Unraveling the Mystery of N_{0} in PSD of Noise

[FrankJ777]

I know the power spectral density, PSD, of noise is:

$\frac{N_{0}}{2}$

from where do we get N$_{0}$ ?

Is that the RMS value of the noise? Is it related to noise power and noise temperature as in the following:

N=kTB, where k = Boltzmann’s constant, T is noise temperature, and B is bandwidth.

I know that PSD is Fourier transform of the autocorrelation function, and that it must have been derived from a random process, but I don't see where we get N$_{0}$ in the first place and what it represents exactly. I'm trying to understand noise better, but the text I'm using jumps around quite a bit on the subject of noise.

Thanks a lot.

 

[f95toli]

The units of the PSD is usually V^2/Hz; since power is proportional to voltage square.
If you want the RMS noise value you need to multiply by the bandwidth (and if you want it is volts then take the square root and divide by the resistance). THe PSD is the noise POWER in BW of 1 Hz.

Note that the expression you've given for N is only valid or white noise: e.g. noise from a resistor. For any other component (for example amplifiers) it will only be approximatelly valid is a certain range.

 

[FrankJ777]

OK, so what does N$_{0}$ signify?
Where do we get that in the first place?

 

[f95toli]

Well, you either measure it or your calculate it

There are lots of ways to measure the PSD (using a spectrum analyzer is the most straightforward way, but is sometimes not very sensitive), and there are even more ways to report the result: noise temperature, noise figure, h0 etc are more or less equivalent ways of stating the level of white noise and you can -as you showed above- write these as N if you want.

Calculating the level of white noise is for all but the simplest system very difficult. The only system where it is easy is a resistor, since you then only need to know the resistance, the temperature of said resistor (not the noise temperature, but the physical temperature) and the bandwidth of your system. Google "Johnson noise"

Btw, the whole point of "noise temperature" is that it makes if possible to compare the noise of your device to the noise of a resistor with the same impedance.

Note also that there are many types of noise, and most types (such as 1/f noise) are more complicated to describe than white noise.

 

[George H]

Hi Frank, I know a bit about noise, but I've never heard of N₀ do you have a link?
For a resistor the (voltage) noise spectral density is 4*k*T*R in units of V^2/Hz. (so the bandwidth is in there.) If you wanted the power density of the resistor that would be V^2/R or just
4*k*T so all resistors have the same noise power spectral density.

 

[f95toli]

N0 is the noise power.
It is the symbol used in e.g. RF engineering, see for example Pozar's book on microwave engineering (which btw has a good chapter on noise)

 

[FrankJ777]

OK. not to beat a dead horse, but where do we get the expression for the PSD of white noise?
Also, I guess I'm specifically talking about zero mean, Gaussian Noise.

PSD$_{White Noise}$ = $\frac{N_{0}}{2}$

I'm assuming it's the Fourier transform of the auto correlation function of a random process?

Also George, here is a link:
Go to page 11, example 8-3.

http://www.ece.uah.edu/courses/ee385/500ch8.pdf

Also, just to let you know, the text I was originally referencing is Leon Couch's Digital and Analog Communications Systems textbook.

Thanks.

 

[f95toli]

Yes, if you want the PSD and you have the autocorrelation function; then it is just the Fourier transform
For white noise you have that the autocorrelation function is something like N δ(t2-t1); where δ is the Dirac delta. It follows that if you do a Fourier transform you get a constant PSD.

I suspect the factor 1/2 comes from the fact that you are only looking at postive frequencies.

Note that is is VERY rare that you have the mathematical description of the noise as a function of time. Mostly, you just end up using the PSD as the "definition" of noise.

 

[FrankJ777]

Thanks. Your explination seems to make sense.

Soimetimes it seems like $\frac{N_{0}}{2}$ is just one of those "magic formulas" that whenever its referenceded its just pulled out of the air with no explanation.