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Comment by pgt | Hacker News Reader

CONTROL THEORY NERD ALERT! This is hands-down the best video on control theory I've ever seen, clearly depicting the relationship of the Laplace Transform to the Fourier Transform.

The Fourier Transform decomposes a signal into its sinusoidal frequency components. It is used in a bunch of everyday appliances like your guitar tuner, JPEG images (wavelet compression) and speech recognition on your smartphone or smart speaker.

The Laplace Transform, on the other hand, decomposes a signal into both its exponential factors (decaying or rising) AND its sinusoidal components. So the FT is just one slice of the Laplace Transform where the input signal has no exponential rise or decay.

In the electrical and mechanical domains, spring mass damper systems are super common, even your car's suspension! And to analyse and control them, engineers apply the Laplace Transform.

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> The Laplace Transform ... decomposes a signal into both its exponential factors .. AND its sinusoidal components

This is the OMG moment from the video. I wish my teachers would have put it this way a few years back. It now feels like an obvious secret hidden in place sight but Laplace was a very abstract concept before

> The Laplace Transform, on the other hand, decomposes a signal into both its exponential factors (decaying or rising) AND its sinusoidal components.

I want to clarify something a tiny bit misleading about this. In general complex exponentials are not orthogonal w.r.t. the relevant inner product. You can't really think of them as independent components that compose a function.

If you think of a function as the impulse response of a linear time invariant system, then the laplace transform of that function tells you the result of an experiment where you drive the system with an exponentially damped sinusoid. This is why the poles of the transformed impulse response tell you about the stability of the system: those are the inputs that cause the system to explode!

Aren't complex exponentials signal rotations? If you multiply a sinusoidal signal by e^(-iwt), you are essentially shifting its phase modulus, no?

Those are purely imaginary exponentials. General complex exponentials also have a real part.

And that's how the FT differs from the LT. LT is complex while FT is purely imaginary, as the video explains.

nitpick for people that like pictures: JPEG uses a 2D discrete cosine transformation (DCT) instead of a fourier transformation. Basically the same principle with a different set of base functions, but suitable for compression, because you generally net fewer real coefficients.

I think DCT is a good entry to understand these kind of transformations. The set of base functions for DCTs is displayed in the image on this site:

https://edoras.sdsu.edu/doc/matlab/toolbox/images/transfo6.h...

I think imaging makes understanding the motivation of frequency dissection plausible and you can imagine any image being a sum of these base images with a weight(R) and a phase/offset(IM). From that understanding LaPlace and FT is far easier. You can even kick one dimension in most applications like audio.

Yes he pretty much ignored the phase output of the FT, which is often very important. Like you said, a minor omission in the name of keeping the video short I expect.

I never thought of it that way. For me Laplace is all about region of convergence and representing things like the Dirac Delta function in a way that’s easy to manipulate.

Basically the video ignores the idea that the X(w) integral may not exist. It doesn’t for many x(t). I don’t want to type a book, but basically the X(s) integral exists (where it does is called the region of convergence) for many more x(t), so there are classes of functions which can now be analyzed where with just Fourier, they could not be.

The other thing is delta functions. With Fourier they are a pain but with Laplace they are a breeze. This is especially important if you want to mathematically model how the control system will respond to a jolt or impulse. I always figured that is why control engineers use Laplace.

What are your thoughts on time-frequency duality, in that time as a variable is able to be abstracted away for our analytical purposes (which works!)? And is the frequency domain a more qualified s-plane?

Consider a point on the edge of a rotating circle which is rotating counter-clockwise at a constant frequency. If you plot the point over time in three-dimensional space (starting from 0rad), you get a spiral or helix (like one half of a double-helix DNA strand).

If you view the spiral from the side, you see a sine wave.

If you view the spiral from the top, you see a cosine wave.

If you look down the barrel of the signal, you will see an infinitely bright circle with a radius equal to the amplitude of the signal. The brightness of the circle relates to the energy in the signal, since by looking down the barrel, we have collapsed time and relinquished all phase information.

Pure sinusoids are infinite, so real-world signals will be windowed (note: windowing always introduces artifacts).

Now, what happens if the signal is a composition of rotating circles all spinning at different frequencies and directions? Well, you'll see a bit of a blurry distribution of where the wavefront spends its time. If you think about this a bit, you can ask yourself what you have to do to "see" the frequency components of the signal :).

I have to go now, but will try to post more later.

I appreciate the insight!

>the brightness of the circle relates to the energy in the signal

What does this “energy” mechanically correspond to in the complex unit circle and how do these mechanics appear in the time-domain?

The ideal low pass filter’s mathematical application - of the frequency domain effects on natural signals - astonishes me and I can’t quite fathom the transformation; ie is the frequency space physical or a purely mathematical abstraction?

Ultimately: is the complex exponential function a natural algorithm?

Do EEs still study this stuff? Lots of people use plugin PID loops for control these days but if you know control theory you just derive them from scratch and know how they're going to behave rather than tweaking endlessly.

Definitely. The various transforms are useful for many things besides control. Although control is an important one.

made an account to ask about this

>The Laplace Transform, on the other hand, decomposes a signal into both its exponential factors

in what sense is this true? yes the s in e^(-st) is complex but that factors the kernel of the transform into e^(-at)e^(-iwt). individually the factors do what you say (decompose over a basis) but as a product they do not do that.

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