# What are the units of $f/f_s$? [closed]

Which is sometimes called the normalized frequency.

Is it unitless?

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

Or not?

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

According to Dimensional Analysis, $$\frac{cycles}{second}$$ (aka Hz) and $$\frac{samples}{second}$$ are both $$sec^{-1}$$ and the normalized frequency is dimensionless, so it is silent on the issue.

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.

I've elaborated a little bit on this in a question:

DFT exercise in the book Understanding digital signal processing 3 Ed

Well, actually taking the point a little bit further. I think it is damaging to teach the DFT strictly in the context of Seconds and Hertz as units. One should not expect a first time learner to be able to differentiate the tool from its application. There have been too many "Where do I plug in the Hertz when looking for repeat patterns in ...." (Paraphrasing).

It doesn't help that most the used MATLAB calls seem to Hertz and Seconds oriented either.

So, if you are an educator of this material, in any setting, please keep this in mind.

Response to Fat32's comment in RB-J's answer.

Actually I am coming at this form an epistemological direction. I am well aware of the goal of DA (Dimensional Analysis) to achieve conceptual harmony to the natural physical constants (and approaching this goal is contrary to integrity checking of equations).

My objection is to RB-J's role as an acolyte insisting that DA is "the only and true way" (and his combative manner). Particularly since it was formalized by a "standards committee" (those with experience, insert running jokes here). Whether rotation should be a dimension was and probably still is a big issue to wank over. DA's definitions of "Dimension" and "Unit" are the pedantic ones and I would think minority ones. The carrying-the-units/math/common man definitions are much more down-to-earth (more concrete than abstract). A dimension is a coordinate in a vector space, and a unit defines a measure along that coordinate.

So, if you want to use DA to find "meaning in Physics", fine. But if you are looking for integrity checking to prevent calculation errors in formulas, you aren't using the best tool. The lander crashed because "velocity" wasn't defined in units. And I challenge you to assert the following equation is nonsense with the carry the units definition of a unit:

$$2 \frac{\text{Semicircles}}{\text{Circle}} \cdot 1 \frac{\text{Circle}}{\text{Cycle}} \cdot \pi \frac{\text{Radians}}{\text{Semicircle}} = 2 \pi \frac{\text{Radians}}{\text{Cycle}}$$

Two kilos of apples is different than two kilos of oranges, so if you are weighing them, I kilo of apples weighs a kilo (conversion factor), and I kilo of oranges weighs a kilo (conversion factor), so you can add them (common units). Note, no need to mention dimension. If you are counting them, well you have so many apples (units) and so many oranges (units). If you want to compare items of different units, you have to convert them to the same scale (units) using what is known as a "Value Function". The value function can be as simple as multiplying by a conversion factor (even if that factor is numerically 1). "Pricing" is the most common value function there is. Yes, carrying the units was developed mainly for use in "rate problems" to keep your units straight and it works well.

The difference between carrying the units vs DA is like weak typing vs strong typing in programming languages.

So, deepster/fancy question du jour: If the Universe didn't exist, would the values of $$\pi$$ and $$e$$ still exist?

I hope everybody agrees that this is a definitional issue, so I'd like to attempt to take the "my definition is right" aspect out of this, and shift to the implications of accepting either definition as the correct one. To do so, I'm going to introduce two working definitions with the intention that the scope not exceed the extent of the this question and its answers, and that the nomenclature not favor one over the other.

t-units = Traditional units, using the root meaning of "one", capable of any level of specificity by the act of naming.

s-units = Standardized units, as defined by the various ISO standards or other governing bodies

There should be no disagreement about the following statements:

1. s-units are a subset of t-units

2. In t-units, the answer to the title question is cycles per sample

3. In s-units, the answer to the title question is unitless

4. Converting from t-units to s-units can be, and often is, lossy

Three questions arise from these observations:

A. Should t-units be stamped out?

B. Was it the intent of standardization to weaken the integrity validation capability of Dimensional/Unit Analysis?

C. How does this relate to "political correctness" in general?

Indeed, this is a pedantic (in a non-perjorative way) exercise, done in pedantic style. That's the point of it being placed in Meta and tagged with "discussion".

Okay, everybody can see the setup, right?

Check-mate.

Observation 4 validates Question B. Question B is a yes/no question. So what can be answered?

No, it is an unintended consequence. Okay, you admit that it is detrimental to integrity checking.

Yes, it is intentional as it keeps things simple [A strong aversion to complexity has repeatedly been expressed here] and any loss of functionality is a small price to pay for the greater good of the benefits of standardization.

In either case, it is clear that a loss of validity has occurred. Therefore the answer to Question A has to be a firm no. Furthermore, those who attempt to do so are firmly in the wrong and should desist immediately, starting with RB-J.

Question C takes us to a whole 'nuther level. This is totalitarianism in one of its many forms. I will fight it where ever I see it. This is called the "good fight".

It is natural that engineers be attracted to technocracy, but you have to resist its siren song or you will surely crash on the rocks.

I believe in the dignity of critters, including biped furless chimps. It is institutions of all forms that need to have restrictive behavioral binds, not individuals.

I told you this would be deep, and yes, I am a "lousy judge of my own expertise", but isn't everyone?

• 🤯<<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 '20 at 0:14
• @MBaz, frequencies are rates. the two notions are one-and-the-same. – robert bristow-johnson Sep 13 '20 at 23:37
• @robertbristow-johnson The way I understand it, frequency is a rate, but not all rates are frequencies. Frequencies are measured in Hz (cycles per second), but rates can have any units. – MBaz Sep 14 '20 at 2:16
• @MBaz, if it's a rate in time then it's the number of occurrences of something per unit time. The dimension is $\mathrm{T}^{-1}$ and the common unit is $\mathrm{s}^{-1}$. – robert bristow-johnson Sep 15 '20 at 4:48
• @robertbristow-johnson So, from a practical engineering persepective, and with my limited understanding, measuring anything (except cycles) per second using herz makes little sense. At least this is my perspective, which I acknowledge may be limited by my personal way of interpreting the world. – MBaz Sep 15 '20 at 15:45
• So, my approach is to make up the units I need (radians, frames, samples, bits, bit errors, frame errors, etc; send complaints to the SI) and divide them by time. I reserve Hz exclusively for signal frequency, and bandwidth. – MBaz Sep 15 '20 at 15:51
• I concur with @MBaz approach and find it very confusing the notion that we should NOT use radians and cycles as units (still trying to get my head around that) but as Cedron says for purpose of DA and even more specifically computer checking of unit consistency it makes a lot of sense to include and not say it it unitless. This was an interesting read: iopscience.iop.org/article/10.1088/0026-1394/52/1/40/pdf – Dan Boschen Sep 18 '20 at 14:50
• If the question was stated "What are the SI Units of f/fs" then there would be no ambiguity in the answer, and it would be unitless. – Dan Boschen Sep 18 '20 at 14:55
• @DanBoschen That paper is a good read, thatnks for the pointer! – MBaz Sep 18 '20 at 17:01
• @Fat32 not sure of that (yet not qualified to really answer; I am still getting my head around this so observing with technical interest). Your last point really doesn’t make sense to me since fs is indeed samples/second in contrast to a sine wave that is cycles/second. What I wouldn’t need is to assign something else like impulses but I do see the utility of samples. The fact that radians are unitless is at the core of my own confusion (if SI says that so be it but I would tend to agree that it is an unfortunate conclusion for my use of it), otherwise a cycle can be a derived unit scaled – Dan Boschen Sep 19 '20 at 12:12
• So then aren't the "units" of radians/sec radians and seconds, and then how can we say that radians/sample is unitless, if anyone is saying that (for my education)? Or is anything that isn't one of the fundamental units is not allowed to be called a "unit"? Is this all semantics or is there an actual problem is using radians as a unit to ensure accurate computations/scaling etc? If it can't be a unit, what do we then call it to avoid the other problem whatever it is? This is definitely interesting. – Dan Boschen Sep 19 '20 at 13:21
• @CedronDawg No I take it as a hobby ;-))) – Fat32 Sep 19 '20 at 19:12
• @DanBoschen sorry I missed your comments... it's not only an impulse you could also say a point too! Just like N-point DFT (to mean N-sample DFT)... point, sample, impulse the same thing in this context :-) I caused a lot of confusions in this therad. I'm sorry for that. And repeating once more I believe that this threat should be deleted in its entirety. Then re-opened if OP wants to ask it again... But only cited answers should be given instead of opinions... That's the best way imho to resolve the issue. We're not gonna invent the units of f/fs here in dsp.se, are we ??? – Fat32 Sep 19 '20 at 19:44

cycles/sample is correct, and it is not unitless. In those units, the Nyquist frequency is 0.5.  In units of radians/sample, the Nyquist frequency is $$\pi$$. The Matlab functions I'm familiar with use units of cycles/half-sample, and the Nyquist frequency is 1.0.

• Indeed, this is a rant against the use of the term "sampling frequency" and specifying the units as Hz, which is frequently (pun intended) done. I recommend you follow the link where I elaborate a bit. You may also find my answer here dsp.stackexchange.com/questions/69430/… interesting as well. BTW, welcome to DSP.SE. You might also find my blog interesting: dsprelated.com/blogs-1/nf/Cedron_Dawg.php It's chock full of formulas you have probably never seen before. – Cedron Dawg Aug 25 '20 at 23:15
• This is absurd. f/fs does not have any units. It's just a number. No need to introduce absurd nomenclature. Now take a simple math example: consider a set of N=50 people with heights between 150 cms and 200 cms. Each person has a height of $x_i$ cms. (i is the order of person in the set). Let's compute the average height (say it's 175 cms.) $$\mu = \frac{x_1 + x_2 +...+x_N}{N} = 175 cms$$ What's the unit of average height, of course cms. What's the unit of N ? It's a counting number without a unit. Hence, the unit of $\mu$ is in cms (or converted to feet, inch, yard). – Fat32 Sep 15 '20 at 11:35
• (cont.) Now lets introduce a new metric, $\bar{x}_i$ for each person defined on the following :$$\bar{x}_i = \frac{x_i}{\mu}$$ Now, what is the unit of $\bar{x}_i$? Both $x_i$ and $\mu$ has the units of cms so unit of $\bar{x}_i$ is nothing; it's unitless.Lets say for some $x_{7} = 175$ cms, then $\bar{x}_7= 1$; what is this unit of $1$? 1 what? 1 cms? Of course not? you may denote it as 1 per mean-height but NO! mean-height is just another name for the unit of cms. So, $f/f_s$ has no units but an association with $f_s$. – Fat32 Sep 15 '20 at 11:53
• (cnt) This same convention is used everywhere in physics and engineering practice. We have quantities with units, and certain quantities (such as ratios of the same kind) without units. THere's no need to introduce awkward units for untless quantities (aka. numbers). This same thing is used in Relativistic Calculations where every speed is divided by $c$ (light-speed in m/s) to get normalised speeds without units. So if your speed is 0.5, then it's half light speed. But that 0.5 does not have a unit. If you have problems with unitless quantities, then go some physics department to argue. – Fat32 Sep 15 '20 at 11:58
• @Fat32 I'm in Bob K.'s camp on this one. It is entirely consistent with the "The factor-label method for converting units" section of en.wikipedia.org/wiki/Dimensional_analysis. Presenting a fallacious argument (en.wikipedia.org/wiki/Argument_from_authority) that Physicists are the "Authoritay" isn't very convincing. I have no problem with unitless quantities, I know what a Rayleigh number is. I am sort of struggling with the Index of Refraction right now and if it is truly unitless. – Cedron Dawg Sep 15 '20 at 14:58
• (con't) The specificity of unit designation is kind of a subjective thing. As long as you use two different names, you have two different units. If there is a one-to-one correspondence, then there is a conversion factor with a magnitude of 1. Whether this should be explicit or implicit is also subjective and context dependent. – Cedron Dawg Sep 15 '20 at 15:01
• @CedronDawg I think there's no need to make the problem bigger or wider than it actually is [if that's not to be used as tactic for blurring the truth]. The question was simple: What is the unit of $f/f_s$ where $f$ is the frequency (of some signal) and $f_s$ is the frequency of sampling (device). Both are designated with $Hz$ hence $f/f_s$ is unitless. Now, if you mix it by attaching $f$ with cycles/second, and $f_s$ with samples per second. And reach the conclusion that $f/f_s$ has a unit of cycles/sample; it may seem consistent but neither a cycle nor a sample is term for a unit. – Fat32 Sep 15 '20 at 15:10
• @CedronDawg [cont]. Furthermore, attaching $f_s$ with samples/second should also be approached with care. On a typical drawing we draw a frequency axis onto which both arbitrary frequency f and the sampling frequqncy $f_s$ are placed (as is done ona Nyquist analysis of sampling). In this regard $f_s$ is no more different than just an ordinary frequency $f$ on that same axis. They both have th (physical) units of 1/s = Hz. Perhaps the samples per second usage is a sort of technical jargon leaked through mathematical designation of $f_s$ as a sampling rate. I mean not a big deal. – Fat32 Sep 15 '20 at 15:15
• @CedronDawg related rates ;-)) – Fat32 Sep 15 '20 at 15:17
• @Fat32 See my followup. I'm still not resentful, hope you aren't either. I would argue strongly that it is (not just seems) consistent. Compared to "it seems you can prove the angle addition formulas from Euler's formula". It seems you can, but you can't since Euler's formula is dependent on the angle addition formulas. Then there's this: en.wikipedia.org/wiki/Cycle_per_second and this: en.wikipedia.org/wiki/Normalized_frequency_(unit) – Cedron Dawg Sep 16 '20 at 9:51
• @CedronDawg why do you use cycle/sec for $f$, but sample/sec for $f_s$ while both are just two different frequencies? You have to answer this, and look at my average height computation example in th eabove comment. Read it carefully and disprove it. And clearly explain: 1-what a cycle is, 2-what a sample is, 3- why you chose them over standard SI Hz unit. Clearly explain them... – Fat32 Sep 16 '20 at 9:59
• @Fat32 First observation since it arises from the true purpose of my question: Notice your language ("You have to answer this"). Mandates tend to come from the PC side in any arena. In context of the DFT, the numerator and denominator pertain to quite different concepts. The denominator (the namer) designates the spacing of the sampling in the application framework (attribute of the tool). The numerator (the counter) designates a propertiy of the signal being measured (target of the tool). I don't see these things as the same at all. – Cedron Dawg Sep 16 '20 at 13:16
• (con't) Labeling units is subjective. The desire for objectivity is a driving motivation for standardization, but at what cost? In your example (and I had been waiting for Bob K. to make an appearance to let him take a stab at it), I would say there is no utility in explicity (no contextual distinction needs to be made) trying to label any unit other than your height unit. You could label "N" with persons. Then $\mu$ would have units of "cms/person". The units of $\bar{x}_i$ [Unconventional use of bar, BTW, 50 lashes] would be "unitless" or "person" respectively. – Cedron Dawg Sep 16 '20 at 13:17
• (con't) In the former case, it would be okay to also say "non-dimensional height", just as Reynolds Number is called the non-dimensional velocity in Fluid Dynamics (Engineering College), or "normalized height" (unit) would be better. I'm about 1.09 persons in the latter case, but without context that isn't very meaningful, or as I like to say "I'm five foot fifteen." – Cedron Dawg Sep 16 '20 at 13:23
• The very names used for a fraction, or rate, literally mean counter of named things, and $\frac{a}{b}$ can always be split into $a \cdot \frac{1}{b}$. Understanding that makes the rules of adding and comparing fractions much easier to grasp. (Sorry for the chat overrun). – Cedron Dawg Sep 16 '20 at 13:31

Hertz particularly refers to time rate of some repetitive event such as rotation. Whereas Cycles per length refer to space rate of some repetitive event such as a picture line bar pattern. Cycle simply means a repeating event, mostly a whole rotation of something, or a $$2\pi$$ increment in a sinusoidal argument. In most contexts, the word cycles can be replaced with the word rotations or repetition, or even just with tics.

So if you are sampling a continuous-time signal then your sampling rate is Fs Hz, or Fs samples per second.

Note that these two are exactly the same things. Since samples do not have a (physical) unit associated with it, (it's just a counting number) then Fs samples per second is dimensionally Fs per second, which is abbreivated as Fs Hz.

So they are the same things stated differently... Be consistent, and both should work well.

Furthermore note that, practically speaking sampling is carried out by ADC devices which are operated at different clock frequencies to achive the desired rate of sampling. Hence the Hz, usage is also quite sound way of describing it.

Finally, the Hz usage is linked more naturally with the analog (continuous-time) signal, whereas the sample per second usage is more linked with the discrete-time signal (sequence) nature.

• Now a days I am using RPM, Hz is just too fast for me. $$;-)$$ The issue of the actual question in the link in this question, is about whether the duration of the sampling interval should be first sample to last sample (N-1) or frame size (N). Coincidently, if you care to follow, the same issue arises in different but related contexts here dsp.stackexchange.com/questions/69430/… and here dsprelated.com/thread/11973/… – Cedron Dawg Aug 4 '20 at 1:23
• Then use 1/60 Hz ;-)) – Fat32 Aug 4 '20 at 2:19
• Actually, the Hz/60/60/24 fly by too fast, even the Hz/60/60/24/365.25 are going by too fast. Did you read the link? It is about keeping units straight in their domains and Hz does not belong in the DFT as that is an application level unit, not tool level. Calling samples per second Hz, as is frequently done (pun!), carries an implied "1 cycle per sample" conversion factor. In the ADC or DAC arena, this is appropriate and known/mentioned explicitly. – Cedron Dawg Aug 9 '20 at 12:55
• @CedronDawg Can you write me a Matlab program that computes the elevation and azimuth angles of the Sun and the Moon on a given day-hour-minute (local hour corrected) at any place on Earth with a given latitude and longitude (and may be altitude too)..? – Fat32 Aug 9 '20 at 15:04
• I don't do Matlab. I've posted something similar in a Gambas forum: forum.gambas.one/viewtopic.php?f=4&t=720 In researching that I found all the formulas you are looking for. I can't find the awesome Swedish site with all the very accurate coefficients I found before, but a search on "azimuth sun moon calculation coefficients" will give you plenty of material. – Cedron Dawg Aug 9 '20 at 15:28
• @CedronDawg Seems like a nice program thank you! I have downloaded the zip file, and will try to convert it to Matlab... I may return to you when things are done... ;-) I'm not interested in the "display" logic, but just calculations. – Fat32 Aug 9 '20 at 16:03
• @CedronDawg Hi cedron, I've converted that code into matlab and it works! Great surprise! It finds sun set and sun rise and noon times wrt local time and time-zone. Yet the code has absolutely no explanations and introduces so many awkard variables without a definition. And mostly hardcoded constant literals that hardly make any sense. So I was looking for something more with an explanation of how it's calculated and definitions of all variables. Some geometric calculations I was looking for. I'm pretty it's doing it but the code is so much condensed and reduced into being helpful... – Fat32 Sep 15 '20 at 9:27
• @CedronDawg Actuallt I've already written some codes based on Sun-Earth-Moon geometry and kinematics of planetary motion, I needed a lot of variables based on the astronomical parameters (such as axis tilt, orbit eccentricity, radius of Earth etc.) and utilized some subtle vector operations to get azimuth-elevation angles of Sun / Moon. I'm looking for a refined means of calculating it and some calibrations. Your code (dunno who wrote it and how) calculates quite fine (not very accurate though). I cannot find a way to improve it though as it's quite peculiar to its owners.. ;-) any helps ? – Fat32 Sep 15 '20 at 12:16
• I'm sorry, Fat32, this was long ago enough that I don't remember the specifics. This is the link I cited in the code (comment at the top) math.stackexchange.com/questions/2186683/how-to-calculate-sunrise-and-sunset-times. IIRC, the Swedish site I referenced had coefficients based on empirical data, but I do vaguely remember reading about some derivations. That's the best I can do right now without further searching. – Cedron Dawg Sep 15 '20 at 13:22
• @CedronDawg ok many thanks for pointing into the resources... Btw. $f/f_s$ is of course unitless :-)) See my comments to BobK down ;-) [I want to stay out of this discussion but no it's hard to do so...] – Fat32 Sep 15 '20 at 14:47
• I've added my own answer and gave the check to Bob K. Neither RB-J's answer or mine address the title question directly. My preference is everybody take this discussion to their respective watercoolers. I would think it more interesting than "It sure is getting cold fast", or "How 'bout them Tigers?" You should definitely stay away from "Wow, did you see the blouse the receptionist was wearing yesterday." Nowadays, that can get you fired. Will procreation drop to zero? – Cedron Dawg Sep 18 '20 at 11:52

This question wasn't really about the units of frequency.

The notion that a standards committee could hijack a common word, claim it as their own, and forbid other uses of it, is so arrogant and hubristic that it boggles my mind. I don't care if it happened 60 years ago (about my age) or yesterday, what is wrong is wrong and when that is recognized if should be corrected. And as they say, "I wasn't born yesterday."

Properly, "units" should remain as general as possible. If you want to give a special meaning linguistically, it requires its own modifier. Like "SI Units", or "standard units", or "basis units", etc. Nobody, or no body, gets to say "these are the officially approved 'units' and everything else you used to consider a unit is now a 'factor label'" Once that kind of control is ceded, it is never recovered and the control freaks get to have a field day. And they won't stop there. Whether OCD is a mental disease or a personality type doesn't matter, it is no excuse, and it isn't a hall pass.

Any quantity you measure, is measured on a scale and some reference is the unit with a "1" value and there is a "0" value. Normalizing is simply a rescaling by picking a representative value and declaring it to be the new "1" on the new scale. Thus, you are still measuring along the same dimension, all that has happened is a rescaling conversion factor has been defined and applied. It does not all the sudden lose its dimensionality. Even if you define a new origin, a new "0", you still just have a rescaling, but now with a bias. It is still along the same dimension.

The notion that only officially approved "Dimensions" are allowable, is Orwellian to say the least. Dimension is a scale type answering the quality (as in property or attribute) of that which is being measured.

A normalized frequency is still measuring frequency, just on a different scale.

A normalized height is still measuring height, just on a different scale.

Neither are unitless (a null concept, every scale has a unit) nor dimensionless (another null concept, every scale has a type). Counting numbers are the tic marks on a scale, they are still in the same units in the same dimension.

Height is a different dimension than length.

It isn't this way because "Ced said so.", it is this way because this is the way it is. If you think I got it wrong, then you need to argue on the merits, not appeal to authorities like governing bodies or textbook authors, or appeal to the crowd by saying this is the way everybody does it in our club. You don't get to shoot the messenger either because you disagree with the message.

Words have meaning. Sometimes the same word has a colloquial meaning and a technical meaning. When the technical meaning diverges too far from the colloquial meaning you can tell a guild power play is in motion, and if you search further you'll find a control freak, or a gaggle of them, who want to rule their world.

Standards have benefits there is no doubt. They also have the ability to be the foundation of tyranny. They are a tool, a policy, a pre-defined decision, nothing more. If you don't understand the rationale behind a policy you are likely to misapply it. Law is crafted carefully (or should be) so the "spirit of the law" is not lost in "the letter of the law". Policy is generally not that carefully wrought.

Now we have people running around with rulers measuring six feet down to the inch. Pathetic.

And this ain't my first time at the rodeo.

I believe the confusion can be explained as semantics, specifically the word "cycle" or "cycles". In the context of signal frequency, such as the Nyquist frequency, it refers to one period of a continuous periodic signal. And in the context of sampling, such as the Nyquist rate, Fat32 equates it to $$T$$, the sampling interval. If we were to rename those units respectively as periods and samples, the units of normalized signal frequency, $$f/f_s$$, are periods/sec / samples/sec = periods/sample, which is not unitless (they don't cancel). Therefore, the only disagreement here comes down to the common practice of substituting cycles for periods, and the uncommon (IMO) practice of substituting cycles for sample-intervals.

It might also be useful to recall how Oppenheim & Schafer, 2nd ed, 1999, pp 140-141 introduce sampling.

$$T$$ is the sampling period, and its reciprocal, $$f_s=1/T$$, is the sampling frequency, in samples per second. [Note they never say Hz in relation to sampling, although (as Fat32 said) other authors do.]. We also express the sampling frequency as $$\Omega_s = 2\pi/T$$ when we want to use frequencies in radians per second.

The implied conversion factor from $$f_s$$ to $$\Omega_s$$ is $$2\pi$$ radians per sample, which only makes sense when sample-interval is synonymous with cycle (Fat32's semantic)... bringing us back to the semantic conflict.

Bottom line: Hz/Hz = cycles/sec / cycles/sec is not unitless when the numerator "cycles" has a different definition than the denominator "cycles". Just having the same overloaded name is not mathematically sufficient for cancellation.

• The problem is not about using cycles or samples sort of a unit designator; it's ok; it's used in practice for example in image processing where cpm (cycles per meter) refers to spatial frequency of image patterns. There we connot use Hz which strictly refers to time periodicity of an event. The problem is about making this a problem? The SI dimension of $f/f_s$ is none. Yet you can assign anything with $f$ and $f_s$ such as f = periods/second and f_s = impulses / second hence you will have f/f_s = periods / impulse as your new invented unit :-) we don't need it :-) – Fat32 Sep 19 '20 at 10:38
• "Just having the same overloaded name is not mathematically sufficient for cancellation." Bam, hits the target dead center. The appeal to authority is supportive and sure doesn't hurt either. – Cedron Dawg Sep 19 '20 at 11:34
• Assuming Wiki is accurate, the SI model is definitionally incomplete. In en.m.wikipedia.org/wiki/SI_derived_unit, the table's right most column references "1" as a base unit (the equivalent in the header doesn't cover this) The "official dimensions list" en.m.wikipedia.org/wiki/International_System_of_Units does not include "1". So a new dimension is needed, maybe call it "Ordinal" or "Natural", with a base unit of "1". This would obviate the term "dimensionless". – Cedron Dawg Sep 19 '20 at 11:49
• @Fat32 The previous comment is a separate question from "Should 'rotation' be an official dimension with 'radians' as the base unit?" In which I agree whole heartedly with Mohr and Phillips that it should. This would eliminate the immediate problem being highlighted by the title question. Interestingly, talking about "need" vs "want", introducing this dimension would eliminate the need for an "Ordinal" dimension. – Cedron Dawg Sep 19 '20 at 12:08
• @CedronDawg Don't assume wiki as accurate for a fundamental research ;-) radian is already included in the SI system as a unit of angle (as a geomteric construct). It's not a base unit but a supplementary one. Radian itself is considered as unitless due its definition being ratio of two lengths. Note that SI also recognises K (Kelvin), cd (candela) ,or Mole number as base-units, but it's known that they can be expressed in terms of other units and actually not true innovations. So these units are not that sort of fixed things, nor they are necessarily irreducible postulates of physics. – Fat32 Sep 19 '20 at 12:50
• @Fat32 But it was on the internet so I know it must be true. ;-) According to the article "supplementary" was retired, and is now "derived". Oh, it's so hard to keep up. I disagree with its definition being the ratio of two lengths. The definition is from polar coordinates which are (radii,radians). – Cedron Dawg Sep 19 '20 at 13:00
• @Fat32 Furthermore, the ratio you speak of can be described as a normalization. So the statement "Radians are the units of normalized arc lengths (where the normalization factor is based on the radius)" is true and accurate in my understanding. Personally, the classification/categorization to the dimensional level is unimportant. I also think the major error in "SI thinking" is extending the algebraic manipulation of the units up to the dimensional level. – Cedron Dawg Sep 19 '20 at 13:45
• Agree or disagree? "Unit conversions are done within a dimension, they don't change the dimension." For anybody, not just Fat32. This is supposed to be a discussion. – Cedron Dawg Sep 19 '20 at 13:55
• @CedronDawg f/fs is a normalization between the members of the same class/category (by definition of normalization they should be of comparable -same- type) therefore its' both dimensionless and unitless in the physical (egnineering) science. You associate it a unit based on defining a ratio between two different kinds of quantities. Just as length/time , mass/time, mass/dollar.. these are for ratios they are not normalizations. You mix normalization with a ratio. So if $f/f_s$ is called a normalized frequency, then it's unitless and dimensionless... I agree :-) – Fat32 Sep 19 '20 at 14:20

I speak with no authority on this subject but interest in what the answer is- I found this table pasted below at this link https://www.sciencedirect.com/topics/engineering/units-and-dimension#:~:text=Dimensions%20are%20physical%20quantities%20that,relative%20unit%20that%20describes%20length).

Which lists the radian as a unit, but doesn't have a fundamental dimension. That said, are not the units of $$f/f_s$$ radians/sample if $$f$$ can be radians per second? (And is therefore dimensionless but not unitless?) Where in SI does it say that the units of radian frequency are not radians/second (and if it does say that, why does it if radians are a unit?). I understand how the dimension can be 1/s based on this but don't quite see how the units are lost.

• Ahh this fundamental research! Just reminds me of the claims to invent time-machine sorta things but will never show it to anyone because a patent was pending :-))) Have a look at physics with vectors. Let $\vec{A}$ be an electric field vector in units of V/m. We can express it like $\vec{A} = |A| \hat{a}$ where $|A|$ is its magnitude (in units of V/m) and $\hat{a}$ is a unitless unit vector along the direction of $\vec{A}$. Now the unit-vector $\hat{a}$ is a ratio $\hat{a} = \frac{\vec{A}}{|A|}$ hence it's normalised and therefore unitless just as unitless as f/fs should be – Fat32 Sep 19 '20 at 13:53
• A normalisation by definition requires the comprasion (and division) of two similar things which should have same dimensions and units. Therefore the result will alwyas be dimensionless and unitless. Normalized Frequency f/fs is of no exception to this rule.... – Fat32 Sep 19 '20 at 14:02
• Just as unitless as decibells. Note that engineers assign units to dB scale too.. Such as dB mV, but that's a conventional usage and strictly speaking wrong... That only means that the reference voltage is 1 mV, but the ratio is unitless. something very helpful. – Fat32 Sep 19 '20 at 14:09
• f/fs is a normalization between the members of the same class/category (by definition of normalization they should be of comparable -same- type) therefore its' both dimensionless and unitless in the physical (egnineering) science. If we associate it a unit based on defining a ratio between two different kinds of quantities. Just as length/time , mass/time, mass/dollar.. these are for ratios they are not normalizations. It's a mix of normalization with a ratio. So if $f/f_s$ is called a normalized frequency, then it's unitless and dimensionless... – Fat32 Sep 19 '20 at 14:23
• @Fat32 Normalization is the process of defining a new scale based on a reference value on the old scale. It is a ratio, and can be considered a conversion factor. The new scale is not unit less, it has tic marks. The same source scale can be normalized on different reference values. So, it is "a normalized frequency" not "the normalized frequency", otherwise fistfights might break out on which reference value is the "True One". – Cedron Dawg Sep 19 '20 at 14:40
• @Fat32 but clearly there is a practical benefit of distinguishing between radians/sample and cycles/sample so using those terms as units-- how is that reconciled otherwise? Is this just semantics or is someone saying it is wrong to describe fractional frequency in units of radians/sample (or what do we use if "unit" is verboten?) – Dan Boschen Sep 19 '20 at 14:42
• The conversion/normalization factor is "unitless/dimensionless", the result is not. – Cedron Dawg Sep 19 '20 at 14:45
• @CedronDawg it's better to consult a measurement book ;-) I believe you should delete this whole post (with answers too) and if necessay re-ask it in a way clear and focused manner. Then we shall collect the best answers with their convincing and consistent explanations... f/f_s is a normalisation and it produces unitless numbers to be processed by mathematical algorithms... Anyway. this is waay too much for such a topic... I don't understand why you reject the unitless definition..? or why you associate f and fs with different sets. They belong to same set. Same units. – Fat32 Sep 19 '20 at 14:54

I hadn't seen this before, but why not have an argument (or a "discussion") about dimensional analysis in the mainspace? rather than the metaspace?

It looks like we're going down that way regarding DFT periodicity again.

@PeterK what do you think we should do about some contentious topics about hard-core mathematics and physics? Engineering practice is another thing with legit differences of opinion, but I find it sorta goofy when it's math and classical physic.

• i don't know for sure what you're thinking but if my guess is correct, the "dimensionless units" such as *percent (%) or degree (°) or even the decibel ($\mathrm{dB}$), all those "units" are, are a convention in that the symbol is equal to a dimensionless conversion factor. ... "%"$=\frac{1}{100}$. ... "°"$=\frac{\pi}{180}$. ... and "$\mathrm{dB}$" $=\frac{\log(10)}{20}$ and those are just numbers. – robert bristow-johnson Sep 11 '20 at 18:02
• the term "radian" is a semantic. angles measured as "radian" are dimensionless quantities that are the ratio of two like-dimensioned lengths: the arc-length divided by the radial arm length. there is no unit. – robert bristow-johnson Sep 11 '20 at 18:53
• @robertbristow-johnson yes I was unfortunately restrictive about units there. However when I said they are physical, I didin't mean they have phsyical existance :-) but instead I meant they are used to scale physical quantities. So we have non-physical units of bits, bytes, and MACs, samples, etc. And yes units is an abstract (anthropometric) construct without a physical existance on their own :-) – Fat32 Sep 14 '20 at 0:23
• @CedronDawg actually a dimension is more general than a unit. So length is a dimension, but it has many different units such as meter, foot, yard, inch etc. Mass is a dimension and it also had many units throughout history. Time also had different units untill the second was adopted universally. So dimensional analysis is more general than the one with explicit units and conversion factors. I'm reading these from Ohanian Physics 2E expanded,ch1: Measurement of Space, Time , and Mass... I highly suggest this book for any scientist or engineer. – Fat32 Sep 14 '20 at 0:27
• @CedronDawg yes I realized it after posting my comment, sorry for that. Now my final words on units/dimensions is this : $2\pi$ is a number (quantifically 6.28..) without a unit. Angles do have units such as Degree, Grad, Radian. When your numbers refer to an angle, then their unit (actually a scale) is, say, radian: 3 radians, 5 radians or $2\pi$ radians. I find the cycles thing as quite unnecessary. Never seen it in any math,physics, eng book. But as long as it works for you. Thats ok. – Fat32 Sep 14 '20 at 0:42
• @Fat32 , i think we are on the same page. "cycles" or "samples" or "ticks" are things that we count. when we count them in time, there is a rate. whether it's "Hz" (or "cycles/sec") or "samples/sec" or "ticks/sec", the dimension is $\mathrm{T}^{-1}$ and the common unit is "$\frac{1}{\mathrm{s}}$" or "$\mathrm{s}^{-1}$". it may be wrong, but it is meaningful to add "ordinary frequency" to "angular frequency" because they are the same dimension. it would be like adding the number of km to the number of miles. but it's usually wrong unless you put in a conversion factor. – robert bristow-johnson Sep 14 '20 at 3:01
• but where Ced is wrong and always had been (we had argued about stuff like this on comp.dsp as recently as 2017) is that there are units attached to dimensionless numbers and that sample rate is not compatible with Hz. whenever you can add, subtract, or compare commensurate quantities, they are the same dimension. and if you can correctly add them without a numerical conversion factor, then they have exactly the same units. – robert bristow-johnson Sep 14 '20 at 3:05
• @robertbristow-johnson yeah we agree here... don't know about your past with Ced on comp.dsp :-)). I find his way of attaching units to everything in DSP as making things more complex than they actually are... At one point I fear if units of units will appear such as : velocity = meter / s = meter per coordinate scale / second per coordinate scale... where coordinate scale is a relativistic concern :-) – Fat32 Sep 14 '20 at 8:52
• @Fat32, //making things more complex than they actually are// -- it's actually worse than making things more complex than they actually are. it's erroneous which becomes obvious when Ced says that pure mathematical numbers like $2$ or $2\pi$ have units. or that $f_\mathrm{s}$ has different units than $f$ when we are adding them directly together. that pendantry will lead other pendants down the wrong path and they will later have to unlearn crap they picked up from Ced. – robert bristow-johnson Sep 14 '20 at 14:37
• @Fat32 Brief explanation: (6) is part of the E-L definition. (7) Is the "slick trick", (18) is the crown jewel and is the result of a unit conversion from (13). (13) and (18) say that the acceleration of the particle occurs in a plane (or line) defined by the gradient of the index of refraction and the velocity of the particle. I was motivated by considering that Snell's law was physically impossible, there had to be a little curvature to that bend. So you can think of (13) and (18) as the vector differential form of Snell's law. "n" is assumed to be real as a simplifying assumption. – Cedron Dawg Sep 14 '20 at 15:29
• (con't) I had to throw away half of the article due to the concept of anisotropism. The speed of the particle under the definition "c/n" is isotropic, the same in all directions. I derived the solution for the Schwarzschild Solution in the radial direction (and it is similar) and thought I had it made, but it turned out not to be that simple. Still a work in progress. The equation can also be applied, with some tweaking, to the propagation of sound waves in water with varying salinity conditions. So it is a general principle. I don't know if it could be considered solving anything unsolved. – Cedron Dawg Sep 14 '20 at 15:32
• Oh, the potential function is the value of the index of refraction (fluff density on the log scale) at every point. It has a gradient which is a conservative vector field. – Cedron Dawg Sep 14 '20 at 15:53
• @CedronDawg so yo call "n" as a potential function ! :-) Never heard that before. Let me visit my electromagnetic books for that :-). I cannot get the slick trick on wikipedia. I have no idea of why it should be that way true or useful...:-)) – Fat32 Sep 14 '20 at 16:02
• @CedronDawg I'm out of this discussion. I will use the notation found in the standard DSP texts, and IEEE standards. You know it's all about standards. Units do not have absolute meanings. Only relative. And I think that's the onto-epistemological-metaphysical mistake we are making here: trying to attach absolute meanings to things which only have binding-relations to objects/purpose they represent. What's the absolute meaning of the word apple? The only meaning is it represents a fruit which we agreed to call an apple. So keep up the standards and avoid useless discussions :-)). – Fat32 Sep 14 '20 at 19:51
• @CedronDawg answer: 1+0=1 is the first step in Real Analysis. 0/0 is covered extensively, from all directions. No need for me to explain it. If this accounts for an answer , then f/fs is first step in DSP and covered across all textbooks as unitless is just a legitimate answer too :-)) So the topic is closed! well-done; -) have a nice day too Ced. – Fat32 Sep 17 '20 at 12:02

Let's leave dimensions out of it, as I intended with the question, and leave it at the unit level. Let's also leave Hz out of it, since there is a non-SI definition (cycles per second) and a SI definition ('one' per second).

Let the number of samples taken in one second be $$f_s$$.

$$f_s \text{ samples } = 1 \text{ second }$$

Both sides can be divided by the LHS:

$$1 = \frac{ 1 \text{ second } }{ f_s \text{ samples } } = \frac{ 1 }{ f_s } \; \frac{ \text{ seconds } }{ \text{ sample } }$$

This is now a conversion factor, it converts samples to seconds.

There is a signal, suppose it has a frequency in cycles per second of $$f$$. Applying the conversion factor yields:

\begin{aligned} f \frac{ \text{ cycles } }{ \text{ second } } &= f \frac{ \text{ cycles } }{ \text{ second } } \cdot 1 \\ &= f \frac{ \text{ cycles } }{ \text{ second } } \cdot \frac{ 1 }{ f_s } \; \frac{ \text{ seconds } }{ \text{ sample } } \\ &= \frac{ f }{ f_s } \; \frac{ \text{ cycles } }{ \text{ sample } } \\ \end{aligned}

The signal's frequency has now been converted to a new unit of cycles per sample with a value of $$f/f_s$$

The boss comes around and says the implementation team needs the units to be radians per sample so they can plug it into the wondrous cycle machine.

No problem, just introduce another conversion factor:

$$2 \pi \text{ radians } = 1 \text{ cycle }$$

Both sides can be divided by the RHS and reversed:

$$1 = \frac{ 2 \pi \text{ radians } }{ 1 \text{ cycle } } = 2 \pi \frac{ \text{ radians } }{ \text{ cycle } }$$

\begin{aligned} \frac{ f }{ f_s } \; \frac{ \text{ cycles } }{ \text{ sample } } \cdot 1 &= \frac{ f }{ f_s } \; \frac{ \text{ cycles } }{ \text{ sample } } 2 \pi \frac{ \text{ radians } }{ \text{ cycle } } \\ &= \frac{ f }{ f_s } 2 \pi \; \frac{ \text{ radians } }{ \text{ sample } } \\ \end{aligned}

Report back to the boss:

The signal's frequency is $$2 \pi f/f_s$$ in units of radians per sample, just as you wanted, boss.

Meanwhile, Roger, does the calculation the SI way, and reports:

The frequency is a unitless $$f/f_s$$

The boss says to Roger, I need that in radians per sample.

What does Roger do?

• He gets promoted to HR, of course. Happy Equinox everybody. Any one want to hear the one about the family recipe for "Seagull in a Boot"? (Please don't upvote this answer, I want it to stay at the bottom.) – Cedron Dawg Sep 22 '20 at 16:43