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<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Hi Samuel,<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Fyi</span></span></font></span>, many
<span class="SpellE">DFT</span> references recommend averaging the values at endpoints and discontinuities in general. See for instance:<o:p></o:p></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Briggs, W. L. and V. E. Henson [1005] The
<span class="SpellE">DFT</span>: an owner's manual for the Discrete Fourier Transform, SIAM, Philadelphia.<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">I think my solution is simple and it looks harmless to me. At least the results look reasonable for different amounts of fractional
shifts tested.<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">But I will keep studying the subject.<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Maybe some expert on the matter can jump in.<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Note sure about your last question as many scenarios can be thought of, of continuous functions with discontinuities and their sampled
versions.<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">I can think of cases where the slope can be < |u($)-u(1)|/1<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Regards,<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Rafael<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">-----Original Message-----<br>
From: users [mailto:users-bounces@lists.scilab.org] On Behalf Of sgougeon@free.fr<br>
Sent: Thursday, June 07, 2018 2:32 PM<br>
To: Users mailing list for Scilab <users@lists.scilab.org><br>
Subject: Re: [Scilab-users] <span class="SpellE">circshift</span>() : Scilab Enhancement Proposal</span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">>Samuel,<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">><o:p> </o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">>The idea and the implementation of the end-points interpolation is that
<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">>this is done on the input data (big slope), not on the
<span class="SpellE">FFT</span> output.<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">This is clear<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">>Maybe be it needs some further work. Tbc.
<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">I don't think so.<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">>Noting also that between the endpoints of the
<span class="SpellE">DFT</span> periodic input, there is the interval of one sample.<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">That is to say: Let u(1:$) be the original signal.<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Since it is virtually periodic, the
<span class="SpellE">FFT</span> sees u($+1)=u(1).<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Do we agree about this?<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Then: Do we agree on that the |slope| across the shifted edges of u is necessarily greater or equal to |u($)-u(1)|/1 ?<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
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