Noise Spectral Denity

From Wikipedia: Noise spectral density:

In communications, noise spectral density \(N_0\) is the noise power per unit of bandwidth; that is, it is the power spectral density of the noise.
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If the noise is white noise, i.e., constant with frequency, then the total noise power \(N\) in a bandwidth \(B\) is \(BN_0\). This is utilized in SNR calculations.

The definition above seems under the assumption of the single-sided power spectral density. Under the definition of two-sided power spectral density, the AWGN spectral density is more often referred to as \(N_0/2\). However, when calculating the total noise power, both the positive and negative bandwidth should be accounted for, and thus the total noise power remains \(BN_0\).

When the signal is sampled by a sampling frequency of \(f_s\), the bandwidth \(B=f_s\), and the total noise power, or, equivalently, the variance of the AWGN at each sample, is \(f_s \cdot N_0\) (assuming the anti-aliasing filter has been applied to eliminate the noise power beyond the sampling rate).