Digital Signal Processing Reference
In-Depth Information
Closed-loop third-order frequency dependent distortion
part 1/2
harm
3
,y
(j
3
ω)
fund
y
(jω)
hd
3,cl
(jω)
=
(A.47)
2
1
F(j
3
ω)
1
F(
j
ω
)
=
hd
3,ol
·
·
·
1
+
F(
j
ω)a
1
H
1
+
F(j
3
ω)a
1
H
(
3
)
fund
z
(jω)
fund
y
(jω)
(
2
)
tf
(j
3
ω)
d
→
y
(
1
)
tf
(jω)
v
in
→
x
This time, the only noticeable difference with the hd
2
calculations is the
quadratic dependency of hd
3
on the input signal level. This is taken into ac-
count by the square of the transfer function from the input of the system
v
in
to the input node
x
of the gain stage. Intuitively, one can predict that the extra
factor of distortion suppression will only be of any significant importance in
the lower frequency region of the transfer function. Starting from the first pole
ω
p
1
in the open-loop system, third-order harmonic suppression will degrade
much faster than it is the case for hd
2
. This effect can easily be verified if
F(jω)
in the generic formula of (A.47) is replaced by a first-order filter. Three
concurrent zeroes appear at the 3 dB cut-off frequency
ω
p
1
of the gain stage in
the forward path (A.48):
1
F(jω)
=
(A.48)
1
+
jω/ω
p
1
2
·
hd
3,ol
1
+
jω/ω
p
1
1
+
jω/ω
p
1
hd
3,cl
(jω)
=
a
1
H)
3
·
jω/
ω
p
1
(
1
a
1
H)
jω/
ω
p
1
(
1
+
a
1
H)
(
1
+
+
+
1
1
+
3
The attentive reader may have noticed that there is a difference between fre-
quency dependent harmonic distortion and frequency dependent intermodu-
lation. Depending on the particular harmonic component under examination,
Formula (A.47) should be adjusted to the correct frequency.
For example, in case of third-order distortion, the harmonic frequency compo-
nent is located at three times the fundamental frequency. In Formula (A.47),
this is already taken into account by the transfer function from node
d
to out-
put node
y
, which is evaluated at three times the fundamental frequency. This
makes that the expression for hd
3
is not entirely correct if employed to cal-
culate third-order intermodulation distortion (im
3
). Suppose the input signal
contains frequency components at
ω
1
and at
ω
2
. Third-order intermodulation
will generate spurious frequencies at both 2
ω
1
−
ω
2
and 2
ω
2
−
ω
1
. If the two
