Opened 9 years ago

Closed 8 years ago

#147 closed question (fixed)

unit of amplitude spectral densities?

Reported by: georg@tencknet.de Owned by: Gunther Schadow
Priority: minor Milestone: Version 2.0
Component: Keywords: spectral density
Cc: georg@…

Description (last modified by Gunther Schadow)

While energy spectral density and power spectral density conform to the UCUM specification in having integer exponents only, amplitude spectral density does not as it is the root of the PSD. For acceleration spectral density e.g. the PSD unit is [(m/s2)2/Hz], the ESD is [(m/s2)2.s/Hz] but the ASD would have [m/s2/Hz1/2]? How can this contradiction be solved? Regards Georg Tenckhoff

Change History (10)

comment:1 Changed 8 years ago by Gunther Schadow

Yes, this is a vexing problem. I must admit, while having seen this mentioned from the start of UCUM, it seemed so odd and without any other example of the same issue, and personally I do not know this particular field, that it has been deferred until some expert could opine.

One possibility would be to define this as a "special unit" with a function pair. Is there some firm conversion one can define from this ASD to ESD? If so, we could define a symbol for m/s2 /Hz1/2 with these conversion funcitions.

comment:2 Changed 8 years ago by Gunther Schadow

Description: modified (diff)

Georg replied by email.

BTW, now you should be able to reply directly to the ticket, as you have been verified. This is a spam prevention measure we had to implement, and which turned out quite effective. Sorry for the inconvenience.

Here is his reply:

Puzzling indeed as I am aware of several normative documents affirming the integer character of the unit exponents. I am therefore not sure how the solution should be, really. Do yo know an authority we might ask? I am neither aware of any other similar examples.

With respect to your proposal of the definition of a "special unit" with a function pair, what kind of function do you mean? Do you have an example that gives me the idea? As mentioned in my ticket the link is, that ASD is the square root of the PSD and the square of the ASD gives the PSD. Does that match your view of a function?

Certainly I would hope that Christof Gessner might be able to shed some light.

If you understand this area, Gerorg, as it looks like you do, we can perhaps find a good solution ourselves. You gave some equations relating ASD and PSD. Let's make that explicit.

PSD = ASD2

is that true?

I am associating via P = R * I2, so there is an R here, so my question is if the conversion between PSD and ASD is so simple really? Could you give an example with some real case (with numbers) where we can see this conversion applied in reality?

Meanwhile, please correct my mistakes in the following dimensional analysis. Based on what you wrote initially, I can say:

ESDunit is 1(m/s2)2.s/Hz m2 s-4 s Hz-1 m2 s-4 s s m2 s-2
PSDunit is 1(m/s2)2/Hz m2 s-4 Hz-1 m2 s-4 s m2 s-3
ASDwould be 1m/s2/Hz1/2 m s-2 Hz-0.5 m s-2 s.5 m s-1.5

I can see there clearly how the unit of [PSD] is [ASD]2 indeed.

We could then define ASD using the function pair x2 and sqrt(x) and define a single unit atom for m/s2/Hz^1/2, may be we just wrap this term into [m/s2/Hz(1/2)] or just [m.s(1.5)].

comment:3 Changed 8 years ago by georg@tencknet.de

Your dimensional analysis and also the conversion is totally correct for my understanding. I should know as I hold a master degree in physics... But what still keeps irritating me, is the contradiction to what I once learned and what has been fixed in several normative documents: that unit exponents should be integers! Only one of them can be true!? Will you please ask Christof Gessner? I will (once again ...) consult the documents mentioned.
At the moment I do not oversee the consequences of a functional mapping, as you propose. How would such a mapping be documented within the UCUM standard and it's XML representation? What about the software implementing the standard? I am somewhat familiar with JScience. But I still have to give it a try what happens if I "throw" it the square root of some base units or that of (m/s2)2/Hz = m2 s-3 ?

comment:4 Changed 8 years ago by georg@tencknet.de

I just consulted "the UCUM" and found what you are relating to at chapter 3.1 "SPECIAL UNITS ON NON-RATIO SCALES", §21 - §23. Truely a "Solomon-style" escape to the problem. I'd agree on solving it like that.

comment:5 Changed 8 years ago by Gunther Schadow

Thanks Georg. Would you still be so kind please to give me a real world example of an actual instance of measurement or measurements that does this conversion? I would like that to test the new function pair and double-check my understanding. If you could post some document that describes this, it would be awesome. Thanks.

comment:6 Changed 8 years ago by georg@tencknet.de

Hi Gunther,
indeed I can supply you with a real world example. They are so common in signal processing, that it is rather the task to select a simple enlightening one. Spectra and their densities (energy, power and amplitude "linear") are beeing commonly calculated in the engineering, electronics, acoustic, etc. sciences from data taken in the time or in the spacial domain. http://www.ap.com/kb/show/170 shows an example. As you see there you will not find single value examples, but "spectra" in the sense of maps of energy/ power/ amplitude and their densities over frequency. Thus if you need a calculated example I could supply you with a (short) table of values calculated with a signal analysis tool we use at my employer. Is it this what you need? Or does the link suffice already? Presumably the spectral character of the data, the transformation is applied to, explains, why it itself is that simple ...
Cheers

Georg

comment:7 Changed 8 years ago by Gunther Schadow

Milestone: Version 2.0
Owner: set to Gunther Schadow

In mile stone.

comment:8 Changed 8 years ago by Gunther Schadow

Summary of what we'll do:

  • ESD - energy spectral density - unit is [(m/s2)2.s/Hz] - m2.s-2
  • PSD - power spectral density - unit is [(m/s2)2/Hz] - m2.s-3
  • ASD - amplitude spectral density - unit is [m/s2/Hz1/2] - m.s-1.5
  • PSD = ASD2 holds true.
  • We could then define ASD using the function pair f(x) = x2 and f-1(x) = x1/2
  • Define a single unit atom for [m/s2/Hz(1/2)] or just [m.s(1.5)], or just [ASD'U]

Question: Such special units, can they be raised to powers?

According to §22, operations on special units, they can not. No operations other than multiplication with a scalar are possible.

So, this throws a major monkey-wrench in, does it not?

Since we cannot raise the special unit [ASD'U] to a power of 2 to obtain m2/s3, the benefit of defining the special unit seems to be absent.

Let us do it anyway, so we have done something and describe the remaining issue.

comment:9 Changed 8 years ago by Gunther Schadow

<u:unit id="600523" Code="[m/s2/Hz(1/2)]" CODE="[M/S2/HZ(1/2)]" isMetric="no"><name>meter per square seconds per square root of hertz</name><printSymbol></printSymbol><property>amplitude spectral density</property><value Unit="sqrt(1 m2/s4/Hz)" UNIT="sqrt(1 m2/s4/Hz)"><function name="sqrt" value="1" Unit="m2/s4/Hz"/></value></u:unit>

The question is, is the conversion function square or square-root? I think it is square root since we put in the number expressed in PSD unit, then take the square root to get the number in the ASD unit.

public final class sqrt implements FunctionPair {    
  static final REAL TWO = ValueFactory.getInstance().REALvalueOfLiteral("2");
  static final REAL HALF = TWO.inverted();
  public REAL f_to  (REAL x) { return x.power(HALF); }
  public REAL f_from(REAL x) { return x.power(TWO); }
}

comment:10 Changed 8 years ago by Gunther Schadow

Resolution: fixed
Status: newclosed
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