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From: dbd on 17 Jul 2010 18:10
On Jul 17, 12:03 pm, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:
>...
>     Hi Dale, thanks for those references.  I've only
> had a look at the Gasior and Gonzalez paper.  However,
> I'm suspicious of their Figure 1.  I wonder what is
> the meaning of the dashed curve in their Figure 1.
>
> If that dashed curve is the *true* spectrum (the
> discrete-time Fourier transform) of a finite-duration
> input sinewave, then the mainlobe null-to-null width
> of that dashed curve should be only two bin-widths.
> However, their dashed curve has a mainlobe width of
> roughly five bin-widths.  That doesn't seem
> correct to me.
>
> Also, if the input sinewave's freq is located exactly
> on a bin center, then the max spectral magnitude component
> will be large and its two neighboring spec magnitude
> components should be *VERY* small (ideally zero).
> But that's not what they show in their Figure 1(a).
>
> Am I missing something here?
>
> See Ya',
> [-Rick-]

Hi Rick

The red-dashed spectrum in figure 1 if the G&G paper is the shape of
the frequency response of the window function. You may note that they
do not include the rectangular window in their collection of windows.
In their first paragraph of conclusions they suggest that parabolic
interpolation does not start to have "significant" accuracy until the
window frequency mainlobe width is at least 3 bins.

We could have a long discussion of whether the assumption that the
choice of fft size equal to the number of samples and a rectangular
window is "true" or just lazy and ignorant, but comp.dsp does not
serve beer, so we should discuss that issue elsewhere.

The parabolic interpolation is a quick and dirty "know nothing but
that there are 3 points about a peak" approach. To use it with a
rectangular window, the transform size is usually increased by zero-
filling to improve accuracy. The bandwidth of the frequency mainlobe
of the window doesn't change, but the samples in the frequency domain
become closer together: more samples per mainlobe width. The parabolic
interpolater is an exact match for the log of the power spectrum with
a Gaussian window, but some error sneaks back in when we sample and
truncate the window to finite extent. The errors so introduced are
smaller for Gaussian windows with high sidelobe rejection (and thus
small tails to truncate).The bottom entries in Table 2 illustrate
this.

"Interpolated FFT" results are common in the Transactions on
Instrumentation and Measurement. Other extensions include estimators
of greater than 3 points and estimators designed for specific windows.

Dale B. Dalrymple
From: Ron N. on 18 Jul 2010 03:51
On Jul 17, 3:10 pm, dbd <d...(a)ieee.org> wrote:
> We could have a long discussion of whether the assumption that the
> choice of fft size equal to the number of samples and a rectangular
> window is "true" or just lazy and ignorant, but comp.dsp does not
> serve beer, so we should discuss that issue elsewhere.
>
> The parabolic interpolation is a quick and dirty "know nothing but
> that there are 3 points about a peak" approach. To use it with a
> rectangular window, the transform size is usually increased by zero-
> filling to improve accuracy.

A couple variations: Since zero-padding is roughly equivalent
to Sinc interpolation, you could just locally interpolate a
few samples before using parabolic interpolator. The other
trick it to pick a window whose transform is very similar to
a parabola in shape, which, of course, should reduce the error
of a parabolic interpolator.


IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M
http://www.nicholson.com/rhn/dsp.html
From: dbd on 18 Jul 2010 11:39
On Jul 18, 12:51 am, "Ron N." <rhnlo...(a)yahoo.com> wrote:

> ...
> A couple variations:  Since zero-padding is roughly equivalent
> to Sinc interpolation, you could just locally interpolate a
> few samples before using parabolic interpolator.  

When the mainlobe shape is simple and well enough known for easy
accurate local interpolation, the same characteristics can be applied
to estimating the position of the peak directly. That is what the
window specific interpolators do.

>The other
> trick it to pick a window whose transform is very similar to
> a parabola in shape, which, of course, should reduce the error
> of a parabolic interpolator.
>

This is done with the log of the Gaussian windowed power spectrum, as
shown in the G&G paper.

Some authors have taken this a different direction by designing a
window that has a triangular mainlobe and using a peak estimator
designed for the triangle shape.

Dale B. Dalrymple

From: Rick Lyons on 18 Jul 2010 12:27
On Sat, 17 Jul 2010 15:10:56 -0700 (PDT), dbd <dbd(a)ieee.org> wrote:

[Snipped by Lyons]
>
>Hi Rick
>
>The red-dashed spectrum in figure 1 if the G&G paper is the shape of
>the frequency response of the window function. You may note that they
>do not include the rectangular window in their collection of windows.
>In their first paragraph of conclusions they suggest that parabolic
>interpolation does not start to have "significant" accuracy until the
>window frequency mainlobe width is at least 3 bins.
>
>We could have a long discussion of whether the assumption that the
>choice of fft size equal to the number of samples and a rectangular
>window is "true" or just lazy and ignorant, but comp.dsp does not
>serve beer, so we should discuss that issue elsewhere.
>
>The parabolic interpolation is a quick and dirty "know nothing but
>that there are 3 points about a peak" approach. To use it with a
>rectangular window, the transform size is usually increased by zero-
>filling to improve accuracy. The bandwidth of the frequency mainlobe
>of the window doesn't change, but the samples in the frequency domain
>become closer together: more samples per mainlobe width. The parabolic
>interpolater is an exact match for the log of the power spectrum with
>a Gaussian window, but some error sneaks back in when we sample and
>truncate the window to finite extent. The errors so introduced are
>smaller for Gaussian windows with high sidelobe rejection (and thus
>small tails to truncate).The bottom entries in Table 2 illustrate
>this.
>
>"Interpolated FFT" results are common in the Transactions on
>Instrumentation and Measurement. Other extensions include estimators
>of greater than 3 points and estimators designed for specific windows.
>
>Dale B. Dalrymple

Hi Dale,
Thanks a lot for your detailed explanation.
I saw those words: "...for efficient interpolation the
minimum width of the spectral peak is 3 bins.", but it
didn't sink into my head that what they really meant was:

"For accurate (not efficient) interpolation, we need
at least three samples within the mainlobe of the
windowed signal's "true" spectrum."

Ya' know, this morning (before I read this post of yours)
I thought, "Maybe those guys were zero-padding their original
input time sequence to have more samples within the
signal's mainlobe." But now, thanks to you, I'm reminded
that the original input sinusoid can also be windowed in
order to widen its mainlobe.

See Ya',
[-Rick-]

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