RF Mixing Theory: RF Multiplication Mathematics

An understanding of the basic theory of RF mixing or RF multiplication can be enhanced by looking at the mathematics.

RF Mixers & Mixing Tutorial Includes:
RF mixing basics     Theory & math     Specs & data     Transistor mixer     FET mixer     Double balanced mixer     Gilbert cell mixer     Image reject mixer

RF mixers are classic RF and microwave building blocks. They enable RF signals to be translated from one frequency to another, ideally with no effect on the amplitude and frequency components of the signal, i.e. with no distortion of the required elements.

Basic RF mixer / multiplication theory

RF mixers are non-linear circuits and in this way the effect of one input signal affects the other and vice versa.

The non-linear response results in new signals being generated. There will be a series of signals at the output that will contain multiples of the input signals, i.e. harmonics plus the sum and difference signals of all the frequencies.

$fout=|nf1±mf2|$

The mathematical equation for the mixing / multiplication can seen in a more visual manner in the diagram below.

As a result the RF multiplication shows that the is an infinite series of discrete outputs extending out either side.

The higher order mixing output frequencies are lower in amplitude - the actual amplitude of the higher order signals is determined by the mixing circuit. Better mixers produce lower high order outputs. However in every case, the second order responses will have the highest amplitudes.

Ideal RF mixer operation & theory

In an ideal mixer, the outputs obtained would only be (f1 + f2) and (f1 – f2). We can look at the mathematics behind the theory to see how this would come about.

The two inputs to the RF mixer can be considered to be two cosine waves with angular frequencies of ω1 and ω2.

Using basic mathematical expansions, the two cosines can be multiplied together and the resultant formula expanded:

$\mathrm{cos}\left(\omega 1\right)\mathrm{cos}\left(\omega 2\right)=\frac{\mathrm{cos}\left(\omega 1+\omega 2\right)}{2}+\frac{\mathrm{cos}\left(\omega 1-\omega 2\right)}{2}$

These mathematics and theory above refer to a perfect multiplying environment within the RF mixer. In reality mixers to not conform to perfection and therefore other terms exist meaning that additional frequencies are generated beyond the sum and difference frequencies.

The levels are dependent on the particular device or circuit, and often high performance RF mixers have levels of the unwanted mix products specified within the datasheet. This can be critical for some applications.