# Re: Human hearing beats the Fourier uncertainty principle: Research (Judi Lapsley Miller )

```Subject: Re: Human hearing beats the Fourier uncertainty principle: Research
From:    Judi Lapsley Miller  <judi@xxxxxxxx>
Date:    Sat, 23 Feb 2013 08:18:34 +1300
List-Archive:<http://lists.mcgill.ca/scripts/wa.exe?LIST=AUDITORY>

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A million years ago (it seems), I investigated the acoustical =
uncertainty
principle for my PhD thesis. I found that humans did a pretty good job =
at
optimally resolving narrow-band short-duration noise stimuli, and did =
best
for stimuli with a bandwidth-duration product of around 2. By optimally, =
I
mean compared to an ideal observer theory. There was no indication my
observers were able to beat the uncertainty principle, but they came =
close
to operating as well as an ideal observer for some bandwidth-duration
conditions.=20

=20

Any study showing observers =93beating=94 the uncertainty principle has =
subjects
who are solving the task differently to what the experimenter expected, =
or
there are issues with defining bandwidth and duration. IMHO.

=20

http://psychophysics.org/judi_thesis.htm

=20

Judi

=20

From: AUDITORY - Research in Auditory Perception
[mailto:AUDITORY@xxxxxxxx On Behalf Of corey
Sent: Friday, 22 February 2013 05:33
To: AUDITORY@xxxxxxxx
Subject: Re: Human hearing beats the Fourier uncertainty principle: =
Research

=20

I do think this article is misleading.  In my opinion the uncertainty
principle is not something to be "overcome".  It simply relates the
bandwidth of a waveform to its duration (which is an unassailable fact). =
In
Fourier theory frequency is defined for periodic signals (which are of
infinite duration).  Any finite duration signal can be viewed equally as =
an
infinite duration periodic signal that has been multiplied by a finite
duration window.  It is this windowing process that introduces a spread =
of
frequencies (after all the window is a waveform too).  It is thus not
physically possible to create stimuli that "beat" the uncertainty =
principle,
which seems to negate the conclusions in the remainder of the article.

=20

On 2013-02-20, at 10:27 AM, James Johnston wrote:

Well, yes. Of course, and the FFT is not a minimum-phase filter, while =
the
ear is very close to such. Well, there goes a large speed factor =
already,
eh?=20

On Wed, Feb 20, 2013 at 8:48 AM, Bastian Epp <bepp@xxxxxxxx> =
wrote:

Hi list!

without having read the paper in detail, I just think the discussion =
should
be done with fair weapons:

The description of the ear via a FFT is a rather poor model, but a FFT =
(or
rather the math behind that) can be interpretet as a bank of overlapping
bandpass filters (see the Oppenheim Schaefer DSP book)....so this point
raised is not valid.=20

I would be surprised if this result would not have been reported before =
in
the solid psychoacoustics literature before.

Bests

BAstian

On 02/18/2013 05:00 PM, James Johnston wrote:=20

I must admit some frustration with this particular paper. First, the
Gabor limit does not apply to the task, and never did. The only limit
here is SNR_based, since there is already expectation of a given set
of frequencies, this is not a task requiring arbitrary detection.
=20
Then, the fact that the ear is a leading edge detector has been
understood for roughly 100 years now, making "1/100 th of a
wavelength" perhaps not such a big deal.
=20
It is nice that this performance ability by the human has been clearly
demonstrated, but the headline is inexcusably misleading, and is
already providing fodder for the audiophile "I told you so" bunch who
simply doesn't understand what it means.
=20
And, in any case, who would use an FFT to detect such a thing? Rather
use a set of bandpass filters, eh?
=20
On Sun, Feb 17, 2013 at 12:02 PM, Peter Meijer
<mailto:feedback@xxxxxxxx> <feedback@xxxxxxxx> =
wrote:
=20

Indeed this relates to a discussion that we had 9 years ago,
=20
http://www.auditory.org/mhonarc/2004/msg00145.html
=20
and that formed the basis of my old web page on beating the
frequency-time uncertainty principle,
=20
http://www.seeingwithsound.com/freqtime.htm
=20
Best regards,
=20
Peter Meijer
=20
=20
Seeing with Sound - The vOICe
http://www.seeingwithsound.com <http://www.seeingwithsound.com/>=20
=20
=20
=20

Date: Sun, 17 Feb 2013 07:43:35 +0000
From: "Beerends, J.G. (John)"  <mailto:john.beerends@xxxxxxxx>
<john.beerends@xxxxxxxx>
To: AUDITORY@xxxxxxxx
Subject: Re: Human hearing beats the Fourier uncertainty principle:
Research
=20
For discrimination the uncertainty limit does not exist, one can build
discriminator devices that go below the uncertainty limit in both the =
time
and frequency domain, the uncertainty limit is only a measure for the =
spread
(Delta) in both domains (DfDt>1), it is not a limit to what extent they =
can
be discriminated. One can also build a device that measures the =
frequency of
a sine wave with an accuracy below the uncertainty limit by exploiting
a-priori knowledge, i.e. if I know  that the signal I am measuring is a
short cut out of an infinite duration sine wave of a certain amplitude I =
can
measure the frequency as accurate as I want.
=20
John Beerends
=20

=20

-----Original Message-----
From: AUDITORY - Research in Auditory Perception
[mailto:AUDITORY@xxxxxxxx On Behalf Of Kevin Austin
Sent: Saturday, February 16, 2013 5:07 PM
To: AUDITORY@xxxxxxxx
Subject: Human hearing beats the Fourier uncertainty principle: Research
=20
Comments?
=20
=20
=20

http://phys.org/news/2013-02-human-fourier-uncertainty-principle.html
=20
=20
(Phys.org)-For the first time, physicists have found that humans can
discriminate a sound's frequency (related to a note's pitch) and timing
(whether a note comes before or after another note) more than 10 times
better than the limit imposed by the Fourier uncertainty principle. Not
surprisingly, some of the subjects with the best listening precision =
were
musicians, but even non-musicians could exceed the uncertainty limit. =
The
results rule out the majority of auditory processing brain algorithms =
that
have been proposed, since only a few models can match this impressive =
human
performance.
=20

=20

Read more at:
http://phys.org/news/2013-02-human-fourier-uncertainty-principle.html#jCp=

=20

Thanks
=20
Kevin
This e-mail and its contents are subject to the DISCLAIMER at
http://www.tno.nl/emaildisclaimer
=20

=20

=20

--=20

=20

Bastian Epp=20

Assistant Professor=20

ELEK VIP=20

DTU Electrical Engineering=20

=20

Technical University of Denmark

<DTU_email_logo_01.gif>=20

Department of Electrical Engineering=20

=D8rsteds Plads=20

Building 352=20

2800 Kgs. Lyngby=20

Direct +45 45253953=20

bepp@xxxxxxxx=20

www.elektro.dtu.dk/=20

=20

=20

--=20

James D. (jj) Johnston

Independent Audio and Electroacoustics Consultant

=20

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</o:shapelayout></xml><![endif]--></head><body lang=3DEN-NZ link=3Dblue =
vlink=3Dpurple><div class=3DWordSection1><p class=3DMsoNormal><span =
style=3D'font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497=
D'>A million years ago (it seems), I investigated the acoustical =
uncertainty principle for my PhD thesis. I found that humans did a =
pretty good job at optimally resolving narrow-band short-duration noise =
stimuli, and did best for stimuli with a bandwidth-duration product of =
around 2. By optimally, I mean compared to an ideal observer theory. =
There was no indication my observers were able to beat the uncertainty =
principle, but they came close to operating as well as an ideal observer =
for some bandwidth-duration conditions. <o:p></o:p></span></p><p =
class=3DMsoNormal><span =
style=3D'font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497=
D'><o:p>&nbsp;</o:p></span></p><p class=3DMsoNormal><span =
style=3D'font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497=
D'>Any study showing observers &#8220;beating&#8221; the uncertainty =
principle has subjects who are solving the task differently to what the =
experimenter expected, or there are issues with defining bandwidth and =
duration. IMHO.<o:p></o:p></span></p><p class=3DMsoNormal><span =
style=3D'font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497=
D'><o:p>&nbsp;</o:p></span></p><p class=3DMsoNormal><span =
style=3D'font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497=
D'><a =
href=3D"http://psychophysics.org/judi_thesis.htm">http://psychophysics.or=
g/judi_thesis.htm</a><o:p></o:p></span></p><p class=3DMsoNormal><span =
style=3D'font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497=
D'><o:p>&nbsp;</o:p></span></p><p class=3DMsoNormal><span =
style=3D'font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497=
D'>Judi<o:p></o:p></span></p><p class=3DMsoNormal><a =
name=3D"_MailEndCompose"><span =
style=3D'font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497=
D'><o:p>&nbsp;</o:p></span></a></p><div><div =
style=3D'border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0cm =
0cm 0cm'><p class=3DMsoNormal><b><span lang=3DEN-US =
style=3D'font-size:10.0pt;font-family:"Tahoma","sans-serif"'>From:</span>=
</b><span lang=3DEN-US =
style=3D'font-size:10.0pt;font-family:"Tahoma","sans-serif"'> AUDITORY - =
Research in Auditory Perception [mailto:AUDITORY@xxxxxxxx <b>On =
Behalf Of </b>corey<br><b>Sent:</b> Friday, 22 February 2013 =
05:33<br><b>To:</b> AUDITORY@xxxxxxxx<br><b>Subject:</b> Re: =
Human hearing beats the Fourier uncertainty principle: =
Research<o:p></o:p></span></p></div></div><p =
class=3DMsoNormal><o:p>&nbsp;</o:p></p><p class=3DMsoNormal>I do think =
this article is misleading. &nbsp;In my opinion the uncertainty =
principle is not something to be &quot;overcome&quot;. &nbsp;It simply =
relates the bandwidth of a waveform to its duration (which is an =
unassailable fact). &nbsp;In Fourier theory frequency is defined for =
periodic signals (which are of infinite duration). &nbsp;Any finite =
duration signal can be viewed equally as an infinite duration periodic =
signal that has been multiplied by a finite duration window. &nbsp;It is =
this windowing process that introduces a spread of frequencies (after =
all the window is a waveform too). &nbsp;It is thus not physically =
possible to create stimuli that &quot;beat&quot; the uncertainty =
principle, which seems to negate the conclusions in the remainder of the =
article.<o:p></o:p></p><div><p =
class=3DMsoNormal><o:p>&nbsp;</o:p></p><div><div><p class=3DMsoNormal>On =
2013-02-20, at 10:27 AM, James Johnston wrote:<o:p></o:p></p></div><p =
class=3DMsoNormal><br><br><o:p></o:p></p><p class=3DMsoNormal =
style=3D'margin-bottom:12.0pt'>Well, yes. Of course, and the FFT is not =
a minimum-phase filter, while the ear is very close to such. Well, there =
goes a large speed factor already, eh? <o:p></o:p></p><div><p =
class=3DMsoNormal>On Wed, Feb 20, 2013 at 8:48 AM, Bastian Epp &lt;<a =
href=3D"mailto:bepp@xxxxxxxx" =
target=3D"_blank">bepp@xxxxxxxx</a>&gt; =
wrote:<o:p></o:p></p><div><p class=3DMsoNormal>Hi list!<br><br>without =
having read the paper in detail, I just think the discussion should be =
done with fair weapons:<br><br>The description of the ear via a FFT is a =
rather poor model, but a FFT (or rather the math behind that) can be =
interpretet as a bank of overlapping bandpass filters (see the Oppenheim =
Schaefer DSP book)....so this point raised is not valid. <br><br>I would =
be surprised if this result would not have been reported before in the =
solid psychoacoustics literature =
before.<br><br>Bests<br><br>BAstian<br><br><br>On 02/18/2013 05:00 PM, =
James Johnston wrote: <o:p></o:p></p><pre>I must admit some frustration =
with this particular paper. First, the<o:p></o:p></pre><pre>Gabor limit =
does not apply to the task, and never did. The only =
limit<o:p></o:p></pre><pre>here is SNR_based, since there is already =
expectation of a given set<o:p></o:p></pre><pre>of frequencies, this is =
not a task requiring arbitrary =
detection.<o:p></o:p></pre><pre><o:p>&nbsp;</o:p></pre><pre>Then, the =
fact that the ear is a leading edge detector has =
been<o:p></o:p></pre><pre>understood for roughly 100 years now, making =
&quot;1/100 th of a<o:p></o:p></pre><pre>wavelength&quot; perhaps not =
such a big deal.<o:p></o:p></pre><pre><o:p>&nbsp;</o:p></pre><pre>It is =
nice that this performance ability by the human has been =
clearly<o:p></o:p></pre><pre>demonstrated, but the headline is =
inexcusably misleading, and is<o:p></o:p></pre><pre>already providing =
fodder for the audiophile &quot;I told you so&quot; bunch =
who<o:p></o:p></pre><pre>simply doesn't understand what it =
means.<o:p></o:p></pre><pre><o:p>&nbsp;</o:p></pre><pre>And, in any =
case, who would use an FFT to detect such a thing? =
Rather<o:p></o:p></pre><pre>use a set of bandpass filters, =
eh?<o:p></o:p></pre><pre><o:p>&nbsp;</o:p></pre><pre>On Sun, Feb 17, =
2013 at 12:02 PM, Peter Meijer<o:p></o:p></pre><pre><a =
href=3D"mailto:feedback@xxxxxxxx" =
target=3D"_blank">&lt;feedback@xxxxxxxx&gt;</a> =
wrote:<o:p></o:p></pre><pre>=A0 <o:p></o:p></pre><blockquote =
style=3D'margin-top:5.0pt;margin-bottom:5.0pt'><pre>Indeed this relates =
to a discussion that we had 9 years =
ago,<o:p></o:p></pre><pre><o:p>&nbsp;</o:p></pre><pre>=A0=A0 <a =
href=3D"http://www.auditory.org/mhonarc/2004/msg00145.html" =
target=3D"_blank">http://www.auditory.org/mhonarc/2004/msg00145.html</a><=
o:p></o:p></pre><pre><o:p>&nbsp;</o:p></pre><pre>and that formed the =
basis of my old web page on beating =
the<o:p></o:p></pre><pre>frequency-time uncertainty =
principle,<o:p></o:p></pre><pre><o:p>&nbsp;</o:p></pre><pre>=A0=A0 <a =
href=3D"http://www.seeingwithsound.com/freqtime.htm" =
target=3D"_blank">http://www.seeingwithsound.com/freqtime.htm</a><o:p></o=
:p></pre><pre><o:p>&nbsp;</o:p></pre><pre>Best =
regards,<o:p></o:p></pre><pre><o:p>&nbsp;</o:p></pre><pre>Peter =
Meijer<o:p></o:p></pre><pre><o:p>&nbsp;</o:p></pre><pre><o:p>&nbsp;</o:p>=
</pre><pre>Seeing with Sound - The vOICe<o:p></o:p></pre><pre><a =
href=3D"http://www.seeingwithsound.com/" =
target=3D"_blank">http://www.seeingwithsound.com</a><o:p></o:p></pre><pre=
><o:p>&nbsp;</o:p></pre><pre><o:p>&nbsp;</o:p></pre><pre>=A0=A0=A0 =
<o:p></o:p></pre><blockquote =
style=3D'margin-top:5.0pt;margin-bottom:5.0pt'><pre>Date: Sun, 17 Feb =
2013 07:43:35 +0000<o:p></o:p></pre><pre>From: &quot;Beerends, J.G. =
(John)&quot; <a href=3D"mailto:john.beerends@xxxxxxxx" =
target=3D"_blank">&lt;john.beerends@xxxxxxxx&gt;</a><o:p></o:p></pre><pre>T=
o: <a href=3D"mailto:AUDITORY@xxxxxxxx" =
target=3D"_blank">AUDITORY@xxxxxxxx</a><o:p></o:p></pre><pre>Subje=
ct: Re: Human hearing beats the Fourier uncertainty =
principle:<o:p></o:p></pre><pre>Research<o:p></o:p></pre><pre><o:p>&nbsp;=
</o:p></pre><pre>For discrimination the uncertainty limit does not =
exist, one can build<o:p></o:p></pre><pre>discriminator devices that go =
below the uncertainty limit in both the time<o:p></o:p></pre><pre>and =
frequency domain, the uncertainty limit is only a measure for the =
spread<o:p></o:p></pre><pre>(Delta) in both domains (DfDt&gt;1), it is =
not a limit to what extent they can<o:p></o:p></pre><pre>be =
discriminated. One can also build a device that measures the frequency =
of<o:p></o:p></pre><pre>a sine wave with an accuracy below the =
uncertainty limit by exploiting<o:p></o:p></pre><pre>a-priori knowledge, =
i.e. if I know=A0 that the signal I am measuring is =
a<o:p></o:p></pre><pre>short cut out of an infinite duration sine wave =
of a certain amplitude I can<o:p></o:p></pre><pre>measure the frequency =
as accurate as I =
want.<o:p></o:p></pre><pre><o:p>&nbsp;</o:p></pre><pre>John =
Beerends<o:p></o:p></pre><pre>=A0=A0=A0=A0=A0 =
<o:p></o:p></pre></blockquote><pre>=A0=A0=A0=A0<o:p></o:p></pre><blockquo=
te style=3D'margin-top:5.0pt;margin-bottom:5.0pt'><pre>-----Original =
Message-----<o:p></o:p></pre><pre>From: AUDITORY - Research in Auditory =
Perception<o:p></o:p></pre><pre>[<a =
href=3D"mailto:AUDITORY@xxxxxxxx" =
target=3D"_blank">mailto:AUDITORY@xxxxxxxx</a>] On Behalf Of =
Kevin Austin<o:p></o:p></pre><pre>Sent: Saturday, February 16, 2013 5:07 =
PM<o:p></o:p></pre><pre>To: <a href=3D"mailto:AUDITORY@xxxxxxxx" =
target=3D"_blank">AUDITORY@xxxxxxxx</a><o:p></o:p></pre><pre>Subje=
ct: Human hearing beats the Fourier uncertainty principle: =
Research<o:p></o:p></pre><pre><o:p>&nbsp;</o:p></pre><pre>Comments?<o:p><=
/o:p></pre><pre><o:p>&nbsp;</o:p></pre><pre><o:p>&nbsp;</o:p></pre><pre>=A0=
=A0=A0=A0=A0 <o:p></o:p></pre><blockquote =
style=3D'margin-top:5.0pt;margin-bottom:5.0pt'><blockquote =
style=3D'margin-top:5.0pt;margin-bottom:5.0pt'><pre><a =
href=3D"http://phys.org/news/2013-02-human-fourier-uncertainty-principle.=
html" =
target=3D"_blank">http://phys.org/news/2013-02-human-fourier-uncertainty-=
principle.html</a><o:p></o:p></pre><pre><o:p>&nbsp;</o:p></pre><pre><o:p>=
&nbsp;</o:p></pre><pre>(<a href=3D"http://Phys.org">Phys.org</a>)-For =
the first time, physicists have found that humans =
can<o:p></o:p></pre><pre>discriminate a sound's frequency (related to a =
note's pitch) and timing<o:p></o:p></pre><pre>(whether a note comes =
before or after another note) more than 10 =
times<o:p></o:p></pre><pre>better than the limit imposed by the Fourier =
uncertainty principle. Not<o:p></o:p></pre><pre>surprisingly, some of =
the subjects with the best listening precision =
were<o:p></o:p></pre><pre>musicians, but even non-musicians could exceed =
the uncertainty limit. The<o:p></o:p></pre><pre>results rule out the =
majority of auditory processing brain algorithms =
that<o:p></o:p></pre><pre>have been proposed, since only a few models =
can match this impressive =
human<o:p></o:p></pre><pre>performance.<o:p></o:p></pre><pre>=A0=A0=A0=A0=
=A0=A0=A0=A0=A0 =
<o:p></o:p></pre></blockquote></blockquote></blockquote><pre>=A0=A0=A0=A0=
<o:p></o:p></pre><blockquote =
style=3D'margin-top:5.0pt;margin-bottom:5.0pt'><blockquote =
style=3D'margin-top:5.0pt;margin-bottom:5.0pt'><blockquote =
style=3D'margin-top:5.0pt;margin-bottom:5.0pt'><pre>Read more =
at:<o:p></o:p></pre><pre><a =
href=3D"http://phys.org/news/2013-02-human-fourier-uncertainty-principle.=
html#jCp" =
target=3D"_blank">http://phys.org/news/2013-02-human-fourier-uncertainty-=
principle.html#jCp</a><o:p></o:p></pre><pre>=A0=A0=A0=A0=A0=A0=A0=A0=A0 =
<o:p></o:p></pre></blockquote></blockquote><pre>Thanks<o:p></o:p></pre><p=
re><o:p>&nbsp;</o:p></pre><pre>Kevin<o:p></o:p></pre><pre>This e-mail =
and its contents are subject to the DISCLAIMER =
at<o:p></o:p></pre><pre><a href=3D"http://www.tno.nl/emaildisclaimer" =
target=3D"_blank">http://www.tno.nl/emaildisclaimer</a><o:p></o:p></pre><=
pre>=A0=A0=A0=A0=A0 =
<o:p></o:p></pre></blockquote></blockquote><pre>=A0=A0<o:p></o:p></pre><p=
class=3DMsoNormal =
style=3D'margin-bottom:12.0pt'><o:p>&nbsp;</o:p></p><div><p =
class=3DMsoNormal>-- <o:p></o:p></p><table class=3DMsoNormalTable =
border=3D0 cellspacing=3D3 cellpadding=3D0><tr><td =
style=3D'padding:.75pt .75pt .75pt .75pt'><div><table =
class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 =
width=3D400 style=3D'width:240.0pt;border-collapse:collapse'><tr><td =
width=3D400 colspan=3D2 valign=3Dtop style=3D'width:240.0pt;padding:0cm =
0cm 0cm 0cm'><p class=3DMsoNormal><span =
style=3D'font-size:8.0pt;font-family:"Arial","sans-serif";color:black'>&n=
bsp; <o:p></o:p></span></p></td></tr><tr><td width=3D400 colspan=3D2 =
valign=3Dtop style=3D'width:240.0pt;padding:0cm 0cm 0cm 0cm'><p =
class=3DMsoNormal><b><span =
style=3D'font-size:8.0pt;font-family:"Arial","sans-serif";color:black'>Ba=
stian Epp <o:p></o:p></span></b></p></td></tr><tr><td width=3D400 =
colspan=3D2 valign=3Dtop style=3D'width:240.0pt;padding:0cm 0cm 0cm =
0cm'><p class=3DMsoNormal><span =
style=3D'font-size:8.0pt;font-family:"Arial","sans-serif";color:#9B9B9B'>=
Assistant Professor <o:p></o:p></span></p></td></tr><tr><td width=3D400 =
colspan=3D2 valign=3Dtop style=3D'width:240.0pt;padding:0cm 0cm 0cm =
0cm;min-height:0px'><p class=3DMsoNormal><span =
style=3D'font-size:8.0pt;font-family:"Arial","sans-serif";color:black'>EL=
EK VIP <o:p></o:p></span></p></td></tr><tr><td width=3D400 colspan=3D2 =
valign=3Dtop style=3D'width:240.0pt;padding:0cm 0cm 0cm 0cm'><p =
class=3DMsoNormal><span =
style=3D'font-size:8.0pt;font-family:"Arial","sans-serif";color:black'>DT=
U Electrical Engineering <o:p></o:p></span></p></td></tr><tr><td =
width=3D400 colspan=3D2 valign=3Dtop =
style=3D'width:240.0pt;border:none;border-bottom:solid #AEAEAE =
1.0pt;padding:0cm 0cm 0cm 0cm'><p class=3DMsoNormal =
style=3D'mso-line-height-alt:0pt'><span =
style=3D'font-size:8.0pt;font-family:"Arial","sans-serif";color:black'>&n=
bsp; <o:p></o:p></span></p></td></tr><tr><td width=3D350 valign=3Dtop =
style=3D'width:210.0pt;padding:6.0pt 0cm 0cm 0cm'><p =
style=3D'margin:0cm;margin-bottom:.0001pt'><b><span =
style=3D'font-size:8.0pt;font-family:"Arial","sans-serif";color:black'>Te=
chnical University of Denmark<o:p></o:p></span></b></p></td><td =
width=3D50 rowspan=3D8 valign=3Dtop style=3D'width:30.0pt;padding:6.0pt =
0cm 0cm 0cm'><p class=3DMsoNormal align=3Dright =
style=3D'text-align:right'><span =
style=3D'font-size:8.0pt;font-family:"Arial","sans-serif";color:black'>&l=
t;DTU_email_logo_01.gif&gt; <o:p></o:p></span></p></td></tr><tr><td =
style=3D'padding:0cm 0cm 0cm 0cm'><p class=3DMsoNormal><span =
style=3D'font-size:8.0pt;font-family:"Arial","sans-serif";color:black'>De=
partment of Electrical Engineering =
<o:p></o:p></span></p></td></tr><tr><td style=3D'padding:0cm 0cm 0cm =
0cm'><p class=3DMsoNormal><span =
style=3D'font-size:8.0pt;font-family:"Arial","sans-serif";color:black'>=D8=
rsteds Plads <o:p></o:p></span></p></td></tr><tr><td =
style=3D'padding:0cm 0cm 0cm 0cm'><p class=3DMsoNormal><span =
style=3D'font-size:8.0pt;font-family:"Arial","sans-serif";color:black'>Bu=
ilding 352 <o:p></o:p></span></p></td></tr><tr><td style=3D'padding:0cm =
0cm 0cm 0cm'><p class=3DMsoNormal><span =
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00 Kgs. Lyngby <o:p></o:p></span></p></td></tr><tr><td =
style=3D'padding:0cm 0cm 0cm 0cm'><p class=3DMsoNormal><span =
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rect +45 45253953 <o:p></o:p></span></p></td></tr><tr><td =
style=3D'padding:0cm 0cm 0cm 0cm'><p class=3DMsoNormal><span =
style=3D'font-size:8.0pt;font-family:"Arial","sans-serif";color:black'><a=
href=3D"mailto:bepp@xxxxxxxx" =
target=3D"_blank">bepp@xxxxxxxx</a> =
<o:p></o:p></span></p></td></tr><tr><td style=3D'padding:0cm 0cm 0cm =
0cm'><p class=3DMsoNormal><span =
style=3D'font-size:8.0pt;font-family:"Arial","sans-serif";color:black'><a=
href=3D"http://www.elektro.dtu.dk/" =
target=3D"_blank">www.elektro.dtu.dk/</a> =
<o:p></o:p></span></p></td></tr></table><div><p =
class=3DMsoNormal><o:p>&nbsp;</o:p></p></div><div><p =
class=3DMsoNormal>&nbsp;<o:p></o:p></p></div></div></td></tr></table></ta=
ble></div></div></div><p class=3DMsoNormal><br><br clear=3Dall><br>-- =
<o:p></o:p></p><div><p class=3DMsoNormal>James D. (jj) =
Johnston<o:p></o:p></p></div><div><p class=3DMsoNormal>Independent Audio =
and Electroacoustics Consultant<o:p></o:p></p></div></div><p =
class=3DMsoNormal><o:p>&nbsp;</o:p></p></div></div></body></html>
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Electrical Engineering Dept., Columbia University