Digital Signal Processing Reference
In-Depth Information
components are subsequently transferred to the output of the closed-loop sys-
tem, which is described by the transfer characteristic from node
d
to node
y
(A.38):
F(jω)
tf
(jω)
d
→
y
=
(A.38)
1
+
F(jω)a
1
H
Continuing the example of the single-pole amplifier, the transfer function of the
injected distortion to the output shows a single pole at the cut-off frequency of
the closed-loop system (A.39):
1
=
F(jω)
1
+
jω/ω
p
1
1
1
tf
(jω)
z
→
y
=
(A.39)
1
+
a
1
H
1
+
jω/
[
ω
p
1
(
1
+
a
1
H)
]
Important remark
It is of vital importance to recognize that, starting from the point
where distortion is injected at node
z
in the loop, the transfer function
to the output should be evaluated at the particular frequency of that
distortion component.
For example, if the fundamental frequency of the input signal is
ω
fund
,
nonlinearities will produce a second-order harmonic spur at 2
ω
fund
which is injected at node
d
. From that moment on, all calculations
must be performed at frequency 2
ω
fund
. The frequency conversion
makes the modelling of the system with a transfer function unfeasible,
without falling back on more complicated algorithms such as the
harmonic transfer matrices (htm) method.
Second-order frequency dependent distortion
Putting all pieces together, the calculation of frequency dependent distortion in
the closed-loop system is obtained in a two-step approach. In order not to over-
complicate matters, the calculation for hd
2
is performed first, then followed by
the slightly more complex hd
3
derivation. Using Figure A.12, the first step is
to determine the amplitude of the fundamental frequency at node
z
, which is
the output of the active gain element. The amplitude is easily determined us-
ing the transfer function from the input to node
x
(A.36), multiplied by the
first-order linear gain
a
1
(A.40):
1
fund
z
(jω)
=
v
in
·
+
F(jω)a
1
H
·
a
1
(A.40)
1
