# Definition of $\mu$

I have a question as follows. This is not a homework question I just need to clarify my doubt on how this modulation index is defined.

Suppose a 2kHz audio tone having 2V amplitude is to be amplitude modulated on a carrier $x_c(t) = 5\cos(6\pi 10^5t)$ with a modulation index of 0.8. For the resulting AM signal

1. Derive the mathematical expression $$x(t) = A_c[1+\mu x_m(t)]\cos(\omega_c t)$$ $$x(t) = 5[1 + \frac{0.8 * 2}{5}x_m(t)]\cos(2\pi*3 *10^{5}t)$$

is this correct? Isn't modulation index made from $\frac{2}{5}$ so do I have to use it like this?

$$x(t) = 5[1 + (0.8 * 2)x_m(t)]\cos(2\pi*3 *10^{5}t)$$

I carried on with the first formula

1. Sketch the frequency spectrum

I calculated the amplitudes as follows
Carrier Amplitude = $\frac{A_c}{2}$ Side band amplitude = $\frac{\mu A_c a}{4}$ where a = 2

1. Find the bandwidth

$2 \times f_m$ = 4 kHz

2. Find the power of the carrier frequency component

$\frac{A_c^2}{2} = \frac{5^2}{2}$ = 12.5 W

1. Express the total sideband power as a ratio to the carrier power

$(A_c[1+\mu x_m(t)]\cos(\omega_c t))^2$ simplifies into $\frac{A_c^2[1+\mu ^2 x_m^2(t)]}{2}$

so the carrier power is $\frac{A_c^2}{2}$ and total sideband power is $\frac{A_c^2\mu^2x_m^2(t)}{2}$

so as a ratio to the carrier power, it is $\mu^2x_m^2(t)$ which simplifies as $\frac{2^2*0.8^2}{2}$ (Because amplitude of modulating signal is 2V)

Is this assumption correct?

• No problem! Inline MathJAX on EE.SE needs backslashes before the dollars signs. Jan 5 '16 at 4:04
• but not before the double dollar signs. and if you go to DSP.SE, or physics.se or math.se, the backslashes are not there. Jan 5 '16 at 4:47
• I am willing to admit (0.8*2)/5 in equ 2 looks incorrect. It would be more accurate if at just (0.8/2) so that audio tone is scaled to same amplitude as carrier before applying the modulation index Jan 5 '16 at 6:08
• It is difficult to continue with the question, based on the (possibly) bad equation at the start, but let's try... the spectrum simply must be incorrect because it is showing over 100% modulation. In voltage terms, when each sideband is 0.5 of carrier amplitude, you have 100% modulation. Jan 5 '16 at 6:30
• Well, I kinda found a solution. As we are supposed to normalize the signal before modulating, the amplitude of the modulating signal doesn't involve in the equation. So the modulated signal will be $$x(t) = 5[1 + 0.8x_m(t)]\cos(\omega_ct)$$ Jan 5 '16 at 12:52

This formula seems to be misinterpreted by you:
$$x(t) = A_c[1+\mu x_m(t)]\cos(\omega_c t)$$ $x_m(t)$ is any message signal, not necessarily a sine wave. Therefore,
$$x(t)=[A_c + A_m x_m(t)]cos(\omega_c t)$$ $A_c$ : carrier amplitude
$A_m$ : message amplitude

Then the carrier amplitude $A_c$ term is taken common to yield:
$$x(t)=A_c[1+\mu x_m(t)]cos(\omega _ct)$$ Where $\mu$ = modulation index $\dfrac{A_m}{A_c}$ and $0 \leq\mu \leq 1$.

So according to this link you cannot give both the carrier and message amplitude and expect a signal with a modulation index within reality.

• This is an examination question so it has to be possible :) Jan 5 '16 at 10:43
• As a professor, it is entirely possible that a) an unintentional error has been made in the exam question, and/or b) it was intentionally introduced. :) Always check your answers. Jan 5 '16 at 12:19
• Yes this is a tricky question if we don't know the definition properly. I got to know that the modulating signal is normalized before modulating. So the modulating signal amplitude is 1. So we can use $\mu$ directly Jan 5 '16 at 12:55