Reply by Steven G. Johnson●October 5, 20042004-10-05
"Clay Turner" <email@example.com> wrote...
> The method I proposed is equiamplitude whereas Reinsch's method is not. And
> I believe this in addition to the orthogonality at low frequency is what
> helps minimize the errors. I predict other structures would be better near
> either 1/4 or 1/2 of the sampling rate.
It would be nice to have an "optimal" Goertzel-like method (or a
systematic way to come up with such a method) that for *any* given
frequency will achieve O(sqrt(N)) mean error.
One option, of course, is to just use the naive DFT formula (which has
O(sqrt(N)) mean error if the twiddle factors are accurate), but with
an accurate trig. recurrence instead of precomputed twiddle factors.
There are trig. recurrences that have O(sqrt(log N)) mean error, and
require only O(log N) storage, and I suspect they would do the trick.
I tried using the naive DFT formula with Singleton's trig. recurrence
(which requires O(1) storage), but the results (for test 2, see
earlier) were a bit weird -- the errors jumped back and forth between
the O(N) line and the O(sqrt(N)) line for different N's, and I
couldn't detect the pattern; maybe there is some fix, though. In any
case, such algorithms (implemented straightforwardly) are much more
expensive than Goertzel's algorithm: 6/4 or 6/6 mults/adds per input,
vs. 1/2 mults/adds per input for Goertzel.
Steven G. Johnson
PS. You can do even better than O(sqrt(N)) error with the naive DFT,
and in fact should be able to get the FFT's O(sqrt(log N)) mean error,
by using e.g. cascade summation (with negligible added computational
cost and O(log N) storage). However, this loses the nice property of
going through the data in order, which is important for many
applications. There might instead be some Kahan-like summation trick
to get similar accuracy, but of course this adds more computational
cost per input.
Reply by Aaron45●August 4, 20162016-08-04
>"Clay Turner" <firstname.lastname@example.org> wrote...
>> I predict other structures would be better
>> near either 1/4 or 1/2 of the sampling rate.
>It would be nice to have an "optimal" Goertzel-like method (or a
>systematic way to come up with such a method) that for *any* given
>frequency will achieve O(sqrt(N)) mean error.
OK, I realize this is major zombie thread, but I wanted to post some
results from some experiments on this topic. If this has already been
done, please let me know.
What I did was create a general purpose application to plug in new
resonators (specifically the ones from here:
http://www.claysturner.com/dsp/ResonatorTable.pdf). You can plug in
resonators in matrix form (for non-staggered update) and execute without
having to modify the general algorithm specified here:
http://www.claysturner.com/dsp/digital_resonators.pdf. I also implemented
the traditional forms of Reinsch as staggered update. See an example
Besides curiosity, I am motivated by a need to avoid unacceptable errors
in another application I'm developing.
I don't know that I found anything terribly promising, but was surprised
by some things. Maybe they're bugs in my code. If anyone's interested I
published it here: https://bitbucket.org/hexamer/testgoertzelresonators
A binary is available here:
It is coded to run on a machine with AVX2 (Haswell). If you don't
configure FMA3, it may work on pre-AVX2 machines.
1) With the exception of Reinsch(pi) all resonators show poor error growth
(looks like N^2) near pi
2) The matrix form of Reinsch that Clay Turner published in this paper
seems to have poor error performance near 0, and “reasonable” (i.e.
not N^2) error performance near pi/2, which is contrary to all other
publications, including results in this thread. Perhaps because it's a
parallel, not staggered form?
3) The standard staggered forms of Reinsch don't seem to have N^2 error
performance near pi/2 as the literature would suggest. Admittedly I
haven't tried to fit the data yet.
4) In fact, I haven't yet seen any of the resonators exhibit what looks
like N^2 error growth near pi/2
5) Use of FMA3 does bring a modest performance benefit to Goertzel, but
surprisingly seems to have no benefit to numerical error.
6) The Coupled resonator produces normal errors up to pi/2, but then just
starts producing strange results from pi/2 to pi.
Resonator Type 6 is Reinsch(0)
N, log(err) theta ~ 0, log(err) theta ~ pi/2, log(err) theta ~ pi
256, -14.1091, -13.5427, -12.0575
512, -13.7318, -13.019, -11.4822
1024, -13.9116, -12.6285, -11.2495
2048, -12.5309, -11.9725, -10.5061
4096, -12.8532, -11.7525, -10.3899
8192, -11.8121, -11.6945, -8.96993
16384, -12.913, -11.1648, -8.31467
Posted through http://www.DSPRelated.com
Reply by PRAVEEN M P●August 19, 20162016-08-19
On Tuesday, 21 September 2004 11:00:31 UTC+5:30, amara vati wrote:
> which is the best fft method if want the spectrum only for a selective
> set of points, say only 10-20% of the points in the entire spectrum.
FFT is highly optimized with total computations of O(NlogN). For computing DFT for a set of M points,
you need atleast O(N.M)(with some scaling constant). I once tried various algorithms, including Goetzels and all of them were
worse compared to fft in time. One way to reduce time is to reduce FFT size (from N to say N/2), if the indices
on which you want FT are 'special'. For example, they are all even or odd or multiples of 4.. etc