How Do Stability and Linearity Determine System Behavior?

[tina_singh]

confused!..system stability and linearity

1)-Is y(n)=cos{x(n)} a stable system??
and is the condition s=Ʃ|h(k)|<∞ for stability valid only for LTI systems?
actually my book solves the given problem using the above method..but according to me the given system is not LTI SINCE ZERO I/P does not lead to zero O/P...so i m really confused


2)to prove system to be linear is it enough to pove that zero i/p leads to zero o/p??
the system y(t)=[{x(t)}^2] also gives zero o/p on zero i/p but it is not linear...

 

[rude man]

No point in subjecting your function cos[x(n)] to a Nyquist test (unit circle stability test) since it's obviously nonlinear.

You cannot apply the usual expression for the output at discrete multiples of time T. On the other hand, cos[x(nT)] is clearly stable since it can never go beyond +/-1.

Finally. I must confess with great chagrin that I don't know what the sufficiency test for linearity is with a z transfer function. But clearly you are right in assuming y = cos[x(n)] is nonlinear. I would assume that if the function is a plynomial fraction in z that it is linear. But that is just a not-so-educated guess.

 

[reddvoid]

if the system is linear
out put for "sum of two different signals" as input should be same as " sum of outputs got when two signals are given separately as input

 

[rude man]

if the system is linear
out put for "sum of two different signals" as input should be same as " sum of outputs got when two signals are given separately as input

That's exactly right and is the right answer. So it can be applied equally well to discrete systems, obviously. Thanks reddvoid! Funny how sometimes one doesn't see the woods for the trees!

 

[truva]

if the system is linear
out put for "sum of two different signals" as input should be same as " sum of outputs got when two signals are given separately as input

Actually if the signal satisfy above property the system is "additive". Which is f(x1+x2)=f(x1)+f(x2).

Additionally the following property, f(λx)=λf(x) is the "homogenity".

If a system is both "additive" and "homogen" it is said that the system is "linear".