# Is there a way to calculate modulation depth after the modulation process?

Suppose I have some amplitude modulated carrier wave $x$. If I have no access to the original message wave nor the unmodulated carrier, is it some how possible to calculate what is the modulation depth of $x$?

If I have understood this correctly, amplitude modulation depth is given by $M/A$, where $M$ is peak amplitude of the message wave and $A$ is the peak amplitude of the carrier wave. From $x$ I will know that $M/A$ equals to some $\lambda$ (when I check its peak amplitude), but M and A could be anything.

Please correct me if I have understood something incorrectly.

• You can do this quite easily, but only if you know that the modulating function reaches its peak value at some point within your sample interval. Jul 13, 2013 at 14:49
• Well, say that the modulating signal is a sine wave with an unknown frequency. Jul 13, 2013 at 15:05

For a sinusoidal signal $x(t) = B\sin(\omega t + \theta)$, the envelope of the AM signal has maximum and minimum values $$A_{\max} = A_c(1+mB), \qquad A_{\min} = A_c(1-mB)\tag{1}$$ where $m \leq B^{-1}$ to ensure that $A_{\min} \geq 0$, and the modulation index (commonly expressed as a percentage) is $M = 100mB\%$. It follows straightforwardly that $$A_c = \frac{A_\max + A_\min}{2}, \qquad mB = \frac{A_\max -A_\min}{A_\max + A_\min}. \tag{2}$$ If $x(t)$ is a sum of sinusoids, then $(1)$ and $(2)$ still apply with the obvious change that $B$ is replaced by $x_\max$, the maximum value of $x(t)$. Note that $x_\min = -B$.
The above is all fine and dandy for the typical modulating signals that are encountered in AM applications. However, things get more complicated if we consider arbitrary (not necessarily sums-of-sinusoids) signals $x(t)$ as modulating signals. If $x(t)$ has a DC component, then there is effectively an increase (or perhaps worse, a decrease) in the nominal carrier amplitude and carrier power. Similarly, if $x_\max \neq |x_\min|$, or $x_\max < 0$ or $x_\min > 0$ (note that the last two cases also mean that $x(t)$ has a DC component), the meaning of modulation index is less clear. Note also that for arbitrary signals, it is not necessarily true that the DC component has value $\frac{x_\max+x_\min}{2}$, and so measuring just $A_\max$ and $A_\min$ is not necessarily going to provide all the information needed to determine the modulation index.