Digital Signal Processing Reference
In-Depth Information
The most noticeable difference with the hd
3,CL
distortion formula based on the
a
3
-coefficient in (A.47) is the appearance of an additional frequency dependent
factor: tf
(j
2
ω)
d
→
x
. Suppose
F(jω)
is once again represented by a single-
pole low-pass filter with cut-off frequency
ω
p
1
, as shown in (A.57):
1
F(jω)
=
(A.57)
1
+
jω/ω
p
1
1
+
jω/ω
p
1
3
2hd
2,ol
·
a
1
H
1
hd
3,CL
(jω)
=
·
ω
p
1
(
1
+
a
1
H)
2
·
·
1
+
a
1
H )
4
j
2
ω
ω
p
1
(
1
j
3
ω
ω
p
1
(
1
(
1
1
+
1
+
jω
+
+
a
1
H
)
+
a
1
H)
tf
(j
2
ω)
d
→
x
The extra transfer function will introduce a new pole located at
ω
p
1
a
1
H/
2.
Intuitive reasoning on Figure A.15 learns that for increasing frequencies, the
second-order harmonics is indeed increasingly suppressed by the filter, before
being fed back to the input of the system. The net result is that the system con-
tains three zeroes at frequency
ω
p
1
, against four poles around the closed-loop
cut-off frequency. As shown in Figure A.16, the bode plot of the hd
3,cl
(jω)
curve will show a maximum somewhere between the open-loop (
ω
p
1
)and
the closed-loop cut-off frequency (
ω
p
1
a
1
H
). Interestingly enough, the level
of maximum distortion does not depend on the feedback factor
H
, but only on
IM
2
HD
2
|
[
HD
3,CL
(f) [dB]
distortion
y
in
open-loop
ω
p1
a
1
1/H
HARM
2
H
closed-loop
a
1
H
ω
p1
20
HD
2,OL
. ΙΜ
2,OL
0
a
1
H/(1+a
1
H)
4
-22.8dB (at peak level)
First-order decrease of
HD
3
is due to the reduction
of second-order harmonics.
−
20
−
40
10
−1
10
0
10
1
frequency [GHz]
Figure A.16.
Frequency dependent third-order harmonic distortion, induced by
second-order intermodulation beat products between the input signal
and second-order distortion fed back to the input of the system. Note
that the third-order distortion caused by the active element itself is
ignored in this graph.
