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

as:
B.1.b “Duo-tone” input signal
Now, still considering a non-linear system with the polynomial characteristic defined
2 3
y = k + k x + k x + k x + ...
(B.8)
0
1
Assuming the case of a signal composed of two sinusoidal waves:
2
3
x t)
= X cos( ω t)
+ X cos( ω t)
= X cos( φ ) + X cos( φ ) . (B.9)
( 1 1 2 2 1 1 2 2
The output signal is obtained combining these last two equations. Let’s have a look at
the result term by term. The linear term is given by:
k x = k X φ ) + k X cos( φ ) . (B.10)
1
1
1 cos( 1 1 2 2
Then, the amplitudes of the sinusoidal waves are merely multiplied by the gain k1.
Thus,
[ ] 2
X cos( φ ) X cos( φ )
2
k 2 x k2
1 1 +
= (B.11)
2 2
2 2
[ X ( φ ) + 2X
X cos( φ ) cos( φ ) X cos ( φ ) ]
2
k 2 x = k2
1 cos 1 1 2 1 2 + 2
2
2
- 190 -
2
(B.12)
2
2
2
2
2 k2
X 1 k2
X 2 k2
X 1 k2
X 2
k 2x
= + + cos( 2φ1
) + cos( 2φ2
) + k2
X 1X
2[
cos( φ1
+ φ2
) + cos( φ1
−φ
2 ) ] (B.13)
2 2 2
2
constant term second harmonics Second order intermodulation
products
These different terms are plotted versus frequency in Figure 199.
It is worth noticing that a constant term is present at DC. There are also second order
harmonic frequencies, but what is interesting to highlight is the presence of second order
intermodulation products. Indeed, the output signal contains two equal amplitude components
for which the frequencies are the sum and the difference of the input frequency. They are
called second order intermodulation products (IM2).