Frequency mixer

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Frequency Mixer Symbol.

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In telecommunication, a mixer is a nonlinear or time-varying circuit or device that accepts as its input two different frequencies and presents at its output a mixture of signals at several frequencies:

the sum of the frequencies of the input signals
the difference between the frequencies of the input signals
both original input frequencies — these are often considered parasitic and are filtered out in subsequent filter stages.

This nonlinear effect can be created by using a nonlinear electrical component, such as a diode. The time-varying effect can be created using a multiplier circuit such as a Gilbert Cell or passive switches.

The manipulations of frequency performed by a mixer can be used to move signals between bands, or to encode and decode them. One other application of a mixer is as a product detector.

1 Mathematical description
- 1.1 Multiplication implementation
- 1.2 Output
2 See also
3 External links

[edit] Mathematical description

The input signals are, in the simplest case, sinusoidal voltage waves, representable as

$v_i(t) = A_i \sin 2\pi f_i t\,$

where each A is an amplitude, each f is a frequency, and t represents time. (In reality even such simple waves can have various phases, but that does not enter here.) One common approach for adding and subtracting the frequencies is to multiply the two signals; using the trigonometric identity

$\sin(A) \cdot \sin(B) \equiv \frac{1}{2}\left[\cos(A-B)-\cos(A+B)\right]$

we have

$v_1(t)v_2(t) = \frac{A_1 A_2}{2}\left[\cos 2\pi(f_1-f_2)t-\cos 2\pi(f_1+f_2)t\right]$

where the sum ( $f 1 + f 2$ ) and difference ( $f 1 - f 2$ ) frequencies appear. This is the inverse of the production of acoustic beats.

[edit] Multiplication implementation

It has been suggested that this article or section be merged with electronic mixer. (Discuss)

There are various ways of multiplying voltages, many of them quite sophisticated. However, as an example, a simple technique involving a diode can be described. The importance of the diode is that it is non-linear (or non-Ohmic), which means its response (current) is not proportional to its input (voltage). The diode therefore does not reproduce the frequencies of its driving voltage in the current through it, which allows the desired frequency manipulation. Certain other non-linear devices could be utilized similarly.

The current I through an ideal diode as a function of the voltage V across it is given by

$I=I_\mathrm{S} \left( e^{qV_\mathrm{D} \over nkT}-1 \right)$

where what is important is that V appears in e's exponent. The exponential can be expanded as

$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$

and can be approximated for small x (that is, small voltages) by the first few terms of that series:

$e^x-1\approx x + \frac{x^2}{2}$

Suppose that the sum of the two input signals $v 1 + v 2$ is applied to a diode, and that an output voltage is generated that is proportional to the current through the diode (perhaps by providing the voltage that is present across a resistor in series with the diode). Then, disregarding the constants in the diode equation, the output voltage will have the form

$v_\mathrm{o} = (v_1+v_2)+\frac12 (v_1+v_2)^2 + \dots$

The first term on the right is the original two signals, as expected, followed by the square of the sum, which can be rewritten as $(v_1+v_2)^2 = v_1^2 + 2 v_1 v_2 + v_2^2$ , where the multiplied signal is obvious. The ellipsis represents all the higher powers of the sum which we assume to be negligible for small signals.

[edit] Output

As every multiplication produces sum and difference frequencies, from the quadratic term of the series we expect to find signals at frequencies $2 f 1$ and $2 f 2$ from $v_1^2$ and $v_2^2$ , and $f 1 + f 2$ and $f 1 - f 2$ from the $v 1 v 2$ term. Often $f_1,f_2\gg|f_1-f_2|$ , so the difference signal has a much lower frequency than the others; extracting this distinct signal is often the principal purpose of using a mixer in such devices as radio receivers.

The other terms of the series give rise to a number of other, weaker signals at various frequencies which act as noise for the desired signal; they may be filtered out downstream to an extent, but sensitive applications will require cleaner output and thus a more complicated design.