Is the system whose output y(n) is cos(x(n)) stable?

I have read that a system is stable if its impulse response approaches
zero as time approaches infinity. Also another definition says that a
system is stable if for every bounded input, we get a bounded output.
What I understand for a bounded input is that the Input should have its
value between -M and +N where M and N are some real numbers. Similarly
is the definition for bounded output.
Which one of the above two definitions for system stability hold good?
If we go by the first definition, cos(x(n)) has an impulse response=
cos(del(n)) which is zero for n=0 and one for n>0, which means impulse
response doesn't approach zero as time approaches infinity.So, the
system is unstable.
If we go by the second definition, then for every bounded input, we get
a bounded output which is always between -1 and +1. So, the system is
stable.
Now..Which definition and reasoning is correct and why/why not?
Bye

Abhishek wrote:
> I have read that a system is stable if its impulse response approaches
> zero as time approaches infinity. Also another definition says that a
> system is stable if for every bounded input, we get a bounded output.
> What I understand for a bounded input is that the Input should have its
> value between -M and +N where M and N are some real numbers. Similarly
> is the definition for bounded output.
> Which one of the above two definitions for system stability hold good?
> If we go by the first definition, cos(x(n)) has an impulse response=
> cos(del(n)) which is zero for n=0 and one for n>0, which means impulse
> response doesn't approach zero as time approaches infinity.So, the
> system is unstable.
> If we go by the second definition, then for every bounded input, we get
> a bounded output which is always between -1 and +1. So, the system is
> stable.
> Now..Which definition and reasoning is correct and why/why not?
> Bye

A constant amplitude sinusoidally oscillating impulse response
(continuous or discrete time) is characteristic of a simple ideally
un-damped resonating system, ex. mass-spring, or inductor-capacitor
system. If such a system is driven, for example, by a constant
amplitude (i.e. bounded) sinusoidal excitation at the same resonating
frequency, then the response will grow linearly in amplitude without
bound, proving it to be BIBO unstable. A BIBO stable system's impulse
response h(t) must not only decay towards zero as t --> infinity (as
would h(t) = 1/t), but exponentially, such as a mass-spring system with
friction, or R-L-C circuit. A system with h(t) = 1/t , driven, for
example, by a step input (bounded) would result in an un-bounded
response.

The impulse response statements pertain to linear systems.  Your system
is NOT linear.

Dirk

The impulse response of the system you
Abhishek wrote:
> I have read that a system is stable if its impulse response approaches
> zero as time approaches infinity. Also another definition says that a
> system is stable if for every bounded input, we get a bounded output.
> What I understand for a bounded input is that the Input should have its
> value between -M and +N where M and N are some real numbers. Similarly
> is the definition for bounded output.
> Which one of the above two definitions for system stability hold good?
> If we go by the first definition, cos(x(n)) has an impulse response=
> cos(del(n)) which is zero for n=0 and one for n>0, which means impulse
> response doesn't approach zero as time approaches infinity.So, the
> system is unstable.
> If we go by the second definition, then for every bounded input, we get
> a bounded output which is always between -1 and +1. So, the system is
> stable.
> Now..Which definition and reasoning is correct and why/why not?
> Bye

Woops - Ignore the last half sentenceAFTER 'Dirk'.

Woops - Ignore the last half sentence AFTER 'Dirk'.

Abhishek wrote:
> I have read that a system is stable if its impulse response approaches
> zero as time approaches infinity. Also another definition says that a
> system is stable if for every bounded input, we get a bounded output.
> What I understand for a bounded input is that the Input should have its
> value between -M and +N where M and N are some real numbers. Similarly
> is the definition for bounded output.
> Which one of the above two definitions for system stability hold good?
> If we go by the first definition, cos(x(n)) has an impulse response=
> cos(del(n)) which is zero for n=0 and one for n>0, which means impulse
> response doesn't approach zero as time approaches infinity.So, the
> system is unstable.
> If we go by the second definition, then for every bounded input, we get
> a bounded output which is always between -1 and +1. So, the system is
> stable.
> Now..Which definition and reasoning is correct and why/why not?

Neither one is correct -- they refer to *linear* time-invariant
systems (well, the linear part at least).

The criterion of impulse response approaching 0 as n -> infinity
goes with the fact that the output is that impulse response
repeated infinitely many times (convolution!).  All that refers
to linear systems, so applying it to a non-linear system is
meaningless.

HTH,

Carlos
--

Carlos Moreno wrote:

> The criterion of impulse response approaching 0 as n -> infinity
> goes with the fact that the output is that impulse response
> repeated infinitely many times (convolution!).  All that refers
> to linear systems, so applying it to a non-linear system is
> meaningless.

You see, Robert.  This is what results from making people think that the 
DFT and fast convolution are circular.  :-)

Carlos, the impulse response needn't repeat infinitely many times.  A 
naive interpretation of the DFT, or convolution based on it, just tempts 
one to believe that.


Bob
-- 

"Things should be described as simply as possible, but no simpler."

                                              A. Einstein

Carlos Moreno wrote:

   ...

> The criterion of impulse response approaching 0 as n -> infinity
> goes with the fact that the output is that impulse response
> repeated infinitely many times (convolution!).  All that refers
> to linear systems, so applying it to a non-linear system is
> meaningless.

Are you confusing impulse response with sampled spectrum? When I bang a 
bell once with a hammer, It peals only once. How about for you?

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Jerry Avins <jya@ieee.org> writes:

> Are you confusing impulse response with sampled spectrum? When I bang
> a bell once with a hammer, It peals only once. How about for you?

It tolls for thee, thee, thee, thee, thee, thee, thee, thee ..... :-)

Ciao,

Peter K.

-- 
"And he sees the vision splendid 
of the sunlit plains extended
And at night the wondrous glory of the everlasting stars."

 

Jerry Avins wrote:

>> The criterion of impulse response approaching 0 as n -> infinity
>> goes with the fact that the output is that impulse response
>> repeated infinitely many times (convolution!).  All that refers
>> to linear systems, so applying it to a non-linear system is
>> meaningless.
> 
> Are you confusing impulse response with sampled spectrum? When I bang a 
> bell once with a hammer, It peals only once. How about for you?

Really don't understand what you guys are referring to;  let me
clarify what I meant:

For a linear, time-invariant system, the output when the system
is fed with an input with infinite support (i.e., it is non-zero
for an infinite amount of time/samples) is obtained as an infinite
sum of shifted impulses -- that's the most basic of the concepts
in linear time-invariant systems, I know;  but that's exactly
what I was talking about.

Intuitively, if you add an infinite amount of things that converge
to a non-zero value as n -> infinity, then the value as n->oo
diverges ===> the system is unstable;  for a bounded input, we're
getting an unbounded output.

But the OP was using the above criterion (impulse response's value
as n -> oo) for a *non-linear* system.

Am I making sense?  Does this clarify what I was trying to say?
Or am I still not understanding what you and Bob are trying to
tell me?

Carlos
--