# maximum output of a LTI system within linear limits

In book , 'fundamental of signal and system'by M.J ROBERT it is written that

Since no practical system can ever produce an unbounded response, strictly speaking, all practical systems are stable. The ordinary operational meaning of BIBO instability is a system described approximately by linear equations that would develop an unbounded response to a bounded excitation if the system remained linear. Any practical system will become nonlinear when its response reaches some large magnitude and can never produce a truly unbounded response. So a nuclear weapon is a BIBO-unstable system in the ordinary sense but a BIBO-stable system in the strict sense. Its energy release is not unbounded even though it is extremely large compared to most other artificial systems on earth

For a given LTI system, what parameter of system along with input which determines the maximum magnitude of response after which it becomes non linear and how to calculate this maximum magnitude of response mathematically?

what parameter of system along with input which determines the maximum magnitude of response

That's highly dependent on the actual implementation of the system. For example, most electrical systems can't exceed the voltage of the power supply or amplifier rails. Sound cannot exceed 194 dBSPL because the smallest pressure you can generate is 0 Pascal: there is no less pressure than vaccuum.

how to calculate this maximum magnitude of response mathematically

By doing a detailed analysis of your actual system including all individual components and all physical mechanisms that are best "assumptions". These need to be compared to whatever your specific requirements for "linear enough" are.

There are no systems that are truly LTI. LTI is simply an approximation that's convenient to work with and in many practical cases "good enough".

For a given LTI system, …, after which it becomes non linear

never. If you're modeling a system to be LTI, it's LTI: Linear (and time-invariant).

You can of course model a system to be non-linear in general, but linear for a range of input. But then the answer to your question is "that's in your own model, and we can't tell you"!

• Any practical system will become nonlinear when its response reaches some large magnitude,then what does this statement suggest ? – user215805 May 30 at 14:27
• that real systems aren't LTI for the whole range of reality. – Marcus Müller May 30 at 14:40
• Think about it: say your real world system is your headphone amplifier. If you put in 1 mV, it gives out 100 mV. If you put in 20 mV, it gives out 2 V. Linear. Now, you put in 20000 V, and it gives out smoke, while electrocuting your head (instead of generating 2 MV). That's non-linear. Also, it totally doesn't matter for the application as amplifier: You will never do that, but always strive to operate it in a range where it's linear. And then, the LTI description works. – Marcus Müller May 30 at 14:51