Retrotechtacular: The Fourier Series


Here’s a really quick video which takes a different approach to understanding the Fourier Series than we’re used to. If you’re a regular reader we’re sure you’ve heard of the Fourier Series (often discussed as FFT or Fast Fourier Transform), but there’s a good chance you know little about it. The series allows you to break down complex signals (think audio waves) into combinations of simple sine or cosine equations which can be handled by a microcontroller.

We’ve had that base level of understanding for a long time. But when you start to dig deeper we find that it becomes a math exercise that isn’t all that intuitive. The video clip embedded after the break changes that. It starts off by showing a rotating vector. Mapping the tip of that vector horizontally will draw the waveform. The Fourier Series is then leveraged, adding spinning vectors for the harmonics to the tip of the last vector. The result of summing these harmonics produces the sine-based square wave approximation seen above.

That’s a mouthful, and we’re sure you’ll agree that the video demo is much easier to understand. But the three minute clip just scratches the surface. If you’re determined to master the Fourier Series give this mammoth Stanford lecture series on the topic a try.

[via Reddit]


  1. Gustavo says:

    Unbelievable how they could explain so good without modern graphic resources, really useful explanation

  2. Alex Rossie says:

    Love these educationals, this one was too short though!

    • Ren says:

      Agreed, too short, it would also been nice if they would have shown more about the Sum of the vectors. But it looks like a good starting point for the larger lecture (series).

  3. Tom the Brat says:

    Keep finding these!

  4. Lloyd Atkinson says:

    “The series allows you to break down complex signals (think audio waves) into combinations of simple sine or cosine equations which can be handled by a microcontroller.”

    That simple single sentence has succeeded in explaining what FFT is and why you use it where every other source I’ve read has failed. Thank you!

    • Dax says:

      The other neat thing about fourier transform is that it turns the signal you’re analyzing from an amplitude-time domain to frequency-time domain because the fourier terms represent different frequencies. That means you can do effects like pitch-shifting, vocoder effects, or passband filtering digitally.

    • krylenko says:

      I’m legitimately curious – how did that sentence explain why you’d want to use an FFT? I read it and thought, “that’s too bad, that totally leaves out the real point – that the sine and cosine equations allow you to process frequency bands separately” (as Dax points out).

      I felt like if I were new to the FFT, I wouldn’t have any better idea why I should care after reading that than before. Care to elaborate on how it helped you?

      • Lloyd Atkinson says:

        Well, finding out you can use FFT to process audio signals in microprocessors by using maths which I’m guessing is the basis of DSP, I’d say that helped me understand one of FFT’s uses.

  5. draeath says:

    Damn, that is a very “intuitive” way to visualize harmonics!

  6. Kevin N. Haw says:

    I found a similar (non animated) treatment a few years back. This one actually shows the math behind the Fourier Transform. A good read.

  7. Pablo says:

    The Fourier serie is for periodic waves. The FFT is an algorithm used to calculate the Fourier transformation for aperiodic signals. They aren’t the same..

    • Dax says:

      You cannot calculate a fourier transform for aperiodic signals. What you will find is that a sample of a signal is treated as if it was a period of a periodic signal and the fourier transform performed on that. You can do that for arbitrarily long samples, but restrictions on memory and processing speed apply.

      • Pablo says:

        You have erroneous concepts, you use fourier transformation to calculate aperiodic signals (continuous variable) on paper.
        Fourier Serie “” only periodical signals.
        Fourier Transformation “” both. I mean aperiodical like a single pulse (continuous variable).
        FFT is an algorithm to implement Discrete Fourier transform or DFT “” In that case you are right, using samples of the signal. But my point was that Fourier series and FFT aren’t the same.
        Sorry about my english :S

        • I completely agree with Pablo. The article should be changed in that portion, although at this point I might just be being anal. Also, it should state FT not FFT where the FFT is a algorithmic representation of the FT (Fourier Transform). Also, the FT can be periodic and can also be validated (proved) by the FS.

  8. DeKay says:

    Complex to Real also has some great tutorials on FFTs and many other topics related to digital communications.

  9. Ross Patterson says:

    This sounds like yet another Paul Frees narration. What series did it come from? If you like this, you might find The Mechanical Universe cool too.

  10. I love these videos!

  11. Hollis Mason says:

    that video is worth watching just for the graphics, even if you already understand fourier series.

  12. Al says:

    I love this shit. Brings me back to my engineering dayz.

  13. Will Morris says:

    Also, remember that a Fourier TRANSFORM is not quite the same as a Fourier SERIES.

    A fourier transform outputs a FUNCTION of frequency, a Fourier series outputs the TAYLOR POLYNOMIAL of that function.

  14. João says:

    FFT != Fourier Series

  15. Will Morris says:

    No, one is series of a discrete set and the other is a continous function of an arbirtrary function.

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