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I have a F system and a input of this system is e x(n) = e^iπn/4 and output is y(n)=cos(πn/4)

So is the F system is LTI or it would be ?

I try to using addivity and Homogeneity but I cant do it

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You don't have enough information to know if the system is LTI or not.

To know a system is linear, you have to know that for any input \$x(n)\$ producing an output \$y(n)\$, then a scaled input \$A\cdot{}x(n)\$ produces the output \$A\cdot{}y(n)\$, no matter what input was chosen for \$x(n)\$. Since you only know the output for one particular input, you don't know if your system is linear or not.

However, the point is largely moot. We normally don't try to prove a system is LTI or not. We just assume a system is (approximately) LTI in order to model its behavior. Real systems are usually nonlinear (because a large enough input will produce saturation or damage the system or cause numerical overflow) and time-variant (because they are turned on and off from time to time). So LTI-ness is just an approximation to the real behavior, or a given of a pedagogical problem, to allow modeling the system.

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  • \$\begingroup\$ Thanks, one more question LTI systems doesnt produce a new frequency right ? and this example input frequency is pi/4 but there is a two output frequency in there(because (cosw=e^iw+e^-iw)/2 and for this reason this system is not LTI What you think about this ı am very confused about this things :D thanks again \$\endgroup\$ Dec 7, 2014 at 19:22

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