# basic function, coefficient, wavey line explenations please

Started by February 12, 2005
```hello,

from a book ('the world according to wavelets' by barbara hubbard) i've
just started to read:

first, why on earth is the temperature measurement graph in the top
left graph wavey? if that was in a non-fourier/non-wavelets book that'd
be a smooth curve, not wavey i'm sure. i can't help feel it's a case of
initial data being completely tailored to suit. unless you heated a
metal bar in a very particular way (some kind of multiple bar heater
with decreasing heat in each bar maybe) you simply wouldn't get results
like that?

the word function and its use is confusing, not just on that linked to
page, but so far in the book. to me a function is a set of rules that
takes an input and manipulates it and outputs the result. curved lines
in a graph keep being called a function. surely, at best, a curvey line
like that can be called output from a function. it isn't a function in
itself is it? the wavey line that i mentioned in the above paragraph,
that is a direct result from measuring temperature along a metal bar at
the same time, but is described as a function. a graph made from
measurements. how is that a function?

what is a coefficient? what does coefficient mean?

at the bottom of the text that accompanies the graphs on the page i
linked to, it says for each such time (1, 5, 10 and 50) the
coefficients are the same for the entire bar. this probably comes down
to me not knowing what coefficient means, but judging from the bottom
right graph and the various little dots on the verticle lines, the
coefficients are not all the same for the entire bar at each time at
all. if you follow the dots accross (and it's a bit unclear because of
the illustration/graph size) the dots are at different hieghts. in fact
it looks like the graph lines in the graph on the bottom left of the
four graphs are following what are pointed to and described as
coeficients (the little dots) and they curve -- they're not horizontal
lines, so that says that the coeficients are not the same for each
time. so that doesn't make sense to me. i've made what i mean hopefully
clear in a graph on the bottom of that page i linked to. it says
"obviously not in a straight line", at the same height, so not the same
coefficient.

any help on the above questions much appreciated.

thanks, ben.
```
```maybe i made the questions too involved/long winded. here's another
attempt, in a shorter, clearer form hopefully:

1.
the top left graph on this page
http://www.hdbatik.co.uk/temp/waveletsbookpage.html shows the results
of measuring the temperatures at various points along a metal bar at
the same time. the graph is, as well as curving, wavey. why isn't it a
smooth, non-wavey curve as you'd (well, at least i would) expect? it
seems unatural, odd and unlikely that the heating of a metal bar
(unless you heated the bar in a very unusual, particular way) would
result in blotchy, varying temperatures as depicted in the graph --
surely it'd be smooth and have a gradual change, and not be wavey?

2.
the word function is used in the book to describe various curves in
graphs. for example, the wavey line that i'm asking about in the
previous question is described as a function -- the line itself, a
function. surely a function is something that takes an input and gives
an output -- a set of rules/a piece of logic with input and output?
surely a line in a graph isn't a function, the closest a line in a
graph could get to a function is being the result of function
operations?

3.
what's meant by a coefficient? what does "coefficient" mean?

any answers to any of those would be much appreciated.

thanks, ben
```
```ben wrote:

> maybe i made the questions too involved/long winded. here's another
> attempt, in a shorter, clearer form hopefully:
>
> 1.
> the top left graph on this page
> http://www.hdbatik.co.uk/temp/waveletsbookpage.html shows the results
> of measuring the temperatures at various points along a metal bar at
> the same time. the graph is, as well as curving, wavey. why isn't it a
> smooth, non-wavey curve as you'd (well, at least i would) expect? it
> seems unatural, odd and unlikely that the heating of a metal bar
> (unless you heated the bar in a very unusual, particular way) would
> result in blotchy, varying temperatures as depicted in the graph --
> surely it'd be smooth and have a gradual change, and not be wavey?

I think the graph is cooked -- a plot of an oversimplified equation
rather than actual measurements. It is what I would expect if the true
curve were approximated by harmonic analysis with higher (but still
relevant) harmonics omitted.

> 2.
> the word function is used in the book to describe various curves in
> graphs. for example, the wavey line that i'm asking about in the
> previous question is described as a function -- the line itself, a
> function. surely a function is something that takes an input and gives
> an output -- a set of rules/a piece of logic with input and output?
> surely a line in a graph isn't a function, the closest a line in a
> graph could get to a function is being the result of function
> operations?

It's all semantics. f(x) is the notation for a function of x. In the
equation y = f(x), since y and f(x) are declared equal, y is also a
function of x. In particular, let y = f(x) = x^2. It's easy to plot
y = x^2, and we say that the plot represents that particular function.
It's a small stretch to say that the plot IS the function, rather than
that it IS OF the function.

> 3.
> what's meant by a coefficient? what does "coefficient" mean?

All parabolas with vertical axes can be represented by the function f(x)
= y = ax^2 + bx + c. The quantities a, b, and c determine exactly which
parabola the function represents. They are the function's coefficients.

When a function is subjected to Fourier analysis, the result is of the
form A*sin(w) + B*cos(w) + C*sin(2w) + D*cos(2w) + E*sin(3w) + ...
The letters A-E... are the coefficients. Since the shape of the curve
changes with time, so do the Fourier coefficients.

> any answers to any of those would be much appreciated.
>
> thanks, ben

You're welcome, Ben

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```In article <379ncbF5av8okU1@individual.net>, Jerry Avins <jya@ieee.org>
wrote:

> ben wrote:

> > 1.
> > the top left graph on this page
> > http://www.hdbatik.co.uk/temp/waveletsbookpage.html shows the results
> > of measuring the temperatures at various points along a metal bar at
> > the same time. the graph is, as well as curving, wavey. why isn't it a
> > smooth, non-wavey curve as you'd (well, at least i would) expect? it
> > seems unatural, odd and unlikely that the heating of a metal bar
> > (unless you heated the bar in a very unusual, particular way) would
> > result in blotchy, varying temperatures as depicted in the graph --
> > surely it'd be smooth and have a gradual change, and not be wavey?
>
>
> I think the graph is cooked
that's what i thought just from common sense, without really knowing.
> -- a plot of an oversimplified equation
> rather than actual measurements. It is what I would expect if the true
> curve were approximated by harmonic analysis with higher (but still
> relevant) harmonics omitted.

i'm not sure what harmonic analysis is, but why wouldn't actual
measurements be used? am i right in saying that fourier transforms need
wavelike data to operate on? that's what they require. so a step has
been omitted in the process that those graphs are describing? a step
between actual temperature measurements and those temperature
measurements exhibitting waveyness? why's that been omitted? -- simply
because the author wanted to make things as simple as possible in order
to explain things. does that sound right?

so basically, when fourier was doing whatever he did with his
temperature based work, he took temperature readings and transformed
them somehow (involving "harmonic analysis" from what you say maybe)
and after that transformation the temperature readings then had wave
properties -- which is essential to fourier's main bit. having got data
into a wave from, *then* his main bit could kick in -- the fourier
transform stuff as the graphs show? is that roughly right? (quite
possibly not as i'm doing a lot of guessing)

> > 2.
> > .... surely a function is something that takes an input and gives
> > an output ... ? .... surely a line in a graph isn't a function ? ...
>
> It's all semantics. ...... It's a small stretch to say that the plot IS the
> function, rather than that it IS OF the function.

right, yup, that makes sense. just seemed a bit odd to start with.

another use of the word 'function' i'm having trouble with: what does
it mean to say that "something is a function of something else"
exactly? that phrase is used all over the place and just doesn't make
sense to me. the book says this about the page i linked to, which uses
that phrase:

As shown [on the linked to page], we start by establishing the initial
temperature -- at time "zero" -- which we consider as a function of
distance along the bar.

could anyone translate "as a function of distance along the bar" into
another way of saying the same thing please? or any other version of
that phrase.  the general meaning of "x is a function of y" isn't
making sense to me.
"... is a function of time" is another common one i think. if anyone
could explain that it'd be great (i know it probably seems very simple
but it'd be much appreciated)

> > 3.
> > what's meant by a coefficient? what does "coefficient" mean?
>
> All parabolas with vertical axes can be represented by the function f(x)
> = y = ax^2 + bx + c. The quantities a, b, and c determine exactly which
> parabola the function represents. They are the function's coefficients.
>
> When a function is subjected to Fourier analysis, the result is of the
> form A*sin(w) + B*cos(w) + C*sin(2w) + D*cos(2w) + E*sin(3w) + ...
> The letters A-E... are the coefficients. Since the shape of the curve
> changes with time, so do the Fourier coefficients.

i'm going to have to think about that one.

thanks very much Jerry -- didn't think anyone was going to bother with
such silly questions :)

thanks, ben.
```
```ben wrote:

> In article <379ncbF5av8okU1@individual.net>, Jerry Avins <jya@ieee.org>
> wrote:
>
>
>>ben wrote:
>
>
>>>1.
>>>the top left graph on this page
>>>http://www.hdbatik.co.uk/temp/waveletsbookpage.html shows the results
>>>of measuring the temperatures at various points along a metal bar at
>>>the same time. the graph is, as well as curving, wavey. why isn't it a
>>>smooth, non-wavey curve as you'd (well, at least i would) expect? it
>>>seems unatural, odd and unlikely that the heating of a metal bar
>>>(unless you heated the bar in a very unusual, particular way) would
>>>result in blotchy, varying temperatures as depicted in the graph --
>>>surely it'd be smooth and have a gradual change, and not be wavey?
>>
>>
>>I think the graph is cooked
>
> that's what i thought just from common sense, without really knowing.
>
>>-- a plot of an oversimplified equation
>>rather than actual measurements. It is what I would expect if the true
>>curve were approximated by harmonic analysis with higher (but still
>>relevant) harmonics omitted.
>
>
> i'm not sure what harmonic analysis is, but why wouldn't actual
> measurements be used? am i right in saying that fourier transforms need
> wavelike data to operate on? that's what they require. so a step has
> been omitted in the process that those graphs are describing? a step
> between actual temperature measurements and those temperature
> measurements exhibitting waveyness? why's that been omitted? -- simply
> because the author wanted to make things as simple as possible in order
> to explain things. does that sound right?
>
> so basically, when fourier was doing whatever he did with his
> temperature based work, he took temperature readings and transformed
> them somehow (involving "harmonic analysis" from what you say maybe)
> and after that transformation the temperature readings then had wave
> properties -- which is essential to fourier's main bit. having got data
> into a wave from, *then* his main bit could kick in -- the fourier
> transform stuff as the graphs show? is that roughly right? (quite
> possibly not as i'm doing a lot of guessing)

Not right, sorry. Any shape can be represented by by a sum of sines and
cosines. Usually, an infinite number of them. For example a square wave
of amplitude 1 and frequency w -- w = 2pi/period -- is the sum
4/pi[sin(wt) + sin(3wt)/3 + sin(5wt)/5 + sin(7wt)/7 + ...]
When some of the harmonics are omitted, the flats become wavy.  See
pictures at http://cnx.rice.edu/content/m0041/latest/

I think you need a book. Look for recommendations at http://www.dspguru.com/

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```In article <37algnF5a2679U1@individual.net>, Jerry Avins <jya@ieee.org>
wrote:

> ben wrote:
>
> > In article <379ncbF5av8okU1@individual.net>, Jerry Avins <jya@ieee.org>
> > wrote:
> >
> >
> >>ben wrote:
> >
> >
> >>>1.
> >>>the top left graph on this page
> >>>http://www.hdbatik.co.uk/temp/waveletsbookpage.html shows the results
> >>>of measuring the temperatures at various points along a metal bar at
> >>>the same time. the graph is, as well as curving, wavey. why isn't it a
> >>>smooth, non-wavey curve as you'd (well, at least i would) expect? it
> >>>seems unatural, odd and unlikely that the heating of a metal bar
> >>>(unless you heated the bar in a very unusual, particular way) would
> >>>result in blotchy, varying temperatures as depicted in the graph --
> >>>surely it'd be smooth and have a gradual change, and not be wavey?
> >>
> >>
> >>I think the graph is cooked
> >
> > that's what i thought just from common sense, without really knowing.
> >
> >>-- a plot of an oversimplified equation
> >>rather than actual measurements. It is what I would expect if the true
> >>curve were approximated by harmonic analysis with higher (but still
> >>relevant) harmonics omitted.
> >
> >
> > .....am i right in saying that fourier transforms need
> > wavelike data to operate on?.......
> > ....
> > ....
> > ....

> Not right, sorry. Any shape can be represented by by a sum of sines and
> cosines. Usually, an infinite number of them.

right. i was just trying to imagine why the author didn't use actual
temperature results in the graph, as she stated they were, but used
data that's processed that's added waveyness.

the fourier coefficients, in the corresponding fourier space graph,
correlate to each wave (to each crest and dip), so making the original
data line wavey, making it repeatedly, uniformly bumpy, makes the
corresponding fourier space graph easier to show i suppose (keeps to a
minimum the number of coefficients), maybe.

the graph could have been smoth curve, and not wavey (as i'd have
expected it), but that would have made the fourier space graph less
obvious. but then i'm still thinking how can you take data which isn't
wavey, force it to have waves in it, then proceed from there. seems a
bit unreasobable.

another imagined guess: maybe temperature results are infact inherently
wavey (but in a very subtle, unoticeable way) and the graph in question
is an exaggerated version, exaggerated to show up the waves?

basically why would the author have based an important part of her
explenation of fourier analysis on a cooked graph? (and how or why (if
it is) ok to do that?)

> For example a square wave
> of amplitude 1 and frequency w -- w = 2pi/period -- is the sum
>    4/pi[sin(wt) + sin(3wt)/3 + sin(5wt)/5 + sin(7wt)/7 + ...]
> When some of the harmonics are omitted, the flats become wavy.  See
> pictures at http://cnx.rice.edu/content/m0041/latest/

yup ok thanks.

>
> I think you need a book. Look for recommendations at http://www.dspguru.com/

well the book i'm reading at the moment (hubbard 'the world according
to wavelets' (where those graphs are from)) is supposed to be a good
book for non-technical/non-methematical people but as soon as i sniff a
contradiction or something that doesn't add up i find it hard to
continue with it until i get it straitened out -- that's what i'm
trying to do.

thanks-a-lot, ben.
```
```"ben" <x@x.x> wrote in message news:140220051131477629%x@x.x...
> In article <37algnF5a2679U1@individual.net>, Jerry Avins <jya@ieee.org>
> wrote:
>
>> ben wrote:
>>
>> > In article <379ncbF5av8okU1@individual.net>, Jerry Avins <jya@ieee.org>
>> > wrote:
>> >
>> >
>> >>ben wrote:
>> >
>> >
>> >>>1.
>> >>>the top left graph on this page
>> >>>http://www.hdbatik.co.uk/temp/waveletsbookpage.html shows the results
>> >>>of measuring the temperatures at various points along a metal bar at
>> >>>the same time. the graph is, as well as curving, wavey. why isn't it a
>> >>>smooth, non-wavey curve as you'd (well, at least i would) expect? it
>> >>>seems unatural, odd and unlikely that the heating of a metal bar
>> >>>(unless you heated the bar in a very unusual, particular way) would
>> >>>result in blotchy, varying temperatures as depicted in the graph --
>> >>>surely it'd be smooth and have a gradual change, and not be wavey?
>> >>
>> >>
>> >>I think the graph is cooked
>> >
>> > that's what i thought just from common sense, without really knowing.
>> >
>> >>-- a plot of an oversimplified equation
>> >>rather than actual measurements. It is what I would expect if the true
>> >>curve were approximated by harmonic analysis with higher (but still
>> >>relevant) harmonics omitted.
>> >
>> >
>> > .....am i right in saying that fourier transforms need
>> > wavelike data to operate on?.......
>> > ....
>> > ....
>> > ....
>
>> Not right, sorry. Any shape can be represented by by a sum of sines and
>> cosines. Usually, an infinite number of them.
>
> right. i was just trying to imagine why the author didn't use actual
> temperature results in the graph, as she stated they were, but used
> data that's processed that's added waveyness.
>
> the fourier coefficients, in the corresponding fourier space graph,
> correlate to each wave (to each crest and dip), so making the original
> data line wavey, making it repeatedly, uniformly bumpy, makes the
> corresponding fourier space graph easier to show i suppose (keeps to a
> minimum the number of coefficients), maybe.
>
> the graph could have been smoth curve, and not wavey (as i'd have
> expected it), but that would have made the fourier space graph less
> obvious. but then i'm still thinking how can you take data which isn't
> wavey, force it to have waves in it, then proceed from there. seems a
> bit unreasobable.
>
> another imagined guess: maybe temperature results are infact inherently
> wavey (but in a very subtle, unoticeable way) and the graph in question
> is an exaggerated version, exaggerated to show up the waves?
>
> basically why would the author have based an important part of her
> explenation of fourier analysis on a cooked graph? (and how or why (if
> it is) ok to do that?)
>
>
>> For example a square wave
>> of amplitude 1 and frequency w -- w = 2pi/period -- is the sum
>>    4/pi[sin(wt) + sin(3wt)/3 + sin(5wt)/5 + sin(7wt)/7 + ...]
>> When some of the harmonics are omitted, the flats become wavy.  See
>> pictures at http://cnx.rice.edu/content/m0041/latest/
>
> yup ok thanks.
>
>>
>> I think you need a book. Look for recommendations at
>> http://www.dspguru.com/
>
> well the book i'm reading at the moment (hubbard 'the world according
> to wavelets' (where those graphs are from)) is supposed to be a good
> book for non-technical/non-methematical people but as soon as i sniff a
> contradiction or something that doesn't add up i find it hard to
> continue with it until i get it straitened out -- that's what i'm
> trying to do.
>
> thanks-a-lot, ben.
Hi Ben,
To try to be fair to Mr. or Ms. Hubbard (I haven't read the book and
know nothing about the author) a book called 'the world according to
wavelets'  should have some wavey drawings or functional descriptions in it.
In further support of the author it seems that, when you look in shorter
lengths/times at things, it is very difficult to actually observe any abrupt
transitions or straight lines and what can be observed/measured is maybe
more accurately described as the summation or product of probabilities that
are apparently well described by wave-like distributions without having to
resort to mathematical inventions like transform domains stretching to +/-
infinity.

Best of Luck - Mike

```
```In article <cuqar3\$pcd\$1@hercules.btinternet.com>, Mike Yarwood
<mpyarwood@btopenworld.com> wrote:

> "ben" <x@x.x> wrote in message news:140220051131477629%x@x.x...
> > In article <37algnF5a2679U1@individual.net>, Jerry Avins <jya@ieee.org>
> > wrote:
> >
> >> ben wrote:
> >>
> >> > In article <379ncbF5av8okU1@individual.net>, Jerry Avins <jya@ieee.org>
> >> > wrote:
> >> >
> >> >
> >> >>ben wrote:
> >> >
> >> >
> >> >>>1.
> >> >>>the top left graph on this page
> >> >>>http://www.hdbatik.co.uk/temp/waveletsbookpage.html shows the results
> >> >>>of measuring the temperatures at various points along a metal bar at
> >> >>>the same time. the graph is, as well as curving, wavey. why isn't it a
> >> >>>smooth, non-wavey curve as you'd (well, at least i would) expect? it
> >> >>>seems unatural, odd and unlikely that the heating of a metal bar
> >> >>>(unless you heated the bar in a very unusual, particular way) would
> >> >>>result in blotchy, varying temperatures as depicted in the graph --
> >> >>>surely it'd be smooth and have a gradual change, and not be wavey?
> >> >>
> >> >>
> >> >>I think the graph is cooked
> >> >
> >> > that's what i thought just from common sense, without really knowing.
> >> >
> >> >>-- a plot of an oversimplified equation
> >> >>rather than actual measurements. It is what I would expect if the true
> >> >>curve were approximated by harmonic analysis with higher (but still
> >> >>relevant) harmonics omitted.
> >> >
> >> >
> >> > .....am i right in saying that fourier transforms need
> >> > wavelike data to operate on?.......
> >> > ....
> >> > ....
> >> > ....
> >
> >> Not right, sorry. Any shape can be represented by by a sum of sines and
> >> cosines. Usually, an infinite number of them.
> >
> > right. i was just trying to imagine why the author didn't use actual
> > temperature results in the graph, as she stated they were, but used
> > data that's processed that's added waveyness.
> >
> > the fourier coefficients, in the corresponding fourier space graph,
> > correlate to each wave (to each crest and dip), so making the original
> > data line wavey, making it repeatedly, uniformly bumpy, makes the
> > corresponding fourier space graph easier to show i suppose (keeps to a
> > minimum the number of coefficients), maybe.
> >
> > the graph could have been smoth curve, and not wavey (as i'd have
> > expected it), but that would have made the fourier space graph less
> > obvious. but then i'm still thinking how can you take data which isn't
> > wavey, force it to have waves in it, then proceed from there. seems a
> > bit unreasobable.
> >
> > another imagined guess: maybe temperature results are infact inherently
> > wavey (but in a very subtle, unoticeable way) and the graph in question
> > is an exaggerated version, exaggerated to show up the waves?
> >
> > basically why would the author have based an important part of her
> > explenation of fourier analysis on a cooked graph? (and how or why (if
> > it is) ok to do that?)
> >
> >
> >> For example a square wave
> >> of amplitude 1 and frequency w -- w = 2pi/period -- is the sum
> >>    4/pi[sin(wt) + sin(3wt)/3 + sin(5wt)/5 + sin(7wt)/7 + ...]
> >> When some of the harmonics are omitted, the flats become wavy.  See
> >> pictures at http://cnx.rice.edu/content/m0041/latest/
> >
> > yup ok thanks.
> >
> >>
> >> I think you need a book. Look for recommendations at
> >> http://www.dspguru.com/
> >
> > well the book i'm reading at the moment (hubbard 'the world according
> > to wavelets' (where those graphs are from)) is supposed to be a good
> > book for non-technical/non-methematical people but as soon as i sniff a
> > contradiction or something that doesn't add up i find it hard to
> > continue with it until i get it straitened out -- that's what i'm
> > trying to do.
> >
> > thanks-a-lot, ben.
> Hi Ben,
>     To try to be fair to Mr. or Ms. Hubbard (I haven't read the book and
> know nothing about the author) a book called 'the world according to
> wavelets'  should have some wavey drawings or functional descriptions in it.

well whatever you call your book, it seems to me that that shouldn't
allow you to tailor data, full stop. if you're writing a book with
wavelets in the title it'd be a reasonable to expect the author to
choose data that fits in with what you're writing about -- not just
tailor any old data to suit. that's really not a reasobable reason imo.

> In further support of the author it seems that, when you look in shorter
> lengths/times at things, it is very difficult to actually observe any abrupt
> transitions or straight lines and what can be observed/measured is maybe
> more accurately described as the summation or product of probabilities that
> are apparently well described by wave-like distributions without having to
> resort to mathematical inventions like transform domains stretching to +/-
> infinity.

hmm, but the graph is of temperature readings taken simultaneously of a
normal sized metal bar -- that's what the graph is of -- there's
nothing infinite or microscopic or whatever about that. the only
uncertainty would be absolute exact readings, but that wouldn't result
in waveyness. i'm unsure why  you've mentioned abrupt changes and
straight lines -- they're not in the graph nor my expectation of how
the graph should be. you wouldn't expect abrupt changes in a heated
metal bar right?

another guess: hubbard is explaining fourier transforms and wants to do
that as simply as possible. fourier's original work was based on
temperature readings, so that was what she's used -- she wanted to be
historically correct. to illustrate the basics of a fourier transform
it's easier with wavey data. fourier transforms work with non-wavey
data perfectly well (as Jerry pointed out) but it's much easier to see
what's going on with obvious repeating waves rather than non-wavey data
-- using wavey data makes the explenation of a fourier transform
simpler.

so the wave aspect of the graph really is cooked. but the data could
have been non-wavey and fourier's transform would still work just as
well with that data. but it'd make the explenation of the fourier
transform more complicated and harder to see. (phew. that sounds
reasonably correct. possibly.)

sorry to go on, but once troubled by something like that it really gets
in the way of continuing with the book for me, and it is supposed to be
a pretty good book. and i think questioning like that helps understand
better anyway.

( the graph in question is the top left graph on this page:
http://www.hdbatik.co.uk/temp/waveletsbookpage.html  )

thanks, ben.
```
```ben wrote:

...

> right. i was just trying to imagine why the author didn't use actual
> temperature results in the graph, as she stated they were, but used
> data that's processed that's added waveyness.

If you ask me, she got confused. I see it like this: She did a harmonic
analysis* of the original curve, the one you would have liked to see.
Then she lopped of the higher harmonics that wouldn't contribute enough
to the subsequent discussion to justify carrying them around. Finally,
in the misguided pursuit of intellectual honesty, she replaced the
original curve with the wavy graph of her truncated series.

> the fourier coefficients, in the corresponding fourier space graph,
> correlate to each wave (to each crest and dip), so making the original
> data line wavey, making it repeatedly, uniformly bumpy, makes the
> corresponding fourier space graph easier to show i suppose (keeps to a
> minimum the number of coefficients), maybe.

That's backwards. The waviness is the result of discarding coefficients.

...

> another imagined guess: maybe temperature results are infact inherently
> wavey (but in a very subtle, unoticeable way) and the graph in question
> is an exaggerated version, exaggerated to show up the waves?

You're too generous.

> basically why would the author have based an important part of her
> explenation of fourier analysis on a cooked graph? (and how or why (if
> it is) ok to do that?)

First came the real curve, then the Fourier coefficients derived from
it. Oy Veh! Too many coefficients for a tutorial discussion! Get rid of
some! Proceed. Wait! We need the curve to refer to, so put it in. Now
the error: instead of the real curve that the the math approximates, a
fake curve that exactly matches the math. Bad form!

>>For example a square wave
>>of amplitude 1 and frequency w -- w = 2pi/period -- is the sum
>>   4/pi[sin(wt) + sin(3wt)/3 + sin(5wt)/5 + sin(7wt)/7 + ...]
>>When some of the harmonics are omitted, the flats become wavy.  See
>>pictures at http://cnx.rice.edu/content/m0041/latest/
>
>
> yup ok thanks.
>
>
>>I think you need a book. Look for recommendations at http://www.dspguru.com/
>
>
> well the book i'm reading at the moment (hubbard 'the world according
> to wavelets' (where those graphs are from)) is supposed to be a good
> book for non-technical/non-methematical people but as soon as i sniff a
> contradiction or something that doesn't add up i find it hard to
> continue with it until i get it straitened out -- that's what i'm
> trying to do.

Swell for wavelets, but short shrift for harmonic (Fourier) analysis.

Jerry
______________________________________
* In 1878, Kelvin completed a mechanical analog computer to to harmonic
analysis. His impetus was tide and weather prediction, but the principle
is more general.
http://zapatopi.net/kelvin/papers/harmonic_analyzer.html
http://www.ingenious.org.uk/See/Scienceandtechnology/Mathematics/?target=SeeMedium&ObjectID={55206993-14B7-3C73-2A95-4C24D701F88E}&s=S1&viewby=images&
In 1897, Michelson and Stratton built an analyzer that read out the
first 40 sine and cosine terms.
http://www-03.ibm.com/ibm/history/exhibits/attic3/attic3_157.html
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```ben wrote:

...

>  i think questioning like that helps understand better anyway.

Right on the mark.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```