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

Question

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.

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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.

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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?

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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".

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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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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.

Answer 7

This is my second answer and it addresses the title question.

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?