Non relaxed systems linear?

Hello all,

Can non relaxed systems be linear? Why is the condition of linearity(superposition) defined only for relaxed systems?

System like y(n) = Ax(n) + B is linear in its equation but does not satisfy the linearity test of superposition. What can be said about such systems?

Thanks
Pallavi Start a New Thread

As far as I can recall the definition, an F(x) is linear if

F(ax + by) = aF(x) + bF(y)

Your equation below seem to satisfy to these conditions, hence
is linear.

--
Andrew

> Subject: Non relaxed systems linear?
> Posted by: "g...@yahoo.com" g...@yahoo.com gpallavi2
> Date: Tue Apr 7, 2009 5:01 am ((PDT))
>
> Hello all,
>
> Can non relaxed systems be linear? Why is the condition of
> linearity(superposition) defined only for relaxed systems?
>
> System like y(n) = Ax(n) + B is linear in its equation but does not satisfy
> the linearity test of superposition. What can be said about such systems?
>
> Thanks
> Pallavi
> Start a New Thread

Hi Andrew,
The equation in my question does not satisfy the linearity test.
I am posting one of the responses I received stating that all non-relaxed systems are non-linear, which answers my original question.

Regards
Pallavi
________________________________

----- Forwarded Message ----
From: karthik
To: pallavi
Sent: Tuesday, April 7, 2009 9:44:51 AM
Subject: Re: [audiodsp] Non relaxed systems linear?
Hi Pallavi,

Strictly speaking it is not a Linear system. Its called Incrementally Linear System.

To prove that it is not linear:

consider the input x(t) = x1(t) + x2(t)
---> o/p y(t) = Ax(t) + B = A( x1(t) + x2(t) ) + B

If we apply the inputs seperately,
y1(t) = A*x1(t) + B
y2(t) = A*x2(t) + B

Sine y(t) not equal to y1(t) + y2(t) system is not Linear.

It is called incrementally linear because:

for an increment of input signal, output also increases in a linear fashion
that is, for input x(t) + del, o/p yy(t) = A*( x(t) + del ) + B = y(t) + del
where y(t) is the o/p for input x(t) alone.

Hope I am clear.

Regards,
R. Karthik
________________________________

From: Pallavi
To: karthik
Sent: Tuesday, April 7, 2009 10:43:48 AM
Subject: Re: [audiodsp] Non relaxed systems linear?

Thanks Karthik. I understood the part of it being incrementally linear.
So can it be safely established that all non relaxed systems are non linear?
This is the question that bothers me.

Regards
Pallavi

----- Forwarded Message ----
From: karthik
To: Pallavi
Sent: Wednesday, April 8, 2009 3:56:23 AM
Subject: Re: [audiodsp] Non relaxed systems linear?
Hi Pallavi,

Non - relaxed system are definitely non - linear
for the very reason that initial conditions are non - zero. That is,
the term 'B' contributes to the non zero initial condition.

Regards,
R. Karthik Start a New Thread