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In our engineering academics, there is one subject named "signals and systems". In this subject only LTI system is discussed.

I want to know what are the advantages of LTI systems over other combinations like system which is stable and causal or system which is Linear and stable ? Also,don't they have any disadvantages?

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  • $\begingroup$ @Marcus I don't understand what is the similarity between the above question and the other question. The above question is related to "in general" advantages and disadvantages of LTI systems over other systems rather than only concern of LTI system with Fourier analysis. $\endgroup$ – user3559780 Jun 5 '16 at 12:57
  • $\begingroup$ ... read the answers. $\endgroup$ – Marcus Müller Jun 5 '16 at 12:58
  • $\begingroup$ @Marcus but Why do you think the above question is only concerned with Fourier analysis ? $\endgroup$ – user3559780 Jun 5 '16 at 13:03
  • $\begingroup$ it's not, but neither are the answers given to the question I referred to. also, your answers here show a clear connection to the topic of Fourier transforms $\endgroup$ – Marcus Müller Jun 5 '16 at 13:07
  • $\begingroup$ @Marcus also answers given to the above question and the other one are totally different $\endgroup$ – user3559780 Jun 5 '16 at 13:09
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It's important to realize that in practice many types of systems are used, and only some of them can be regarded as (approximately) LTI. The (didactical) advantage of treating LTI systems in a basic signals and systems course is the elegance and relative simplicity of the underlying theory. Stability and causality are easily checked, and the input-output relation is conveniently described by convolution (in the time domain) or multiplication (in the frequency domain). The Fourier transform is a powerful tool for analyzing LTI systems.

Another advantage is the relative ease with which such systems can be designed. Just think of the vast number of filter design methods.

Even though LTI systems are often used in practice, you will find many situations where other systems are needed. A simple example of a linear but time-varying system is a modulator, which multiplies a signal with a given function. A decision circuit in a receiver is a non-linear system. An adaptive equalizer is a time-varying and non-linear system, since the adaptation of the coefficients depends on the input signal. It would be easy to continue this list of practical examples of non-LTI systems. You will probably learn about such systems in more advanced signal processing courses. However, a deep understanding of LTI system theory will help in grasping these more advanced concepts.

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One of the keys to system analysis in the notion of invariance.

A system can be seen as a box with inputs and outputs. To be able to understand outputs from inputs, or to predict outputs from inputs, you need to understand the rationality under the system.

Some crucial questions can be:

  • are there "invariant" signals that are not modified, or only moderately so, through the system?
  • can these special signals (sometimes called "root signals") be used to understand the behavior of arbitrary signals?

Linear and time-invariant systems use quite basic assumptions. Time-invariance is important in many cases: you don't want a sensor whose outputs, in the same conditions, change depending on the time of the day. Linearity or additivity is not respected everywhere, but many equations in physics are linear, or can be approximated, locally, by linear ones.

As said by @Matt L., only these two properties unfold a both "theoretically sound" and "practically efficient" framework. Root signals appear to be complex exponentials, they form bases of the space of signals. They can be used to interpret periodicity, the convolution inherent to the system can be turned into a simpler product in the dual frequency domain. Weighted averages become a central tool for defining filters.

And many more: you get a framework to discretized continuous signals and still deal with them easily (discrete Fourier transform), even faster implementations (FFT) and a wealth of related tools to deal with noise, nonstationarity, etc.

Disadvantages are LTI does not handle so well discontinuity and non-linearity (eg amplitude quantization, saturation). And since they are heavily related to least-squares and Gaussian noise, LTI is not the most robust to faults and outliers.

So far, I have no knowledge of many other systems so rich and so "simple" as the same time. The notion of root signals and structures has been well developed for median-like systems for instance, but might be not used a lot, except for median-filter design.

In image processing, linearity is often not verified. Mathematical morphology is another example of a powerful framework, not based on linear algebra, but on lattices. Apparently, they are somehow related through a Cramér transform. But this is a whole different story.

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Before vast amounts of cheap computing power was readily available to practitioners, the best and easiest to use tools available for analyzing and designing many kinds of systems were the mathematics for domains that were or could be well approximated by LTI systems. Often, simple closed form equations (that could fit on a professor's chalkboard) regarding LTI systems were excellent approximations to easily realizable (circuit, etc.) designs.

It was a form of searching for ones lost keys under the lamppost. Elsewhere, it was too dark to search without tripping.

The disadvantage of LTI sytems is that in the real world, almost nothing is absolutely precisely linear, and any real systems will most likely change their behavior during ones lifetime (rust, decay, etc.). And supercomputers as powerful as the ones used for weather and climate modeling a few years back now fit in ones pocket, allowing the behavior of some non-linear systems to be more readily estimated.

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