Proving a system is LTI based on input and output

In summary, the conversation is about whether a given system could be LTI. The system is x(t)=-5cos(2t) --> y(t)=exp(-2tj) and the question is whether it could be LTI. The chapter is about eigenfunctions of LTI systems, which are of the form exp(st). The suggested solution is to find a transfer function H(s) so that H(s)*x(s)=y(s), but since y and x are still functions of t, it is unclear what to do next. The given answer is 'yes (H(s=2j)=0)' and it seems that the next step would be to manipulate the transfer function in some way. The final question is whether
  • #1
robijnix
4
0
edit: you aren't proving it's LTI, you proving it COULD be lti

Homework Statement


the question:
could the following system be LTI?
x(t)=-5cos(2t) --> y(t)=exp(-2tj)

Homework Equations


the chapter is about eigenfunctions of LTI systems, which are of the form exp(st).

The Attempt at a Solution


So my guess for what i had to do, was find a transfer function H(s), so that H(s)*x(s)=y(s).
so i wrote x(t) as follows:
x(t)=-5/2(exp(2*t*j)+exp(-2*t*j)
so
x(2)=-5/2(exp(st)+exp(-st)) with s=2j.

so than i thought, H(s)=y(s)/x(s), but y and x are still functions of t, so i don't know what to do now...

and btw laplace isn't explained until the next chapter so I'm not supposed to use that.

the given answer is 'yes (H(s=2j)=0)'.
so i think i do indeed need to do something with the transfer function.
 
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  • #2
If the output is of the same form as the input but scaled by a constant is that an LTI system? Is the output the same form as the input? (coefficient of sin could be 0)
 

FAQ: Proving a system is LTI based on input and output

1. How do you determine if a system is LTI based on input and output?

To determine if a system is LTI (linear time-invariant) based on input and output, you need to perform two tests: linearity and time-invariance. The linearity test involves checking if the system follows the principle of superposition, which means that the output of the system for a sum of two inputs is equal to the sum of the individual outputs for each input. The time-invariance test involves checking if the output of the system is independent of when the input is applied.

2. What is the importance of proving that a system is LTI based on input and output?

Proving that a system is LTI based on input and output is essential because it helps us understand the behavior of the system. LTI systems have a predictable and consistent response to different inputs, which makes them easier to analyze and control. Additionally, many real-world systems can be approximated as LTI systems, making this analysis technique widely applicable.

3. Can a system be LTI based on input and output if it is not linear or time-invariant?

No, a system cannot be LTI based on input and output if it is not linear or time-invariant. LTI systems must satisfy both linearity and time-invariance properties to be classified as such. If a system fails either of these tests, it cannot be considered LTI.

4. How can you test for linearity in a system based on input and output?

To test for linearity in a system based on input and output, you can perform a superposition test. This involves applying two different inputs to the system and measuring the corresponding outputs. Then, you can add the two inputs and observe if the output is equal to the sum of the individual outputs. If it is, the system is linear.

5. What is an example of a system that is not LTI based on input and output?

An example of a system that is not LTI based on input and output is a system with a memory element, such as a capacitor or an inductor. These systems are not time-invariant because their output depends on the past inputs or the current state of the system. Therefore, they cannot be classified as LTI based on input and output.

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