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. I want to clarify something a tiny bit misleading about this. In general complex exponentials are not orthogonal w. 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? 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. 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. 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. Basically the video ignores the idea that the X w integral may not exist. 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! 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. The various transforms are useful for many things besides control. Although control is an important one. I actually used this math at home! The flicker was due to oscillation, due to a pole too close to the right half plane. I was about to stick a bigger capacitor in the feedback compensation circuit, but a bit of maths told me that that would make the problem worse and probably blowing up my LED's - so instead I used a smaller resistor value, and yay - perfect flicker free light!
I noticed outdoor LED lamps flickering too, but only at dusk, and for what I thought was a feedback issue. These lamps are solar powered and turn on at night. The cool part is that at twilight, when the light turned on, the little bit of reflected light bounced back to the photovoltaics, adding just enough light to turn the light off, which brought the system full circle. Amazingly, this behavior only occured for about 1 minute each day. The normal solution to this is hysteresis.
In most cases, it doesn't cost any extra to add hysteresis. Enginerrrd on Nov 5, root parent next [—]. But remember previously we plotted time vs number of meteors. This graph is fixed. Forget about that graph. We want a new graph that tells us what is the frequency corresponding to time values.
We would like to plot time vs frequency. What you asked can be plotted in one graph and understood easily. You get corresponding frequency value. Don't ask where are the meteors in this graph, that was just for explaining what time values actually are. You can latter plot frequency vs number of meteors. Each point in the s-plane is to be evaluated in the Laplace Transform Eq, i. What I still dont understand is e. Sign up to join this community. The best answers are voted up and rise to the top.
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Viewed k times. My question is how does Laplace Transform give frequency view? Show 1 more comment. Active Oldest Votes. Add a comment. Rod Carvalho Rod Carvalho 2, 14 14 silver badges 13 13 bronze badges. In other words, the LT is useful not only for analysis, but also for synthesis.
However, you seem to suggest that the Laplace transform is more general than the Fourier transform. This is not the case. There are many important function for which only the Fourier transform exists, but not the Laplace transform.
Think of any periodic function, or impulse responses of ideal brick-wall filters such as ideal low-pass, band-pass etc. However, at this point, the amount of work required for Laplace transforms is starting to equal the amount of work we did in those sections. Laplace transforms comes into its own when the forcing function in the differential equation starts getting more complicated. It is these problems where the reasons for using Laplace transforms start to become clear.
We will also see that, for some of the more complicated nonhomogeneous differential equations from the last chapter, Laplace transforms are actually easier on those problems as well. The Definition — In this section we give the definition of the Laplace transform. We will also compute a couple Laplace transforms using the definition.
Laplace Transforms — In this section we introduce the way we usually compute Laplace transforms that avoids needing to use the definition. We discuss the table of Laplace transforms used in this material and work a variety of examples illustrating the use of the table of Laplace transforms.
Inverse Laplace Transforms — In this section we ask the opposite question from the previous section. In other words, given a Laplace transform, what function did we originally have? We again work a variety of examples illustrating how to use the table of Laplace transforms to do this as well as some of the manipulation of the given Laplace transform that is needed in order to use the table.
Step Functions — In this section we introduce the step or Heaviside function.
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