the power spectrum:
fl-
271.8’ .0.5 ns E=
1.29
(6) from 0 to co,
C(50.25 ns atan(tu. 50.25 ns))l$
- (49.75 ns atan(01.49.75
ns))l$]
= 117.66~0.5~10~~=9.24x10-*Ws.
(9)
In order to measure the energy with a certain precision the required bandwidth for the system can be calculated combining (7) and (9). The energy in the spectrum below the 3 dB point is given by E’ = 117.66[(50.25 ns atan(w.50.25 - (49.75 ns atan(w.49.75
ns))I$“d”
ns))l;‘d”]
= 117.66(2.8865 x 10m8 - 2.8352 x lo-‘)
1.29 3dB - 27r(T1 + r2)
= 2rt(49.75 ns + 50.25 ns)
n.lOOns
by integrating
= 6.04 x lo-” = 2.05 MHz
(8)
This result can also be graphically obtained as the result of the FFT (see Fig. 4). To estimate the error introduced by the limited bandwidth the total energy of the pulse dissipated in a 1 R
Ws.
(10)
The results show that the spectrum below the - 3 dB point contains more than 65% of the total signal energy. Further calculations resulted in a required bandwidth of 24 MHz to transfer the pulse energy with an accuracy of 1% (10 bit) and showed that 40 MHz are sufficient for 12 bit accuracy.
204
:: (
Pulse Energy
Integrated Power Specbum
;
I :
: : : : : : : : i :___:_ :
:
/
:
:
:__:_
:
: :
: :
,,
Fig. 5. Integrated pulse energy in time and frequency domain
For a coarse estimation of the bandwidth needed to transmit the pulse energy up to a fixed fraction Fig. 5 can be used. It shows the integrated power in time and frequency domain. Eqs. (11) and (12) show the formulas to calculate the energy using the discrete samples in time and frequency domain.
3.3. PSD
qf cfymtnic comp,ussed pulses
To calculate the PSD of the output signal of the compressor. the analytical method is difficult to apply, because the transfer function of a nonlinear circuit is dependent on the (input) signal level. As the input signal is a function of time also the gain becomes a function of time and therefore the spectrum becomes a function of signal level. Linear system:
(12)
Time domain:
.x(t) = A’ Vi(t) + /co.
Frequency
domain:
Nonlinear
system:
X(jo)
= A’ F [V,(t)),
is the number of samples, xi are the samples wherensampte of the signal in time domain and yi are the amplitude spectrum samples of the FFT. Any calculation of the PSD with a finite number of data samples may only be considered as an estimate having a certain error. One of the most important parameters is the record length of the samples. A small record length can produce large errors because ‘ghost’ frequencies are introduced which interact to produce distortions. It can be shown that the error can be kept sufficiently small, when the product of record length and time step is large 17). The difference between the numerical and symbolical solution is smaller than 0.1 yild,using the FFT with given parameters. The good correspondence between the results of the symbolical calculation and the FFT makes the numerical way preverable to obtain the PSD of pulses with complex shapes (e.g. the output of the nonlinear dynamic compression circuit)
Time domain: Frequency
.x(t) = ~(v/,(r)). v,(r) + k,,
domain:
X(jw) = F [A( vi(t)); @ F [Vi(r))
Thus for a nonlinear transfer function the spectrum for different input amplitudes is not only changing by a constant factor as in the case of a linear system, but is the PSD of the input pulse convoluted with the Fouriertransformed gain function. Fig. 6 shows the pulse shape of the double exponential pulse after compression using the transfer function of Fig. 2. The pulse amplitues at the input of the compressor are the clamping voltages of the gain segments (30mV. 300 mV, 1.2 V, 2 V). The PSD of each of the output pulses has been calculated using a 215 point FFT. In order to be able to compare the resulting power density spectra, the values have been
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100
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400
500
205
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Time [ns] Fig. 6. Pulse shape after compression.
normalized to the pulse peak value. The results are shown in Fig. 7.3 From Fig. 7 it can be seen, that the 3 dB point moves to lower frequencies for higher pulse amplitudes, but the slope becomes more flat. Due to edges in the compressed pulses the PSD is not any more a simple polynom but shows a ripple at higher frequencies. It can be seen that at a frequency of 100 MHz the difference between the PSD of the small (linearly amplified) pulse and the nonlinear large pulse is in the order of 15 dB. The reason for this is the faster rising edge of the compressed signals compared to the linear pulse. As a consequence the compressed pulse requires a slightly higher system bandwidth. The normalized PSD of the compressed pulse can be used to derive bandwidth requirements for the individual stages in the compressor circuit, for the transmission lines and the ADC. The energy of the compressed pulse respectively the integral of its PSD has no linear relation to the original pulse energy and is not suitable to calculate bandwidth requirements. The overall result of this investigation shows that the bandwidth demands on the electronics to transmit this kind of detector pulses with the necessary accuracy respecting the pulse energy are within feasible boundaries.
3 The PSD for the 1.2 and 2 V pulse are virtually are shown as one line in the diagram.
the same and
However, in the case of a nonlinear system and a low sampling rate compared to the required bandwidth the bandwidth demands are higher. as will be shown in the next section.
4. Bandwidth and Slew rate requirements 4.1. Bandwidth estimation for the compressor Ideally, when no bandwith limit is present, the relation between input and output voltage should be independent of the rate of change dV/dt of the signal. However, every amplifier has a finite bandwidth and the output cannot immediately follow the input signal; this causes settling effects and attenuation of the amplitude. If the shape of the input signal is known and the settling constraints are defined. the bandwidth of the system can be calculated. For a nonlinear amplifier the absolute amplitude error due to bandwidth limitation is not necessarily growing monotonically with the signal level. It is therefore necessary to determine the signal level, which causes the largest deviation between the ideal and bandwidth limited signal (worst case condition). To establish this condition for a piecewise linear circuit, the behaviour of the internal stages of the circuit has to be investigated, with detector pulses of various amplitudes applied.
Frequency
[MHz]
Fig. 7. PSD of pulses in Fig. 6.
For the circuit in Fig. 1 different input functions can be approximated for each gain segment, depending on the signal level of the pulse. If the gain segments (gml-gm4) are located after the clamping amplifiers and the gains of the clamping amplifiers (AllA3) are assumed to be unity, this leads to the following approximations: ~ a step function is seen by the first clamping stages, when large pulses are applied, ~ a ramp function is seen by limiting stages, when moderate pulses are applied, ~ an exponential function is seen by all stages which are not clamping. The bandwidth limitation of the compressor can be modelled by modifying Fig. 1 under the assumption that every gain segment introduces one dominant pole at w,, = 2rrf, (Fig. 8). Furthermore it has been assumed, that all gain segments have the same cut-off frequencyf,. As it will be shown later. this presumption does not affect the worstcase calculation, because the gain segments with lower gain usually have higher cut-off frequencies and their error contribution becomes smaller for larger signal levels. Simulations of the building blocks and measurements on prototypes [S] also confirmed this assumption. If bandwidth limitation is introduced, the amplitude error at an arbitrary sampling time t,, according to the input function of the gain stage can be calculated: To perform these calculations the exponential pulse of Eq. (1) has to be further analysed.
The peaking time is calculated by setting derivative of (1) equal to zero giving T, =
the first
In ri - lnz2 1
1
r2
ri
(13)
The peak voltage at t = T, can be obtained
by
(14) If a gain stage is clamping, the input signal seen by this stage can be approximated by either a ramp or a step function. If the input signal is a ramp from r = 0 to to. Eq. (15) has to be used to calculate the deviation V,,,, from the ideal value A,, at a time t, > to (see appendix for explanations).
= ,+ vdirf..,,.,
_
e”qe-‘,‘,
(15)
For the case that r,>s to, Vdirf can be approximated using the step response of a lowpass. This is given by V,irr.,_ = A0 e-‘v’
(16)
If a gain segment is not clamping, then Vdirr can be found using the Laplace transform of the exponential
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207
Vin
Fig. 8. Bandwidth
limited compression
pulse (see Eq. (2)), which leads to
As the lowpass characteristics implies a time delay this has to be considered in the calculations in that the observation time of the output has to be delayed. The delay time ~~~~~~has to be chosen such, that the peak values of the input and output signal appear at the same location (t, = T, + ~~~~~~~ ) This means no restriction as the phase of the detector pulse is unknown and the absolute delay is of no importance. To calculate the error of the entire circuit. the error contribution of every gain stage at t, has to be summed up. If the difference between ideal peak value and output voltages at time t, = T, + ~~~~~~is calculated over the whole input voltage range, the largest possible deviation depending on the input signal level can be found. For this calculation the value of the time constant T = l/co,, is arbitrary but should be chosen much smaller than the peaking time T, to obtain clear results. For the sample delay, the choice to ~~~~~~= T is a useful approximation.
circuit.
The calculations have been carried out using MATLAB” and MicroSim” PSpice software tools and lead to the decision, that the worst case arises just before the first gain segment starts to clamp (V, = V/limitl). It should be noted that although this result might be predictable, it is strongly dependent on the shape of the transfer characteristics. If the ratio between the gains of the linear sections is different, the worst case can also arise at voltages larger than V,i,,,itl. However, for the given application this means that the maximum error occurs when none of the stages is clamping and the contribution of every gain stage is proportional to its gain, again provided that all amplifiers have the same bandwidth. For the calculations this means that the system can considered to be linear and Eq. (17) can be used to calculate the error. As a specification guideline for a detector read-out system the difference between the ideal (no bandwidth limitation) and the actual output voltage due to limited bandwidth should not exceed 0.5 LSB of the ADC around the peaking time T, of the exponential pulse. For a given sampling interval At, the maximum distance between the peak and its closest sample is At,/2. This means that in the region T, k At,/2 the sum of the deviations of all gain stages has to be smaller than 0.5 LSB to guarantee the accuracy of the sample which s closest to the peak value (see Fig. 9).
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r
8.6
0 ,
bandwidth
IO
20
limited pulse
30
40
50
60
70
60
100
90
Time [ns] Fig. 9. Error due to bandwidth limitation. 0
-
1 J
.?I+-________________
-0.02
0.5LSB,2at
-0.04 a, 3 ;: T -0.06 LL + 0 -0.01
-0:
-0.1:
I
I
SO
100
Cutoff frequency
fo [h
200
44
Fig. 10. Peaking error versus cutoff frequency.
To obtain the required bandwidth of the compressor, the sum of the deviations of all gain stages (T/di‘r)must be set to the desired maximum value (0.5 LSB) and (17) has to be solved for Z. The maximum error has to be related
to the full scale value respectively to the LSB size of the employed ADC. The error in % ofjh// scale at the peaking time T, as a function of the cut-off frequency is shown in Fig. 10.
W. Kuchauer,
Table 1 Input signal: experimental pulse f, = 49.75 ns, ~~ = 50.25 ns. k = 271.8 Error in % [FS] Error in LSBs
3 dB cut-off frequency /;, (MHz)
2 1 0.5 0.25
1 LSB,om
0.1 0.05 0.025 0.0115
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54 75 106 148
0.5 LSB,Osi, 0.5 LSB, j Ri, 0.5 LSBIZRi,
“Conditions: P’fullqEA,r = 2 V. t, = T,, rdrlav= l/too.
The sampling delay T,,~,=~ has been chosen with q,elay = l/o~~ which is very close to optimum value for a minimum deviation at the peaking time T,. Table 1 summarizes the result shown in Fig. 10 for a maximum error of 0.5 LSB at T, for different ADC resolutions. The cut-off frequencies and the corresponding errors are shown. It can be seen, if the bandwidth of the gain segments is doubled, the error expressed in % offi& scale at the peaking time T, becomes l/4. This means that a resolution of 12 bits requires about twice the bandwidth of a 10 bit system. In an ideal sampling system (no jitter present) the maximum deviation between the ideal signal and the
bandwidth limited signal occurs at T,. This result can be seen from Fig. 11, which shows the difference in % o~,/iiii scule as a function of time. The four curves show the deviation for four different cut-off frequencies. For each frequency, Ar,,+ (see Fig. 9) has been numerically optimized to minimize the error. If jitter is present the time distance between the samples varies introducing additional errors in the sampled values. This is important, because in a sampling system phase noise and uncertainties in the synchronization between sampling clock and detector signal cause such effects. To model the error due to jitter for the peak value. the sample at T, can be assumed to have a nonideal delay time Q,_,~~in Fig. 9. As mentioned before, the error caused by jitter strongly depends on the location of the sample on the pulse. Here only the effect at the peaking time will be investigated. Fig. 12 shows the difference between the peak value and sampled value at T, with a maximum sampling jitter of + 2.5 ns, which corresponds to + 10% of the sample interval At,. The error is expressed in % $l:fiJ/ .sca/e. The parameters for the curves are the different cut-off frequencies of the gain segments. The values for the bandwidth limits in the diagram are the numbers of Table 1 (54, 75, 106 and 148 MHz). If a time jitter is present the error grows depending on the deviation from the nominal sampling time. The error in Fig. 12 consists of two parts: one part is the
0
-0.01 -0.02
s -0.04 r% 5 -0.05 LL ’ -0.06
-0.1
0
I
20
40
60
80
100 120 Time [ns]
I D
160
Fig. I I. Error versus sample time for different cutoff frequencies
180
:
210
.I_
-0.18 -2.5
I -2
-1.5
-1
1 -0.5
Jitter at
peal&
I 1
tig,”
I 1.5
i
2
5
[ns]
Fig. 12. Peaking error due to sample jitter.
non-frequency dependent error due the pulse shape, the other part is the error due to limited bandwidth. If no jitter is present the deviation arises only due to limited bandwidth (marked as dots in Fig. 12) but it can be seen, that the contribution of the nonfrequency dependent error becomes quickly dominant for larger jitter amplitudes. Thus it is important to assure a clean sample timing in order to benefit from a large bandwidth. It can be seen from Fig. 11, that the error due to bandwidth limitation is a maximum at T,, but because of the uniformity of the pulse in this region the sensitivity to jitter is small compared to the rising and falling flanks. Especially if samples are taken on the fast rising edge of compressed pulses, the error due to jitter will be dominant. 4.2. Dejinition of Slew rate In the previous section, the analysis was done under the assumption that the nonlinearity of the output signal is only caused by changing the gain for different input signal levels. However, the nonlinearity of the signal will also depend on how fast the output of the compressor can change in order to represent the transformed input signal. For input signals with a large amplitude and a small rise time every amplifier has a limit above which the output signal cannot linearly follow the input signal. The
reason for this effect, which is referred to as slew rate limiting, is the finite speed with which the internal capacitances can be charged or discharged. Compared to the wanted nonlinearities according to the compression principle the distortion caused by slew rate limitation is smaller but the fact that it is not time independent implies that it is not possible to retransform the distorted signal to its original shape by means of a first-order system. This retransformation, referred to as expansion of the signal, is achieved by applying the inverse transfer function of the compressor to the compressed signal. This is done in a digital way, e.g. the output values of an ADC. which converts the signal into digital samples are corrected by means of a lookup table. Such a look-up table can only perform a first-order transformation, and thus, slew rate errors cause nonlinearities in the expanded (re-linearized) signal. The slew rate of every amplifier is defined by the fastest possible dV/ldt at its output. In an open loop configuration the input signal level has always to be within the linear input range of the amplifier. The finite rise time at the output is therefore caused by the inherent lowpass characteristics of the gain and limitations in the current driving capability due to high current effects in the transistors. However, in a feedback configuration, with high open loop gain, or, in general, when an amplifier is driven outside of its linear region the situation is different. Parasitic capacitances or compensation capacitors limit the
211
W. Rurzhauer, B. Lsfitedt / Nucl. Instr. and Meth. in Phys. Res. A 396 (19971 198-213
voltage rise time and the current to charge the capacitances is not anymore proportional to the input voltage4 and the slew rate of the output is determined by the maximum available current to charge the capacitances. In the afterward application this is of particular interest, because to drive the gain stages outside of their linear region is the basic principle of this kind of compressor. 4.3. Slew rate requirements The fastest possible dV/dt occurs at the beginning the rising edge, when a full scale pulse is applied. The maximum rate of change of the pulse occurs t = 0 and is
dV(t) dt
max
Error in LSBs
3dB cut-off frequencyf,
Slew rate
(ohs)
(MHz) 1 LSB,osit 0.5 LsB,o~i, 0.5LSBIIRit 0.5LSBIZBi,
54 75 106 148
310 450 600 850
of at
This can be used to derive the theoretical required slew rate if no bandwidth limit would be present in that this maximum rate of change is assumed to be constant up to ~/limitl. By multiplying (18) with the gain of the first gain stage, the necessary slew rate for the system is obtained. For the values (ri = 49.75 ns, TV = 50.25 ns, k = 271.8 V) the factor k is chosen such, that the pulse is normalized to 1 V peak voltage. Multiplication with the full scale factor yields the maximum slew rate of
SL = gain, FS-
Table 2
All these considerations to derive the bandwidth and slew rate constraints for the analog compressor have been made under the assumption that a specified absolute precision of the signal has to be guaranteed and other parts of the chain like transmission lines and line receivers have minor influence. The assumption that the shaped pulse has a minimum rise time of 50 ns should reflect a worst-case estimation to cover pile-up effects and other deviations from the nominal pulse shape. Furthermore, as mentioned in the beginning, apart from higher technology costs. a large bandwidth is accompanied by a number of additional drawbacks like noise and power dissipation. It might therefore be necessary to sacrifice bandwidth for the sake of improving other system requirements and to rise the overall system performance. In addition to this theoretical considerations practical measurements and testbeam result should help to reach this aim.
= 38OOJ!. us 5. Conclusion
This value might be feasible with fast technologies but is, as mentioned before, of theoretical interest, as a bandwidth limitation will always exist. If the necessary bandwidth is known, the slew rate can be calculated using the relation
f.
2
0.35/t,,,,
(20)
where,/; is the cut-off frequency and triseis the rise time of the circuit (see appendix for explanations). For a worstcase estimation this relation should be valid for full scale signals (V,,, = 2V,,) leading to
(21) Table 2 shows the required frequencies of Table 1.
slew rates for the cutoff
4 This is of course depending on the circuit topology, but is valid for all input structures using a differential (long tailed) pair.
The idea of dynamic signal compression has been explained and a specification method for the bandwidth for the very front end electronics of a detector has been presented. The detector signal has been analysed in frequency domain and the relation between pulse energy and in time and frequency domain has been worked out. The slew rate and bandwidth requirements to measure the pulse energy of a unipolar shaped pulse with a certain precision have been calculated. Also the demands and constraints for dynamic range compression of detector pulses have been shown. The calculation results showed, that the frequency spectrum of compressed pulses is in the same range as for the original signal and does not require substantially faster circuits, The dynamic compression principle for CMS ECAL has been treated in detail and an estimation of the bandwidth requirements for different read out system resolutions was presented. The results predict that, with existing analog technologies, an adequately fast compression circuit is feasible.
Appendix
by combining (A.4), (A.6) and (A.7) the final equation the transfer function is given by
A. I. ,Idapting the LSB size to u simple c,onstant.fiuctiorlal resol~~tiatz
(2” - 7)(ln Vi - In V,,,,,,t) il(“j)
f2”-
= In
If the input signal V/i has a range from 0 to VimaXwith an accuracy x,, this can be expressed as ~( I’;) = ViX,.
(A.1)
To obtain a constant has to be set to
LSB( I’;) = li
significance
of bits the LSB size
(A.2)
=
where k is constant and defining the contribution of the quantization error to the resolution of the input signal. Eq. (A.2) can be related to the transfer function n( Vi) of the system, such that
-
In
1.
iA.8)
“c ,,,,”
This shows that, under these assumptions for the signal resolution, an input conditioning amplifier with a logarithmic transfer characteristics is the natural choice. It should be noted. that for an ideal logarithmic amplifier small input voltages in (AX) would result in negative ADC counts. This can be prevented, when the logarithmic amplifier has no gain below its intercept voltage [9], In reality, noise sets a lower limit below which the logarithmic relation is no longer valid. It should be further noted that the reason for the logarithmic transfer characteristics of the nonlinear amplifier is the simplified assumption, that the fractional energy resolution consists only a constant term. A.2
dlz( Vi) __=-= dV,
“L
for
Ramp response qf‘a jirst order lo~cpuss
(A.3)
(A.4)
To obtain the response in the case of a ramp as input signal the Laplace transformed ramp has to be multiplied with the Laplace transform of a (first order) lowpass and then retransformed into time domain. The Laplace transform of a ramp starting at t = 0 and increasing until t,, with A, as final amplitude is given by
To set up the condition for the lower end point, the minimum input signal Vim,,,,has to be chosen to set the treshold for the first ADC count. If Vim,,_is chosen, such that
Multiplication with the Laplace transform of a firstorder lowpass with cut-off frequency o0 leads to
Integration fer function.5
of (A.3) yields the desired composite
trans-
(A.51
fl(Vi”,,,,)= 1, then k is given by k
=
x,(2” - 2) l) = ln Vi“,,, j/ - ln VLn,xnln Vi,,,,,- ln VL,~ -ye(n(vin,.,)
-
b4.6) (A.10)
where n is the number of Bits of the ADC. It can be seen from (A.6) that either the number of bits of the ADC has to be sufficiently high or the threshold for the first ADC count has to be raised to maintain a large factor k. The constant C in (A.4) can be found by substitution of Vi by V;~~~~ leading to
C=
n(vi,,,,,)
-
”x, In Vi,,,,I = 2” -
Retransformation of (A.lO) results in the time response of the lowpass. given by
j’(t) =
2 [t ~~(1 -
e-l!‘)
_ o(t - f,)((t - to) - t(1 - e-‘r-t~l)‘Z))] 1 - ” In Vi,,, x,
(A.ll)
(A.7)
’ The composite transfer function represents the transfer function of the analog Input conditioner together with the transfer function of the ADC.
where 7 = l/w, and a(t) is the unity step function. For to approaching 0 using l’hopital’s rule the ramp response results in the step response of a lowpass: lim A,( - a(r - fO)(e~t’~‘~,‘:’- 1)) = Ao(l ~ em”‘) 10-O (A.12)
W. Kurzbauer. B. Lofitedt / Nucl. Instr. and Mrth. in Ph>a. Rex .4 396 (1997) 198-213 References
A.3. Relation between minimum rise time and 3 dB bandwidth To obtain the minimum rise time of a bandwidth limited system a unity step function is applied to the input. The response of the first-order system is then given by Viout(f) = a(f)A(l
- eeriz).
(A.13)
where z = l/w,,. o(t) is the unity step function and A is the final amplitude the gain. To calculate the rise time of the signal from 10% to 90% of the final amplitude (A.13) can be used: 0.1 A = A(1 - e-‘l”), 0.9 A = A( 1 - e-‘Zj’),
113
(A.14)
where f, and f2 are the delay times at which the signal reaches respectively, 10% and 90% of A. Combining the two expressions in (A. 14) yields the rise time as a function of r: frisc = t, - tr = s(ln 0.9 - lnO.1) = t ln9. (A.15) Rewriting (A.15) and replacing T by& finally results in (A.16)
111 V.G. Goggi. B. Lofstedt, Digital front-end electronics for calorimetry at LHC, Proc. ECFA Large Hadron Collider Workshop, ECFA 90-133. vol., Aachen. 3, 1990. 94-38, PI Cern CMS, Technical Proposal, CERN’LHCC LHCC/Pl, 1994. 131 V.G. Goggi, Dynamic range compression for calorimetry at LHC, FERMI Note #2, 1991. c41 W. Kurzbauer. A software tool to optimize the error matching of a dynamic compression function, programmed in MATLAB” 1996. not published. Circuits Handbook, 2nd 151 Analog Devices. Inc. Nonlinear ed.. Norwood Massachusetts 02061. USA 1976. Processing the Signals from 161 E. Gatti, P.F. Manfredi. Solid-State Detectors in Elementary-Particle Physics. La rivista del nuovo cimento. vol. 9, serie 3, Bologna. 1986. Signal Processing using Analog and 171 K.G. Beauchamp. Digital Techniques. 1st ed.. George Allen & Unwin LTD.. London, 1973. PI
S. Berglund, W. Kurzbauer, Evaluation and Testresults of the Analog Compressor in DMILL Technology, FERMI Note #47. 1995. [9]B. Gilbert. B. Clark, Monolithic DC-to-120 MHz Log-Amp is Stable and Accurate, Analog Dialogue. vol. 23, no. 3. 1889.
ELSF!VIER
Investigations
NUCLEAR INSTRUMENTS 8 METHODS IN PHYSICS RESEARCH section A
of the dynamic compression principle for fast detector pulses Werner
KurzbauePb,*,
Bo Lofstedt”
Received 14 January 1997 Abstract For detectors at LHC, fast and accurate data acquisition systems are required. The method of analog dynamic range compression in order to overcome the technological limits of quantization circuits is presented and shown for an application at the electromagnetic calorimeter (ECAL) for CMS. A model of the detector pulses and an analysis of the compressed pulses in time and frequency domain is given. As a consequence the demands on the electronics are derived. A method to specify the bandwidth and the slew rate requirements for the very front end electronics is presented. For nonlinear circuits approximated by piecewise-linear circuits a detailed analysis is worked out and applied to the dynamic range compressor. This work is meant to be a guideline to determine bandwidth requirements also for other topologies and should help to optimize overall system performance.
1. Introduction To analyse the suitability of electronic circuits for high energy physics experiments, tests are usually carried out in a testbeam arrangement using test signals with typical shapes of detector pulses. To evaluate the performance of the electronics statistical methods in the time domain are applied on the test data, whilst in order to specify a circuit electrically, unity steps or delta pulses are used and tests are accomplished in an electronics laboratory environment. However, investigations in frequency domain can also be very helpful to optimize circuits for specific applications. One well known example is the bode diagram which offers a relatively simple way of analysing the stability of feedback systems. Especially in analog systems where pulses are to be processed, an analysis in frequency domain can be used to elaborate the bandwidth requirements for the various components. This is important to minimize noise and can help to find the cheapest solution fulfilling the requirements.
* Corresponding author 016%9002/97/$17.00 Copyright PI1 SO1 68-9002(97)00747-X
The restrictions for performing the necessary transformations in an analytical way are driven by the difliculties to set up mathematical models for complex pulses and to calculate the transfer characteristics in frequency domain, especially for nonlinear circuits. One approach is to simplify as required the models for the pulses and the circuit and to calculate the power spectral density (PSD) using Laplace- and/or Fourier transformation. One drawback of this method is the problem of estimating the error due to the simplification of the models. A useful method to circumvent a symbolical analysis is to apply Fast Fourier Transformation (FFT) on which allows to perform the simulation results, transformations fast and with high precision. The precision mainly depends on the computation power available and has to be considered when the results are interpreted. A direct application of this kind of analysis is given by the method of dynamic range compression by means of a nonlinear amplifier as used in the FERMI system [ 11. The goal of this investigation is to analyse the compressed signal in time and frequency domain and to work out a specification method to define the bandwidth requirements of the dynamic compressor circuit.
‘1 1997 Elsevier Science B.V. All rights reserved
CT Kurzhauer.
B. LqfStedt / Nucl. hstr.
2. Principle of analog dynamic compression -3.1. General considerutions To measure signals using a read out system four factors are driving the design and the effort of the system: ~ precision provided by the signal to be measured. - precision needed by the application, _ technological possibilities, ~ cost of the system. To design a read out system for a specific application these factors have to be weighted against each other to achieve an optimized solution. In the field of high energy physics the precision which is needed is theoretically infinity. Thus the system limiting factors are (neglecting the cost factor) the accuracy of the signals to be measured e.g. the detector resolution and. especially for experiments planned at LHC [2], technological constraints. One current constraint is the limited product of speed and resolution of present Analog to Digital Converters (ADC), as both of these factors are essential for high energy physics applications. In order to overcome this problem and nevertheless keep the required performance of the system a method of fitting the properties of the read out system to the error characterisitcs’ of the detector is being used. To accomplish this, an analog signal conditioner in front of the ADC is used to perform the matching. Such a circuit, located in the front part of a data acquisition system, should perform signal manipulation by keeping the precision of the signal as close as possible to the resolution, which is provided by the subsequent stages of the acquisition chain. For example in a system, where the resolution is determined by the number of quantization intervals, it is a waste of performance (and money) to use electronics which provides a much higher precision than for what the digital system is built. In other words the ratio between the error contribution of the electronics and the uncertainty of the signal being processed should not exceed a specified limit. Because the error contributions of all components in a chain sum up, it is up to the system designer to determine a maximum factor of merit for every part in order to meet a given overall performance. This factor of merit defines the error contribution of every component with regard to the precision of the signal it processes. In general, every manipulation of a signal can only decrease its information content. However, by doing this
’ In the following sections the error characteristics signal IS referred to as the fractional energy resolution detector.
of the of the
and Met/l. in Phys. Rex A 396 (IY97) I%‘,713
199
manipulation in an appropriate way the information of interest can be separated from the uninteresting part (noise, etc.), thereby improving the signal quality. The goal is to build a read out system which is optimized to extract only the relevant information of the input signal by matching the resolution of the system to the precision of the signal. For example, assume a detector signal having a very simple fractional energy resolution’ which consists only of a calibration term a(E)/E = c. The constant c is supposed to be k 0.5% (In reality a certain lower limit exists for this kind of accuracy, caused by noise, etc., but this is of no relevance in this example). This signal is assumed to be represented by a voltage with a maximum of 1 V,. To measure this voltage a 10 bit read out system is supposed to be used. A common way would be to use linear quantization leading to an LSB of - 1 mV: For a signal level equal to 0.1 V the error of the measured signal equals f 0.5 LSB (0.5% of 0.1 V = 0.5 mV). At this signal level the system can be considered matched. For signals between 0.1 V and 1 V the error of the measured signal will occupy between 1 and 10 LSBs; thus the system is slightly overdesigned. For signals < 0.1 V the resolution of the ADC is too small to get a digital replica with the required precision. By matching the resolution of the read out system to the precision of the signal using a nonlinear amplifier the significance of all bits can be made equal over the whole input voltage range. In other words the LSB size is varied according to the absolute accuracy of the value to be measured. The exact calculation is shown in the appendix. Despite of this improvement it has to be taken into account that any amplifier has a deviation from its ideal model. Error sources like noise, limited bandwidth, distortion, etc. have to be considered when such a input conditioning circuit is put in front of the system. 2.1. Application ECAL
of dwamic
range cotnpTession at CiMS
In the read out system, which is required for the electromagnetic calorimeter in CMS 121, one of the most challenging problems is to handle pulses with a dynamic range of - 100 dB at a sampling rate of 40 Msamples/s. Investigations lead to a system, where the dynamic range of the original signal is being compressed before analog
‘This example should just illustrate the principle. in reality the fractional energy resolution of a detector consists of more terms 121.
to digital conversion and afterwards digitally reconstructed by means of a look-up table. The decisive factors have been [t]: ~ the absolute accuracy of the detector signal is decreasing at higher signal levels. ~ especially the low energy range (small signal amplitudes) should be measured with high resolution. ~ technological limit of analog to digital converters is at the moment far from the demanded performance, _ fast analog electronics devices (ASICs) are available at reasonable costs, _ system should be flexible and easy to be adapted to new technological possibilities. The principle of the dynamic compression is to map the dynamic range of the detector signal to the resolution of the ADC. If the error introduced by this input conditioning is kept below a defined fraction of the finite accuracy of the detector, the overall resolution and the fractional energy resolution of the detector have a well defined relation [3]. This can be achieved by applying the mathematical method shown in the previous section on the fractional energy resolution of the detector [2]. The resulting function is approximated by the input conditioning circuit using a piecewise-linear function. One way to approximate a function using piecewiselinear transfer function fitting is to apply the method of least mean squares. In this application the slopes and
Fig. 1. Basic block diagram
breakpoints of the transfer function have been found empirically using mathematical software tools [4]. This is required, because the energy resolution of the read out system has to emphasize the precision in some energy regions which are of special interest for the experiment
PI. Fig. 1 shows the basic block diagram of such a circuit. The circuit performs a piecewise-linear function fitting [S] with 4 linear regions, which is accomplished by summing the output currents of gain segments with different clamping voltages. The knee voltages and gains are calculated to optimize the error contribution at various signal levels. Fig. 2 shows the SPICE simulation of the (ideal) transfer characteristics and the required values for the gains and breakpoints. For investigations, which lead to the shape of the transfer curve and to the choice of four linear section refer to 131. In Fig. 1 the blocks are assumed to be ideal. However, each cell has a limited bandwidth, clamping stages have a finite response time. etc. These effects have to be kept sufficiently small to render the applicability of the compression technique in the high frequency domain. The following chapters will show how to determine the performance requirements of each stage for the given application.
of a compressing
circuit
II/ Kwzhauer. B. Lofstedt / Nucl. Insb. and Meth. in Phw. Rex. A 396 (1997) 19C-713
section
I I I
0
0.5
Ylimit
3
1.2v
4
201
I
-
I
1.0 Input Voltage [V]
a.31
1 I
0.085
I
1 J
1.5
2.c
Fig. 2. Transfer curve of the nonlinear circuit.
3. Analysis of the signals
In CMS ECAL the detector signals are short charge pulses coming from an avalanche photo diode (APD), converted into voltage pulses by means of a charge amplifier [I]. The signal then passes a CRC shaper which reduces the bandwidth and improves the signal-tonoise ratio [6]. The pre-amplifier and the shaper comprise a unit which is referred to as the very front end electronics (VFE) of the detector. The compressor is located together with the VFE close to the APD and gets its input signal from this amplifier/shaper part of the chain. As it is the main task of the VFE section to amplify and filter the signal, the transfer functions of these sections can be assumed to be well controlled. If these functions are chosen appropriately a reconstruction of the original signal shape from the filtered signal is possible. As the compressor section performs a nonlinear operation insufficient bandwidth causes signal distortion in such a way, that a reconstruction is not possible by means of a first order system. One of the goals of this investigation is to specify the bandwidth requirements for the compressor such, that the distortion due to bandwidth limitation is smaller than the required resolution and it can be regarded as being ideal within the read out system. This approximation can only be valid with some constraints and it is
therefore important to model the boundary given by the application.
conditions
A typical shaped pulse of a scintillating calorimeter can be modelled by a double exponential pulse according to (I). The shape of such a pulse is shown in Fig. 3. This pulse will be used to derive a specification for the compressor electronics and it will also be utilized to compare the analytical method which the discrete Fourier Transformation method (DFT) and to estimate the error due to the time discrete sampling method. .u(t) = k((1 - e-“7l) - (1 - e-“‘2)) where
(I)
t = time in sec. T, = rise time constant in set, constant in sec. k = amplitude factor in
T- = fall time
volts. 3.2. Power spectral densi@ @shaped
detector pulses
The power spectral density (PSD) of a pulse describes its energy characteristics in frequency domain. It provides a powerful method to estimate bandwidth requirements and to yield valuable information about the energy involved in a process.
202
W. Kurzbauer,
B. Lqfjtedt J Nucl. Instr. and Meth. in PhJs. Rex A 396 (1997) 19X-213
100
150
200
250
300
350
400
450
!
Time [ns] Fig. 3. Model of a shaped detector pulse.
One technique for the derivation of the PSD is to use direct Fourier transform respectively to use the direct FFT method [7]. To transform the exponential pulse into frequency domain Laplace transformation on (1) can be applied which leads to
s co
L{x(t))
=
.x(t)e-“‘dt
0
= k ~
1
1 - ~
s + (l/t21
s + (l/Zl)
.
approximation
The total energy of the pulse in time domain must equal the total energy in frequency domain (Parseval’s theorem (5)) [7]. This relation can be used to estimate the error introduced by bandwidth limitation of the electronics, when the energy of the pulse is of importance:
(2) Substituting s in (2) by jw and integrating from - x to + CC results in the Fourier integral (3) which can be directly used to obtain the PSD of the pulse:
F{.w(t)} =
s
cc
x(t)e-’ mot
2k(r2 - tJ
Jw’df =(1 + jwrr)(l
+ jwz,)
(3)
Note that the factor 2k(zz - TV) represents a peak value and has to be multiplied by l/G in later calculations in order to obtain the RMS value. The two poles of the polynom in (3) create two break points in the loglog diagram of the PSD. After the first pole the curve drops with 20 dB/decade, after the second pole with 40 dB/decade. In the case when the time constants or and x2 are close to each other the - 3 dB point of the spectral density can be estimated using the
EL 2n
;
IX(w)l’dw ,u
s
= j”‘(t)dt. 0
To calculate the needed bandwidth for a given percentage of the total energy of the pulse the integral in frequency domain (6) has to be solved. It yields the energy in the power spectrum within the bandwidth limit w1 and CII~.It should be noted, that w1 and w2 are treated as being ideal bandwidth limits. Usually the bandwidth characteristics of a circuit has a rolloff rate, so the result of the integral represents a lower limit for the energy.
E’=;
s
w2/X(w)12dw
WC
2k2(T2 - zl)’ 1 + oJ”(T: + T:) + w”#
do.
(6)
W. Kurzhaurr.
203
B. Lqfitedt ! Nucl. Instr. and Meth. in Phys. Rex A 396 (1997) 198-213
-8O-----
Frequency [MHz] Fig. 4. PSD of the double exponential pulse in Fig. 3
The solution E,
=
of this integral
k2(r2~ ~1)(TVatan(cut,) n(z: +
resistor can be calculated which leads to
is given by
- 7L atan(oJ~,))i:;;.
(7)
Tl)
As an example the calculation of the PSD of a typical physical pulse with a peaking time of 50 ns and a peak amplitude of 1 V, is used. The constants for the double exponential pulse are: Z, = 49.75 ns, ?Z = 50.25 ns, k = 271.8 V. The pulse shape is according to Fig. 3 and the PSD has been calculated using an FFT with 215 (32768) points. The sampling frequency was 3.28 GHz with a time window of 10 ps. This leads to a frequency resolution of 100 kHz. Fig. 4 shows the normalized PSD in dB. The time constants 7, and 52 differ very little, therefore (4) can be used to calculate the - 3dB points of the power spectrum:
fl-
271.8’ .0.5 ns E=
1.29
(6) from 0 to co,
C(50.25 ns atan(tu. 50.25 ns))l$
- (49.75 ns atan(01.49.75
ns))l$]
= 117.66~0.5~10~~=9.24x10-*Ws.
(9)
In order to measure the energy with a certain precision the required bandwidth for the system can be calculated combining (7) and (9). The energy in the spectrum below the 3 dB point is given by E’ = 117.66[(50.25 ns atan(w.50.25 - (49.75 ns atan(w.49.75
ns))I$“d”
ns))l;‘d”]
= 117.66(2.8865 x 10m8 - 2.8352 x lo-‘)
1.29 3dB - 27r(T1 + r2)
= 2rt(49.75 ns + 50.25 ns)
n.lOOns
by integrating
= 6.04 x lo-” = 2.05 MHz
(8)
This result can also be graphically obtained as the result of the FFT (see Fig. 4). To estimate the error introduced by the limited bandwidth the total energy of the pulse dissipated in a 1 R
Ws.
(10)
The results show that the spectrum below the - 3 dB point contains more than 65% of the total signal energy. Further calculations resulted in a required bandwidth of 24 MHz to transfer the pulse energy with an accuracy of 1% (10 bit) and showed that 40 MHz are sufficient for 12 bit accuracy.
204
:: (
Pulse Energy
Integrated Power Specbum
;
I :
: : : : : : : : i :___:_ :
:
/
:
:
:__:_
:
: :
: :
,,
Fig. 5. Integrated pulse energy in time and frequency domain
For a coarse estimation of the bandwidth needed to transmit the pulse energy up to a fixed fraction Fig. 5 can be used. It shows the integrated power in time and frequency domain. Eqs. (11) and (12) show the formulas to calculate the energy using the discrete samples in time and frequency domain.
3.3. PSD
qf cfymtnic comp,ussed pulses
To calculate the PSD of the output signal of the compressor. the analytical method is difficult to apply, because the transfer function of a nonlinear circuit is dependent on the (input) signal level. As the input signal is a function of time also the gain becomes a function of time and therefore the spectrum becomes a function of signal level. Linear system:
(12)
Time domain:
.x(t) = A’ Vi(t) + /co.
Frequency
domain:
Nonlinear
system:
X(jo)
= A’ F [V,(t)),
is the number of samples, xi are the samples wherensampte of the signal in time domain and yi are the amplitude spectrum samples of the FFT. Any calculation of the PSD with a finite number of data samples may only be considered as an estimate having a certain error. One of the most important parameters is the record length of the samples. A small record length can produce large errors because ‘ghost’ frequencies are introduced which interact to produce distortions. It can be shown that the error can be kept sufficiently small, when the product of record length and time step is large 17). The difference between the numerical and symbolical solution is smaller than 0.1 yild,using the FFT with given parameters. The good correspondence between the results of the symbolical calculation and the FFT makes the numerical way preverable to obtain the PSD of pulses with complex shapes (e.g. the output of the nonlinear dynamic compression circuit)
Time domain: Frequency
.x(t) = ~(v/,(r)). v,(r) + k,,
domain:
X(jw) = F [A( vi(t)); @ F [Vi(r))
Thus for a nonlinear transfer function the spectrum for different input amplitudes is not only changing by a constant factor as in the case of a linear system, but is the PSD of the input pulse convoluted with the Fouriertransformed gain function. Fig. 6 shows the pulse shape of the double exponential pulse after compression using the transfer function of Fig. 2. The pulse amplitues at the input of the compressor are the clamping voltages of the gain segments (30mV. 300 mV, 1.2 V, 2 V). The PSD of each of the output pulses has been calculated using a 215 point FFT. In order to be able to compare the resulting power density spectra, the values have been
W. Kwzhauel: B. Lo@dt
“0
/ Nucl. Instr. and Meth. in Phvs. Rex A 396 (1997) 19G-113
200
100
300
400
500
205
600
Time [ns] Fig. 6. Pulse shape after compression.
normalized to the pulse peak value. The results are shown in Fig. 7.3 From Fig. 7 it can be seen, that the 3 dB point moves to lower frequencies for higher pulse amplitudes, but the slope becomes more flat. Due to edges in the compressed pulses the PSD is not any more a simple polynom but shows a ripple at higher frequencies. It can be seen that at a frequency of 100 MHz the difference between the PSD of the small (linearly amplified) pulse and the nonlinear large pulse is in the order of 15 dB. The reason for this is the faster rising edge of the compressed signals compared to the linear pulse. As a consequence the compressed pulse requires a slightly higher system bandwidth. The normalized PSD of the compressed pulse can be used to derive bandwidth requirements for the individual stages in the compressor circuit, for the transmission lines and the ADC. The energy of the compressed pulse respectively the integral of its PSD has no linear relation to the original pulse energy and is not suitable to calculate bandwidth requirements. The overall result of this investigation shows that the bandwidth demands on the electronics to transmit this kind of detector pulses with the necessary accuracy respecting the pulse energy are within feasible boundaries.
3 The PSD for the 1.2 and 2 V pulse are virtually are shown as one line in the diagram.
the same and
However, in the case of a nonlinear system and a low sampling rate compared to the required bandwidth the bandwidth demands are higher. as will be shown in the next section.
4. Bandwidth and Slew rate requirements 4.1. Bandwidth estimation for the compressor Ideally, when no bandwith limit is present, the relation between input and output voltage should be independent of the rate of change dV/dt of the signal. However, every amplifier has a finite bandwidth and the output cannot immediately follow the input signal; this causes settling effects and attenuation of the amplitude. If the shape of the input signal is known and the settling constraints are defined. the bandwidth of the system can be calculated. For a nonlinear amplifier the absolute amplitude error due to bandwidth limitation is not necessarily growing monotonically with the signal level. It is therefore necessary to determine the signal level, which causes the largest deviation between the ideal and bandwidth limited signal (worst case condition). To establish this condition for a piecewise linear circuit, the behaviour of the internal stages of the circuit has to be investigated, with detector pulses of various amplitudes applied.
Frequency
[MHz]
Fig. 7. PSD of pulses in Fig. 6.
For the circuit in Fig. 1 different input functions can be approximated for each gain segment, depending on the signal level of the pulse. If the gain segments (gml-gm4) are located after the clamping amplifiers and the gains of the clamping amplifiers (AllA3) are assumed to be unity, this leads to the following approximations: ~ a step function is seen by the first clamping stages, when large pulses are applied, ~ a ramp function is seen by limiting stages, when moderate pulses are applied, ~ an exponential function is seen by all stages which are not clamping. The bandwidth limitation of the compressor can be modelled by modifying Fig. 1 under the assumption that every gain segment introduces one dominant pole at w,, = 2rrf, (Fig. 8). Furthermore it has been assumed, that all gain segments have the same cut-off frequencyf,. As it will be shown later. this presumption does not affect the worstcase calculation, because the gain segments with lower gain usually have higher cut-off frequencies and their error contribution becomes smaller for larger signal levels. Simulations of the building blocks and measurements on prototypes [S] also confirmed this assumption. If bandwidth limitation is introduced, the amplitude error at an arbitrary sampling time t,, according to the input function of the gain stage can be calculated: To perform these calculations the exponential pulse of Eq. (1) has to be further analysed.
The peaking time is calculated by setting derivative of (1) equal to zero giving T, =
the first
In ri - lnz2 1
1
r2
ri
(13)
The peak voltage at t = T, can be obtained
by
(14) If a gain stage is clamping, the input signal seen by this stage can be approximated by either a ramp or a step function. If the input signal is a ramp from r = 0 to to. Eq. (15) has to be used to calculate the deviation V,,,, from the ideal value A,, at a time t, > to (see appendix for explanations).
= ,+ vdirf..,,.,
_
e”qe-‘,‘,
(15)
For the case that r,>s to, Vdirf can be approximated using the step response of a lowpass. This is given by V,irr.,_ = A0 e-‘v’
(16)
If a gain segment is not clamping, then Vdirr can be found using the Laplace transform of the exponential
W. Kurzbauer,
B. Lofstedt / Nucl. Instr. and Meth. it1 Php.
Rex A 396 (1997) 198-213
207
Vin
Fig. 8. Bandwidth
limited compression
pulse (see Eq. (2)), which leads to
As the lowpass characteristics implies a time delay this has to be considered in the calculations in that the observation time of the output has to be delayed. The delay time ~~~~~~has to be chosen such, that the peak values of the input and output signal appear at the same location (t, = T, + ~~~~~~~ ) This means no restriction as the phase of the detector pulse is unknown and the absolute delay is of no importance. To calculate the error of the entire circuit. the error contribution of every gain stage at t, has to be summed up. If the difference between ideal peak value and output voltages at time t, = T, + ~~~~~~is calculated over the whole input voltage range, the largest possible deviation depending on the input signal level can be found. For this calculation the value of the time constant T = l/co,, is arbitrary but should be chosen much smaller than the peaking time T, to obtain clear results. For the sample delay, the choice to ~~~~~~= T is a useful approximation.
circuit.
The calculations have been carried out using MATLAB” and MicroSim” PSpice software tools and lead to the decision, that the worst case arises just before the first gain segment starts to clamp (V, = V/limitl). It should be noted that although this result might be predictable, it is strongly dependent on the shape of the transfer characteristics. If the ratio between the gains of the linear sections is different, the worst case can also arise at voltages larger than V,i,,,itl. However, for the given application this means that the maximum error occurs when none of the stages is clamping and the contribution of every gain stage is proportional to its gain, again provided that all amplifiers have the same bandwidth. For the calculations this means that the system can considered to be linear and Eq. (17) can be used to calculate the error. As a specification guideline for a detector read-out system the difference between the ideal (no bandwidth limitation) and the actual output voltage due to limited bandwidth should not exceed 0.5 LSB of the ADC around the peaking time T, of the exponential pulse. For a given sampling interval At, the maximum distance between the peak and its closest sample is At,/2. This means that in the region T, k At,/2 the sum of the deviations of all gain stages has to be smaller than 0.5 LSB to guarantee the accuracy of the sample which s closest to the peak value (see Fig. 9).
208
W. Kurzhauer.
B. Lojitedt / Nucl. Itntr. arld Mrth. in P&s. Rex A 39fi (1 YY7i lY&-713
r
8.6
0 ,
bandwidth
IO
20
limited pulse
30
40
50
60
70
60
100
90
Time [ns] Fig. 9. Error due to bandwidth limitation. 0
-
1 J
.?I+-________________
-0.02
0.5LSB,2at
-0.04 a, 3 ;: T -0.06 LL + 0 -0.01
-0:
-0.1:
I
I
SO
100
Cutoff frequency
fo [h
200
44
Fig. 10. Peaking error versus cutoff frequency.
To obtain the required bandwidth of the compressor, the sum of the deviations of all gain stages (T/di‘r)must be set to the desired maximum value (0.5 LSB) and (17) has to be solved for Z. The maximum error has to be related
to the full scale value respectively to the LSB size of the employed ADC. The error in % ofjh// scale at the peaking time T, as a function of the cut-off frequency is shown in Fig. 10.
W. Kuchauer,
Table 1 Input signal: experimental pulse f, = 49.75 ns, ~~ = 50.25 ns. k = 271.8 Error in % [FS] Error in LSBs
3 dB cut-off frequency /;, (MHz)
2 1 0.5 0.25
1 LSB,om
0.1 0.05 0.025 0.0115
209
B. LoJStedt j Nucl. Itwtr. and Meth. in Phvs. Rex A 396 (1997) IYX-,713
54 75 106 148
0.5 LSB,Osi, 0.5 LSB, j Ri, 0.5 LSBIZRi,
“Conditions: P’fullqEA,r = 2 V. t, = T,, rdrlav= l/too.
The sampling delay T,,~,=~ has been chosen with q,elay = l/o~~ which is very close to optimum value for a minimum deviation at the peaking time T,. Table 1 summarizes the result shown in Fig. 10 for a maximum error of 0.5 LSB at T, for different ADC resolutions. The cut-off frequencies and the corresponding errors are shown. It can be seen, if the bandwidth of the gain segments is doubled, the error expressed in % offi& scale at the peaking time T, becomes l/4. This means that a resolution of 12 bits requires about twice the bandwidth of a 10 bit system. In an ideal sampling system (no jitter present) the maximum deviation between the ideal signal and the
bandwidth limited signal occurs at T,. This result can be seen from Fig. 11, which shows the difference in % o~,/iiii scule as a function of time. The four curves show the deviation for four different cut-off frequencies. For each frequency, Ar,,+ (see Fig. 9) has been numerically optimized to minimize the error. If jitter is present the time distance between the samples varies introducing additional errors in the sampled values. This is important, because in a sampling system phase noise and uncertainties in the synchronization between sampling clock and detector signal cause such effects. To model the error due to jitter for the peak value. the sample at T, can be assumed to have a nonideal delay time Q,_,~~in Fig. 9. As mentioned before, the error caused by jitter strongly depends on the location of the sample on the pulse. Here only the effect at the peaking time will be investigated. Fig. 12 shows the difference between the peak value and sampled value at T, with a maximum sampling jitter of + 2.5 ns, which corresponds to + 10% of the sample interval At,. The error is expressed in % $l:fiJ/ .sca/e. The parameters for the curves are the different cut-off frequencies of the gain segments. The values for the bandwidth limits in the diagram are the numbers of Table 1 (54, 75, 106 and 148 MHz). If a time jitter is present the error grows depending on the deviation from the nominal sampling time. The error in Fig. 12 consists of two parts: one part is the
0
-0.01 -0.02
s -0.04 r% 5 -0.05 LL ’ -0.06
-0.1
0
I
20
40
60
80
100 120 Time [ns]
I D
160
Fig. I I. Error versus sample time for different cutoff frequencies
180
:
210
.I_
-0.18 -2.5
I -2
-1.5
-1
1 -0.5
Jitter at
peal&
I 1
tig,”
I 1.5
i
2
5
[ns]
Fig. 12. Peaking error due to sample jitter.
non-frequency dependent error due the pulse shape, the other part is the error due to limited bandwidth. If no jitter is present the deviation arises only due to limited bandwidth (marked as dots in Fig. 12) but it can be seen, that the contribution of the nonfrequency dependent error becomes quickly dominant for larger jitter amplitudes. Thus it is important to assure a clean sample timing in order to benefit from a large bandwidth. It can be seen from Fig. 11, that the error due to bandwidth limitation is a maximum at T,, but because of the uniformity of the pulse in this region the sensitivity to jitter is small compared to the rising and falling flanks. Especially if samples are taken on the fast rising edge of compressed pulses, the error due to jitter will be dominant. 4.2. Dejinition of Slew rate In the previous section, the analysis was done under the assumption that the nonlinearity of the output signal is only caused by changing the gain for different input signal levels. However, the nonlinearity of the signal will also depend on how fast the output of the compressor can change in order to represent the transformed input signal. For input signals with a large amplitude and a small rise time every amplifier has a limit above which the output signal cannot linearly follow the input signal. The
reason for this effect, which is referred to as slew rate limiting, is the finite speed with which the internal capacitances can be charged or discharged. Compared to the wanted nonlinearities according to the compression principle the distortion caused by slew rate limitation is smaller but the fact that it is not time independent implies that it is not possible to retransform the distorted signal to its original shape by means of a first-order system. This retransformation, referred to as expansion of the signal, is achieved by applying the inverse transfer function of the compressor to the compressed signal. This is done in a digital way, e.g. the output values of an ADC. which converts the signal into digital samples are corrected by means of a lookup table. Such a look-up table can only perform a first-order transformation, and thus, slew rate errors cause nonlinearities in the expanded (re-linearized) signal. The slew rate of every amplifier is defined by the fastest possible dV/ldt at its output. In an open loop configuration the input signal level has always to be within the linear input range of the amplifier. The finite rise time at the output is therefore caused by the inherent lowpass characteristics of the gain and limitations in the current driving capability due to high current effects in the transistors. However, in a feedback configuration, with high open loop gain, or, in general, when an amplifier is driven outside of its linear region the situation is different. Parasitic capacitances or compensation capacitors limit the
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voltage rise time and the current to charge the capacitances is not anymore proportional to the input voltage4 and the slew rate of the output is determined by the maximum available current to charge the capacitances. In the afterward application this is of particular interest, because to drive the gain stages outside of their linear region is the basic principle of this kind of compressor. 4.3. Slew rate requirements The fastest possible dV/dt occurs at the beginning the rising edge, when a full scale pulse is applied. The maximum rate of change of the pulse occurs t = 0 and is
dV(t) dt
max
Error in LSBs
3dB cut-off frequencyf,
Slew rate
(ohs)
(MHz) 1 LSB,osit 0.5 LsB,o~i, 0.5LSBIIRit 0.5LSBIZBi,
54 75 106 148
310 450 600 850
of at
This can be used to derive the theoretical required slew rate if no bandwidth limit would be present in that this maximum rate of change is assumed to be constant up to ~/limitl. By multiplying (18) with the gain of the first gain stage, the necessary slew rate for the system is obtained. For the values (ri = 49.75 ns, TV = 50.25 ns, k = 271.8 V) the factor k is chosen such, that the pulse is normalized to 1 V peak voltage. Multiplication with the full scale factor yields the maximum slew rate of
SL = gain, FS-
Table 2
All these considerations to derive the bandwidth and slew rate constraints for the analog compressor have been made under the assumption that a specified absolute precision of the signal has to be guaranteed and other parts of the chain like transmission lines and line receivers have minor influence. The assumption that the shaped pulse has a minimum rise time of 50 ns should reflect a worst-case estimation to cover pile-up effects and other deviations from the nominal pulse shape. Furthermore, as mentioned in the beginning, apart from higher technology costs. a large bandwidth is accompanied by a number of additional drawbacks like noise and power dissipation. It might therefore be necessary to sacrifice bandwidth for the sake of improving other system requirements and to rise the overall system performance. In addition to this theoretical considerations practical measurements and testbeam result should help to reach this aim.
= 38OOJ!. us 5. Conclusion
This value might be feasible with fast technologies but is, as mentioned before, of theoretical interest, as a bandwidth limitation will always exist. If the necessary bandwidth is known, the slew rate can be calculated using the relation
f.
2
0.35/t,,,,
(20)
where,/; is the cut-off frequency and triseis the rise time of the circuit (see appendix for explanations). For a worstcase estimation this relation should be valid for full scale signals (V,,, = 2V,,) leading to
(21) Table 2 shows the required frequencies of Table 1.
slew rates for the cutoff
4 This is of course depending on the circuit topology, but is valid for all input structures using a differential (long tailed) pair.
The idea of dynamic signal compression has been explained and a specification method for the bandwidth for the very front end electronics of a detector has been presented. The detector signal has been analysed in frequency domain and the relation between pulse energy and in time and frequency domain has been worked out. The slew rate and bandwidth requirements to measure the pulse energy of a unipolar shaped pulse with a certain precision have been calculated. Also the demands and constraints for dynamic range compression of detector pulses have been shown. The calculation results showed, that the frequency spectrum of compressed pulses is in the same range as for the original signal and does not require substantially faster circuits, The dynamic compression principle for CMS ECAL has been treated in detail and an estimation of the bandwidth requirements for different read out system resolutions was presented. The results predict that, with existing analog technologies, an adequately fast compression circuit is feasible.
Appendix
by combining (A.4), (A.6) and (A.7) the final equation the transfer function is given by
A. I. ,Idapting the LSB size to u simple c,onstant.fiuctiorlal resol~~tiatz
(2” - 7)(ln Vi - In V,,,,,,t) il(“j)
f2”-
= In
If the input signal V/i has a range from 0 to VimaXwith an accuracy x,, this can be expressed as ~( I’;) = ViX,.
(A.1)
To obtain a constant has to be set to
LSB( I’;) = li
significance
of bits the LSB size
(A.2)
=
where k is constant and defining the contribution of the quantization error to the resolution of the input signal. Eq. (A.2) can be related to the transfer function n( Vi) of the system, such that
-
In
1.
iA.8)
“c ,,,,”
This shows that, under these assumptions for the signal resolution, an input conditioning amplifier with a logarithmic transfer characteristics is the natural choice. It should be noted. that for an ideal logarithmic amplifier small input voltages in (AX) would result in negative ADC counts. This can be prevented, when the logarithmic amplifier has no gain below its intercept voltage [9], In reality, noise sets a lower limit below which the logarithmic relation is no longer valid. It should be further noted that the reason for the logarithmic transfer characteristics of the nonlinear amplifier is the simplified assumption, that the fractional energy resolution consists only a constant term. A.2
dlz( Vi) __=-= dV,
“L
for
Ramp response qf‘a jirst order lo~cpuss
(A.3)
(A.4)
To obtain the response in the case of a ramp as input signal the Laplace transformed ramp has to be multiplied with the Laplace transform of a (first order) lowpass and then retransformed into time domain. The Laplace transform of a ramp starting at t = 0 and increasing until t,, with A, as final amplitude is given by
To set up the condition for the lower end point, the minimum input signal Vim,,,,has to be chosen to set the treshold for the first ADC count. If Vim,,_is chosen, such that
Multiplication with the Laplace transform of a firstorder lowpass with cut-off frequency o0 leads to
Integration fer function.5
of (A.3) yields the desired composite
trans-
(A.51
fl(Vi”,,,,)= 1, then k is given by k
=
x,(2” - 2) l) = ln Vi“,,, j/ - ln VLn,xnln Vi,,,,,- ln VL,~ -ye(n(vin,.,)
-
b4.6) (A.10)
where n is the number of Bits of the ADC. It can be seen from (A.6) that either the number of bits of the ADC has to be sufficiently high or the threshold for the first ADC count has to be raised to maintain a large factor k. The constant C in (A.4) can be found by substitution of Vi by V;~~~~ leading to
C=
n(vi,,,,,)
-
”x, In Vi,,,,I = 2” -
Retransformation of (A.lO) results in the time response of the lowpass. given by
j’(t) =
2 [t ~~(1 -
e-l!‘)
_ o(t - f,)((t - to) - t(1 - e-‘r-t~l)‘Z))] 1 - ” In Vi,,, x,
(A.ll)
(A.7)
’ The composite transfer function represents the transfer function of the analog Input conditioner together with the transfer function of the ADC.
where 7 = l/w, and a(t) is the unity step function. For to approaching 0 using l’hopital’s rule the ramp response results in the step response of a lowpass: lim A,( - a(r - fO)(e~t’~‘~,‘:’- 1)) = Ao(l ~ em”‘) 10-O (A.12)
W. Kurzbauer. B. Lofitedt / Nucl. Instr. and Mrth. in Ph>a. Rex .4 396 (1997) 198-213 References
A.3. Relation between minimum rise time and 3 dB bandwidth To obtain the minimum rise time of a bandwidth limited system a unity step function is applied to the input. The response of the first-order system is then given by Viout(f) = a(f)A(l
- eeriz).
(A.13)
where z = l/w,,. o(t) is the unity step function and A is the final amplitude the gain. To calculate the rise time of the signal from 10% to 90% of the final amplitude (A.13) can be used: 0.1 A = A(1 - e-‘l”), 0.9 A = A( 1 - e-‘Zj’),
113
(A.14)
where f, and f2 are the delay times at which the signal reaches respectively, 10% and 90% of A. Combining the two expressions in (A. 14) yields the rise time as a function of r: frisc = t, - tr = s(ln 0.9 - lnO.1) = t ln9. (A.15) Rewriting (A.15) and replacing T by& finally results in (A.16)
111 V.G. Goggi. B. Lofstedt, Digital front-end electronics for calorimetry at LHC, Proc. ECFA Large Hadron Collider Workshop, ECFA 90-133. vol., Aachen. 3, 1990. 94-38, PI Cern CMS, Technical Proposal, CERN’LHCC LHCC/Pl, 1994. 131 V.G. Goggi, Dynamic range compression for calorimetry at LHC, FERMI Note #2, 1991. c41 W. Kurzbauer. A software tool to optimize the error matching of a dynamic compression function, programmed in MATLAB” 1996. not published. Circuits Handbook, 2nd 151 Analog Devices. Inc. Nonlinear ed.. Norwood Massachusetts 02061. USA 1976. Processing the Signals from 161 E. Gatti, P.F. Manfredi. Solid-State Detectors in Elementary-Particle Physics. La rivista del nuovo cimento. vol. 9, serie 3, Bologna. 1986. Signal Processing using Analog and 171 K.G. Beauchamp. Digital Techniques. 1st ed.. George Allen & Unwin LTD.. London, 1973. PI
S. Berglund, W. Kurzbauer, Evaluation and Testresults of the Analog Compressor in DMILL Technology, FERMI Note #47. 1995. [9]B. Gilbert. B. Clark, Monolithic DC-to-120 MHz Log-Amp is Stable and Accurate, Analog Dialogue. vol. 23, no. 3. 1889.