# AWGN interpretation of bit error rate in time domain

I'll consider a 20 MHz bandwidth between 1.99-2.01 GHz. The carrier has been modulated by an OFDM signal (I/Q modulation) and is passed through a narrowband filter at the receiver.

The noise that is actually received after the input filter is narrowband. So, it is not white anymore. The noise looks like a sinewave of 2 GHz with varying amplitude and phase, which has a Rayleigh distribution. In time domain, something like this:

This is something like the aliasing measured when undersampling a signal, but it has a Rayleigh distribution.

This noise is overimposed on the OFDM signal, upconverted on the carrier at 2 GHz, which in the time domain looks like this:

BER is expressed in the simplest case for an AWGN channel. I've seen that the procedure is to add AWGN of a certain SNR to the signal. The representation of this in time domain is not how the signal measured after the filter would appear, in my understanding.

An example of AWGN addition is this (https://www.mathworks.com/help/comm/ug/analog-passband-modulation.html)

Looking at the picture above, considering that the signal is sampled at the Nyquist rate, we can observe that there is an uncertainty in the values that are sampled. Could the same uncertainty appear with a bandlimited gaussian noise of the same power? Is that the baseband received signal that is represented above? Are there cases when the white gaussian noise is not equivalent to the real bandlimited gaussian noise?

BER in AWGN appears to be the most common metric of noise performance, so there must be an explanation for this method. What is it?

• When you filter the noise, you don't change the noise itself, but limit its effect to the signal bandwidth. This is the part of the noise that we care about. The noise remains white. – BlackMath Jan 14 at 1:05
• What do you by "band-limited Gaussian noise"? White noise means it has a constant PSD. Actually, AWGN has a constant PSD over frequencies of interest in digital communication, but it changes in other frequencies, but they are not of interest. But still, it is considered white. So, when you filter the noise, you just reject the parts of the noise that are irrelevant to the signal, but it remains white, because it has a constant PSD within the signal bandwidth. Do you have any resources about band-limited Gaussian noise? – BlackMath Jan 14 at 12:22
• Now I understand why noise can be random at any sample, it's because of the sampling. In most cases we measure only two samples per cycle, the Nyquist rate, so the noise is just an uncertainty in the measured samples. What if I oversampled a lowpass filtered signal? This will reveal correlations in the noise, so if I simulate the same uncertainty at every sample, the result will be different. So I'm thinking about the concept of white noise in the case when I'm interested in the shape of the signal (oversampling). – user10010101 Jan 14 at 14:09
• White noise by definition has the auto-correlation function $R_n(\tau)=\frac{N_0}{2}\delta(\tau)$. This means that any two different samples of the noise has zero correlation, regardless of oversampling or otherwise. – BlackMath Jan 14 at 14:17