# What is the physics behind mixing two radio frequencies to produce the sum and difference frequencies?

In studying for my amateur exam I see the general equation $$f = n*f1 +- m*f2$$ where $$m$$ and $$n$$ are integers and $$f$$, $$f1$$, and $$f2$$ are radio frequencies. What are the physics behind that equation?

This is called intermodulation distortion. It occurs when the sum of signals of frequencies $$\omega_1$$ and $$\omega_2$$ is input to a nonlinear system, which in practice could model e.g. a diode or transistor. Suppose the output of the system is simply $$y = f(x)$$, where $$x$$ is the input and $$f$$ is a nonlinear function. Consider the signals $$x_1 = A_1 \cos(\omega_1t)$$ $$x_2 = A_2 \cos(\omega_2t).$$ Using the Taylor expansion of $$f$$, the output of the system for $$x=x_1+x_2$$ is $$f(x_1 + x_2)=f(0)+f'(0)(x_1+x_2)+\frac{f''(0)}{2!}(x_1+x_2)^2+\frac{f'''(0)}{3!}(x_1+x_2)^3+...$$ Notice that once fully expanded, the sum will consist of terms proportional to $$x_1^mx_2^n\propto \cos^m(\omega_1t)\cos^n(\omega_2t)$$ where $$m$$ and $$n$$ are non-negative integers. $$\cos^m(a)$$ can be written in the form $$C_1\cos(a)+C_3\cos(3a) +...C_m(ma)\ \ \ \ \ (\text{for odd }m)$$ $$C_0+C_2\cos(2a) +...C_m(ma)\ \ \ \ \ (\text{for even }m)$$ Consequently $$x_1^mx_2^n\propto \cos^m(\omega_1t)\cos^n(\omega_2t)$$ consists of terms proportional to $$\cos(m'\omega_1t)\cos(n'\omega_2t)=\frac{1}{2}\left[\cos(m'\omega_1+n'\omega_2)t+\cos(m'\omega_1-n'\omega_2)t\right]$$ where $$m'$$ and $$n'$$ are non-negative integers less than or equal to $$m$$ and $$n$$, respectively. In general, all frequencies $$M\omega_1\pm N\omega_2$$ for all non-negative integers $$M$$ and $$N$$ will be present in the output, unless their coefficients happen to be zero for the function $$f$$.