One sided noise spectral density VS double sided noise spectral density

[unseensoul]

No is the one sided noise spectral density in communication systems. N is the double sided noise spectral density.

The relationship between them is given by 2N = No but I don't understand why it is like that. Why isn't it N = 2No instead?!

 

[marcusl]

A complex signal like $$\exp{i\omega_0 t}$$ has a one-sided spectrum, in this case $$\delta(\omega-\omega_0)$$. Its real counterpart $$\cos{\omega_0 t}$$ has the two-sided spectrum

$$\frac{1}{2}[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)]$$

The same power is spread in the real-signal case over positive and negative frequencies, each of which is half as large as the complex spectrum.

When you talk of the power spectral density (PSD) of thermal noise, it again matters whether you are using a real or a complex representation. Audio engineers often use the former, for instance, communications engineers generally the latter. For white noise where the PSD is a constant, and using your notation where N is two-sided and N0 is one-sided, they are related by N0 = 2N.