Comments on: The DFT “à Pied”: Mastering The Fourier Transform in One Day

By: Bernsee

Bernsee — Thu, 25 Feb 2010 06:05:19 +0000

Thank you for your feedback. In our example listing 1.2 bin #0 contains only the DC component.

Some FFT implementations put both DC and fs/2 into bin #0 (as real and imaginary part) because both are real-valued and that way you can use N/2 complex transform bins instead of N/2+1.

By: shabtronic

shabtronic — Wed, 24 Feb 2010 11:53:03 +0000

great web page – starting to understand the fft now!!

what’s in Bin 0 – is that a DC offset or overall magnitude?

By: Mark

Mark — Sat, 20 Feb 2010 15:20:40 +0000

Awesome! This has helped me a lot. Thank you!

By: Kris Bishop

Kris Bishop — Fri, 19 Feb 2010 13:22:16 +0000

I have preiously studied this at University, passed the exam, but never understood it until I read your explanation.

very well done

Thanks

By: Bernsee

Bernsee — Fri, 15 Jan 2010 09:14:57 +0000

Thank you, I appreciate your feedback! I am working on more tutorials at the moment – if you have a specific topic that you would like to read about please let me know (either by emailing or posting here).

By: Jack Kinsella

Jack Kinsella — Thu, 14 Jan 2010 16:50:21 +0000

Fantastically well written.

By the way the internet already recognizes this article’s merit – it’s one of the most popular articles tagged “math” on delicious.

Please do more tutorials in future.

By: yose

yose — Thu, 14 Jan 2010 14:09:18 +0000

This is an excellent article. Finally, I can get a clear sense of what is happening and what a fourier transform involves of. Was being confused with the convoluted ways most other sources try to explain things (by going to deep into the math-sides of things).

By: Paul

Paul — Mon, 21 Dec 2009 05:51:49 +0000

Yes, but I’m not sure if scientists have been able to measure them yet… they are still in the process of repairing the LHC.

-P

By: Aran

Aran — Mon, 21 Dec 2009 01:11:52 +0000

im sure these limits have been established, the universe does exist…

By: Paul

Paul — Sun, 20 Dec 2009 18:32:03 +0000

The Nyquist limit is a result of sampling, not quantization. Not sure what you’re getting at with the minimum mass/maximum frequency limit of the universe but I think these limits have not been established yet…

-P

By: Loki Clock

Loki Clock — Sun, 20 Dec 2009 18:16:21 +0000

Note, this half-period resolution limit is what’s known as the Nyquist or cut-off frequency. This is a reflection of quantization itsself, which need not be digital. If the universe had a minimum unit of mass, but not a maximum frequency, you could potentially create aliasing by letting a wave with a wavelength twice or less than the length of that minimum unit propagate through it. I have no idea what that would do, or if there are any problems with that sentiment, but it’s still crazy to think about!

To the author, I am looking forward to reading this over break. It shall surely be a treat, as I’ve been needing an explanation of a Fourier transform for the very purpose of programming one in C++.

By: Bernsee

Bernsee — Thu, 29 Oct 2009 19:09:08 +0000

Yes, this is indeed possible. However, such a signal cannot be correctly described by our digital representation. If you have several signal periods between adjacent sampling points then you will not be able to reconstruct that signal and aliasing will occur. Aliasing is an effect that causes frequencies that are higher than the highest frequency that can be represented to appear as low frequencies. The following picture should make this clear:

The black waveform has multiple periods between the sampling points (blue lines). So if we sample that waveform (= measure it at the blue intervals) it will effectively look like a different waveform that has a much lower frequency (red dotted waveform).

In essence what this means is that once you get to play with your sampled signal all frequencies that are higher than the highest possible frequency (the one with adjacent peaks and valleys located on the sample grid) have either been removed, or appear as lower frequencies due to aliasing.

By: Ed

Ed — Tue, 27 Oct 2009 20:33:53 +0000

Very nice tutorial.

However I don’t quite understand why “we see that the upper frequency of a sampled sine wave is every other sample being a peak and a valley and the lower frequency bound is half a period of the sine wave which is just fitting in the number of samples we are looking at.”

Isn’t it possible that a partial sinusoid can have a very high frequency that there can be several periods between two sampling points?

By: Christian Brumm

Christian Brumm — Thu, 17 Sep 2009 08:14:24 +0000

This is a VERY good and gentile introduction to the Fourier Transform. The sample code helps a lot and it is a good exercise to derive the math from it. Please keep this article up, I think it will still be useful to many people in the years to come!

By: Jai

Jai — Fri, 07 Aug 2009 16:33:49 +0000

It’s probably not necessary since there are so many kudos here, but I just wanted to add a heart thanks for a fantastic “executive summary”. I am sure that was a lot of work to condense. Not only is this a Fourier Transform for Dummies, it’s also a great Fourier Transform for Dummies with ADHD.

By: Bernsee

Bernsee — Thu, 06 Aug 2009 18:30:45 +0000

I don’t think that I understand your question. The DFT per se does not introduce a phase shift into the signal – if you iDFT a transformed block of data you will get back your original signal (a very good approximation of it, that is).

By: Said

Said — Thu, 06 Aug 2009 16:03:35 +0000

It’s a great tutorial about the DFT, however I cannot see any details about the phase correction (compensation) when DFT is dealing with sine wave signal, as it introduce a phase shift to the signal. I read many books where they said you have to compensate for 90 degree phase shift and i dont thing it is a general way to do it.

By: Gabriel

Gabriel — Thu, 06 Aug 2009 01:32:21 +0000

Wonderful tutorial, I read before another excellent tutorial for Fourier transforms by Robi Polikar, but it deals more with Wavelets (a better way than Short Time Fourier Transforms).

The only place where I’ve been confused on this one is the computation of Magnitude and the Phase in the code. Where does it come from?

Thanks again for the tutorial.

By: Chris Smith

Chris Smith — Sun, 12 Jul 2009 08:39:37 +0000

Thank you for this. Fantastic article.

Having encountered FFT’s in the past and never truly understanding them (thanks to some poor tuition), this hits the spot. If only university text books and Knuth were as concise as this

By: Gustav

Gustav — Sat, 11 Jul 2009 17:13:21 +0000

Thanks a lot for taking the time to dumb this down and make it “programmatic”.

By: Arzon

Arzon — Sat, 11 Jul 2009 17:13:07 +0000

Fantastic tutorial, but this sentence doesn’t make any sense, to me:

“Summarizing what we have just learned, we see that the upper frequency of a sampled sine wave is every other sample being a peak and a valley and the lower frequency bound is half a period of the sine wave which is just fitting in the number of samples we are looking at.”

By: Gustavo Leig

Gustavo Leig — Sat, 11 Jul 2009 14:18:37 +0000

Thats great… what would be beautiful..at least for 1st readers would be nice to see what happens to a sin wave after the transfourm.

By: Bernsee

Bernsee — Sat, 11 Jul 2009 09:50:04 +0000

Sure!

By: Tato

Tato — Thu, 25 Jun 2009 10:32:03 +0000

Can I build a multi effects processor for my electric guitar using a DSP with the Fourier stuff and the C++ programming?

By: Kyle Mew

Kyle Mew — Fri, 29 May 2009 21:08:10 +0000

very good – i’ve been looking for a jargon free explanation of dft and fft for ages – and this is just that – thank you