Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
V.4. Pulse Shaping
CR-RC shaper with unequal time constants
For simplicity, the preceding discussion assumed equaldifferentiator and integrator time constants.
Furthermore, the detector capacitance was the only capacitancepresent at the input. In reality this is never the case, but additionalcapacitance can easily be accounted for by replacing CD by thetotal input capacitance Ci, i.e. the sum of all capacitances presentat the input node.
1. Noise voltage at shaper output
ωω dGvVk
knino )( 2
0
2,
2 ⋅= ∫ ∑∞
( )∫∞
⋅+++=0
222222 )( ωω dGvvvvV nanrnpnbno
ωτωτωπ
τωd
id
d )1)(1(2 2222
22
++
⋅
⋅
++
++= ∫
∞
0
222
2 4) (1
4
) (
2 naS
iP
P
i
beno vkTR
CR
kTR
C
IqV
ωω
++
+=
id
dbe
Pino Iq
R
kT
CV
τττ 2
22 2
4
4
1
+
++
++i
d
id
df
idi
dnaS AvkTR
ττ
τττ
ττττ
ln )(
)4( 22
22
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
2. Peak signal voltage at the shaper output
Assume the detector signal is a pulse of constant current withduration tc (i.e. the signal charges ramps up linearly until t= tc ).
At times t < tc
At times t > tc
The peak amplitude of the signal is
Assume that the shaper has no additional voltage gain. Then Vo isequal to the input signal voltage
In the limit that the collection time is negligible, tc<< τd , τi
( )
−
−−−= −−− )(
)(1 )( //
2/ icdi tt
idc
dt
c
doso ee
te
tVtV τττ
ττττ
( )
−
−−−
−= −− icicdcic tt
idc
idtt
idc
doso ee
tee
tVtV ττττ
ττττ
τττ ////
2
)1()(
1)(
)(
( )( ) )/( /
)/( /
1
1 idiic
idddc
t
t
c
doso
e
e
tVV ττττ
τττττ−
−
−
−=
i
sio C
QVV ==
),( idi
sso f
C
QV ττ=
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Furthermore, since the peak signal depends only on the ratios oftime constants, i.e. it is independent of the absolute time scale
The noise at the output, however, does depend on the time scale(as it determines the noise bandwidth).
First, to simplify the expression for the output noise, neglect the 1/fterm and combine the terms for
current noise
and voltage noise
and introduce a characteristic time T, which initially will be setequal to the peaking time tp . With these conventions
)/( idsi
sso F
C
QV ττ=
Pben R
kTIqi
422 +=
22 4 naSn vkTRv +=
++
+=
id
dbe
Pino Iq
R
kT
CV
τττ 2
22 2
4
4
1
+
++
++i
d
id
df
idi
dnaS AvkTR
ττ
τττ
ττττ
ln )(
)4( 22
22
id
d
i
pn
id
d
p
d
ino
t
Tv
tTi
CV
τττ
τττττ
++
+=
1
4
1 22n2
2
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Using the above results and reintroducing 1/f noise,
and
the equivalent noise charge
can be written as
where
sso
non Q
V
VQ =
+
++
++
=i
d
id
df
id
d
i
pn
id
d
p
d
ino A
t
Tv
tTi
CV
ττ
τττ
τττ
τττττ
ln 1
4
122
222
n22
)/( idsi
sso F
C
QV ττ=
vffivniinn FACFT
vCFTiQ 1
22222 ++=
)/(
1/1
14 idsdip
di Ft
Fττττ
τ+
≡
)/(1
ln)/(1
12
idsi
d
divf F
Fτττ
τττ
+
≡
)/(1
/11
idsdii
pv F
tF
τττττ +≡
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
The noise indices or “form factors” Fi , Fv and Fvf characterizea type of shaper, for example CR-RC or CR-(RC)4.
They depend only on the ratio of time constants τd /τi, rather thantheir absolute magnitude.
The noise contribution then scales with the characteristic time T.The choice of characteristic time is somewhat arbitrary. so anyconvenient measure for a given shaper can be adopted in derivingthe noise coefficients F.
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
CR-RC Shapers with Multiple Integrators
a. Start with simple CR-RC shaper and add additional integrators(n= 1 to n= 2, ... n= 8) with the same time constant τ .
With additional integrators the peaking time Tp increases
Tp = nτ
0 5 10 15 20T/tau
0.0
0.1
0.2
0.3
0.4
SH
AP
ER
OU
TP
UT
n=1
n=2
n=4
n=6
n=8
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
b) Time constants changed to preserve the peaking time(τn= τn=1 /n)
Increasing the number of integrators makes the output pulse moresymmetrical with a faster return to baseline.
⇒ improved rate capability at the same peaking time
Shapers with the equivalent of 8 RC integrators are common.Usually, this is achieved with active filters (i.e. circuitry thatsynthesizes the bandpass with amplifiers and feedback networks).
0 1 2 3 4 5TIME
0.0
0.2
0.4
0.6
0.8
1.0
SH
AP
ER
OU
TP
UT
n=8
n=1
n=2
n=4
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Summary
Two basic noise mechanisms: input noise current ininput noise voltage vn
Equivalent Noise Charge:
where Ts Characteristic shaping time (e.g. peaking time)
Fi , Fv "Form Factors" that are determinedby the shape of the pulse.
Ci Total capacitance at the input node(detector capacitance + input capacitance of preamplifier + stray capacitance + … )
Typical values of Fi , Fv
CR-RC shaper Fi = 0.924 Fv = 0.924
CR-(RC)4 shaper Fi = 0.45 Fv = 1.02
CR-(RC)7 shaper Fi = 0.34 Fv = 1.27
CAFE chip Fi = 0.4 Fv = 1.2
Note that Fi < Fv for higher order shapers.Shapers can be optimized to reduce current noise contributionrelative to the voltage noise (mitigate radiation damage!).
Q i T F C vFTn n s i i n
v
s
2 2 2 2= +
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
The noise analysis of shapers is rather straightforward if thefrequency response is known.
On the other hand, since we are primarily interested in the pulseresponse, shapers are often designed directly in the time domain, soit seems more appropriate to analyze the noise performance in thetime domain also.
Clearly, one can take the time response and Fourier transform it tothe frequency domain, but this approach becomes problematic fortime-variant shapers.
The CR-RC shapers discussed up to now utilize filters whose timeconstants remain constant during the duration of the pulse, i.e. theyare time-invariant.
Many popular types of shapers utilize signal sampling or change thefilter constants during the pulse to improve pulse characteristics, i.e.faster return to baseline or greater insensitivity to variations indetector pulse shape.
These time-variant shapers cannot be analyzed in the mannerdescribed above. Various techniques are available, but someshapers can be analyzed only in the time domain.
Example:
A commonly used time-variant filter is the correlated double-sampler.
This shaper can be analyzed exactly only in the time domain.
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Correlated Double Sampling
1. Signals are superimposed on a (slowly) fluctuating baseline
2. To remove baseline fluctuations the baseline is sampled prior tothe arrival of a signal.
3. Next, the signal + baseline is sampled and the previous baselinesample subtracted to obtain the signal
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Noise Analysis in the Time Domain
What pulse shapes have a frequency spectrum corresponding totypical noise sources?
1. voltage noise
The frequency spectrum at the input of the detector system is“white”, i.e.
This is the spectrum of a δ impulse:
inifinitesimally narrow,but area= 1.
2. current noise
The spectral density is inversely proportional to frequency, i.e.
This is the spectrum of a step impulse:
const. =df
dA
fdf
dA 1∝
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
• Input noise can be considered as a sequence of δ and step pulseswhose rate determines the noise level.
• The shape of the primary noise pulses is modified by the pulseshaper:
δ pulses become longer,
step pulses are shortened.
• The noise level at a given measurement time Tm is determined bythe cumulative effect (superposition) of all noise pulses occurringprior to Tm .
• Their individual contributions at t= Tm are described by theshaper’s “weighting function” W(t).
References:
V. Radeka, Nucl. Instr. and Meth. 99 (1972) 525V. Radeka, IEEE Trans. Nucl. Sci. NS-21 (1974) 51F.S. Goulding, Nucl. Instr. and Meth. 100 (1972) 493F.S. Goulding, IEEE Trans. Nucl. Sci. NS-29 (1982) 1125
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Consider a single noise pulse occurring in a short time interval dtat a time T prior to the measurement. The amplitude at t= T is
an = W(T)
If, on the average, nn dt noise pulses occur within dt, the fluctuation oftheir cumulative signal level at t= T is proportional to
The magnitude of the baseline fluctuation is
For all noise pulses occurring prior to the measurement
wherenn determines the magnitude of the noise
and
describes the noise characteristics of theshaper – the “noise index”
dtnn
[ ]∫∞
∝0
22 )( dttWnnnσ
[ ] dttWnT nn2
2 )( )( ∝σ
[ ] dttW∫∞
0
2 )(
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
The Weighting Function
a) current noise Wi (t) is the shaper response to a steppulse, i.e. the “normal” output waveform.
b) voltage noise
(Consider a δ pulse as the superposition oftwo step pulses of opposite polarity andspaced inifinitesimally in time)
Examples: 1. Gaussian 2. Trapezoid
current(“step”)noise
voltage(“delta”)noise
Goal: Minimize overall area
⇒ For a given pulse duration a symmetrical pulse provides the best noise performance.
)(')()( tWtWdt
dtW iv ≡=
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Time-Variant Shapers
Example: gated integrator with prefilter
The gated integrator integrates the input signal during a selectabletime interval (the “gate”).
In this example, the integrator is switched on prior to the signal pulseand switched off after a fixed time interval, selected to allow theoutput signal to reach its maximum.
Consider a noise pulse occurring prior to the “on time” of theintegrator.
occurrence of contribution of the noise pulse noise pulse to
integrator output
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
For W1 = weighting function of the time-invariant prefilter
W2 = weighting function of the time-variant stage
the overall weighting function is obtained by convolution
Weighting function for current (“step”) noise: W(t)
Weighting function for voltage (“delta”) noise: W’(t)
Example
Time-invariant prefilter feeding a gated integrator(from Radeka, IEEE Trans. Nucl. Sci. NS-19 (1972) 412)
∫∞
∞−
−⋅= ' )'()'( )( 12 dtttWtWtW
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Comparison between a time-invariant and time-variant shaper(from Goulding, NIM 100 (1972) 397)
Example: trapezoidal shaper Duration= 2 µsFlat top= 0.2 µs
1. Time-Invariant Trapezoid
Current noise
Voltage noise
Minimum for τ1= τ3 (symmetry!) ⇒ Ni2= 0.8, Nv
2= 2.2
3 )1( )]([ 31
20 0
2
3
2
2
1
221 2
1
3
2
τττττ
τ τ
τ
τ
τ
++=
++
== ∫ ∫ ∫ ∫
∞
dtt
dtdtt
dttWN i
31
2
30 0
2
1
22 111
1 )]('[
3
2
1
ττττ
τ
τ
τ
+=+
+
== ∫∫ ∫∞
dtdtdttWNv
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Gated Integrator Trapezoidal Shaper
Current Noise
Voltage Noise
⇒ time-variant shaper Ni2= 1.4, Nv
2= 1.1
time-invariant shaper Ni2= 0.8, Nv
2= 2.2
time-variant trapezoid has more current noise, less voltage noise
∫ ∫ −=+
=
−T
I
TT
Ti
TTdtdt
T
tN
I
0
22
2
3)1( 2
∫ =
=
T
v Tdt
TN
0
22 21
2
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Interpretation of Results
Example: gated integrator
Current Noise
Increases with T1 and TG ( i.e. width of W(t) )
( more noise pulses accumulate within width of W(t) )
Voltage Noise
Increases with the magnitude of the derivative of W(t)
( steep slopes → large bandwidth determined by prefilter )
Width of flat top irrelevant(δ response of prefilter is bipolar: net= 0)
∫∝ dttWQnv22 )]('[
∫∝ dttWQni22 )]([
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Quantitative Assessment of Noise in the Time Domain
(see Radeka, IEEE Trans. Nucl. Sci. NS-21 (1974) 51 )
↑ ↑current noise voltage noise
Qn= equivalent noise charge [C]
in= input current noise spectral density [A/√Hz]
vn= input voltage noise spectral density [V/√Hz]
Ci= total capacitance at input
W(t) normalized to unit input step response
or rewritten in terms of a characteristic time t → T/ t
∫∫∞
∞−
∞
∞−
+= dttWvCdttWiQ ninn222222 )]('[
21
)]([ 21
∫∫∞
∞−
∞
∞−
+= dttWT
vCdttWTiQ ninn222222 )]('[
1
21
)]([ 21
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Correlated Double Sampling
1. Signals are superimposed on a (slowly) fluctuating baseline
2. To remove baseline fluctuations the baseline is sampled prior tothe arrival of a signal.
3. Next, the signal + baseline is sampled and the previous baselinesample subtracted to obtain the signal
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
1. Current Noise
Current (shot) noise contribution:
Weighting function (T= time between samples):
Current noise coefficient
so that the equivalent noise charge
∫∞
∞−
= dttWiQ nni222 )]([
21
τ
τ
/)(
/
)( :
1)( : 0
0)( : 0
Tt
t
etWTt
etWTt
tWt
−−
−
=>
−=≤≤
=<
∫∞
∞−
= dttWFi2)]([
( ) ∫∫∞
−−− +−=T
TtT
ti dtedteF ττ /)(2
0
2/1
2
22/2/ τττ ττ +
−+= −− TT
i eeTF
( )
+−+= −− 1
2
2
1 /2/22 τττ TT
nni eeTiQ
+−+= −− 1
2
4
1 /2/22 ττ
ττ TT
nni eeT
iQ
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Reality Check 1:
Assume that the current noise is pure shot noise
so that
Consider the limit Sampling Interval >> Rise Time, T >> τ :
or expressed in electrons
where Ni is the number of electrons “counted” during the samplinginterval T.
Iqi en 22 =
TIqQ eni ⋅≈2
ee
eni q
TI
q
TIqQ
⋅=⋅≈ 22
ini NQ ≈
+−+= −− 1
2
2
1 /2/2 ττ
ττ TT
eni eeT
IqQ
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
2. Voltage Noise
Voltage Noise Contribution
Voltage Noise Coefficient
so that the equivalent noise charge
∫∞
∞−
= dttWvCQ ninv2222 )]('[
21
∫∞
∞−
= dttWFv2)]('[
∫∫∞
−−−
+
=
T
TtT
tv dtedteF
2/)(2
0
2/
1
1 ττ
ττ
( )ττ
τ
21
1 21 /2 +−= − T
v eF
( )τ
τ/2222 2
41
1
Tninv evCQ −−=
( )τ
τ/22
21 T
v eF −−=
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Reality Check 2:
In the limit T >> τ :
Compare this with the noise on an RC low-pass filter alone (i.e. thevoltage noise at the output of the pre-filter):
(see the discussion on noise bandwidth)
so that
If the sample time is sufficiently large, the noise samples taken at thetwo sample times are uncorrelated, so the two samples simply add inquadrature.
τ21222 ⋅⋅= ninv vCQ
τ41
)( 222 ⋅⋅= nin vCRCQ
2 )(
sample) double( =RCQ
Q
n
n
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
3. Signal Response
The preceding calculations are only valid for a signal response ofunity, which is valid at T >> τ.
For sampling times T of order τ or smaller one must correct for thereduction in signal amplitude at the output of the prefilter
so that the equivalent noise charge due to the current noise becomes
The voltage noise contribution is
and the total equivalent noise charge
τ/1 / Tis eVV −−=
2/
/2222
)1( 4
2
1 τ
τ
τ T
T
ninve
evCQ −
−
−−=
22nvnin QQQ +=
( )2 /
/2/
22
1 4
12
τ
ττ
ττT
TT
nnie
eeT
iQ−
−−
−
+−+=
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Optimization
1. Noise current negligible
Parameters: T= 100 nsCd= 10 pFvn= 2.5 nV/√Hz
→ in= 6 fA/√Hz (Ib= 0.1 nA)
Noise attains shallow minimum for τ = T .
0
200
400
600
800
1000
1200
0 0.5 1 1.5 2 2.5 3
tau/T
Eq
uiv
alen
t N
ois
e C
har
ge
Qni [el]Qnv [el]Qn [el]
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
2. Significant current noise contribution
Parameters: T= 100 nsCd= 10 pFvn= 2.5 nV/√Hz
→ in= 0.6 pA/√Hz (Ib= 1 µA)
Noise attains minimum for τ = 0.3 T .
0
1000
2000
3000
4000
5000
0 0.5 1 1.5 2 2.5 3
tau/T
Eq
uiv
alen
t N
ois
e C
har
ge
Qni [el]Qnv [el]Qn [el]
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Parameters: T= 100 nsCd= 10 pFvn= 2.5 nV/√Hz
→ in= 0.2 pA/√Hz (Ib= 100 nA)
Noise attains minimum for τ = 0.5 T .
0
500
1000
1500
2000
0 0.5 1 1.5 2 2.5 3
tau/T
Eq
uiv
alen
t N
ois
e C
har
ge
Qni [el]Qnv [el]Qn [el]
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
3. Form Factors Fi, Fv and Signal Gain G vs. τ / T
Note: In this plot the form factors Fi, Fv are not yet corrected bythe gain G.
The voltage noise coefficient is practically independent of τ / T .
Voltage contribution to noise charge dominated by Ci /τ .
The current noise coefficient increases rapidly at small τ / T .
At large τ / T the contribution to the noise charge increases
because of τ dependence.
The gain dependence increases the equivalent noise charge withincreasing τ / T (as the gain is in the denominator).
0
5
10
0 0.5 1 1.5 2 2.5 3
tau/T
Fi
FvG
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Summary
Two basic noise mechanisms: input noise current ininput noise voltage vn
Equivalent Noise Charge:
↑ ↑ ↑ ↑ front shaper front shaper end end
where Ts Characteristic shaping time (e.g. peaking time)
Fi, Fv "Form Factors" that are determinedby the shape of the pulse.
They can be calculated in the frequency or time domain.
Ci Total capacitance at the input node(detector capacitance + input capacitance of preamplifier + stray capacitance + … )
• Current noise contribution increases with T
• Voltage noise contribution decreases with increasing T
Only for “white” voltage noise sources + capacitive load
“1/f ” voltage noise contribution constant in T
s
vniisnn T
FvCFTiQ 2222 +=
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
1. Equivalent Noise Charge vs. Pulse Width
Current Noise vs. T
Voltage Noise vs. T
Total Equivalent Noise Charge
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
2. Equivalent Noise Charge vs. Detector Capacitance (Ci = Cd + Ca)
If current noise in2FiT is negligible
↑ ↑ input shaper stage
Zero intercept
1
)( 222
TFvCCTFiQ vnadinn ++=
1
)(
12
222
2
TFvCCTFi
TFvC
dC
dQ
vnadin
vnd
d
n
++=
2 T
Fv
dC
dQ vn
d
n ⋅≈
/ 0
TFvCQ vnaCnd
==
Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping
Noise slope is a convenient measure to compare preamplifiers andpredict noise over a range of capacitance.
Caution: both noise slope and zero intercept depend onboth the preamplifier and the shaper
Same preamplifier, but different shapers:
Caution: Current noise contribution may be negligible at highdetector capacitance, but not for Cd=0.
/ 2220
TFvCTFiQ vnainCnd
+==