Decibels per hertz

05.04.2012 20:10

I promise this isn't turning to a series of math rants. But since I have been lately studying spectrum analyzers and similar machinery let me tell you about a small annoyance that has been bothering me in texts and datasheets covering this topic.

Often when discussing noise that has constant power spectral density over a range of frequencies the level of this noise is given in the logarithmic scale in units of decibels per hertz (for instance thermal noise is often said to be -174 dBm/Hz). This is wrong, as it implies that you need to multiply this value by bandwidth (in hertz) to get the power in dBm when in reality you need to add a logarithm of the bandwidth. Of course, everyone dealing with these equations just knows that. Logarithms turn multiplication into addition right? But then you end up with equations where the two sides of the equal sign have different units and that is just plain sloppy writing.

Here's how you properly convert a formula for Johnson-Nyquist noise into logarithmic units:

P = kT\Delta f

Apply definition of dBm:

P_{dBm} = 10\log{\frac{kT\Delta f}{1\mathrm{mW}}}

See how the value inside the logarithm has no dimension? The value in the numerator is in units of power and that cancels with the milliwatt in the denominator. If you are doing things correctly there should never be any physical unit inside a logarithm or exponential function.

To split off the bandwidth term, multiply and divide by one hertz.

P_{dBm} = 10\log{\frac{kT\Delta f\cdot 1\mathrm{Hz}}{1\mathrm{mW}\cdot 1\mathrm{Hz}}}
P_{dBm} = 10\log{\frac{kT\cdot1\mathrm{Hz}}{1\mathrm{mW}}\cdot\frac{\Delta f}{1\mathrm{Hz}}}
P_{dBm} = 10\log{\frac{kT\cdot1\mathrm{Hz}}{1\mathrm{mW}}}+10\log{\cdot\frac{\Delta f}{1\mathrm{Hz}}}

Note that you still have dimensionless values inside logarithms. There is no need to invent magic multiplication factors "because of milliwatts" or fix the units of the variables in the equation. The division by 1 Hz in the bandwidth part also nicely solves the confusion that happens when you have bandwidth defined in units other than hertz.

So how do you concisely write this value? There is no short notation that I'm aware of that conveys the proper meaning. I would simply write out that noise level is -174 dBm at 1 Hz bandwidth and leave it at that.

Now let the flames come that this is school nonsense and that real engineers with real dead-lines don't do any of this fancy dimensional analysis stuff and that back-of-the-envelope calculations just work since you just know that this here number is in watts and that there is in kilohertz.

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