# The meaning of $\rm mHz$ and $\rm Hz^{1/2}$

If Hertz(Hz) is for once cycle per unit time what does mHz and $$\sqrt{\text{Hz}}$$ represents? I mean what is the physical manifestation of the a fractional power of a unit ?

• Can you provide a link to a document where these are used in context? There are a few things that could be meant. – rob Apr 24 at 21:56
• mHz is literally 0.001 Hz. Can you explain why're asking about mHz and sqrt Hz at the same time? One seems conceptually complex while the other does not. – DanielSank Apr 24 at 22:47
• Context means everything. Please be more specific. – David White Apr 24 at 23:36

this is a good question to ask the electrical engineers. $$\sqrt{\text{Hz}}$$ is a unit you will see in a spec sheet for operational amplifiers and other analog parts.

usually it's about a noise figure expressed as an additive voltage to the input. so the unit is $$\text{V}/\sqrt{\text{Hz}}$$. if you square it, you get $$\text{V}^2/\text{Hz}$$. if you integrate that over the applicable bandwidth, you get $$\text{V}^2$$ and if you apply that to a 1 ohm resistor, you get watts.

• I think this answer goes in the right direction but would be improved by explaining why noise power scales with frequency bandwidth. – DanielSank Apr 24 at 22:49
• Ditto for gravitational wave strain. The amplitude noise is expressed as Hz$^{-1/2}$. Same idea, it's the square root of the noise power per unit frequency. – Rob Jeffries Apr 24 at 22:51
• oh dear. but guys, i only do DSP${}^{*}$ nowadays. i haven't designed or breadboarded an analog circuit for 4 decades. ( ${}^{*}$ in fact, my life is so pathetic i am working solely in MATLAB. i don't even do real DSP anymore. :-( .... ) – robert bristow-johnson Apr 24 at 22:56

The millihertz is simply the inverse of a time period, 1000 s. If a complete oscillation requires 1000 s, then the frequency of the oscillating system is 1 mHz. Or in 1 s the system moves through 0.001 of a complete cycle.

@robertbristow-johnson gives a succint answer for the $$\sqrt{Hz}$$

• from your profile: "Ph. D. in nuclear physics, Navy, first language = FORTRAN-IV, also coded DEC-10 assembly,... motorcycle rider, classic Mustang driver, Bible reader, and getting older."  likely not Bill Nye. :-) – robert bristow-johnson Apr 24 at 22:51
• i'm a failed PhD student - ABD in EE from the 80s, and my first language was also Fortran IV that i punched into a deck of cards. i did a little DEC PDP-11 asm. learned resently that the DEC-10 had an interesting floating-point format that is a little like IEEE-754 but using 2's complement instead of sign-magnitude. kinda cool. – robert bristow-johnson Apr 24 at 22:54
• @robertbristow-johnson Indeed, likely not! LOL. Yeah, the DEC-10 bit me in the rear when I was doing quantum mechanical calculations. $h^2$ is well outside the range of the float-point values it could handle even at double-precision. I had to re-write a lot of calculations using logarithms. – Bill N Apr 24 at 23:04