do you have a simple way of roughly estimate the max frequency? how about this interpolation scheme?

Hi all,

Do you have a simple way of "roughly" estimate the max freuqency in a plot 
of data?

For example, suppse I have collect a trace of data about one week's raining 
amount...  I have to decide how much is the Nyquist sampling rate and how 
many samples do I need... how do I estimate the max frequency in the 
rainning data roughly?

If I used Matlab to do data spectrum plot of the past a few years and find 
the spectrum is between -1 to 1 cycle per day... so I decide the Nyquist 
rate to be 2 cycle per day, i.e. I measure twice data a day... of the 
function x(t)...

But now I want to measure once a day, but sampling both x(t) and x'(t) 
(derivitative) once per day... for what kind of x(t) can I do this kind of 
lazy measurement sampling?

How do I reconstruct the original function x(t) by having the x(t) and x'(t) 
samples for one measurement per day?

Can I reduce the measurements to once per week, by sampling x(t), x'(t) , 
x''(t), ... till the 14th derivative?

What are the reasons why this sampling/interpolation scheme has not been 
used in practice?

Any ideas?

Thanks a lot, 

"kiki" <lunaliu3@yahoo.com> wrote in message
news:cqlenk$bef$1@news.Stanford.EDU...
> Hi all,
>
> Do you have a simple way of "roughly" estimate the max freuqency in a plot
> of data?
>
> For example, suppse I have collect a trace of data about one week's
raining
> amount...  I have to decide how much is the Nyquist sampling rate and how
> many samples do I need... how do I estimate the max frequency in the
> rainning data roughly?
>
> If I used Matlab to do data spectrum plot of the past a few years and find
> the spectrum is between -1 to 1 cycle per day... so I decide the Nyquist
> rate to be 2 cycle per day, i.e. I measure twice data a day... of the
> function x(t)...
>
> But now I want to measure once a day, but sampling both x(t) and x'(t)
> (derivitative) once per day... for what kind of x(t) can I do this kind of
> lazy measurement sampling?

it is less than Nyquist, so you have distorted data

>
> How do I reconstruct the original function x(t) by having the x(t) and
x'(t)
> samples for one measurement per day?
>
> Can I reduce the measurements to once per week, by sampling x(t), x'(t) ,
> x''(t), ... till the 14th derivative?
>
> What are the reasons why this sampling/interpolation scheme has not been
> used in practice?
>

Think of it, using only one sample per week, you can predict the rain amouts
for each day?
hardly.

derivitatives bring up "noise", by the second/third your data is all noise
a derivitative is a high pass filter

> Any ideas?
>
> Thanks a lot,
>
>

"kiki" <lunaliu3@yahoo.com> wrote in message
news:cqlenk$bef$1@news.Stanford.EDU...
> Hi all,
>
> Do you have a simple way of "roughly" estimate the max freuqency in
a plot
> of data?

Double the rciprocal of the mean time between observations is a
reasonable figure to use.

Franz

kiki wrote:

> Hi all,
> 
> Do you have a simple way of "roughly" estimate the max freuqency in a plot 
> of data?
> 
> For example, suppse I have collect a trace of data about one week's raining 
> amount...  I have to decide how much is the Nyquist sampling rate and how 
> many samples do I need... how do I estimate the max frequency in the 
> rainning data roughly?
> 
> If I used Matlab to do data spectrum plot of the past a few years and find 
> the spectrum is between -1 to 1 cycle per day... so I decide the Nyquist 
> rate to be 2 cycle per day, i.e. I measure twice data a day... of the 
> function x(t)...
> 
> But now I want to measure once a day, but sampling both x(t) and x'(t) 
> (derivitative) once per day... for what kind of x(t) can I do this kind of 
> lazy measurement sampling?
> 
> How do I reconstruct the original function x(t) by having the x(t) and x'(t) 
> samples for one measurement per day?
> 
> Can I reduce the measurements to once per week, by sampling x(t), x'(t) , 
> x''(t), ... till the 14th derivative?
> 
> What are the reasons why this sampling/interpolation scheme has not been 
> used in practice?
> 
> Any ideas?
> 
> Thanks a lot, 


You can always find a way to push a system beyond it's applicable domain
by ignoring the details needed to get it there. I'll go you one (or two)
better.

You can't sample rainfall at all because at any instant there's either a
drop or there isn't, and sampling is about what happens at instants.
When a drop hits, the derivative is, at the very least, very large, and
between drops, it is zero. So sampling the derivative isn't much use.

OK: lets get real. What we really do ...

Wait: you tell me. How do you propose to collect your samples? What
limitations do that sampling scheme impose? When you respect those
limitations, is there still a dilemma?

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

kiki wrote:

> Hi all,
> 
> Do you have a simple way of "roughly" estimate the max freuqency in a plot 
> of data?
> 
> For example, suppse I have collect a trace of data about one week's raining 
> amount...  I have to decide how much is the Nyquist sampling rate and how 
> many samples do I need... how do I estimate the max frequency in the 
> rainning data roughly?

How did you collect the data in a way that you're sure you haven't 
already introduced sampling error?
> 
> If I used Matlab to do data spectrum plot of the past a few years and find 
> the spectrum is between -1 to 1 cycle per day... so I decide the Nyquist 
> rate to be 2 cycle per day, i.e. I measure twice data a day... of the 
> function x(t)...

You obviously aren't measuring rain in western Oregon.
> 
> But now I want to measure once a day, but sampling both x(t) and x'(t) 
> (derivitative) once per day... for what kind of x(t) can I do this kind of 
> lazy measurement sampling?

In theory any x(t) for which two measurements per day would be adequate 
-- that means an x(t) which is _strictly_ bandlimited.
> 
> How do I reconstruct the original function x(t) by having the x(t) and x'(t) 
> samples for one measurement per day?

I don't know offhand, so offhand I'd say you actually do some math 
before you ask the question.  Then you'll not only know the answer, 
you'll also understand it.  Start with the fact that x(t) is strictly 
bandlimited, and remember that the Taylor's series expansion figures in 
there somehow.
> 
> Can I reduce the measurements to once per week, by sampling x(t), x'(t) , 
> x''(t), ... till the 14th derivative?

Sure -- if you are getting _really good_ derivatives, and if your x(t) 
is really truly bandlimited.  Of course, if you want to get really good 
estimates of the derivatives from observation and you're not sure about 
the bandlimiting of x(t) then you need to sample it twice each day...
> 
> What are the reasons why this sampling/interpolation scheme has not been 
> used in practice?

Think about it using _your own brain_ and you'll not only know the 
answer, but you'll also understand it.
> 
> Any ideas?
> 
> Thanks a lot, 
> 
> 
Kiki, get out of Matlab, turn off your computer, get out of the lab, go 
outside, stare at the sky and THINK for a while, will you?

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Jerry Avins wrote:

> 
> You can't sample rainfall at all because at any instant there's either a
> drop or there isn't, and sampling is about what happens at instants.

I would disagree. When a farmer wants to know how much rainfall there
has been in order to decide how much irrigation his crops will require
he has a perfectly good bandlimited continuous function called
'rainfall' which he can refer to. And it is indeed a function that is
sampled at instants (points in both time and space) even though the
sampling device may spend all 24/7 collecting data. Also, it is a very
accurate representation of the moisture that is getting to the roots of
his crops which is what the farmer means when he says 'rainfall'. 


> When a drop hits, the derivative is, at the very least, very large, and
> between drops, it is zero. So sampling the derivative isn't much use.
> 

2 rain gauges set a foot apart in the open will have no discernible
difference in their readings so there is no need to measure individual
raindrops. The diameter of the rain gauge (within reason) has no effect
on the measurement of the rainfall function - the function just simply
isn't changing fast enough. 
	What you or the OP mean by measuring the derivatives of rainfall I have
no clue.

-jim

> OK: lets get real. What we really do ...
> 
> Wait: you tell me. How do you propose to collect your samples? What
> limitations do that sampling scheme impose? When you respect those
> limitations, is there still a dilemma?
> 
> Jerry
> --
> Engineering is the art of making what you want from things you can get.
> �����������������������������������������������������������������������


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jim wrote:

> 
> Jerry Avins wrote:
> 
> 
>>You can't sample rainfall at all because at any instant there's either a
>>drop or there isn't, and sampling is about what happens at instants.
> 
> 
> I would disagree. When a farmer wants to know how much rainfall there
> has been in order to decide how much irrigation his crops will require
> he has a perfectly good bandlimited continuous function called
> 'rainfall' which he can refer to. And it is indeed a function that is
> sampled at instants (points in both time and space) even though the
> sampling device may spend all 24/7 collecting data. Also, it is a very
> accurate representation of the moisture that is getting to the roots of
> his crops which is what the farmer means when he says 'rainfall'. 

He doesn't sample the rate of fall at an instant in time. Instead, he
checks his rain gauge for accumulated rainfall. He is rarely interested
in rate, and when he is, he settles for an average rate over some
(probably variable) time.

>>When a drop hits, the derivative is, at the very least, very large, and
>>between drops, it is zero. So sampling the derivative isn't much use.
>>
> 
> 
> 2 rain gauges set a foot apart in the open will have no discernible
> difference in their readings so there is no need to measure individual
> raindrops. The diameter of the rain gauge (within reason) has no effect
> on the measurement of the rainfall function - the function just simply
> isn't changing fast enough. 
> 	What you or the OP mean by measuring the derivatives of rainfall I have
> no clue.

In most places, an inch an hour is considered heavy rain, but it
frequently rains at that heavy rate for brief periods. Once, when it had
already rained copiously during much of the day, we had a total
additional rainfall of 4 inches in a little under an hour. Children used
inflatable wading pools to raft down our main street. The manhole cover
at it low point had been pushed aside by the water pressure and a geyser
of much diluted sewage spewed 12 feet into the air. Although usually
only the total matters, sometimes rate matters too.

The OP's error was treating rainfall as a bandlimited periodic function.
I was trying to get him to think instead of just pushing variables
around. Similar problems arise when sampling stock prices.

  ...

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Jerry Avins wrote:
> 
> 
> The OP's error was treating rainfall as a bandlimited periodic function.

Your error is you have said nothing to support this claim. If the
farmers fields are flat his definition of rainfall is in fact a
bandlimited periodic function and he's not to terribly interested in
what effect it has on your pavement covered environment.
	If your interested in examining rainfall on a cyclic period of 5 to 10
days it doesn't much matter whether you collect data religiously once an
hour or once a day your conclusions based on the data on whether to
irrigate (to artificially maintain the optimal periodic cycle) will be
the same.

-jim


> I was trying to get him to think instead of just pushing variables
> around. Similar problems arise when sampling stock prices.
> 
>   ...
> 
> Jerry
> --
> Engineering is the art of making what you want from things you can get.
> �����������������������������������������������������������������������


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"Franz Heymann" <notfranz.heymann@btopenworld.com> wrote in message 
news:cqm650$dom$3@titan.btinternet.com...
>
> "kiki" <lunaliu3@yahoo.com> wrote in message
> news:cqlenk$bef$1@news.Stanford.EDU...
>> Hi all,
>>
>> Do you have a simple way of "roughly" estimate the max freuqency in
> a plot
>> of data?
>
> Double the rciprocal of the mean time between observations is a
> reasonable figure to use.
>
> Franz

???? If an "observation" is a "sample" then how does this address the 
concern?

Fred 

Some suggestions below:

Fred

"kiki" <lunaliu3@yahoo.com> wrote in message 
news:cqlenk$bef$1@news.Stanford.EDU...
> Hi all,
>
> Do you have a simple way of "roughly" estimate the max freuqency in a plot 
> of data?

>
> For example, suppse I have collect a trace of data about one week's 
> raining amount...  I have to decide how much is the Nyquist sampling rate 
> and how many samples do I need... how do I estimate the max frequency in 
> the rainning data roughly?

You could compute a Fourier Transform of the data.  That would probably mean 
computing an FFT using a very high sample rate.

Under the assumption that you're using a rainfall guage, it integrates the 
individual drops and even some of the small squalls.  So, the individual 
raindrops are not the issue here.

>
> If I used Matlab to do data spectrum plot of the past a few years and find 
> the spectrum is between -1 to 1 cycle per day... so I decide the Nyquist 
> rate to be 2 cycle per day, i.e. I measure twice data a day... of the 
> function x(t)...

How often was the data sampled in the first place?
What is a "cycle" in this context?  I don't know what "-1 cycle" means - 
that the rain was falling up?

> But now I want to measure once a day, but sampling both x(t) and x'(t) 
> (derivitative) once per day... for what kind of x(t) can I do this kind of 
> lazy measurement sampling?

What is x(t)?  What was measured?  Is it a rainfall guage?

>
> How do I reconstruct the original function x(t) by having the x(t) and 
> x'(t) samples for one measurement per day?

It probably isn't too much of a stretch to say: "you can't"
>
> Can I reduce the measurements to once per week, by sampling x(t), x'(t) , 
> x''(t), ... till the 14th derivative?
>
> What are the reasons why this sampling/interpolation scheme has not been 
> used in practice?

If you have a function that is nicely defined mathematically, then in theory 
you might be able to do this.  In that case you'd be using a mathematical 
manipulation to go from one set of numbers to another. Maybe that's properly 
called a mapping.

However, in the real world you don't have such a nice math function and 
using the derivatives don't work for a number of reasons.  One of those 
reasons is that the underlying "data" is from a random process.  Another is 
that there is likely noise in real data - which adds more randomness.