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:

simulated bit error rate

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:

theorical error probability

Also, in my course, I can find this formula for orthogonal signal:

course example

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.


1 Answer 1


2-PPM is an orthogonal modulation scheme just like 2-FSK (i.e. BFSK), and the same BER equation applies to both which is:

BER = 0.5 * exp(-(Eb/N0)/2).

I hope this still helps -- it's been a while since you asked this question.

Edit: Forgot to mention that the above applies to noncoherent reception. In case you consider coherent reception, I believe the equation is:

BER = exp(-(Eb/N0)/2) / sqrt(2pi(Eb/N0)).

Edit #2: I also noticed that the paper you refer to considers optical communication. I'm not fully sure if the exact same equations apply to optical and RF communication. The equations is wrote above apply to radio communication.


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