Frequency mixer

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Image:Frequency mixer.gif In telecommunication, a mixer is a nonlinear circuit or device that accepts as its input two different frequencies and presents at its output (a) a signal equal in frequency to the sum of the frequencies of the input signals, (b) a signal equal in frequency to the difference between the frequencies of the input signals, and, if they are not filtered out, (c) the original input frequencies.

An application of a mixer is a product detector.

1 Mathematical mechanism
- 1.1 Mathematics of the practicalities
2 Spurious signals
3 See also

[edit] Mathematical mechanism

The two frequencies that are to be mixed are, in reality, sinusoidal voltage waves. They can be represented as:

where

<math>v_1, v_2\,</math> represent the two varying voltages
<math>A_1, A_2\,</math> represent the respective maximum voltages (amplitudes)
<math>f_1, f_2\,</math> represent the two frequencies in hertz
<math>t\,</math> represents time

If a way can be found to multiply these two signals by each other at each instant in time, then the following trigonometric identity could be applied:

<math>\sin(A) \cdot \sin(B) \equiv \frac{1}{2}\left[\cos(A-B)-\cos(A+B)\right]\,</math>

We get:

<math>v_1 \cdot v_2 = \frac{A_1 A_2}{2}\left[\cos(2\pi[f_1-f_2]t)-\cos(2\pi[f_1+f_2]t)\right]\,</math>

So, the sum (<math>f_1 + f_2\,</math>) and difference (<math>f_1 - f_2\,</math>) frequencies are visible as required.

[edit] Mathematics of the practicalities

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The next question is, how to achieve this multiplication? There are complex circuits that tackle this question with increasing accuracy, but the simplest answer is so simple that it is also worth some analysis. It is to use a forward-biased semiconductor diode.

A diode is a non-linear device. Almost any device whose output changes non-linearly with respect to changes in its input could form the basis of a mixer. Many other semiconductor devices can also fulfill this criterion in different ways.

From the diode page, the I-V equation for an ideal diode is:

<math>I=I_\mathrm{S} \left( {e^{qV_\mathrm{D} \over nkT}-1} \right)\,</math>

From the Taylor series page, we can expand the exponential function as below:

<math>e^{x} = \sum^{\infin}_{n=0} \frac{x^n}{n!}\,</math> or

Now, we are going to start simplifying things (without forgetting that we have done so!)

First we apply a small voltage to a diode that represents our two sine waves added together: <math>v_1 + v_2\,</math>, then we generate a second voltage proportional to the current that flows through the diode (a simple resistor will do this, according to ohm's law).

According to the Taylor series expansion, the second, output voltage from a diode mixer will be related to the following:

<math>v_\mathrm{o} = 1+(v_1+v_2)+\frac{(v_1+v_2)^2}{2!}+\frac{(v_1+v_2)^3}{3!} + \dots\,</math>

The terms represent

1, a DC shift, which we shall ignore
The original two signals, which we expected and shall ignore
a square-law signal: the square of the sum
signals equivalent to the cube and higher powers.

We said this was going to be a small signal, compared to the other voltages around – like the 0.6 V forward bias that the diode expects, etc. With that in mind, we are going to ignore all cube and higher power terms too for now.

Also ignoring the constant divisor, the square of the sum term expands out to:

So, among other things, we have achieved our goal to multiply the two signals: we have <math>2 v_1 v_2\,</math>.

[edit] Spurious signals

Main article: Spurious emission

As shown in the previous section, every multiplication produces sum and difference frequencies. From the first two terms alone we can expect signal at the following frequencies: <math>f_1, f_2, 2f_1, 2f_2, f_1+f_2\,</math> and <math>f_1-f_2\,</math>.

If <math>f_1\,</math> and <math>f_2\,</math> are both large and relatively close in value, then by far the lowest of these will be the last, the frequency difference signal. This is the one that is almost exclusively selected in modern, low cost radio receivers that are likely to use a simple diode mixer.

It must be remembered that we ignored the cube and all higher order terms earlier. These will produce a plethora of other high frequencies, and a few not so high. Any of these could slip into or break into the passband of the low-cost filters that would follow this diode mixer and it is these that set the main performance limitations of this approach.