Context: I’m studying for my Electrical Engineering Fundamentals of Engineering Exam. We’re working with Laplace Transforms and Transfer Functions in the time and frequency domains. I’m interested to know what makes an LTI system an LTI system and to understand any practical applications or examples. Thank you in advance!
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In this context, a “system” is something that takes an input and turns it into an output. For example, the input might be “bumps in the road”, and the output might be [“jiggles in the cute springy pig dashboard ornament”](https://www.amazon.com.au/Dashboard-Ornaments-Bobblehead-Accessories-Decorations/dp/B09CM9HH1T). The “system” is then the car, especially the tyres, suspension etc.
If the system is “time invariant” it means that you don’t modify the system on the fly – you don’t adjust your speed, say, or the suspension, or put your hand on the pig to stop it jiggling: the *system itself* does not change with time. The bumps in the road, and the jiggles of the springy pig, are allowed to vary with time.
If the system is “linear”, it means, basically, that if you double the strength of the bumps, it just doubles the amount of jiggles in the pig. If you add one set of bumps to another, the jiggles in the pig will be the sum of the jiggles due to each set of bumps. (It will be described to you differently, perhaps, but that’s what’s *important and practical* about the fact that it’s linear – if S is the system and x is the input, then S(ax) = aS(x), and S(x1+x2) = S(x1)+S(x2))
The reason for studying linear systems is:
* The maths is easier and more powerful. For example, a lot of the Laplace Transform stuff you’re about to learn just wouldn’t work with non-linear (or time-varying) systems
* Many nonlinear systems can be approximated by linear ones, so you can use all the easy powerful maths to get an approximate answer.
Some examples:
* An antenna: the input is the radio waves, the output is the signal tot he speaker, the system is the antenna and the electronics. This might not be completely linear, but many important parts are.
* A pendulum: the input is the wind or other driving forces, the output is the swinging of the pendulum. This will be approximately linear unless it starts swinging wildly
* A rocket guidance system: the input is gravity and the wind buffeting the rocket (out of our control) and the force from the rocket engines (within our control), the output is the location and speed of the rocket (this will be linear, since it’s basic newtonian mechanics. The system converting “rocket engine control instructions” to “force from the rocket engine” might be highly nonlinear, but hopefully it’s well understood)
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