Digital Signal Processing Reference
In-Depth Information
a
2
a
1
·
harm
2,x
(j
2
ω)
im
2,z
(j
2
ω)
≈
harm
2,x
(j
2
ω)
v
in
≈
im
2,OL
·
(A.53)
With the knowledge of the second-order intermodulation distortion ratio at
node
d
, the effective amplitude of the intermodulation product at this node can
be calculated. Subsequently, the value of the intermodulation product at the
output of the system is found by using the transfer function from node
d
to the
output
y
. Remember that the latter transfer characteristic should be evaluated
at
j
3
ω
, which is the beat frequency resulting from the second-order intermodu-
lation product between the fundamental signal frequency and the second-order
distortion component which is aimlessly wandering around the loop (A.54):
harm
3,d
(j
3
ω)
=
im
2,z
(j
2
ω)
·
fund
z
(jω)
(A.54)
harm
3,y
(j
3
ω)
=
im
2,z
(j
2
ω)
·
fund
z
(jω)
·
tf
(j
3
ω)
d
→
y
a
1
v
in
F(j
3
ω)
=
im
2,z
(j
2
ω)
·
+
F(jω)a
1
H
·
+
F(j
3
ω)a
1
H
The frequency dependent third-order distortion characteristic in (A.55) is de-
fined as the ratio of the third-order harmonic component (harm
3,y
)totheam-
plitude of the fundamental component (fund
y
).
1
1
harm
3,y
(j
3
ω)
fund
y
(jω)
hd
3,cl
(jω)
=
1
F(jω)a
1
H
F(jω)a
1
v
in
+
=
harm
3,y
(j
3
ω)
·
(A.55)
Finally, the flattened expression for hd
3,cl
(jω)
is deduced by combining
Equations (A.50)-(A.55). The reader should keep in mind that the expression
given below denotes the third-order distortion characteristic of a closed-loop
system which includes an amplifier with only
second-order
nonlinearities. Af-
ter some simplifications formula (A.56) is obtained:
Closed-loop third-order frequency dependent distortion
part 2/2
harm
3,y
(j
3
ω)
fund
y
(jω)
hd
3,cl
(jω)
=
(A.56)
hd
2
,ol
·
im
2
,ol
·
a
1
[1
HF(j
2
ω)
F(j
3
ω)
1
F(
j
ω
)
=
·
·
·
F(jω)a
1
H
]
2
1
+
F(j
2
ω)a
1
H
1
+
F(j
3
ω)a
1
H
+
fund
z
(jω)
fund
y
(jω)
tf
(j
2
ω)
d
→
x
tf
(j
3
ω)
d
→
y
