# 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. – Paul R Jul 13 '13 at 14:49
• Well, say that the modulating signal is a sine wave with an unknown frequency. – wireless Jul 13 '13 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.