# What are the units of $f/f_s$?

Which is sometimes called the normalized frequency.

Is it unitless?

$$\frac{f \; Hz }{f_s \; Hz } = \frac{f}{f_s}$$

Or not?

$$\frac{f \; \frac{cycles}{second} }{f_s \; \frac{samples}{second} } = \frac{f}{f_s} \frac{cycles}{sample}$$

Dimensionally, $$\frac{cycles}{second}$$ (aka Hz) and $$\frac{samples}{second}$$ are both $$sec^{-1}$$ and the normalized frequency is dimensionless.

The problem with that is the other common definition of normalized frequency has units of $$\frac{radians}{sample}$$ which is also dimensionless.

Sampling is a repeated discrete task, it is not cyclical. It has a rate, not a frequency.

If I multiply the sampling rate by a time interval do I get cycles or samples?

$$N \; samples = f_s \; \frac{samples}{second} \cdot t \; seconds$$

I don't think I've ever seen someone say "The DFT frame is 32 cycles long", yet when you call the sampling rate a frequency that is what you are saying.

So, can anyone give a justification for calling it sampling frequency other than ad populum. And, if you can't, why should the deacons of a discipline (I'm talking about the regulars here) propagate a bad practice?

This is more of a deepster question than it appears. Please don't give a fired from the hip response. Think about it a bit first.

• 🤯<<mindblown emoji>> – Peter K. Dec 24 '19 at 16:24
• @PeterK. You're welcome. Merry Christmas ;-) I thought meta would be the right place to throw down the gauntlet. I got a little peeved this morning about a down vote on dsp.stackexchange.com/questions/62812/… which I suspect is due to some one thin skinned not liking my P.S. The deeper question here is should a bad practice be tolerated because it has been the norm? Or should it be corrected? This applies to any discipline. – Cedron Dawg Dec 24 '19 at 16:55
• FWIW, I think sampling has a rate, not a frequency. Sometimes I do say "sampling frequency" out of laziness and/or to avoid dealing with the units (for example, how to properly state Nyquist's sampling theorem, which compares a frequency with a rate?) I tend to just avoid the issue, but I think you're right, we should deal with it. I'm not sure what's the best way. – MBaz Jan 27 at 0:14