Digital Signal Processing Reference
In-Depth Information
Closed-loop second-order frequency dependent distortion
harm
2
,y
(j
2
ω)
fund
y
(jω)
hd
2,cl
(jω)
=
(A.45)
1
F(j
2
ω)
1
F(
j
ω
)
=
hd
2,ol
·
·
·
1
+
F(
j
ω)a
1
H
1
+
F(j
2
ω)a
1
H
(
3
)
fund
z
(jω)
fund
y
(jω)
(
1
)
tf
(jω)
v
in
→
x
(
2
)
tf
(j
2
ω)
d
→
y
Note that when the
F(jω)
filter characteristic is chosen equal to unity, exactly
the same result as for the frequency-independent system in (A.24) is obtained.
Furthermore, three separate factors play an important role in the distortion
characteristic of (A.45). The first factor (A.45-1) is the transfer characteristic
from the input signal
v
in
to the input node
x
of the amplifier. At low frequen-
cies, below the first pole of
F(jω)
, it follows that the signal swing at node
x
is actively suppressed by the loop gain of the system. Secondly (A.45-2),
the closed loop system under consideration provides an additional level of dis-
tortion suppression: harmonic components that still emerge at the output of
the amplifier are additionally suppressed by the transfer function from node
d
to node
y
before they appear at the output of the system. Finally, it is im-
portant to recognize that also the fundamental frequency component plays a
role in the harmonic distortion expression. The last factor (A.45-3), takes into
account that the fundamental frequency component is affected by the
F(jω)
filter characteristic.
1
F(jω)
=
(A.46)
1
+
jω/ω
p
1
hd
2,ol
1
+
jω/ω
p
1
1
+
jω/ω
p
1
hd
2,cl
(jω)
=
a
1
H)
2
·
jω/
ω
p
1
(
1
a
1
H)
·
jω/
ω
p
1
(
1
+
a
1
H)
(
1
+
+
+
1
1
+
2
When the role of the general filter characteristic
F(jω)
is filled in by an el-
ementary single-pole filter (A.46), the second-order distortion characteristic
shows a double zero at the filter's cut-off frequency
ω
p
1
. As a consequence,
the linearity performance of the closed-loop system starts to
degrade
at a
rate of 20 dB
/
dec for frequencies beyond this pole. The hd
2,cl
characteris-
tic also exhibits two non-coinciding poles. One pole is located at half the cut-
off frequency of the closed loop system (
2
a
1
Hω
p
1
) and is due to the fact that
second-order harmonic components have twice the fundamental frequency. As
a result, the corner frequency of the transfer function from node
d
to the output
