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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?

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  • $\begingroup$ Google or youtube "wave superposition". $\endgroup$
    – Kyle B
    Mar 8, 2022 at 4:15
  • $\begingroup$ @KyleB the superposition principle doesn't answer this question. Intermodulation distortion requires nonlinearity. $\endgroup$
    – Puk
    Mar 8, 2022 at 4:43

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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 $f$.

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