Short note: Modulation index

The modulation index tells how much a carrier parameter is changed by the message.

In AM

Ratio of message amplitude to carrier amplitude:

$$\mu = \frac{A_m}{A_c} = \frac{A_{max} - A_{min}}{A_{max} + A_{min}}$$

• $\mu < 1$: under-modulation, envelope follows the message.
• $\mu = 1$: 100% modulation, maximum efficiency (33.3%) without distortion.
• $\mu > 1$: over-modulation; envelope distortion, extra sidebands, envelope detector fails.

For several tones, $\mu_t = \sqrt{\mu_1^2 + \mu_2^2 + \dots}$.

Power relation: $P_t = P_c(1 + \mu^2/2)$. A higher $\mu$ puts more power in the sidebands and gives a stronger, clearer received signal.

Example: $A_{max} = 8$ V, $A_{min} = 2$ V gives $\mu = 6/10 = 0.6$ (60%).

In FM and PM

$$\beta = \frac{\Delta f}{f_m} \ (\text{FM}), \qquad \beta = k_pA_m \ (\text{PM})$$

$\beta$ is the peak phase deviation in radians; it decides the number of significant sidebands and the bandwidth (Carson's rule $B = 2(\beta + 1)f_m$). Unlike AM, $\beta$ can exceed 1. Example: $\Delta f = 75$ kHz, $f_m = 15$ kHz gives $\beta = 5$.