matlab - How to Simulate AWGN (Additive White Gaussian Noise) in Communication Systems for Specific Bandwidth

I am trying to generate a AWGN waveform to add it to the signal of my simulated communication system. The operating bandwidth of the communication system is about B=3GHz and suppose that T=300K (my signal is comprised of very short pulses, I am simulating an IR-UWB system).

What I have done so far is:

N = k*T*B
sigma = sqrt(N./2)
noise = sigma.*randn(1,s_length)

where s_length is the number of samples for my useful signal and noise the awgn waveform.

Is this procedure correct? I have browsed through different questions but was unable to clarify it.

Answer

To create Band Limited AWGN all you need is randn in MATLAB.

The question lies only in how to set its Standard Deviation. To illustrate that, let's assume our AWGN generator has PSD which equals to $ {N}_{0} $.
Namely we have AWGN with zero mean and variance equals to $ \delta(0) {N}_{0} $.

Assuming we have limited bandwidth channel, hence an ideal LPF is applied. Assume its cutoff frequency is $ {F}_{LPF} = \frac{{F}_{S}}{2} $.

Hence the Variance of the band limited AWGN is (Multiplication in frequency) the integral over its PSD multiplied by the Norm of the LPF (The factor 2 is for integration over the range $ -{F}_{LPF}:{F}_{LPF} $):

$$ {Var}_{BandLimitedAWGN} = 2 {F}_{LPF} {N}_{0} = {F}_{S} {N}_{0} $$

Now, generate in MATLAB, using randn noise with the corresponding STD (By the data of your simulation).

Good Luck!

Some Remarks

• The Variance of the noise is independent of the signal (At least in the classic model).


• The Variance of the noise is a function only of the analog channel and the Analog to Digital converter.
  The classic model assumes that if the signal is sampled at $ {F}_{s} $ an ideal LPF with a cut off frequency of $ {F}_{LPF} = \frac{{F}_{S}}{2} $.
  
  As I said, this is the frequency which sets the variance of the Band Limited noise.


• The input signal (Which is the transmitted signal + noise) may have any bandwidth it might have, after the LPF its bandwidth is limited.


• In order to minimize the energy of the noise in the system the LPF band width and the sampling rate should be as low as possible (Namely, the bandwidth of the signal in interest).
  Though if the next step is "Matched Filter" the SNR will be maximized for any finite energy white noise (Or colored if the "Colorization" is known and the Matched Filter is accordingly updated).