Digital Signal Processing Reference
In-Depth Information
v
in
x
z
y
a(x)
F(j
ω)
H
Figure A.10.
Closed-loop system embedding a single-pole amplifier. Remark that the pole
of the active element is located at its output. For other configurations, the
frequency dependent distortion calculations should be adjusted accordingly.
To start with simplicity, suppose that the active element in the forward path
can be split in a frequency dependent part
F(jω)
and a nonlinear part
a(x)
as
is shown in Figure A.10. If only the first-order linear gain coefficient
a
1
of the
amplifier is considered, the problem is reduced to an lti-system of which the
transfer function can be easily determined. Apart from the closed-loop lti dis-
tortionless transfer function from above, two extra transfer characteristics are
important for the derivation of the frequency dependent distortion behaviour.
The transfer characteristic from the input signal
v
in
to the signal at input node
x
of the active element is given by (A.36):
1
=
tf
(jω)
v
in
→
x
(A.36)
1
+
F(jω)a
1
H
It follows that the amplitude at node
x
is reduced by the loop-gain of the sys-
tem. A reduction of the signal swing at the input node of the active element
has a large impact on the distortion produced by the amplifier. For example,
consider the case of a single-pole amplifier with a pole at frequency
ω
0
.The
transfer function from
v
in
to node
x
will show a zero at
ω
z
=
ω
p
1
and a pole at
ω
p
=
ω
p
1
a
1
H
(A.37):
1
F(jω)
=
1
+
jω/ω
p
1
1
1
+
jω/ω
p
1
tf
(jω)
v
in
→
x
=
(A.37)
1
+
a
1
H
1
+
jω/
[
ω
p
1
(
1
+
a
1
H)
]
For a large loop gain and low frequencies, the linearity of the closed-loop sys-
tem is thus better than that of the amplifier in an open-loop setup. Unfortu-
nately, at the first zero in the open-loop transfer characteristic, the excess gain
in the loop is gradually reduced, resulting in a degradation of the linearity
performance of the closed-loop system. When the excess loop gain eventually
drops to 0, it follows from Equation (A.36) that the amplitude at node
x
is equal
to the input amplitude. At this point, the linearity will be no better than this of
the open-loop amplifier. The entire process is visualized in Figure A.11, along
