simultaneous frequence and phase estimation

Whoa. I almost fell for it and was about to type a serious reply,
until I read a bit further. I'm glad I didn't.

Congratulations though, if the laws of physics were to argue with you,
I'm sure they'd lose!

:-)

--smb

Stephan M. Bernsee wrote:
> Whoa. I almost fell for it and was about to type a serious reply,
> until I read a bit further. I'm glad I didn't.
> 
> Congratulations though, if the laws of physics were to argue with you,
> I'm sure they'd lose!

The matter has been under discussion elsewhere. Admittedly, I have to 
confess two amendments.

1) The most serious one concerns the 10th paragraph beginning with the 
phrase 'In other words':
I found a more correct new solution to the very old problem of 
contradiction between symmetry and zero.
My reasoning started a few month ago when i got aware that upward 
counting starts with 'one' while downward counting ends with 'zero'.
The notion of numbers derives from counting of entities. Forward 
measurement of a distance resembles counting peaks along toothes of a 
saw. Therefore zero is not a natural (forward-) number. I applied the 
same idea to toothes of infinitely small size, incidentally including 
Dirac-Delta impulse, too. This led me to a correction: In case of 
forward counting real numbers zero is not included within IR+ but only 
in IR-. So the (directed) ring can be written as follows:
...(-infy, 0](0, +infty](-infty, 0](...    --->
In other words, the fence always belongs to the neighbour to the right.
The opposite is true if one counts in opposite direction.
In other words, progress always includes the fictitious fence.
Symmetry has been preserved to some extent in so far as the chosen 
direction is arbitrary in mathematics and also in space as long as there 
is no natural reference point like e.g. the middle of a sphere.

2) The second one concerns the role of FCT as basis of the natural 
spectrogram in comparison with two variants of FT in IR.

Look at the 3rd and 5th paragraphs. The sentences 'Use of FCT implies 
the possibility to chose a sliding point of time reference' and
'Not so with FCT' are not wrong but slightly misleading.
Proponents of FT were correct when they objected that use of 
singularity functions rather than preference to the FCT is the decisive 
step towards the natural spectrogram. Alternatively to the plausible and 
simple singularity functions in IR+, one can also use the traditional 
but more clumsy ones in IR. FT demands redundancy, i. e., continuation 
into IR of the physically real data which are completely contained in 
IR+. There are at least two possibilities to do so: even continuation 
and Heaviside's zero continuation. In so far, the sentence 'Hermitian 
symmetry is a must' (see the 7th paragraph, beginning with 'Admittedly') 
was also a bit misleading.

Even continuation shows more clearly that the detour from IE+ into IR 
does not offer any advantage over the direct use of the singularity
function in IR+. Even continuation simply adds a symmetrical mirror 
function each, in time domain as well as frequency domain. The added 
negative values of time and frequency, respectively, are known to lack 
any special physical meaning. Any restriction to real values hampers the 
otherwise favourable method of time-shift, and it also alters the 
peculiarities of the result. A real-valued spectrum must not be 
expressed in terms of phase and always positive magnitude but simply as 
an alternating function of frequency.
An other proponent of FT suggested zero continuation by menans of using 
Re[FT of f(t-t') multiplied by Heaviside(t)]. Despite of more effort, 
the result is the same: a symmetrical real function of frequency.

Hopefully you are not unable to accept my results at least in part.
I would like to apologize for perhaps demanding too much at a time.
Please restrict to factual responses and indicate them to me
                  blumschein@et.uni-magdeburg.de
because I will rarely have time for looking into the news within the 
next weeks. Being in a hurry, I also beg your pardon for making 
mistakes and misspellings. Please do not generalize this and my shaky 
command of English as an indication for mediocre work.

Eckard Blumschein

Hi Eckard,

I'm still a bit confused what it is you're trying to say. No pun
intended, but you seem to clutter the vital points with too much extra
information and seem to spread the important bits and pieces over
several messages. Many of them don't make any sense to me right now,
but that might be my being unable to put everything together in the
right order. So here's a proposal: Why don't you leave out the
references to older posts for the moment and try to describe in no
more than ten sentences what exactly your assertion is about the FCT,
and that "natural spectrogram" of yours?

FWIW, the FCT is simply the cosine part of the DFT and per se has no
special property wrt. its definition or output that would relate it to
something that could more or less rightfully be called a "natural
spectrogram" than the straight output of a DFT.

I'm curious - what exactly is the definiton of your spectrogram
property anyway that makes it "natural"? Causality? Constant-Q? Both?

Eckard Blumschein <blumschein@et.uni-magdeburg.de> wrote:
> 1) The most serious one concerns the 10th paragraph beginning with the 
> phrase 'In other words':
> [snipped ref to older post]
> 
> 2) The second one concerns the role of FCT as basis of the natural 
> spectrogram in comparison with two variants of FT in IR.
> [snipped more refs to older post]
> 
> Hopefully you are not unable to accept my results at least in part.

I'd very much like to hear about them first. Just state your point and
I'll be more than happy to let you know if I agree or disagree!

> I would like to apologize for perhaps demanding too much at a time.

The problem is that I had to dig out and read more than 2 of your
older posts (which all are about two pages of text) to figure out what
you're getting at, and I still don't see where you're going.

> Please restrict to factual responses

Well, I always do. At least I try.

> I also beg your pardon for making 
> mistakes and misspellings. Please do not generalize this and my shaky 
> command of English as an indication for mediocre work.

I can see nothing wrong with your language and will certainly never
criticize you for speaking a second (or third, fourth) language better
than I do *)

 :-)

--smb

*) English is my second language as well.

Stephan M. Bernsee wrote:
> Whoa. I almost fell for it and was about to type a serious reply,
> until I read a bit further. I'm glad I didn't.
> 
> Congratulations though, if the laws of physics were to argue with you,
> I'm sure they'd lose!

Haha, I thought Eckhard's post was quite funny! It's a weird mixture
between Sokal's hoax and ELIZA.

This part here 

> > Using Matlab, I managed to demonstrate that ...

reminded me of a story I recently read in the papers. Appearantly a
young Swedish student claimed to have solved a famous mathematical
problem. She submitted her work to a journal which accepted and
promised to publish. She then promptly alerted the media and stirred
up a big fuss. Upon closer inspection of her paper however, it was
noticed that she cooked up some numeric approximations to differential
solutions, found that they looked sinusoid and substituted a sine
function into her further calculations, claiming this to be close
enough for a proof :-).

Regards,
Andor

Stephan M. Bernsee wrote:

(snip)

> FWIW, the FCT is simply the cosine part of the DFT and per se has no
> special property wrt. its definition or output that would relate it to
> something that could more or less rightfully be called a "natural
> spectrogram" than the straight output of a DFT.

DCT is slightly different than the cosine part of a DFT.

DCT has period 2L where DFT has period L.   There are ways to
convert on to the other to allow a DFT subroutine to solve DCT
problems, some of which are in Numerical Recipes.

-- glen

I know, Glen. But we're not talking about the DCT, we're talking about
the FCT here, which is a slightly different thing. Eckard is
apparently assigning it a special property that I do not seem to
understand and which he describes as "natural".

--smb

glen herrmannsfeldt wrote:
> Stephan M. Bernsee wrote:
> 
> (snip)
> 
> > FWIW, the FCT is simply the cosine part of the DFT and per se has no
> > special property wrt. its definition or output that would relate it to
> > something that could more or less rightfully be called a "natural
> > spectrogram" than the straight output of a DFT.
> 
> DCT is slightly different than the cosine part of a DFT.
> 
> DCT has period 2L where DFT has period L.   There are ways to
> convert on to the other to allow a DFT subroutine to solve DCT
> problems, some of which are in Numerical Recipes.
> 
> -- glen

Now I see why you got me wrong: I was talking about the "FCT" as in
"Fourier Cosine Transform", not as in "Fast Cosine Transform". That
should have been obvious from Eckards post, but maybe not.

Damn abbreviations... ;-)

--smb

Here's some more conversation between Eckard and me on his "natural
spectrogram" - he agreed that I may post this here. To avoid
misunderstandings, FCT is meant to be the Fourier Cosine Transform,
and not the Fast Cosine Transform that can be built using the FFT, as
Glen has pointed out in an earlier post...

--smb


Hi Eckard,

no problem with the hurry, I know what you're talking about ;)

Still, I have more questions for you because I'm afraid that your
answers are not really making things better. I assume you have no
objections against my sharing this with the comp.dsp newsgroup, since
the original thread started there?

Am 06.05.2004 um 15:42 schrieb Eckard Blumschein:
> 
> > I'm curious - what exactly is the definiton of your spectrogram
> > property anyway that makes it "natural"? Causality? Constant-Q? Both?
> 
> 1) Well, non-causal behavior is one among several flaws of traditional
> spectrograms.
> 2) While there is a variety of traditional models mimicking cochlear
> function (FT with overlapping windows, wavelet-based models, TW-models,
> filter banks) there is only one natural spectrogram, and only this does not
> need tweaking as to come at least as close to the known data of cochlear
> response as do the latter two.

In what regard? I am a bit puzzled by the absolutism that there is
"only one", without exactly defining what the term "natural" is
supposed to mean. I can think of many "natural spectrograms" as I
choose to define them, so I doubt that this statement is correct
without further explanation, which I still don't see.

Also, the FCT can hardly be used to deliver a constant-Q scale, at
least not without modification (as by coordinate transformation � la
Mellin for example) - but then it wouldn't be a FCT anymore.

> 3) The natural spectrogram does not require any arbitrarily chosen time
> window, and it is not subject to the notorious trade-off between spectral
> and temporal resolution.

Would you care to explain how you plan to achieve this, then? This
answer is still too vague to actually know what you're talking about.
I take it you know that there is *always* a tradeoff beween spectral
and temporal resolution - especially if you really mean *spectral* in
the sense of the word - ie. wrt. frequency.

And without some sort of a time window or weighting, any possible
result from such a transform would be quite meaningless.

> 4) All these achievements are based on simple real-valued frequency
> analysis taking causality into account from the very beginning by using an
> admittedly strange scale of always just elapsed time instead of usual time.

What scale? Can you add some maths to this to make it clear what
you're talking about?

> 5) A second hurdle was the widespread believe that complex FT is always
> better than Fourier-Cosine transform and the latter is merely the real part
> of it.

I don't see that either of them can be better for anything, without
defining for what purpose. And yes, I agree it is a common belief that
the FCT is the cosine part of the DFT. I'd say this is a relatively
sound belief, too - if you have a different definition I'd be
interested to hear it.

> 6) Actually, IR+ in combination with FCT is best suited to currently
> contain all genuine past but exclude the hampering redundancy which is
> necessary for complex calculus, in particular fictitious future time and
> negative frequency.

I agree there is redundancy in the DFT for purely real valued signals.
You can easily fix this by using the Hartley transform (which operates
on real signals only), or the DCT, DST, or "cheat" by creating a
purely real DFT out of a DFT, if you're worried about things like
negative frequency.

"Future time" depends on your point of reference - it has no meaning
except for the phase of the complex exponentials involved in the DFT
so I am wondering what might be wrong with that...

> 7) In particular, FCT-based elementary singularity functions constitute a
> much simpler and more logical and realistic set in IR+ than in IR.

From the information you have presented I can only guess what you're
referring to here when you mention "elementary singularity
functions"... the z transform polynomials perhaps? If that is what
you're referring to, what about your "strange scale" then? How would
that look like?

> 8) Since summed FCTs of elementary singularity functions are not bound to
> an arbitrary temporal window, they may optimally adapt to the structural
> needs of an immediate analysis looking back upon the just passed flow of time.

How far back? Wouldn't this imply some sort of a window, too?

> 9) Traditional spectrograms suffer from permanently growing shift between
> the chosen zero of time and the actual "now" demanding relocation on a
> regular basis.

I'm not sure what you mean. Time is passing for all types of a
spectrogram, otherwise it would be quite useless. And if you were to
constantly accumulate the results without having some kind of
weighting the result would be equally useless. As soon as you do the
weighting, you'd either end up with a window or with something other
than the periodic cosines that stretch out for all eternity in either
direction - and I can't see that you have such a thing at your
disposal as long as you're talking about the FCT.

> 10) The natural spectrogram does not exhibit always positive magnitude and
> phase but a rectifiably alternating amplitude as a function of time
> instead.

The DFT does not readily exhibit always positive magnitude and phase!
First, magnitude by definition is positive and second, you have to do
an explicit Cartesian -> polar coordinate conversion to get to this
particular result.

If you "rectify" the output of your "natural spectrogram" you'd
probably end up with a result that has a similar meaning (although you
would need the imaginary part to get the entire result), so I can't
see what's wrong with that either.

Care to explain a bit more?

Kind regards,
--smb

Hi Stephan

 >Still, I have more questions for you because I'm afraid that your
 >answers are not really making things better. I assume you have no
 >objections against my sharing this with the comp.dsp newsgroup, since
 >the original thread started there?

I don't have anything to hide.

Let me start with suggestions for symbols.
You are certainly familiar with the symbols for transform from real into 
complex domain and vice versa: a filled circle linked with an unfilled 
one, perhaps similar to: o--* and *--o.
Facing limitations of the keyboard I would suggest C>C and C<C instead 
for a+ib while c>c and c<c could denote the same for a-ib.

Accordingly, I suggest symbols for even (E) and zero (Z) continuation 
and pertaining inversions like this:
[f(t) for 0<t<infty, i.e. in IR+] E>E [f(t)=f(-t) for (-infty<t<+infty), 
i.e. in IR]

Of course, zero continuation leads into complex plane with its either 
positive (Z) or negative (z) imaginary part. This arbitrariness is a 
consequence of the missing natural zero in IR impying an arbitrary 
choice of zero.

To be continued

Kind regards,
Eckard

Hi Stephan,

Here I am again.

Just to those who incidentally read my reply, I will in brief explain 
why we need continuation at all.
It's a pity, definition of Fourier transform nurtures an illusion since 
it is as an integral over time from -infty to + infty: the illusion that 
time always means physical time in that case. Actually, half of the time 
axis is fictitious. It would not make sense to include past as well as 
future time. This fact is called causality. All physical relationship 
could be completely expressed within past time. However, use of complex 
calculus requires negative values of time and frequency, too.

 >>> I'm curious - what exactly is the definiton of your spectrogram
 >>> property anyway that makes it "natural"? Causality? Constant-Q? Both?
 >>
 >> 1) Well, non-causal behavior is one among several flaws of traditional
 >> spectrograms.
 >> 2) While there is a variety of traditional models mimicking cochlear
 >> function (FT with overlapping windows, wavelet-based models, TW-models,
 >> filter banks) there is only one natural spectrogram, and only this
 >> does not
 >> need tweaking as to come at least as close to the known data of
 >> cochlear
 >> response as do the latter two.
 >
 >In what regard? I am a bit puzzled by the absolutism that there is
 >"only one", without exactly defining what the term "natural" is
 >supposed to mean. I can think of many "natural spectrograms" as I
 >choose to define them, so I doubt that this statement is correct
 >without further explanation, which I still don't see.

Isn't a definition the more natural the less arbitrariness it implies?
Arbitrariness starts with too much freedom, enforcing incidental choices.
For instance, Cartesian coordinates require a lot of unnatural 
definiton. As a result, we feel uncomfortable within a rectangular 
world. In case of time, I consider the current "now" the only natural 
reference point available. The natural spectrogram is the only one that 
adapts by definition to this reference.
The myriads of arbitrary choices derive from the missing natural basis 
of common time. The obvious good agreement between the natural 
spectrogram and natural behavior of cochlea is uniquely not bound to 
correcting coefficients. In other words, an exactly fitting definition 
is the natural one. A prudent king has all power but no choice.

 >
 >Also, the FCT can hardly be used to deliver a constant-Q scale, at
 >least not without modification (as by coordinate transformation � la
 >Mellin for example) - but then it wouldn't be a FCT anymore.

No no, even if Roy Patterson also suggested to me using Mellin transform.
I do not see any reason for that. In order to avoid possible confusion I 
would like you to clarify what you mean by Q.

 >
 >> 3) The natural spectrogram does not require any arbitrarily chosen time
 >> window, and it is not subject to the notorious trade-off between
 >> spectral
 >> and temporal resolution.
 >
 >Would you care to explain how you plan to achieve this, then? This
 >answer is still too vague to actually know what you're talking about. I
 >take it you know that there is *always* a tradeoff beween spectral and
 >temporal resolution - especially if you really mean *spectral* in the
 >sense of the word - ie. wrt. frequency.

I was puzzling myself with this trade-off for decades before I 
understood the reason.
While I achieved even better resolution than natural hearing, I would 
like to stress that even everybody's ears outperform the 
misinterpretation of uncertainty principle. I consider those who took 
wrongly understood tenets a gospel worse than those without any 
knowledge. The enigmas of hearing are not yet completely resolved. 
Nonetheless it might be helpful to imagine cochlea to perform 
measurement with different trade-offs at a time and benefit from 
subsequent highly parallel signal processing. This can also be 
interpreted as using a priori knowledge. In principle, my approach is 
similar. It uses the knowledge of sinc function and sums a lot of them.

 >
 >And without some sort of a time window or weighting, any possible
 >result from such a transform would be quite meaningless.

Be aware, that this is an unsubstantiated suspicion of you. Perhaps some 
narrow minded teachers of you are to blame for this sort of thinking. 
Try to get more specific.

 >
 >> 4) All these achievements are based on simple real-valued frequency
 >> analysis taking causality into account from the very beginning by
 >> using an
 >> admittedly strange scale of always just elapsed time instead of usual
 >> time.
 >
 >What scale? Can you add some maths to this to make it clear what you're
 >talking about?

Wouldn't it most convenient to you just applying your knowledge? This 
does not work.
Of course, you can try to mathematically relate the scale of elapsed 
time te to the arbitrarily defined ordinary time t. However, recall that 
time does not have a physical meaning from minus infinite to plus 
infinite. The original and physically relevant time is restricted to 
elapsed time filling IR+. Imagine its time scale like an arrow. The even 
continuation generates a second arrow pointing into opposite direction. 
Both arrows are getting continuously longer. Imagine them growing out of 
the origin. Maybe, this kind of mental acrobatics is suspect to you. 
Compare it with Feynman's still accepted notion of time running forward 
and backward simultaneously. Perhaps you will agree that my simpler 
real-valued model is more plausible.

 >
 >> 5) A second hurdle was the widespread believe that complex FT is always
 >> better than Fourier-Cosine transform and the latter is merely the real
 >> part
 >> of it.
 >
 >I don't see that either of them can be better for anything, without
 >defining for what purpose.

You are quite right and not at odds with what I wrote.
If your purpose is merely real-time analysis of an incoming signal then 
I recommend singularity functions based on Fourier cosine transform in IR+.
If your purpose includes complex calculus then it is better to choose 
complex Fourier transform.

 >And yes, I agree it is a common belief that
 >the FCT is the cosine part of the DFT. I'd say this is a relatively
 >sound belief, too - if you have a different definition I'd be
 >interested to hear it.

The difference is the range of definition. The cosine part of FT for a 
real world signal is a symmetrical function extending in frequency from 
minus infinite to plus infinite while there are no negative frequencies 
and no symmetrical functions in IR+ at all. I introduced E>E. If you are 
aware of a better notion, I would be happy to get informed.

Well, the whole world will feel it a cheaky whim of mine if I am humbly 
asking: Aren't FT and FCT equal sisters rather than an important mother 
and a marginal daughter? Mathematicians know that it is likewise 
possible to put IR completely into the seemingly smaller IR+ as vice 
versa. FT does not yield more information than FCT. It just combines 
frequency analysis with complex analysis on the expense of accepting 
redundancy and other flaws.


 >> 6) Actually, IR+ in combination with FCT is best suited to currently
 >> contain all genuine past but exclude the hampering redundancy which is
 >> necessary for complex calculus, in particular fictitious future time
 >> and
 >> negative frequency.
 >
 >I agree there is redundancy in the DFT for purely real valued signals.

Hermitian symmetry is a special kind of redundancy, too, even if we are 
not immediately allowed to omit something. As a complex transform, FT 
would not work without redundancy.

 >You can easily fix this by using the Hartley transform (which operates
 >on real signals only), or the DCT, DST, or "cheat" by creating a purely
 >real DFT out of a DFT, if you're worried about things like negative
 >frequency.

I do not see any reason to fix it, and I am also not worried about 
negative frequency. Your argumentation shows clearly that (not you in 
particular but) the whole practice of teaching theory of signal 
processing considers itself astute while being actually authoritative in 
the sense of arrogant and narrow minded.

 >
 >"Future time" depends on your point of reference - it has no meaning
 >except for the phase of the complex exponentials involved in the DFT so
 >I am wondering what might be wrong with that...

This is perhaps the most basic mistake of almost the whole scientific 
world. You would be right in case of off-line analysis. Once a record 
has been taken, one can manipulate it at will: Reverse it, stretch it, 
whatever. Not so at caudal position as within your inner ears. Forget 
the complex exponentials in that case. All of us rely on more or less 
the same sliding point of reference. Past is known in principle, but we 
can absolutely not change it while we can influence future but not know 
it for sure. Do not misuse Emmy Noether's theorem as a justification for 
an illusory putative time shift in reality.

 >
 >> 7) In particular, FCT-based elementary singularity functions
 >> constitute a
 >> much simpler and more logical and realistic set in IR+ than in IR.
 >
 > From the information you have presented I can only guess what you're
 >referring to here when you mention "elementary singularity
 >functions"... the z transform polynomials perhaps? If that is what
 >you're referring to, what about your "strange scale" then? How would
 >that look like?

I borrowed this term from literature. Among usual singularity functions 
are Heaviside step, sign, rectangle, ramp, triangle, and Dirac delta. 
Some of them are symmetrical relative to t=0. So they include a real and 
a fictitious singularity. I redefined singularity functions within IR+ 
where only the real singularity remains: The rectangle mutates into a 
new step, the triangle into a new ramp being the integral of the new 
step being at its part the integral of Dirac delta.
 >
 >> 8) Since summed FCTs of elementary singularity functions are not bound
 >> to
 >> an arbitrary temporal window, they may optimally adapt to the
 >> structural
 >> needs of an immediate analysis looking back upon the just passed flow
 >> of time.
 >
 >How far back? Wouldn't this imply some sort of a window, too?

I feel your annoying intention to brutally subordinate natural life to a 
clumsy method of windowing reminding me of the comic Harald Nielsen's 
machine for cutting finger nails. Do not people have fingers of 
different length? Yes, but only before using the machine.
Of course, even our ears do not have an unlimited number of hair cells. 
Everybody of us will die. Decay is the only parameter to choose as 
realistically as possibe for the natural spectrogram.

 >
 >> 9) Traditional spectrograms suffer from permanently growing shift
 >> between
 >> the chosen zero of time and the actual "now" demanding relocation on a
 >> regular basis.
 >
 >I'm not sure what you mean. Time is passing for all types of a
 >spectrogram, otherwise it would be quite useless.

I am not sure if I will manage to always correctly retrace your 
reasoning. Elapsed time does not pass. The zero of elapsed time is 
always at the "now". Spectrograms do already take this point of view.

 >And if you were to
 >constantly accumulate the results without having some kind of weighting
 >the result would be equally useless.

This is considered with the chosen decay.

 >As soon as you do the weighting,
 >you'd either end up with a window

Gradual decay is quite different from the usual window.

 >or with something other than the
 >periodic cosines

My sinc functions converge.

 >that stretch out for all eternity in either direction

That is why I prefer unilaterality in IR+ instead of bilaterality in IR.

 >- and I can't see that you have such a thing at your disposal as long
 >as you're talking about the FCT.

That's why I am so keen to distinguish between FCT as you know it as 
real part of FT in IR, and the slighly different FCT immediately 
definded in IR+.


 >> 10) The natural spectrogram does not exhibit always positive magnitude
 >> and
 >> phase but a rectifiably alternating amplitude as a function of time
 >> instead.
 >
 >The DFT does not readily exhibit always positive magnitude and phase!
 >First, magnitude by definition is positive and second, you have to do
 >an explicit Cartesian -> polar coordinate conversion to get to this
 >particular result.

Of course, I did not mean that phase is always positive. Also, I see all 
kinds of a complex representation equivalent to each other. Perhaps you 
didn't get my points compressed with the last sentence:
- Neither the traditional spectrograms nor the natural one show phase. 
This only matters in case of the traditional spectrogram because there 
is no phase at all with FCT.
- The natural spectrogram looks quite different from the usually one in 
that, it does not show a magnitude being always positive by definition 
but a pulsating pattern instead.

 >If you "rectify" the output of your "natural spectrogram" you'd
 >probably end up with a result that has a similar meaning (although you
 >would need the imaginary part to get the entire result), so I can't see
 >what's wrong with that either.

One way rectification is an important subsequent step in cochlear signal 
processing. One can neither rectify magnitude nor an imaginary part.


 >Care to explain a bit more?

While I tried my best within spare time, and I highly appreciate your 
factual style, I fear we will still have to work hard as to agree.

Kind regards,
Eckard

 >
 >Kind regards,
 >--smb
 >
 >