Digital Signal Processing Reference
In-Depth Information
ε
rel
[%]
|TF(f)| [dB]
Α
0
Ηω
p1
ω
p1
30
open-loop
ε
rel
: relative error between
A
0
100
1/H and the closed-loop
transfer function (dashed).
25
80
20
Αbove frequency A
0
Hω
p1
:
1/H
closed-loop
jω
60
15
ε
rel
≈
A
0
H
ε
abs
[dB]
ω
p1
+j
ω
10
40
Below frequency A
0
H
ω
p1
:
5
1+jω/ω
p1
1+A
0
H
20
ε
rel
≈
0
10
−1
10
0
10
1
frequency [GHz]
Figure A.5.
Gain error between the ideal closed-loop transfer function (1
/H
)and
the behaviour of a system with limited gain- and bandwidth-resources
(dashed). The gain error crosses the 50% boundary before the 3 dB fre-
quency of the closed-loop system.
in which the nonlinearities are suppressed is dependent on the frequency and,
as a consequence, are related in some degree to the excess gain of the loop.
As a place to start, consider a generic nonlinear
open-loop
amplifier, of which
the gain factor exhibits both second- and third-order harmonic distortion. The
frequency independent characteristic of the amplifier can be modelled by the
following truncated polynomial (A.19):
a(x)
=
a
0
+
a
1
x
+
a
2
x
2
+
a
3
x
3
+
...
,
(A.19)
where
a
0
is the dc-offset,
a
1
represents the linear gain parameter and coeffi-
cients
a
2
and
a
3
generate the unwanted second- and third-harmonic distortion
components in the output spectrum of the amplifier. When a sinusoidal wave
x
U
cos
(ωt)
is applied to the input, the output signal of the amplifier in
Formula (A.19) is given by (A.20):
=
a
0
+
2
U
2
a
1
+
4
a
3
U
2
U
cos
(ωt)
a
2
3
a(x)
=
+
(A.20)
a
2
a
3
2
U
2
cos
(
2
ωt)
4
U
3
cos
(
3
ωt)
+
+
+
...
The second order harmonic distortion (hd
2
) of this amplifier is defined as the
strength of the second-order component relative to the first-order output signal
