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A little ELI5 for those who haven't had Laplace transforms at school, from someone who only had a Laplace 101 course, so for what's it worth: Laplace transforms

A little ELI5 for those who haven't had Laplace transforms at school, from someone who only had a Laplace 101 course, so for what's it worth: Laplace transforms allow you to convert differential equations into easier equations, and back: the differentials and integrals become multiplications and divisions. So you can take a differential equation, transform it into the Laplace domain, manipulate it, and convert it back. And that's cool because differential equations tend to appear everywhere, for instance to model springs, electrical circuits with caps and coils, the surface of a soap bubble in a metal rod, etc. A sibling is the z-transform, which is like the digital version. This one is used for instance to design digital audio filters. I'm sure some math wizards here can elaborate and correct me.

cm2187 6y ago
And don’t forget those like me for whom school is a distant memory! What are the domains where this new method can be applied? Is it mostly physics simulations and the likes?
- timclark 6y ago
  I got taught them in a course on linear systems which was a pre-requisite course to control theory. Lots of electrical circuits, mechanical systems and electro-mechanical systems can be modelled using laplace transforms if they are linear systems. I did an electrical and electronic engineering degree and we got to skip the tedious differential equation solving lectures that the mechanical, civil and chemical engineers had to attend because of Monsieur Laplace.
- srean 6y ago
  Control systems, signal filters (noise attenuation), modeling epidemics, modeling queues, modeling reliability of repairable systems, modeling recurrent events(such as failures), renewal processes, modeling inventory plans, probability in general (because of the connection with moment generating function) ...
GolDDranks 6y ago
I don't have any real understanding over the Laplace transform, but I understand Fourier transform well enough that it makes sense to me. Back then, I saw an claim that Laplace transform is a generalization of Fourer transform in the sense that it transforms a function not only to a space of frequencies and phases of sine waves, but to a larger space of parameters of exponentials. Note that the parameter space of the sine waves is subset of the (complex) parameter space of exponentials. Is this claim correct?
- beagle3 6y ago
  Yes. The Fourier transform characterizes on a circle (usually the unit circle in the complex domain) at different frequencies. It is properly defined for periodic signals. The Laplace transform takes any exponential spiral in the complex plane, and reduces to the Fourier transform if you only care about the unit circle. I appreciate that doesn’t make things clearer unless you already have some understanding of integral transforms in the complex plane (in which case, you probably know this already). However, I have never met a simple intuitive explanation of the Laplace transform, - and actually no meaningful explanation that doesn’t involve integrals.
  - durbat 6y ago
    With Fourier you can analyze oscillatory characteristics of function (frequency and phase). With Laplace you can also analyze amplification/attenuation.
    - tilt_error 6y ago
      The Fourier transform, well, transforms a time-based phenomenon such as an alternating current sine wave into a frequency spectrum where you can observe the frequency spectrum components of the signal. A pure nice sine becomes a spike (delta function) located at a specific frequency. Music, as we observe it through our ears and can view it on an oscilloscope becomes moving spikes (lots of them :) in the frequency spectrum where the "amplitude" at a given frequency relates to the "amount" of that frequency in the music. The behaviour of a filter is much easier to describe in the frequency spectral domain than it would be in the time domain. Now to the direct current (DC) view. This cannot be handled by the Fourier transform -- at least the DC-part of the signal cannot be transformed to the frequency domain. As shown in the article, there were "steps", "ramps" and such. A typical scenario would be to describe what happens in your amplifier during startup, to describe how electrical circuits are behaving during startup before reaching the "running" state. The Laplace transform will handle these types of scenarios, and can thus be used to study (or describe) systems during other types of transitions than the "steady state" when you are up and running. Regarding filters, the Fourier transform describes things going on at the unit circle, while the Laplace transform can be used to study both the interior and exterior of the plane. In this sense, creating filters relates to locate "poles" and "zeros" in the plane (amplification and attenuation) which can be observed on the unit circle as the behaviour on periodic signals.
taneq 6y ago
Sounds right to my (very very rusty) recollection. Laplace transforms are a magic trick that let you easily solve some kinds of differential equations.
- phkahler 6y ago
  >> Laplace transforms are a magic trick that let you easily solve some kinds of differential equations. To mathematicians I don't think they're so much magic. When I took differential equations class it was frustrating that they went too fast for me to fully digest what was "really" going on. It didn't feel out of reach, but something I needed to look at a couple different ways but didn't have time (or the internet) to do so. Think I'm gonna checkout 3blue1brown after this - he can probably close that gap for me.
  - scythmic_waves 6y ago
    > Think I'm gonna checkout 3blue1brown after this - he can probably close that gap for me. You might like this lecture from MIT's OCW: [1]. It's my favorite source for motivating the Laplace transform. It's a bit difficult to make this concept "simple", and this resource assumes that you already have some familiarity with the following concepts: infinite series, power series, radius of convergence, and (indefinite) integration. The tl;dw is that the Laplace transform is a generalization of a power series. [1] https://www.youtube.com/watch?v=sZ2qulI6GEk https://www.youtube.com/watch?v=sZ2qulI6GEk Edit: I also wrote up a form of this video elsewhere if anyone's interested. It's kinda long though, and I didn't want to spam this thread with it.
kpmah 6y ago
I've had some problems understanding the Laplace transform. Maybe somebody here can point me towards some material. I have an interest understanding how IIR filters are designed, and I always get stuck at this part in DSP books. The Laplace transform is used, but as well as finding the mathematics difficut I don't really understand why it is being used at all. I think it is trying to replicate the effect of an analog circuit?
- Gibbon1 6y ago
  You can describe a circuit by it's time domain behavior. Or you can describe the circuit by it's frequency domain behavior. Both are valid and congruent. The thing is a lot of questions are easy to answer in the frequency domain. For instance, you want to know if a circuit with feedback will oscillate. Hard to answer using time domain equations. But in the frequency domain there is a simple constraint. If for all frequencies where the the gain is greater than one the phase shift is less than 180 degrees, circuit won't oscillate. This is obviously rather useful. Also a point with a lot of 'books' the authors get caught up in describing how something is done that they never explain why something is done. I've found often the answer is simple yet opaque and frustratingly never talked about.
  - pantulis 6y ago
    This. I remember having adequate cursory knowledge of Fourier Transform to the point of understanding the value of FFT algorithms, but the Laplace Transform was explained like hell so I failed my robotics classes.
    - TheOtherHobbes 6y ago
      If you have an electronic circuit, you can model each element with a differential equation. E.g. voltage across a capacitor is modelled as the integral of current, voltage across an inductor is dI/dt. This is a useful fact for a simple circuit in a classroom, but the differential equations for any circuit with more than a few components soon become insanely complex. With the Laplace transform you (more or less) replace an integral with 1/s and a differential with s, plus some constants derived from the component values. Then you can simplify for s, and use the Inverse Laplace Transform to convert the final expression in s into an expression in t. You have now solved an insanely complex differential equation with some basic algebra, and your final expression in t - with component constants, and some exponentials that appear after the inverse transform - accurately models how the circuit responds over time. There's also a related fairly simple trick for converting the s-domain representation into a frequency/phase plot which tells you how the circuit operates in the frequency domain. And another related fairly simple trick for converting the continuous s-domain into the z-domain for DSP calculations over a sampled time series. Because the same theory also applies in other domains - spring/mass systems, and so on - you can use the same technique there too.
barbecue_sauce 6y ago
>from someone who only had a Laplace 101 course Laplace transforms are an entire course?
- jusssi 6y ago
  From my time at the uni, I wish we'd had a proper course that covered Laplace (and the important special cases, e.g. Fourier and z) transforms properly. Instead, the coverage was interspersed to general math courses and to the courses that needed to apply them.
- Koshkin 6y ago
  Appears as an exercise on page 505 of this (excellent and freely downloadable, at least for now) book: https://link.springer.com/book/10.1007%2F978-3-319-01195-0 https://link.springer.com/book/10.1007%2F978-3-319-01195-0.
- steve76 6y ago
  Laplace transforms are an entire industry. They are much more than just a simple trick. To hell with the passivity of myself in nature. Time to grow up. Everything really is PID control. I will have zero surprise if gravitational applications, popping a particle into existence and using that event to get all the data in the universe, use Laplace. Good measurement of - relativity to jolt - momentum to acceleration - capacitance of viscosity - resistance from elasticity is all you just really need for total sovereignty of everything. Quantum theory would give you the odds of not messing up which would be something like 99.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999% Part of my education was done with a nod and wink by people with some pretty high up security clearances, as in, we have all this here on the shelf. Foreign treaties prevent us from announcing it. Please for the love of god wake up and figure it all out yourself it's really not that difficult to do.
contravariant 6y ago
Something also worth mentioning is that it isn't just useful when dealing with differentiation / anti-differentiation but also when dealing with convolution.
steve76 6y ago
I think of Laplace as being the non-passive approach: https://en.wikipedia.org/wiki/Coherent_control#Controllability https://en.wikipedia.org/wiki/Coherent_control#Controllabili...

cm2187 6y ago

And don’t forget those like me for whom school is a distant memory! What are the domains where this new method can be applied? Is it mostly physics simulations and the likes?

timclark 6y ago
I got taught them in a course on linear systems which was a pre-requisite course to control theory. Lots of electrical circuits, mechanical systems and electro-mechanical systems can be modelled using laplace transforms if they are linear systems. I did an electrical and electronic engineering degree and we got to skip the tedious differential equation solving lectures that the mechanical, civil and chemical engineers had to attend because of Monsieur Laplace.
srean 6y ago
Control systems, signal filters (noise attenuation), modeling epidemics, modeling queues, modeling reliability of repairable systems, modeling recurrent events(such as failures), renewal processes, modeling inventory plans, probability in general (because of the connection with moment generating function) ...

GolDDranks 6y ago

I don't have any real understanding over the Laplace transform, but I understand Fourier transform well enough that it makes sense to me. Back then, I saw an claim that Laplace transform is a generalization of Fourer transform in the sense that it transforms a function not only to a space of frequencies and phases of sine waves, but to a larger space of parameters of exponentials. Note that the parameter space of the sine waves is subset of the (complex) parameter space of exponentials. Is this claim correct?

beagle3 6y ago
Yes. The Fourier transform characterizes on a circle (usually the unit circle in the complex domain) at different frequencies. It is properly defined for periodic signals. The Laplace transform takes any exponential spiral in the complex plane, and reduces to the Fourier transform if you only care about the unit circle. I appreciate that doesn’t make things clearer unless you already have some understanding of integral transforms in the complex plane (in which case, you probably know this already). However, I have never met a simple intuitive explanation of the Laplace transform, - and actually no meaningful explanation that doesn’t involve integrals.
- durbat 6y ago
  With Fourier you can analyze oscillatory characteristics of function (frequency and phase). With Laplace you can also analyze amplification/attenuation.
  - tilt_error 6y ago
    The Fourier transform, well, transforms a time-based phenomenon such as an alternating current sine wave into a frequency spectrum where you can observe the frequency spectrum components of the signal. A pure nice sine becomes a spike (delta function) located at a specific frequency. Music, as we observe it through our ears and can view it on an oscilloscope becomes moving spikes (lots of them :) in the frequency spectrum where the "amplitude" at a given frequency relates to the "amount" of that frequency in the music. The behaviour of a filter is much easier to describe in the frequency spectral domain than it would be in the time domain. Now to the direct current (DC) view. This cannot be handled by the Fourier transform -- at least the DC-part of the signal cannot be transformed to the frequency domain. As shown in the article, there were "steps", "ramps" and such. A typical scenario would be to describe what happens in your amplifier during startup, to describe how electrical circuits are behaving during startup before reaching the "running" state. The Laplace transform will handle these types of scenarios, and can thus be used to study (or describe) systems during other types of transitions than the "steady state" when you are up and running. Regarding filters, the Fourier transform describes things going on at the unit circle, while the Laplace transform can be used to study both the interior and exterior of the plane. In this sense, creating filters relates to locate "poles" and "zeros" in the plane (amplification and attenuation) which can be observed on the unit circle as the behaviour on periodic signals.

taneq 6y ago

Sounds right to my (very very rusty) recollection. Laplace transforms are a magic trick that let you easily solve some kinds of differential equations.

phkahler 6y ago
>> Laplace transforms are a magic trick that let you easily solve some kinds of differential equations. To mathematicians I don't think they're so much magic. When I took differential equations class it was frustrating that they went too fast for me to fully digest what was "really" going on. It didn't feel out of reach, but something I needed to look at a couple different ways but didn't have time (or the internet) to do so. Think I'm gonna checkout 3blue1brown after this - he can probably close that gap for me.
- scythmic_waves 6y ago
  > Think I'm gonna checkout 3blue1brown after this - he can probably close that gap for me. You might like this lecture from MIT's OCW: [1]. It's my favorite source for motivating the Laplace transform. It's a bit difficult to make this concept "simple", and this resource assumes that you already have some familiarity with the following concepts: infinite series, power series, radius of convergence, and (indefinite) integration. The tl;dw is that the Laplace transform is a generalization of a power series. [1] https://www.youtube.com/watch?v=sZ2qulI6GEk https://www.youtube.com/watch?v=sZ2qulI6GEk Edit: I also wrote up a form of this video elsewhere if anyone's interested. It's kinda long though, and I didn't want to spam this thread with it.

kpmah 6y ago

I've had some problems understanding the Laplace transform. Maybe somebody here can point me towards some material. I have an interest understanding how IIR filters are designed, and I always get stuck at this part in DSP books. The Laplace transform is used, but as well as finding the mathematics difficut I don't really understand why it is being used at all. I think it is trying to replicate the effect of an analog circuit?

Gibbon1 6y ago
You can describe a circuit by it's time domain behavior. Or you can describe the circuit by it's frequency domain behavior. Both are valid and congruent. The thing is a lot of questions are easy to answer in the frequency domain. For instance, you want to know if a circuit with feedback will oscillate. Hard to answer using time domain equations. But in the frequency domain there is a simple constraint. If for all frequencies where the the gain is greater than one the phase shift is less than 180 degrees, circuit won't oscillate. This is obviously rather useful. Also a point with a lot of 'books' the authors get caught up in describing how something is done that they never explain why something is done. I've found often the answer is simple yet opaque and frustratingly never talked about.
- pantulis 6y ago
  This. I remember having adequate cursory knowledge of Fourier Transform to the point of understanding the value of FFT algorithms, but the Laplace Transform was explained like hell so I failed my robotics classes.
  - TheOtherHobbes 6y ago
    If you have an electronic circuit, you can model each element with a differential equation. E.g. voltage across a capacitor is modelled as the integral of current, voltage across an inductor is dI/dt. This is a useful fact for a simple circuit in a classroom, but the differential equations for any circuit with more than a few components soon become insanely complex. With the Laplace transform you (more or less) replace an integral with 1/s and a differential with s, plus some constants derived from the component values. Then you can simplify for s, and use the Inverse Laplace Transform to convert the final expression in s into an expression in t. You have now solved an insanely complex differential equation with some basic algebra, and your final expression in t - with component constants, and some exponentials that appear after the inverse transform - accurately models how the circuit responds over time. There's also a related fairly simple trick for converting the s-domain representation into a frequency/phase plot which tells you how the circuit operates in the frequency domain. And another related fairly simple trick for converting the continuous s-domain into the z-domain for DSP calculations over a sampled time series. Because the same theory also applies in other domains - spring/mass systems, and so on - you can use the same technique there too.

barbecue_sauce 6y ago

>from someone who only had a Laplace 101 course Laplace transforms are an entire course?

jusssi 6y ago
From my time at the uni, I wish we'd had a proper course that covered Laplace (and the important special cases, e.g. Fourier and z) transforms properly. Instead, the coverage was interspersed to general math courses and to the courses that needed to apply them.
Koshkin 6y ago
Appears as an exercise on page 505 of this (excellent and freely downloadable, at least for now) book: https://link.springer.com/book/10.1007%2F978-3-319-01195-0 https://link.springer.com/book/10.1007%2F978-3-319-01195-0.
steve76 6y ago
Laplace transforms are an entire industry. They are much more than just a simple trick. To hell with the passivity of myself in nature. Time to grow up. Everything really is PID control. I will have zero surprise if gravitational applications, popping a particle into existence and using that event to get all the data in the universe, use Laplace. Good measurement of - relativity to jolt - momentum to acceleration - capacitance of viscosity - resistance from elasticity is all you just really need for total sovereignty of everything. Quantum theory would give you the odds of not messing up which would be something like 99.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999% Part of my education was done with a nod and wink by people with some pretty high up security clearances, as in, we have all this here on the shelf. Foreign treaties prevent us from announcing it. Please for the love of god wake up and figure it all out yourself it's really not that difficult to do.

contravariant 6y ago

Something also worth mentioning is that it isn't just useful when dealing with differentiation / anti-differentiation but also when dealing with convolution.

steve76 6y ago

I think of Laplace as being the non-passive approach: https://en.wikipedia.org/wiki/Coherent_control#Controllability https://en.wikipedia.org/wiki/Coherent_control#Controllabili...