# SNR relation with modulation index and noise power

What is the relation between the AM modulation index and the SNR? (I plotted this and it came as the shape of the exponential graph)

Also what is the relation between the channel noise power and the SNR? I plotted this and I got the shape as this graph:

(ignore the labels of the axis)

• This question is unanswerable because you give no indication of where the noise is and how it is affecting the signal. – Dilip Sarwate Apr 2 '14 at 13:44

If $x(t)$ is your message signal, then you can write the AM signal as

$$s(t)=A[1+mx(t)]\cos\omega_0 t$$

where $A>0$ is a real-valued constant, $m$ is the modulation index, and $\omega_0$ is the carrier frequency (in radians). If you assume that the noise added by the channel is white with power spectral density $N_0$, then the SNR after demodulation is

$$SNR=\frac{A^2m^2\overline{x^2(t)}}{4N_0B_x}$$

where $B_x$ is the bandwidth of the message $x(t)$ and $\overline{x^2(t)}$ is its average power.

• But SNR is not an exponential function of modulation index $m$ as the OP claims his result shows, so the OP must be using a different definition of SNR. – Dilip Sarwate Apr 2 '14 at 14:07
• @DilipSarwate You're right, but this is the most basic case and if the OP is not happy with it, this will hopefully at least start a discussion revealing the exact details of the question. – Matt L. Apr 2 '14 at 15:03
• For SNR I used power of signal/power of noise – user1930901 Apr 2 '14 at 18:55
• Yes, that's what's usually done. – Matt L. Apr 2 '14 at 20:21
• so what you posted is applicable or not? – user1930901 Apr 3 '14 at 6:55