10 Signal Processing for Particle Detectors
459
Since such exact calculations of ENC f for any weighting function can be time
consuming, we emphasize here a simple estimation method, which provides results
sufficiently close to the exact calculations for most purposes. It has been pointed
out by Gatti et al. [28] that the three integrals in Eqs. (10.6, 10.7, and 10.8) have to
satisfy the Cauchy-Schwartz inequality,
∞
−∞
w
(1/2) (t)
2
dt ≤
∞
−∞
w
(t)
2 dt •
∞
−∞
[w(t)]
2 dt
,
(10.19)
that is,
A 2 ≤ (A 1 A 2 )
1/2 .
(10.20)
Thus there is an upper limit to A 2 in relation to A 1 and A 3 which are easily
calculated from Fig. 10.13, or Eqs. (10.6, 10.7, and 10.8). A study of the most
commonly used weighting functions, [28], reveals that A 2 /(A 1 A 3 ) 1/2 falls between
0.64 and 0.87, a spread of less than ±8% in the calculation of rms noise, so that for
the estimation of 1/f noise the following approximate relation can be used,
A 2 ≈ 0.75(A 1 A 3 )
1/2 .
(10.21)
For the trapezoidal weighting function in Fig. 10.19, A 1 = 2 and A 3 = + 2/3,
and the approximation for this case is,
A 2 ≈ 0.75
2
Δ + 2
3
1/2
.
(10.22)
Figure 10.19 shows that this approximation is within a few percent of the exact
analytical solution, Eq. (10.18).
In any noise analysis of charge amplifiers one will have already calculated, or
otherwise determined the values of A 1 and A 3 , so that the information about the
filtering (pulse shaping) effect on the series and parallel white noise will readily
also provide an estimate of the 1/f noise,
ENC
2
f = πC
2
in A f A 2 ≈ πC
2
in A f
0.75
A 1 A 3
.
(10.23)
It is interesting to note that for a Gaussian weighting function A 2 = 1.00, for a
triangular weighting function 0.88, for a fourth order semi-Gaussian 1.02, for CRRC 1.18.
A f is a parameter resulting from a measured spectral density and it does not
contain any specific information about the properties of the amplifying device unless
other parameters are known.
For input transistor optimization a parameter which is to the first order independent of the device dimensions is more useful [29], K f = A f C gs [J]. This constant
459
Since such exact calculations of ENC f for any weighting function can be time
consuming, we emphasize here a simple estimation method, which provides results
sufficiently close to the exact calculations for most purposes. It has been pointed
out by Gatti et al. [28] that the three integrals in Eqs. (10.6, 10.7, and 10.8) have to
satisfy the Cauchy-Schwartz inequality,
∞
−∞
w
(1/2) (t)
2
dt ≤
∞
−∞
w
(t)
2 dt •
∞
−∞
[w(t)]
2 dt
,
(10.19)
that is,
A 2 ≤ (A 1 A 2 )
1/2 .
(10.20)
Thus there is an upper limit to A 2 in relation to A 1 and A 3 which are easily
calculated from Fig. 10.13, or Eqs. (10.6, 10.7, and 10.8). A study of the most
commonly used weighting functions, [28], reveals that A 2 /(A 1 A 3 ) 1/2 falls between
0.64 and 0.87, a spread of less than ±8% in the calculation of rms noise, so that for
the estimation of 1/f noise the following approximate relation can be used,
A 2 ≈ 0.75(A 1 A 3 )
1/2 .
(10.21)
For the trapezoidal weighting function in Fig. 10.19, A 1 = 2 and A 3 = + 2/3,
and the approximation for this case is,
A 2 ≈ 0.75
2
Δ + 2
3
1/2
.
(10.22)
Figure 10.19 shows that this approximation is within a few percent of the exact
analytical solution, Eq. (10.18).
In any noise analysis of charge amplifiers one will have already calculated, or
otherwise determined the values of A 1 and A 3 , so that the information about the
filtering (pulse shaping) effect on the series and parallel white noise will readily
also provide an estimate of the 1/f noise,
ENC
2
f = πC
2
in A f A 2 ≈ πC
2
in A f
0.75
A 1 A 3
.
(10.23)
It is interesting to note that for a Gaussian weighting function A 2 = 1.00, for a
triangular weighting function 0.88, for a fourth order semi-Gaussian 1.02, for CRRC 1.18.
A f is a parameter resulting from a measured spectral density and it does not
contain any specific information about the properties of the amplifying device unless
other parameters are known.
For input transistor optimization a parameter which is to the first order independent of the device dimensions is more useful [29], K f = A f C gs [J]. This constant