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Hi,

    Let

X(n)----->  Input Discreate Time Sequnece.
H(n)----->  Impulse response of the system.
Y(n)----->  Output Discreate time Sequnece.

X(Z), Y(Z) are the z trasfroms of Input & output Discreate time sequnece and
H(Z) is the transfer function  of the system.

Then the mathamatical relation ship holds

 Y(n) = X(n)(+)H(n)  // Convolution Operation.
 Y(Z) = X(Z)*Y(Z)  // Multipilcation Operation.

The Above relationships are only valid of LTI system ?? or they also valid
for Non-Linear, Non-Time Invaraint System ??

Thanks in advance.

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The impulse response, h[n], and the convolution operation are DEFINED based 
on a system being LTI.  Therefore they are in a strict sense only applicable 
to linear, time-invariant systems.  However, you will still probably see 
them used even for systems that aren't LTI.  In this case there is some 
"fudge factor" involved.  For example in a LTV system you could treat it as 
LTI for short periods of time.  In a nonlinear, time-invariant system you 
might treat it as LTI but then throw in some kind of correction factor to 
account for non linearities.

Brad

"Abdul" <a...@yahoo.com> wrote in message 
news:cuslvf$52c$1...@rzsun03.rrz.uni-hamburg.de...
> Hi,
>
>    Let
>
> X(n)----->  Input Discreate Time Sequnece.
> H(n)----->  Impulse response of the system.
> Y(n)----->  Output Discreate time Sequnece.
>
> X(Z), Y(Z) are the z trasfroms of Input & output Discreate time sequnece 
> and
> H(Z) is the transfer function  of the system.
>
> Then the mathamatical relation ship holds
>
> Y(n) = X(n)(+)H(n)  // Convolution Operation.
> Y(Z) = X(Z)*Y(Z)  // Multipilcation Operation.
>
> The Above relationships are only valid of LTI system ?? or they also valid
> for Non-Linear, Non-Time Invaraint System ??
>
> Thanks in advance.
>
>
>
> 

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"Brad Griffis" <b...@hotmail.com> wrote in message
news:AUmQd.5744$N...@newssvr31.news.prodigy.com...
> The impulse response, h[n], and the convolution operation are DEFINED
based
> on a system being LTI.  Therefore they are in a strict sense only
applicable
> to linear, time-invariant systems.  However, you will still probably see
> them used even for systems that aren't LTI.  In this case there is some
> "fudge factor" involved.  For example in a LTV system you could treat it
as
> LTI for short periods of time.  In a nonlinear, time-invariant system you
> might treat it as LTI but then throw in some kind of correction factor to
> account for non linearities.
>
> Brad
>
> "Abdul" <a...@yahoo.com> wrote in message
> news:cuslvf$52c$1...@rzsun03.rrz.uni-hamburg.de...
> > Hi,
> >
> >    Let
> >
> > X(n)----->  Input Discreate Time Sequnece.
> > H(n)----->  Impulse response of the system.
> > Y(n)----->  Output Discreate time Sequnece.
> >
> > X(Z), Y(Z) are the z trasfroms of Input & output Discreate time sequnece
> > and
> > H(Z) is the transfer function  of the system.
> >
> > Then the mathamatical relation ship holds
> >
> > Y(n) = X(n)(+)H(n)  // Convolution Operation.
> > Y(Z) = X(Z)*Y(Z)  // Multipilcation Operation.
> >
> > The Above relationships are only valid of LTI system ?? or they also
valid
> > for Non-Linear, Non-Time Invaraint System ??
> >
> > Thanks in advance.

Actually, I believe that impulse responses and convolutions are applicable
to all linear systems, whether or not they are time invariant or not. It's
just that the formulas are a little more complicated since h[n, k] is not
necessarily expressable as h[n-k]. Transfer functions and other
frequency-domain relationships, however, only apply to linear,
time-invariant systems.

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Brad Griffis wrote:
> The impulse response, h[n], and the convolution operation are DEFINED based 
> on a system being LTI.  Therefore they are in a strict sense only applicable 
> to linear, time-invariant systems.  However, you will still probably see 
> them used even for systems that aren't LTI.  In this case there is some 
> "fudge factor" involved.  For example in a LTV system you could treat it as 
> LTI for short periods of time.  In a nonlinear, time-invariant system you 
> might treat it as LTI but then throw in some kind of correction factor to 
> account for non linearities.
> 
> Brad
> 
> "Abdul" <a...@yahoo.com> wrote in message 
> news:cuslvf$52c$1...@rzsun03.rrz.uni-hamburg.de...

   ...

And Abdul, keep your notation straight. X(a), X[a], x(a), and x[a] have 
different conventional meanings. ('a' doesn't stand for the same symbol 
in all instances.)

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

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in article AUmQd.5744$N...@newssvr31.news.prodigy.com, Brad Griffis at
b...@hotmail.com wrote on 02/15/2005 08:35:

> The impulse response, h[n], and the convolution operation are DEFINED based
> on a system being LTI.

Brad, i have to disagree with this statement.  any system, whether it's LTI
or not, can be banged with a unit impulse and be allowed to respond (or
"ring") and that response, whether it's an LTI system or not, would the be
"impulse response".  if the system is not LTI, i would think that the
impulse response has less meaning than it does for an LTI system where the
impulse response completely characterizes the system from an input-output
perspective.  you could not say as much form the impulse response of a
system that is not LTI.

the convolution operation is *not* defined but is a *consequence* of the
fact that the LTI system is linear and time-invariant.  you can derive it
from the LTI properties and, particularly for a discrete-time system, it's
pretty easy.  the only DEFINED properties of an LTI system that i can think
of is that it is linear and that it is time-invariant.  there are many other
properties, but they can be derived.

-- 

r b-j                  r...@audioimagination.com

"Imagination is more important than knowledge."

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robert bristow-johnson wrote:
> in article AUmQd.5744$N...@newssvr31.news.prodigy.com, Brad Griffis at
> b...@hotmail.com wrote on 02/15/2005 08:35:
> 
> 
>>The impulse response, h[n], and the convolution operation are DEFINED based
>>on a system being LTI.
> 
> 
> Brad, i have to disagree with this statement.  any system, whether it's LTI
> or not, can be banged with a unit impulse and be allowed to respond (or
> "ring") and that response, whether it's an LTI system or not, would the be
> "impulse response".  if the system is not LTI, i would think that the
> impulse response has less meaning than it does for an LTI system where the
> impulse response completely characterizes the system from an input-output
> perspective.  you could not say as much form the impulse response of a
> system that is not LTI.
> 
> the convolution operation is *not* defined but is a *consequence* of the
> fact that the LTI system is linear and time-invariant.  you can derive it
> from the LTI properties and, particularly for a discrete-time system, it's
> pretty easy.  the only DEFINED properties of an LTI system that i can think
> of is that it is linear and that it is time-invariant.  there are many other
> properties, but they can be derived.
> 
And I'll have to disagree with you both (whee!).

Robert:  While any system can be banged with an impulse and it's 
response checked this is meaningless in general with nonlinear systems 
-- what if the system is a block that limits any input to +/- 10V 
followed by a purely linear system?  In that case the impulse will 
disappear entirely, and there'll be no response.

Brad:  This has already been pointed out, but a time-varying linear 
system can be completely defined by its impulse response, the only 
caveat being that the impulse response has two time parameters: the time 
of the output, and the time the impulse happened.  So instead of talking 
about h(n) and assuming that the impulse happened at n=0 with no loss of 
generality you have to talk about h(n_0, n), where the impulse happened 
at n=n_0.

Neither a nonlinear system or a time-varying one can be reduced exactly 
to a transfer function.

Some time-varying systems can be expressed as a combination of 
multiplies and convolutions in the frequency domain -- this is 
particularly useful in analyzing heterodyne radio receivers (either 
superhets or direct-conversion).  The idea of using the time-varying 
impulse response and doing a convolution on it is _very_ useful in 
analyzing communications systems.

Some nonlinear systems can be expressed with a "describing function" 
where a real input signal is chosen, the system is excited with it, and 
the frequency response of the system is extracted.  This technique is 
very useful for control system design.  It can be done directly in the 
case of a swept-sine measurement or indirectly in the case of random 
noise or step excitation, and you will often find yourself taking 
measurements at different amplitudes or waveshapes and seeing a range of 
"frequency responses".

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

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Tim Wescott wrote:
> robert bristow-johnson wrote:
> 
>> in article AUmQd.5744$N...@newssvr31.news.prodigy.com, Brad Griffis at
>> b...@hotmail.com wrote on 02/15/2005 08:35:
>>
>>
>>> The impulse response, h[n], and the convolution operation are DEFINED 
>>> based
>>> on a system being LTI.
>>
>>
>>
>> Brad, i have to disagree with this statement.  any system, whether 
>> it's LTI
>> or not, can be banged with a unit impulse and be allowed to respond (or
>> "ring") and that response, whether it's an LTI system or not, would 
>> the be
>> "impulse response".  if the system is not LTI, i would think that the
>> impulse response has less meaning than it does for an LTI system where 
>> the
>> impulse response completely characterizes the system from an input-output
>> perspective.  you could not say as much form the impulse response of a
>> system that is not LTI.
>>
>> the convolution operation is *not* defined but is a *consequence* of the
>> fact that the LTI system is linear and time-invariant.  you can derive it
>> from the LTI properties and, particularly for a discrete-time system, 
>> it's
>> pretty easy.  the only DEFINED properties of an LTI system that i can 
>> think
>> of is that it is linear and that it is time-invariant.  there are many 
>> other
>> properties, but they can be derived.
>>
> And I'll have to disagree with you both (whee!).

Well Tim, I disagree with you. (Double whee!)

> Robert:  While any system can be banged with an impulse and its 
> response checked, this is meaningless in general with nonlinear systems 
> -- what if the system is a block that limits any input to +/- 10V 
> followed by a purely linear system?  In that case the impulse will 
> disappear entirely, and there'll be no response.

That is true only if you insist that an impulse has zero width and 
infinite height. In that case, no impulse can get through to any real 
system. It will always be clipped, maybe to a megavolt, but what's left 
still disappears entirely. A practical impulse has a width so narrow 
that it doesn't matter and a height that's "reasonable". One of those 
_will_ be seen on the other side of your limiter. Try it.

   ...

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

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in article 1...@corp.supernews.com, Tim Wescott at
t...@wescottnospamdesign.com wrote on 02/15/2005 13:33:

> robert bristow-johnson wrote:
>> in article AUmQd.5744$N...@newssvr31.news.prodigy.com, Brad Griffis at
>> b...@hotmail.com wrote on 02/15/2005 08:35:
>> 
>> 
>>> The impulse response, h[n], and the convolution operation are DEFINED based
>>> on a system being LTI.
>> 
>> 
>> Brad, i have to disagree with this statement.  any system, whether it's LTI
>> or not, can be banged with a unit impulse and be allowed to respond (or
>> "ring") and that response, whether it's an LTI system or not, would the be
>> "impulse response".  if the system is not LTI, i would think that the
>> impulse response has less meaning than it does for an LTI system where the
>> impulse response completely characterizes the system from an input-output
>> perspective.  you could not say as much form the impulse response of a
>> system that is not LTI.
>> 
>> the convolution operation is *not* defined but is a *consequence* of the
>> fact that the LTI system is linear and time-invariant.  you can derive it
>> from the LTI properties and, particularly for a discrete-time system, it's
>> pretty easy.  the only DEFINED properties of an LTI system that i can think
>> of is that it is linear and that it is time-invariant.  there are many other
>> properties, but they can be derived.
>> 
> And I'll have to disagree with you both (whee!).
> 
> Robert:  While any system can be banged with an impulse and it's
> response checked this is meaningless in general with nonlinear systems
> -- what if the system is a block that limits any input to +/- 10V
> followed by a purely linear system?  In that case the impulse will
> disappear entirely, and there'll be no response.

so where is the disagreement?  doesn't "meaningless" have less meaning than
"completely characterizes the system from an input-output perspective"?  i
don't see any disagreement.  :-\

> Brad:  This has already been pointed out, but a time-varying linear
> system can be completely defined by its impulse response, the only
> caveat being that the impulse response has two time parameters: the time
> of the output, and the time the impulse happened.  So instead of talking
> about h(n) and assuming that the impulse happened at n=0 with no loss of
> generality you have to talk about h(n_0, n), where the impulse happened
> at n=n_0.

not a substantive disagreement, but a semantic one:  i wouldn't say a
time-varying linear system can be completely defined (i would say
"described") by its impulse response (emphasis _singular_ tense).  i would
say it's completely defined (or described) by *all* possible impulse
responses of unit impulses applied at all possible times.  for the LTI, h[n]
which is h[n0, n] where n0 is the special case of being 0, is completely
sufficient, whereas it is not sufficient for LTV.  you're saying this, but i
wouldn't use the singular tense.

-- 

r b-j                  r...@audioimagination.com

"Imagination is more important than knowledge."

______________________________
DSPRelated.com's 50,000th member announced! Details Here.

"Abdul" <a...@yahoo.com> wrote in message
news:cuslvf$52c$1...@rzsun03.rrz.uni-hamburg.de...
> Hi,
>
>     Let
>
> X(n)----->  Input Discreate Time Sequnece.
> H(n)----->  Impulse response of the system.
> Y(n)----->  Output Discreate time Sequnece.
>
> X(Z), Y(Z) are the z trasfroms of Input & output Discreate time sequnece
and
> H(Z) is the transfer function  of the system.
>
> Then the mathamatical relation ship holds
>
>  Y(n) = X(n)(+)H(n)  // Convolution Operation.
>  Y(Z) = X(Z)*Y(Z)  // Multipilcation Operation.
>
> The Above relationships are only valid of LTI system ?? or they also valid
> for Non-Linear, Non-Time Invaraint System ??
>
> Thanks in advance.

To reiterate after the discussion...

Linear and Time invariance are two distinct properties, neither one implies
the other. Linearity is all that is required for a system to be
characterized by an impulse response FUNCTION. Normally this is a function
of two variables, the "current" time and the time of the impulse. In the
case of a TIME INVARIANT linear system, the response to an impulse depends
only on the difference between the current time and the time of the impulse
so only a single parameter is needed. I misspoke earlier when I suggested
that convolution can be used for a linear system that is not time invariant.
Both linearity and time invariance are required for convolution or transfer
functions to be usable to calculate the response to an arbitrary input. It
should be noted that even for time-varying linear systems, the response to
an arbitrary input can be calculated by an integral/summation. It is just a
matter of "decomposing" the arbitrary input into a sum of impulses of
different amplitudes at different times and then, by linearity, summing
their responses.

HTH

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"robert bristow-johnson" <r...@audioimagination.com> wrote in message 
news:BE37A0C2.482A%r...@audioimagination.com...
> in article AUmQd.5744$N...@newssvr31.news.prodigy.com, Brad Griffis at
> b...@hotmail.com wrote on 02/15/2005 08:35:
>
>> The impulse response, h[n], and the convolution operation are DEFINED 
>> based
>> on a system being LTI.
>
> Brad, i have to disagree with this statement.  any system, whether it's 
> LTI
> or not, can be banged with a unit impulse and be allowed to respond (or
> "ring") and that response, whether it's an LTI system or not, would the be
> "impulse response".  if the system is not LTI, i would think that the
> impulse response has less meaning than it does for an LTI system where the
> impulse response completely characterizes the system from an input-output
> perspective.  you could not say as much form the impulse response of a
> system that is not LTI.

True enough.  Point taken.

> the convolution operation is *not* defined but is a *consequence* of the
> fact that the LTI system is linear and time-invariant.  you can derive it
> from the LTI properties and, particularly for a discrete-time system, it's
> pretty easy.  the only DEFINED properties of an LTI system that i can 
> think
> of is that it is linear and that it is time-invariant.  there are many 
> other
> properties, but they can be derived.

Convolution itself is a definition.  (See the triangle over the equal sign 
at web page below.)

http://ccrma.stanford.edu/~jos/filters/Convolution_Representation.html

Calculating the output of an LTI filter using convolution is a consequence 
of linearity and time-invariance.

When convolution was originally defined it was not the whim of a madman.  It 
was created methodically based on calculating the output of an LTI filter. 
Hence my statement that its DEFINITION was based on LTI filters.  It's the 
physical correspondence that is a consequence.

Brad

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