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I have a signal, S whose bandwidth is bw Hertz and is sampled at fs Hertz.
To contaminate it with a noise corresponding to SNR x dB, I used the matlab function
out = awgn(S,x);
But I have came across some codes doing it instead like the following.
out = awgn(S,corrected_x);
Where corrected_x = x + 10*log(bw/fs);
Which is the correct method and why?
10*log(bw/fs) to take into account the oversampling operation because the awgn() function specifies the signal-to-noise ratio per sample, in dB.
The discrete time AWGN model is
$$Y = X+N$$
where X is data from continuous time $X(t)$, N is noise sequence from AWGN process $N(t)$ and Y is receive symbols.
If $X(t)$ is characterized by its baseband equivalent limited between $[-W/2, +W/2] \textrm{ (Hz)}$, then we can identify $X(t)$ by observing $Y$ at a rate $W$ symbols per second. See chapter 2, sampling theorem and Theorem of irrelevance.
Call $P$ the average power (joules per second). The sample power $E_s= P/W$ and the noise symbol power is $N_0$. The signal noise ratio per symbol is defined $\mathrm{SNR} = \frac{P}{BW_0} = E_s/N_0$.
If the complex baseband signal is oversampled $\alpha = f_s/W$, the noise sample power is still $N_0$ while the data sample power is reduced $\alpha$ times, thus $E_s/N_0 = \mathrm{SNR} \times W/f_s$.
In dB, $E_s/N_0 = \mathrm{SNR} + 10\log_{10}{(W/f_s)}$.
The awgn() function add AWG noise by the previously defined $E_s/N_0$.
