III. Gm-C Filtering - Epublications - Université de Limoges

Now, let’s have a look to the cubic term.
[ ] 3
3
k 3 x k3
X 1 cos( φ1) + X 2 cos( φ2
)
3 [ X
3
2
( φ ) + 3X
X
2
2
2
cos ( φ ) cos( φ ) + 3X
X cos( φ ) cos ( φ )
3 3
X cos ( φ ) ]
= (B.14)
3
k 3x
= k3
1 cos 1 1 2 1 2 1 2 1
2 + 2 2
Thus,
k
3
x
3
3k
3 ⎛
=
1
2 ⎜ X X
⎝
2
2
+
X
3
k3
X 1 k3
X
+ cos( 3φ1
) +
4
4
2
3k3
X 1 X 2
+
4
1
2
3k3
X 1X
2
+
4
2
3
1
2
3
2
⎞ 3k
⎟
⎟cos(
φ1
) +
⎠ 2
cos( 3φ
)
[ cos( 2φ
+ φ ) + cos( 2φ
− φ ) ]
2
[ cos( 2φ
+ φ ) + cos( 2φ
− φ ) ]
1
2
1
2
3
⎛
⎜ X
⎝
2
1
2
1
X
2
+
- 191 -
X
2
3
2
⎞
⎟
⎟cos(
φ2
)
⎠
These different terms are also plotted versus frequency in Figure 199.
fundamental
3 rd harmonics
(B.15)
(B.16)
3 rd order IM products
This latter equation clearly shows that third order non-linearity produces a
fundamental component as well as third order harmonic frequencies. Moreover, we can notice
the presence of third order intermodulation products at 2f1 + f2, 2f2 + f1, 2f1 - f2 and 2f2 – f1.
When f1 and f2 are close, the components at frequencies 2f1 - f2 and 2f2 – f1 can be
particularly bothersome because they are very close to initial frequencies.
Figure 199 illustrates the spectrum which is composed of second and third order
intermodulation products and also of harmonic frequencies.
Figure 199. Output spectrum of a two-tone input signal transformed by a non-linear system