Let the modulated AM signal $s\left(t\right)=[A_c+A_{m}x\left(t\right)]cos2\pi f_{c}t$ ……………..... (1)

${s\left(t\right)=A}_c[1+\frac{A_m}{A_c}x\left(t\right)]cos2\pi f_{c}t$

$s\left(t\right)=A_c[1+m_{a}x\left(t\right)]cos2\pi f_{c}t$ ………….….. (2)

Where $m_a=\frac{\left|A_m\right|}{A_c}$ for all $A_c\ge A_m$ and it is called modulation index.

If positive and negative amplitudes are not identical, $A_m$ is the absolute negative peak amplitude.

The percentage of modulation is equal to $\frac{\left|A_m\right|}{A_c}\times 100$

The modulation index is defined as the ratio of non negative maximum amplitude of the modulating signal to the amplitude of the carrier signal. Also it is called depth of modulation. The range of modulation index to recovered the original message signal without distortion from the modulated signal is $0\le m_a\le 1$. We shall, for convenience assume here that $x\left(t\right)$ has been so normalized that $\left|x\left(t\right)\right|\le 1$.

$m_a=\frac{A_{max}+A_{min}}{A_{max}-A_{min}}$ …………… (3)

Where, $A_{max}\wedge A_{min}$ are the maximum and minimum amplitude of the AM signal.

Hence from equation (2) to (3) is the relation between modulated signal amplitude and the modulation index.