Power spectral density and SNR for AWGN

Question

I have a GMSK modulator in MATLAB and wish to generate AWGN with a specific SNR, and I'm having some issues figuring out just how.

I have found this step-by-step process to generate zero-mean Gaussian noise with a specified power spectral density W0:

1. Form Gaussian distributed random variable: w = randn(1,N)
2. Map top zero mean: w = w - sum(w)/N
3. Compute average power: Pw = sum(w.^2)/N
4. Form w = w.*sqrt(W0*fs/Pw)

Note: As pointed out by Deve, steps 2 and 3 should not be necessary, since randn() generates a random variable with zero-mean and power 1. However, it is possible that the mean and power may be slightly off, since the vector is of finite length. For my application, neglecting these steps give a slightly less accurate SNR.

So, how do I determine which power spectral density I need to achieve a specified SNR? I understand that SNR is defined as $SNR=P_s/P_n$. Measuring the signal power using $P_s=\frac{1}{N}\sum_{n=0}^{N-1} s[n]^2$ and then isolating $P_n=P_s/SNR$ seems trivial, but how do I relate $P_n$ to $W_0$?

Thanks in advance!

Edit: Using Deves answer, I have written the following code:

s           = signal;
L           = length(s);

% Convert SNR from dB
SNRdB       = 10;
SNR         = 10^(SNRdB/10);

% Measure average power of signal
Ps          = sum(s.^2)/L;

% Calculate wanted noise power
Pn          = Ps/SNR;

% Generate random vector and ensure 0-mean 
w           = randn(1,L);
w           = w-sum(w)/L;

% Scale to wanted power
Pw          = 1/L*sum(w.^2);
w           = sqrt(Pn/Pw).*w;

% Measure resulting SNR
Pw_meas     = 1/L*sum(w.^2);
SNRdB_meas  = 10*log10( Ps/Pw_meas ); % Gives 10.0000

And here's a plot in time domain for anyone interested:

It doesn't seem like a lot of noise - but I guess the calculation of SNRdB_meas is quite foolproof.

[PSD plot removed]

Answer 1

First a comment on your noise-generation process. The Matlab function randn() generates Gaussian noise with zero mean and mean power 1. So steps 2 and 3 are obsolete.

If you'd like to achieve a given SNR, then creating the noise signal with the wanted power is as simple as

w = w .* sqrt(P_n)

Where P_n is the mean noise power and can be calculated by the equation you've already found yourself. Actually you're doing the exact same thing in your step 4, from which you can also derive that

P_n = W0 * fs

assuming that fs is the sampling frequency of your system and thus equal to the bandwidth and using Pw = 1.

Answer 2

This is how I compute the EbN0, you must realize that exist a little difference between it and SNR(depends on modulation type).

%signal energy
calculate the mean symbol energy
% Bit energy for each symbol
Eb   = Es/bit_per_symbol;
EbNo = 10^(EbNo_dB/10);
% Noise variance   
No   = Eb/EbNo;                             
noise = sqrt(No/2)*(randn(1,length(symbol))+j*randn(1,length(symbol))); 
noise_symbol = symbol + noise;

I'm using the pseudo-code above in several systems with success. You may pay attention to the characteristics of the system you are simulating as: Number of bits per symbol, overhead in the symbol etc.

You may wish to take several symbols for a better estimate on each Eb/N0 value.

I don't know GMSK modulators very well so I can't give you a working code, but I guess you can handle the simulation with the code above.