Why the PSDF of white noise is N/2 ?

[magnetra]

White Noise

The psdf of white noise is N/2 when defined for both ±freq where as N when defined for positive freq. Why is it so? once defined for ±freq, why define again for only pos freq?

Magnetra

 

[carpa]

White Noise

Only positive frequency has real meanings in real world. For example, you can only see spectrum for positive frequency in a spectrum analyzer.

 

[claudiocamera]

White Noise

The PSD is an even function, thus Sn(-w) = Sn(w).

For the white noise the PSD is a constant "C" in practical values of frequency. For instance we can call "C" of N/2 , thus we can say that the PSD of the white noise is constant with the value of N/2.

The power over a range of frequency is the integral of the PSD in the specified range. As the PSD of white noise is constant we can say that P = (N/2)* 2* (Delta F), where delta F is the range of frequency ( positive values) . The question raised here is why multiplying by 2 ? The answer is simple for each positive interval of frequency we have a correspondent negative interval of frequency so that we need to consider both areas in the integration: areas due to positive and negative frequency intervals.

Now imagine if you have the frequency interval from 0 to wo , we have P = (N/2)*2*( wo-0) = N*wo, now consider wo goint to infinite. It is just your question... the positive range of frequency . Thus P = N*(positive range of frequency), now can you realize the answer of you question? I hope so.

By the way, there is nothing to do with what you can see in a spectrum analyzer. Negative frequency is a mathematical tool very useful in solving problems in real world.