Time-invariant problem

[arhzz]

Hey guys so im stuck with determining if the following system is time-invariant.The system looks as following

y(t) = Re{sin(t)x(t)}+ Im{jcos(t)x∗(t)}​

I did all of the steps with the sin(t-t0) and jcos(t-t0) and also the y2 = x1 (t-t0). But i cant seem to be able to finish the analysis. Could anyone help?

 

[dick_freebird]

By inspection, the y(t) is full of nontrivial func(t)
expressions so how can it be time invariant?

I think the equation probably has some typos but:

sin(t) is not time invariant
x(t) might or might not be.
jcos(t) is not time invariant
x (as a constant) is undefined
(t) is obviously not time invariant.

Only if x(t) and x both equal zero, could the larger
equation be time invariant.

 

[andre_luis]

Perhaps you should decompose x(t) and its complex conjugate x*(t) into their arbitrary Real and Imaginary parts a±jb and make the necessary algebraic manipulation to see what comes out.

 

[arhzz]

Hmmm I didn't think of that. I will try it and see if it helped, thanks for the help

 

[albbg]

I presume you have a complex output, that is (as said by andre_teprom)

x(t) = a(t) + jb(t)

substituting in you system definition we have:

y(t) = a(t)*sin(t) + a(t)*cos(t)

to check if it's time invariant we have to calculate first the output when the intput [that is x(t)] is time shifted by "to". Let's call it yo(t):

yo(t) = a(t+to)*sin(t) + a(t+to)*cos(t)

now we have to calculate the output when the system is time shifted by the same amount "to"

y(t+to) = a(t+to)*sin(t+to) + a(t+to)*cos(t+to)

since yo(t) <> y(t+to) the system in NOT time invariant

 

[arhzz]

Ohhhh i see i was susposed to consider the entire system within the formula of a+jb.Okay so now a and b should be x1 right? and in the part y= a(t)*sin(t) that indicates that the number a(t) is conjugated? Not multiplication. So now if i would to insert numbers for t and t0 i should be getting diffrent end results?

Thanks for your help u helped me a lot.

 

[andre_luis]

substituting in you system definition we have:

y(t) = a(t)*sin(t) + a(t)*cos(t)

Seems like there is only particular case at which the above function could be time invariant would be if a(t) could elliminate ( sin(t) + cos(t) ) :

a(t) = 1 / ( sin(t) + cos(t) )

Which is not valid at some values of t, since it has discontinuity, so perhaps we could assume that y(t) berhaves more like a time-variant function.

 

[arhzz]

Actually, and this is probably on me, im susposed to determine which one of these is it. It doenst say PROVE that its a time-variant or time-invariant function, but it just say check which one it is.