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In the case of statistics, there are two terms: "Variance" and "Standard Deviation" to - for in the latter case - have one to note the case in which the units are without squares.

Am I correct to understand, that there is no such differentiation in signal processing, and in both cases - taken root or not - it's always termed "Power Spectral Density"?

I am asking as on page 3 of Analog Devices AD8307 there's the unit of $\frac{nV}{\sqrt{Hz}}$ for the "Input Noise Spectral Density".

Is there a specific reason as to why manufacturers are taking the square-root?

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Power spectral density is given in power per $Hz$. "Input Noise Spectral Density" is given in units of volts per $\sqrt{Hz}$. These are related:

$$ PSD = \frac{W}{Hz} $$

$$ W = \frac{V^{2}}{R} $$

$$ Input\ noise\ spectral\ density\ = \frac{V}{\sqrt{Hz}} $$

The $\sqrt{Hz}$ is from the conversion power to volts. For example if you want the noise ($nV$) in $1kHz$ $BW$ you must multiply $\frac{nV}{\sqrt{Hz}}\sqrt{1000}$

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