Editing Heterodyne (section)

===Output of a mixer===
To demonstrate mathematically how a nonlinear component can multiply signals and generate heterodyne frequencies, the nonlinear function <math>F</math> can be expanded in a [[power series]] ([[MacLaurin series]]):

:<math>\ F(v) = \alpha_1 v + \alpha_2 v^2 + \alpha_3 v^3 + \cdots\ </math>

To simplify the math, the higher order terms above {{math|''α''<sub>2</sub>}} are indicated by an ellipsis (<math>\ \cdots\ </math>) and only the first terms are shown. Applying the two sine waves at frequencies {{math|1=''ω''<sub>1</sub> =&nbsp;2{{pi}}''f''<sub>1</sub>}} and {{math|1=''ω''<sub>2</sub> =&nbsp;2{{pi}}''f''<sub>2</sub>}} to this device:

:<math>\ v_\mathsf{out} = F\Bigl(A_1 \cos(\omega_1 t) + A_2 \cos(\omega_2 t) \Bigr)\ </math>

:<math>\ v_\mathsf{out} = \alpha_1 \Bigl( A_1 \cos(\omega_1 t) + A_2 \cos(\omega_2 t) \Bigr) + \alpha_2 \Bigl( A_1 \cos(\omega_1 t) + A_2 \cos(\omega_2 t) \Bigr)^2 + \cdots\ </math>

:<math>\ v_\mathsf{out} = \alpha_1 \Bigl( A_1 \cos(\omega_1 t) + A_2 \cos(\omega_2 t)\Bigr) + \alpha_2 \Bigl( A_1^2 \cos^2(\omega_1 t) + 2 A_1 A_2 \cos(\omega_1 t)\ \cos(\omega_2 t) + A_2^2 \cos^2(\omega_2 t) \Bigr) + \cdots\ </math>

It can be seen that the second term above contains a product of the two sine waves. Simplifying with [[trigonometric identity|trigonometric identities]]:

:<math>
\begin{align}
v_\mathsf{out} = {} & \alpha_1 \Bigl( A_1 \cos(\omega_1 t) + A_2 \cos(\omega_2 t) \Bigr) \\
& {} + \alpha_2 \Bigl( \tfrac{1}{2} A_1^2 [ 1 + \cos(2 \omega_1 t) ] + A_1 A_2 [\cos(\omega_1 t - \omega_2 t ) + \cos (\omega_1 t + \omega_2 t) ] + \tfrac{1}{2} A_2^2 [ 1 + \cos(2 \omega_2 t) ] \Bigr) + \cdots
\end{align}
</math>

Which leaves the two heterodyne frequencies among the many terms:
:<math>\ v_\mathsf{out} = \cdots + \alpha_2 A_1 A_2 \cos (\omega_1 - \omega_2 )t + \alpha_2 A_1 A_2 \cos (\omega_1 + \omega_2 ) t + \cdots\ </math>
along with many other terms not shown.

In addition to components with frequencies at the sum {{math|''ω''<sub>1</sub>&nbsp;+&nbsp;''ω''<sub>2</sub>}} and difference {{math|''ω''<sub>1</sub>&nbsp;−&nbsp;''ω''<sub>2</sub>}} of the two original frequencies, shown above, the output also contains sinusoidal terms at the original frequencies and terms at multiples of the original frequencies {{nobr|{{math|2 ''ω''<sub>1</sub>}} ,}} {{nobr|{{math|2 ''ω''<sub>2</sub>}} ,}} {{nobr|{{math|3 ''ω''<sub>1</sub>}} ,}} {{nobr|{{math|3 ''ω''<sub>2</sub>}} ,}} etc., called ''[[harmonics]]''. It also contains much more complicated terms at frequencies of {{nobr|{{math|''M ω''<sub>1</sub> + ''N ω''<sub>2</sub>}} ,}} called [[intermodulation product]]s. These unwanted frequencies, along with the unwanted heterodyne frequency, must be removed from the mixer output by an [[electronic filter]], to leave the desired heterodyne frequency.