Hello all,
I have been studying about LTI system and Eigen functions. Here is a doubt I like to clarify.
Say I have a LTI system defined by the system equation [y''(t) + 5y'(t) + 6y(t) = x(t) ]. It is evident that for the above system the transient portion will be of the form Aexp(-2t) + Bexp(-3t) [Same as Eigen functions of this system].
My understanding is as follows: that the transients are independent of x(t). Hence no matter what signal I send into the signal via x(t), the transient will persist. Also, by definition of Eigen functions, if I send an Eigen function into the system, I will get the same as output with different gain.
Say, I decide to send the signal exp(-2t) as input via x(t). Then the output has to be, by definition of Eigen functions, C *exp(-2t). I would like to know what has happened to the transient exp(-3t). Has it disappeared? If so how?
My guess is that it has something to do with initial conditions. If so, I don't see how. Kindly let me know.
I have been studying about LTI system and Eigen functions. Here is a doubt I like to clarify.
Say I have a LTI system defined by the system equation [y''(t) + 5y'(t) + 6y(t) = x(t) ]. It is evident that for the above system the transient portion will be of the form Aexp(-2t) + Bexp(-3t) [Same as Eigen functions of this system].
My understanding is as follows: that the transients are independent of x(t). Hence no matter what signal I send into the signal via x(t), the transient will persist. Also, by definition of Eigen functions, if I send an Eigen function into the system, I will get the same as output with different gain.
Say, I decide to send the signal exp(-2t) as input via x(t). Then the output has to be, by definition of Eigen functions, C *exp(-2t). I would like to know what has happened to the transient exp(-3t). Has it disappeared? If so how?
My guess is that it has something to do with initial conditions. If so, I don't see how. Kindly let me know.