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    Michael, could you perhaps comment on that again? I am also a bit
    confused. You stated below that "The units on the FFT are [...] for
    power density.", but later you stated that "The units are those of
    energy not power."<br>
    <br>
    So, does the FFT depend (in amplitude) on the length of a (uniform)
    data vector and, thus, on the total energy of the data subject to
    analysis?<br>
    <br>
    Ole<br>
    <br>
    <br>
    <div class="moz-cite-prefix">Am 17.06.2014 17:37, schrieb Laszlo
      Balazs:<br>
    </div>
    <blockquote cite="mid:53A060B4.7070809@ttk.mta.hu" type="cite">
      <meta content="text/html; charset=ISO-8859-1"
        http-equiv="Content-Type">
      Hi all,<br>
      Now I am getting confused. I believe that we are talking about
      power (density) and not energy (density). PS is the squared
      frequency spectrum (Fourier transform of the signal which has the
      dimension of microvolts). The resulting dimension is uV^2
      (microvolts squared) (or uV^2/Hz for density). Electrical power is
      proportional with U^2 (P=U^2/R). Power is energy per unit of time,
      or in other words, if it were energy than it were dependent of the
      length of the period under question.<br>
      Correct me if I am wrong.<br>
      Best,<br>
      Laszlo<br>
      <br>
      <div class="moz-cite-prefix">On 6/16/2014 9:53 PM, Makoto
        Miyakoshi wrote:<br>
      </div>
      <blockquote
cite="mid:%3CCAEqC+SWbFemEsNvfY8z9_dDKzSzdV9GZ9TnhFGM7NuHqoikb9A@mail.gmail.com%3E"
        type="cite">
        <div dir="ltr">Thanks Ole. It was very informative for me!
          <div><br>
          </div>
          <div>Makoto</div>
        </div>
        <div class="gmail_extra"><br>
          <br>
          <div class="gmail_quote">On Tue, Jun 10, 2014 at 12:28 AM, Ole
            Traupe <span dir="ltr"><<a moz-do-not-send="true"
                href="mailto:ole.traupe@tu-berlin.de" target="_blank">ole.traupe@tu-berlin.de</a>></span>
            wrote:<br>
            <blockquote class="gmail_quote" style="margin:0 0 0
              .8ex;border-left:1px #ccc solid;padding-left:1ex">
              <div text="#000000" bgcolor="#FFFFFF"> HI all,<br>
                <br>
                I forward a short discussion on this topic to the list
                showing that things always are more complicated.<br>
                <br>
                Best,<br>
                Ole<br>
                <div><br>
                  <br>
                  -------- Original-Nachricht --------
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                    <tbody>
                      <tr>
                        <th valign="BASELINE" align="RIGHT"
                          nowrap="nowrap">Betreff: </th>
                        <td>RE: [Eeglablist] Power Spectrum vs. Power
                          Spectral Density</td>
                      </tr>
                      <tr>
                        <th valign="BASELINE" align="RIGHT"
                          nowrap="nowrap">Datum: </th>
                        <td>Thu, 5 Jun 2014 13:06:31 -0400 (EDT)</td>
                      </tr>
                      <tr>
                        <th valign="BASELINE" align="RIGHT"
                          nowrap="nowrap">Von: </th>
                        <td>MICHAEL JOSEPH PIOVOSO <a
                            moz-do-not-send="true"
                            href="mailto:mjp5@psu.edu" target="_blank"><mjp5@psu.edu></a></td>
                      </tr>
                      <tr>
                        <th valign="BASELINE" align="RIGHT"
                          nowrap="nowrap">An: </th>
                        <td>Ole Traupe <a moz-do-not-send="true"
                            href="mailto:ole.traupe@tu-berlin.de"
                            target="_blank"><ole.traupe@tu-berlin.de></a></td>
                      </tr>
                    </tbody>
                  </table>
                  <br>
                  <br>
                  <div>
                    <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">What


                        you say is correct.  The FFT gives what should
                        be called the Energy Density (Not power
                        density).   The units on the FFT are as you say
                        for power density.   Common usage refers to it
                        as a power density and also as a power spectral
                        density.  But power spectral density is
                        incorrect.  It is in fact an energy density not
                        a power density.  The units are those of energy
                        not power.</span></p>
                    <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d"> </span></p>
                    <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d"> </span></p>
                    <div>
                      <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">Michael


                          J. Piovoso, Ph.D., P.E.</span></p>
                      <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">Professor


                          of Electrical Engineering</span></p>
                      <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">Penn


                          State University School of Graduate
                          Professional Studies</span></p>
                      <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">30


                          E. Swedesford Road</span></p>
                      <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">Malvern,


                          PA 19355</span></p>
                    </div>
                    <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d"><a
                          moz-do-not-send="true" href="tel:610-648-3356"
                          value="+16106483356" target="_blank">610-648-3356</a></span><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d"></span></p>
                    <div>
                      <div style="border:none;border-top:solid #b5c4df
                        1.0pt;padding:3.0pt 0in 0in 0in">
                        <p class="MsoNormal"><b><span
style="font-size:10.0pt;font-family:"Tahoma","sans-serif";color:windowtext">From:</span></b><span
style="font-size:10.0pt;font-family:"Tahoma","sans-serif";color:windowtext">
                            Ole Traupe [<a moz-do-not-send="true"
                              href="mailto:ole.traupe@tu-berlin.de"
                              target="_blank">mailto:ole.traupe@tu-berlin.de</a>]
                            <br>
                            <b>Sent:</b> Thursday, June 05, 2014 1:03 PM<br>
                            <b>To:</b> MICHAEL JOSEPH PIOVOSO<br>
                            <b>Subject:</b> Re: [Eeglablist] Power
                            Spectrum vs. Power Spectral Density</span></p>
                      </div>
                    </div>
                    <p class="MsoNormal"> </p>
                    <p class="MsoNormal" style="margin-bottom:12.0pt">I
                      am a psychologist and a programmer, and I clearly
                      see your expertise here. I don't mean that there
                      is a fundamental difference between the two. But
                      there must be some kind of difference from my
                      perspective, as the unit is different. No? In PS
                      (unit^2), the values are power. In PSD
                      (unit^2/Hz), the area under the curve is power. So
                      you could probably say it's basically the same
                      with PSD being somewhat transformed?<br>
                      <br>
                      Ole<br>
                      <br>
                    </p>
                    <div>
                      <p class="MsoNormal">Am 05.06.2014 18:57, schrieb
                        MICHAEL JOSEPH PIOVOSO:</p>
                    </div>
                    <blockquote
                      style="margin-top:5.0pt;margin-bottom:5.0pt">
                      <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">If


                          it is wrong, then  I have been teaching that
                          for over 40 years now and I need to understand
                          the difference.  I check with other electrical
                          engineers  and they agree with me.  I did not
                          realize that it was sent to only you.  </span></p>
                      <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d"> </span></p>
                      <div>
                        <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">Michael


                            J. Piovoso, Ph.D., P.E.</span></p>
                        <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">Professor


                            of Electrical Engineering</span></p>
                        <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">Penn


                            State University School of Graduate
                            Professional Studies</span></p>
                        <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">30


                            E. Swedesford Road</span></p>
                        <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">Malvern,


                            PA 19355</span></p>
                      </div>
                      <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d"><a
                            moz-do-not-send="true"
                            href="tel:610-648-3356" value="+16106483356"
                            target="_blank">610-648-3356</a></span></p>
                      <div>
                        <div style="border:none;border-top:solid #b5c4df
                          1.0pt;padding:3.0pt 0in 0in 0in">
                          <p class="MsoNormal"><b><span
style="font-size:10.0pt;font-family:"Tahoma","sans-serif";color:windowtext">From:</span></b><span
style="font-size:10.0pt;font-family:"Tahoma","sans-serif";color:windowtext">
                              Ole Traupe [<a moz-do-not-send="true"
                                href="mailto:ole.traupe@tu-berlin.de"
                                target="_blank">mailto:ole.traupe@tu-berlin.de</a>]
                              <br>
                              <b>Sent:</b> Thursday, June 05, 2014 6:10
                              AM<br>
                              <b>To:</b> MICHAEL JOSEPH PIOVOSO<br>
                              <b>Subject:</b> Re: [Eeglablist] Power
                              Spectrum vs. Power Spectral Density</span></p>
                        </div>
                      </div>
                      <p class="MsoNormal"> </p>
                      <p class="MsoNormal" style="margin-bottom:12.0pt">Hi

                        Micheal,<br>
                        <br>
                        I think this is wrong. Anyhow, did you realize
                        you replied only to me?<br>
                        <br>
                        Ole<br>
                        <br>
                        <br>
                      </p>
                      <div>
                        <p class="MsoNormal">Am 04.06.2014 21:10,
                          schrieb MICHAEL JOSEPH PIOVOSO:</p>
                      </div>
                      <blockquote
                        style="margin-top:5.0pt;margin-bottom:5.0pt">
                        <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">There


                            really is no difference between power
                            spectral density and power spectrum.  Power
                            spectral density is more correct but many
                            use the term power spectrum for that as
                            well.</span></p>
                        <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d"> </span></p>
                        <div>
                          <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">Michael


                              J. Piovoso, Ph.D., P.E.</span></p>
                          <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">Professor


                              of Electrical Engineering</span></p>
                          <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">Penn


                              State University School of Graduate
                              Professional Studies</span></p>
                          <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">30


                              E. Swedesford Road</span></p>
                          <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">Malvern,


                              PA 19355</span></p>
                        </div>
                        <p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d"><a
                              moz-do-not-send="true"
                              href="tel:610-648-3356"
                              value="+16106483356" target="_blank">610-648-3356</a></span></p>
                        <div>
                          <div style="border:none;border-top:solid
                            #b5c4df 1.0pt;padding:3.0pt 0in 0in 0in">
                            <p class="MsoNormal"><b><span
style="font-size:10.0pt;font-family:"Tahoma","sans-serif";color:windowtext">From:</span></b><span
style="font-size:10.0pt;font-family:"Tahoma","sans-serif";color:windowtext">
                                <a moz-do-not-send="true"
                                  href="mailto:eeglablist-bounces@sccn.ucsd.edu"
                                  target="_blank">eeglablist-bounces@sccn.ucsd.edu</a>
                                [<a moz-do-not-send="true"
                                  href="mailto:eeglablist-bounces@sccn.ucsd.edu"
                                  target="_blank">mailto:eeglablist-bounces@sccn.ucsd.edu</a>]
                                <b>On Behalf Of </b>Ole Traupe<br>
                                <b>Sent:</b> Wednesday, June 04, 2014
                                5:34 AM<br>
                                <b>To:</b> <a moz-do-not-send="true"
                                  href="mailto:eeglablist@sccn.ucsd.edu"
                                  target="_blank">eeglablist@sccn.ucsd.edu</a><br>
                                <b>Subject:</b> Re: [Eeglablist] Power
                                Spectrum vs. Power Spectral Density</span></p>
                          </div>
                        </div>
                        <div class="">
                          <p class="MsoNormal"> </p>
                          <p class="MsoNormal"
                            style="margin-bottom:12.0pt">Hi Brian, hi
                            Makoto,<br>
                            <br>
                            as far as I know, and notwithstanding any
                            considerations regarding a particular
                            software implementation, the PS yields the
                            power of a discrete, predefined set of
                            frequencies interpretable as individual
                            values. In contrast, similar to the
                            probability density of the (standard) normal
                            distribution, the PSD is a density function
                            the (partial) area under which can be
                            interpreted in terms of power. The
                            transformation '10*log10' rescales to dB not
                            affecting the units (i.e. unit^2 and
                            unit^2/Hz, respectively).<br>
                            <br>
                            Therefore, the choice between PS and PSD
                            depends on the question you want to answer.
                            In EEG matters you are typically interested
                            in the power of continuous frequency bands
                            and therefore should consider the PSD (and
                            determine the area within certain limits of
                            interest).<br>
                            <br>
                            Ole<br>
                            <br>
                            <br>
                            <br>
                          </p>
                          <div>
                            <p class="MsoNormal">Am 03.06.2014 23:26,
                              schrieb <a moz-do-not-send="true"
                                href="mailto:eeglablist-request@sccn.ucsd.edu"
                                target="_blank">eeglablist-request@sccn.ucsd.edu</a>:</p>
                          </div>
                        </div>
                        <blockquote
                          style="margin-top:5.0pt;margin-bottom:5.0pt">
                          <div>
                            <p class="MsoNormal">Dear Brian, </p>
                            <div class="">
                              <div>
                                <p class="MsoNormal"> </p>
                              </div>
                              <div>
                                <p class="MsoNormal">Actualy I did not
                                  know the difference. According to
                                  wikipedia, spectral density is
                                  something like spectra represented as
                                  cumulative distribution function.</p>
                              </div>
                              <div>
                                <p class="MsoNormal"> </p>
                              </div>
                              <div>
                                <p class="MsoNormal"><a
                                    moz-do-not-send="true"
                                    href="http://en.wikipedia.org/wiki/Spectral_density"
                                    target="_blank">http://en.wikipedia.org/wiki/Spectral_density</a></p>
                              </div>
                              <div>
                                <p class="MsoNormal"> </p>
                              </div>
                              <div>
                                <p class="MsoNormal">I don't know though
                                  when it is more appropriate to use
                                  power spectra and spectral density...
                                  someone in the list please help us.</p>
                              </div>
                              <div>
                                <p class="MsoNormal"> </p>
                              </div>
                              <div>
                                <p class="MsoNormal">Makoto</p>
                              </div>
                            </div>
                          </div>
                          <div class="">
                            <div>
                              <p class="MsoNormal"
                                style="margin-bottom:12.0pt"> </p>
                              <div>
                                <p class="MsoNormal">On Wed, May 28,
                                  2014 at 8:50 AM, Erickson <<a
                                    moz-do-not-send="true"
                                    href="mailto:ericksonb.eng@gmail.com"
                                    target="_blank">ericksonb.eng@gmail.com</a>>


                                  wrote:</p>
                                <div>
                                  <p class="MsoNormal">List, </p>
                                  <div>
                                    <p class="MsoNormal">The function
                                      "spectopo" produces output in
                                      units of 10log_10 (uV^2/Hz). This
                                      is power spectral density, as
                                      opposed to a power spectrum. Could
                                      anyone comment on the implications
                                      of interpreting, physiologically,
                                      the PS vs. the PSD of a neural
                                      signal? Thank you,</p>
                                  </div>
                                  <div>
                                    <p class="MsoNormal"> </p>
                                  </div>
                                  <div>
                                    <p class="MsoNormal">- Brian</p>
                                  </div>
                                  <div>
                                    <p class="MsoNormal">Applied Brain
                                      and Cognitive Sciences Program</p>
                                  </div>
                                  <div>
                                    <p class="MsoNormal">Drexel
                                      University</p>
                                  </div>
                                </div>
                                <p class="MsoNormal"><br>
_______________________________________________<br>
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                        <p class="MsoNormal"> </p>
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          <br>
          <br clear="all">
          <div><br>
          </div>
          -- <br>
          <div dir="ltr">Makoto Miyakoshi<br>
            Swartz Center for Computational Neuroscience<br>
            Institute for Neural Computation, University of California
            San Diego<br>
          </div>
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      </blockquote>
      <br>
      <pre class="moz-signature" cols="72">-- 

Laszlo Balazs, Ph.D.
MTA Research Centre for Natural Sciences
Institute of Cognitive Neuroscience and Psychology

P O B 398, Budapest, Hungary, H-1394
<a moz-do-not-send="true" class="moz-txt-link-freetext" href="Tel:+36%281%29382-6811">Tel:+36(1)382-6811</a>
<a moz-do-not-send="true" class="moz-txt-link-freetext" href="http://www.cogpsyphy.hu/balazs">http://www.cogpsyphy.hu/balazs</a>
</pre>
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