452
V. Radeka
I 1 , I 2 , I 3 are the noise integrals for the series (voltage) white noise and the 1/f
noise, and for the parallel (current) noise, respectively. The integrals are derived in
the time domain from Campbell’s theorem Eq. (10.3) and expressed in the frequency
domain using Parseval’s theorem [9],
I 1 =
∞
−∞
w
(t)
2 dt =
1
2π
∞
−∞
|H (j ω)|
2 ω
2 dω =
A 1
τ
,
(10.6)
I 2 =
∞
−∞
w
(1/2) (t)
2
dt =
1
2π
∞
−∞
|H (j ω)|
2 ωdω = A 2 ,
(10.7)
I 3 =
∞
−∞
[w(t)]
2 dt =
1
2π
∞
−∞
|H (j ω)|
2 dω = A 3 τ,
(10.8)
where τ is the time width parameter of the weighting function, either the peaking
time of the function, or some characteristic time constant of the filter implemented
in hardware (or software in case of digital filtering).
Noise contributions for both types of white noise due to various segments
(piece-wise linear approximation) of the weighting function are shown in Fig.
10.13 (expressions for integrals I 1 and I 3 ). In these calculations, either the impulse
response h(t) of the system or the weighting function w(t) (the mirror image of the
impulse response) can be used for time-invariant systems. For time-variant (gated or
switched) systems, only a weighting function describes the performance correctly,
while an apparent impulse response (waveform at the output) is not correct and can
be misleading. Steepest parts of the weighting function contribute most to ENC s ,
as they correspond to larger bandwidth. Flat parts do not contribute anything. In
contrast, ENC p is largely proportional to the width of the weighting function where
it has any significant value.
A bipolar weighting function, i.e., impulse response h(k), as shown in Fig. 10.13,
with equal lobes would result in square root of two higher ENC than for a unipolar
function (single lobe). If the amplitude of the second lobe is less than one half, its
rms noise contribution becomes small (<12%).
The half-order integral I 2 for 1/f noise is not amenable to such a simple
interpretation, and it will be discussed in Sect. 10.4.6.
Equations (10.6, 10.7, and 10.8) provide an insight into the general behaviour of
signal processing systems with respect to noise. The ENC due to the series white
amplifier noise is proportional to the slope (~1/t) of the weighting function and
therefore proportional to the bandwidth of the system. The ENC due to parallel
white noise is proportional to the width of the weighting function and therefore to
the overall integration time. If the weighting function form remains constant the
ENC due to 1/f noise is independent of the width of the weighting function, since
the ratio of the high frequency cutoff and low frequency cutoff remains constant,
V. Radeka
I 1 , I 2 , I 3 are the noise integrals for the series (voltage) white noise and the 1/f
noise, and for the parallel (current) noise, respectively. The integrals are derived in
the time domain from Campbell’s theorem Eq. (10.3) and expressed in the frequency
domain using Parseval’s theorem [9],
I 1 =
∞
−∞
w
(t)
2 dt =
1
2π
∞
−∞
|H (j ω)|
2 ω
2 dω =
A 1
τ
,
(10.6)
I 2 =
∞
−∞
w
(1/2) (t)
2
dt =
1
2π
∞
−∞
|H (j ω)|
2 ωdω = A 2 ,
(10.7)
I 3 =
∞
−∞
[w(t)]
2 dt =
1
2π
∞
−∞
|H (j ω)|
2 dω = A 3 τ,
(10.8)
where τ is the time width parameter of the weighting function, either the peaking
time of the function, or some characteristic time constant of the filter implemented
in hardware (or software in case of digital filtering).
Noise contributions for both types of white noise due to various segments
(piece-wise linear approximation) of the weighting function are shown in Fig.
10.13 (expressions for integrals I 1 and I 3 ). In these calculations, either the impulse
response h(t) of the system or the weighting function w(t) (the mirror image of the
impulse response) can be used for time-invariant systems. For time-variant (gated or
switched) systems, only a weighting function describes the performance correctly,
while an apparent impulse response (waveform at the output) is not correct and can
be misleading. Steepest parts of the weighting function contribute most to ENC s ,
as they correspond to larger bandwidth. Flat parts do not contribute anything. In
contrast, ENC p is largely proportional to the width of the weighting function where
it has any significant value.
A bipolar weighting function, i.e., impulse response h(k), as shown in Fig. 10.13,
with equal lobes would result in square root of two higher ENC than for a unipolar
function (single lobe). If the amplitude of the second lobe is less than one half, its
rms noise contribution becomes small (<12%).
The half-order integral I 2 for 1/f noise is not amenable to such a simple
interpretation, and it will be discussed in Sect. 10.4.6.
Equations (10.6, 10.7, and 10.8) provide an insight into the general behaviour of
signal processing systems with respect to noise. The ENC due to the series white
amplifier noise is proportional to the slope (~1/t) of the weighting function and
therefore proportional to the bandwidth of the system. The ENC due to parallel
white noise is proportional to the width of the weighting function and therefore to
the overall integration time. If the weighting function form remains constant the
ENC due to 1/f noise is independent of the width of the weighting function, since
the ratio of the high frequency cutoff and low frequency cutoff remains constant,