Editing Heterodyne (section)

==Mathematical principle==
Heterodyning is based on the [[trigonometric identity]]:

:<math>(\cos\theta_1) (\cos\theta_2) = \tfrac{1}{2}\cos(\theta_1 - \theta_2) + \tfrac{1}{2}\cos( \theta_1 + \theta_2)</math>

The product on the left hand side represents the multiplication ("mixing") of a [[sine wave]] with another sine wave (both produced by [[cosine]] functions). The right hand side shows that the resulting signal is the sum of two [[sinusoidal]] terms, one at the sum of the two original frequencies, and one at the difference, which can be dealt with separately, since their (large) frequency difference makes it easy to cleanly filter out one signal's frequency, while leaving the other signal unchanged.

Using this trigonometric identity, the result of multiplying two cosine wave signals <math>\ \cos\left(2 \pi f_1 t\right)\ </math> and <math>\ \cos\left(2 \pi f_2 t\right)\ </math> at different frequencies <math>\ f_1\ </math> and <math>\ f_2\ </math> can be calculated:

:<math>\cos(2 \pi f_1 t) \cos(2 \pi f_2 t) = \tfrac{1}{2}\cos[2 \pi ( f_1 - f_2) t ] + \tfrac{1}{2}\cos[ 2 \pi ( f_1 + f_2) t ]</math>

The result is the sum of two sinusoidal signals, one at the sum {{math|''f''<sub>1</sub>&nbsp;+&nbsp;''f''<sub>2</sub>}} and one at the difference {{math|''f''<sub>1</sub>&nbsp;−&nbsp;''f''<sub>2</sub>}} of the original frequencies.

===Mixer===
The two signals are combined in a device called a ''[[Frequency mixer|mixer]]''.  As seen in the previous section, an ideal mixer would be a device that multiplies the two signals.  Some widely used mixer circuits, such as the [[Gilbert cell]], operate in this way, but they are limited to lower frequencies.  However, any ''[[linear circuit|nonlinear]]'' electronic component also multiplies signals applied to it, producing heterodyne frequencies in its output—so a variety of nonlinear components serve as mixers. A nonlinear component is one in which the output current or voltage is a [[Linear function|nonlinear function]] of its input. Most circuit elements in communications circuits are designed to be [[Linear circuit|linear]]. This means they obey the [[superposition principle]]; if <math>\ F(v)\ </math> is the output of a linear element with an input of <math>\ v\ </math>:

:<math>\ F(v_1 + v_2) = F(v_1) + F(v_2)\ </math>

So if two sine wave signals at frequencies {{math|''f''<sub>1</sub>}} and {{math|''f''<sub>2</sub>}} are applied to a linear device, the output is simply the sum of the outputs when the two signals are applied separately with no product terms. Thus, the function <math>F</math> must be nonlinear to create mixer products.  A perfect multiplier only produces mixer products at the sum and difference frequencies {{math|(''f''<sub>1</sub>&nbsp;±&nbsp;''f''<sub>2</sub>)}}, but more general nonlinear functions produce higher order mixer products: {{math|''n''&sdot;''f''<sub>1</sub>&nbsp;+&nbsp;''m''&sdot;''f''<sub>2</sub>}} for integers {{math|''n''}} and {{math|''m''}}.  Some mixer designs, such as double-balanced mixers, suppress some high order undesired products, while other designs, such as [[harmonic mixer]]s exploit high order differences.

Examples of nonlinear components that are used as mixers are [[vacuum tube]]s and [[transistor]]s biased near cutoff ([[class C amplifier|class C]]), and [[diode]]s. [[Magnetic core|Ferromagnetic core]] [[inductor]]s driven into [[saturation (magnetic)|saturation]] can also be used at lower frequencies. In [[nonlinear optics]], crystals that have nonlinear characteristics are used to mix [[laser]] light beams to create [[Optical heterodyne detection|optical heterodyne frequencies]].

===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.