I am trying to build a simulation on MATLAB for PPM transmission through AWGN channel. I have a function wrapping the transmitter, channel and receiver that takes a SNR value as input. My simulation program does multiple iterations for different SNR values and plot the bit error probability (BER) as a function of SNR (in dB). My results are extremly low and I find it a bit surprising:
To compare simulation to the theory I decided to derive the bit error probability for my modulation scheme (2-PPM where the basis functions are orthogonal). According to this paper, the probability of error for M-PPM can be written as: $$ P_{err}\approx Q \left ( \frac{s}{\sqrt{2N_{0}}} \right ) $$ Where: $$ s=P\sqrt{\frac{L_{PPM}log_{2}(L_{PPM})}{R_{b}}} $$ Which I modified with my parameters ($L_{PPM}=2$, $P=\sqrt{E_{b}}$, and $R_{b}=10^6$) to get it as function of $ \frac{E_{b}}{N_{0}} $ like this: $$ P_{err}\approx Q \left ( \frac{1}{\sqrt{R_{b}}} \sqrt{\frac{E_{b}}{N_{0}}} \right ) $$ This, when plotted with MATLAB for SNR values frome -40 dB to 10 dB looks like this:
Also, in my course, I can find this formula for orthogonal signal:
Given that my basis functions are also orthogonal, which formula should I use ?
I suspect the noise generation in my simulation to be incorrect and I think that I don't really understand how it works. Here is my code for the AWGN channel:
% AWGN channel Ps = mean(abs(x).^2); SNR_lin = 10^(SNR/10); N0 = Ps / SNR_lin; n = sqrt(N0/2)*randn(size(t))+1i*randn(size(t)); y = x + n; Where x is the transmitted signal and the SNR is given in dB in the parameters.
Thank you for your help.
