# How to find the noise power in my simulation

I am currently working a tricky communication chain that I want to simulate into Matlab.

The chain is like a classic BPSK modulator/demodulator chain, but where the BPSK modulated signal has been merged with an other carrier using PM (Phase modulation) modulation method.

So, after the PM modulation step, the signal can be expressed like this ($$\mathrm{h}$$ is the modulation index and $$d(t)$$ is the NRZ data signal):

$$\cos\left( \omega_c t + \mathrm{h} d(t) \cos(\omega_{sc} t + \varphi_{sc}) + \varphi_c \right)$$

My goal is to find the curve of the $$BER = f\left(\frac{E_b}{N_0}\right)$$ (Which should fit the theorical BPSK BER curve).

Well, I used to work with this kind of chain without the PM step. To simulate it, I usually just use a theorical noise power $$P_n = \frac{\sigma_c^2}{\frac{E_b}{N_0} * \mathrm{BPS}}$$ to create a gaussian noise of standard deviation $$\sqrt{\frac{Pn}{2}}$$. But I have some issue in my current case: I made this chain in Matlab, but I just cannot simulate it because I don't know how to find the standard deviation I have to apply to my gaussian noise when my signal is between the two PM steps.

Actually, I have no idea about how to find it... I just have a big mental breakdown because of the non-linearity of the PM modulation/demodulation step, and I don't know what to do to find $$P_n$$...

I already calculated the theorical signal autocorellation/PSD just after the PM modulation step, which use first kind Bessel functions (Something similar to the expression (9) of this paper). But honestly I don't know what to do know...

• $N_0$ is normally assumed normalized. Therefore, you need to calculate the occupied bandwidth and the symbol energy to derive the energy per bit per Hz, which is $E_b$ in $E_b/N_0$. For PCM, linearization may be needed. Just follow the definition of SNR and EbNo. May 18, 2020 at 9:44
• The full statement is "in simulations, $N_0$ is normally assumed normalized" May 18, 2020 at 10:06