Bandwidth, Spectral Density and Noise Spectral Density $N_0$ definitions

Started by jmarcelold 5 years ago2 replieslatest reply 5 years ago329 views

Please someone correct me if I am wrong.

Baseband and Passband bandwidth definitions are the same for real and complex signal. I mean, baseband bandwidth is the upper cut-off frequency, and passband bandwith is the difference between the upper and lower cut-off frequency, whatever the signal is complex or not, right?

Power spectrum density has multiple definition: the double-side and the single-side, with the double-side being the one defined as $$ S_{xx}(\omega)=\left | X(\omega) \right |^2= \mathcal{F}\left [ \mathcal{R}(\tau) \right ]$$ where, \( \mathcal{R}(\tau) \) is the auto-correlation function of the input signal.

The symbol \( N_0 \) is used only for the white noise spectral density in the single-side definition, or is it also common to use \( N_0 \) as the noise spectral density in the double-side definition?

These doubts about who is who are killing me!


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Reply by SlartibartfastFebruary 27, 2018

Don't worry, stuff like this is brain-bending even for experienced guys.

Generally the double-sided noise is just N, such that 2N = No, where the 2 accounts for the double-sided aspect of the spectrum of a real-valued signal.

People who work primarily with complex-valued signals, e.g., communications, use the single-sided version, No.   People who work primarily with real-valued signals, e.g., audio, use the double-sided N.   Whichever realm you work in, you generally don't have to pay much attention to the other, which leads to confusion when people from different disciplines try to talk about the same thing, e.g., noise power.   It gets even worse when you try to define SNR, as everybody has a different way of looking at it, it seems.

Hope that helps a little.

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Reply by jmarceloldFebruary 28, 2018

Thank you Slartibartfast,

However I am still a little bit confuse. I thought that the single-sided power spectral density definition was

$$ S^{'}_{xx}(\omega)=S_{xx}(\omega)+S_{xx}(-\omega) $$

defined only for \(x(t)\) being a real signal and for \(\omega \geq 0\)