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I have a question about Grandke's method of frequency interpolation. (I suspect this question has been asked before). Using the Hanning window (and Schoukens, 1991 notation), he derives f_interp = (i + delta) * fo, where delta = (2*alpha - 1) / (alpha + 1), and alpha = abs(X(i+1) / X(i)), and {X(i), X(i+1)} includes the peak. By definition, 0 < delta < 1, which implies 1/2 < alpha < 2. Why must alpha satisfy this? ---

On Wed, 20 Apr 2011 14:28:13 -0500, creekmor <u...@compgroups.net/> wrote: >I have a question about Grandke's method of frequency interpolation. (I suspect this question has been asked before). Using the Hanning window (and Schoukens, 1991 notation), he derives > >f_interp = (i + delta) * fo, where > >delta = (2*alpha - 1) / (alpha + 1), and > >alpha = abs(X(i+1) / X(i)), and > >{X(i), X(i+1)} includes the peak. By definition, > >0 < delta < 1, which implies 1/2 < alpha < 2. Why must > >alpha satisfy this? The units on delta are DFT bins, so if delta > 1 then you picked the wrong bin for the maximum. The correction is just to find a finer estimate between bins, so the assumption is that the peak is already determined within one bin. Eric Jacobsen http://www.ericjacobsen.org http://www.dsprelated.com/blogs-1//Eric_Jacobsen.php

>On Wed, 20 Apr 2011 14:28:13 -0500, creekmor <u...@compgroups.net/> >wrote: > >>I have a question about Grandke's method of frequency interpolation. (I suspect this question has been asked before). Using the Hanning window (and Schoukens, 1991 notation), he derives >> >>f_interp = (i + delta) * fo, where >> >>delta = (2*alpha - 1) / (alpha + 1), and >> >>alpha = abs(X(i+1) / X(i)), and >> >>{X(i), X(i+1)} includes the peak. By definition, >> >>0 < delta < 1, which implies 1/2 < alpha < 2. Why must >> >>alpha satisfy this? > >The units on delta are DFT bins, so if delta > 1 then you picked the >wrong bin for the maximum. The correction is just to find a finer >estimate between bins, so the assumption is that the peak is already >determined within one bin. > > >Eric Jacobsen >http://www.ericjacobsen.org >http://www.dsprelated.com/blogs-1//Eric_Jacobsen.php > I agree with that; however, the quantity alpha = abs(X(i+1) / X(i)) clearly could be between 0 and infty. For example, if the peak is located at bin # i, and the next largest bin # is i+1, then 0 < alpha < 1 while if the peak is located at bin # i+1 and the next largest bin # is i, then 1 < alpha < infty However, to ensure that the fractional bin spacing delta satisfies 0 < delta < 1, alpha must then satisfy 1/2 < alpha < 2 which contradicts the above. In comparison, the interpolation formula for the rectangular window is delta = alpha / (alpha+1) which causes no similar problem.