in article z7IQd.6074$TY6.168@newssvr31.news.prodigy.com, Brad Griffis at
bradgriffis@hotmail.com wrote on 02/16/2005 08:45:

> These semantics battles are tiring!  I'm dropping off the thread after this
> post, RBJ.  You've worn me down!

yeah, sometimes i do that.

>  I hope you realize I agree with the meat
> of what you're saying and it is in fact almost exactly what I was trying to
> get across to the original person posting.  Just a tiny discrepancy in
> wording as to what is a definition and what is a consequence.

often it's when people who are in the same wing of the same party that what
seems to be the littlest differences get blown all up.  sorta like, in the
conservative "evangelical" community, when the pre-tribulationist
pre-millenialists have a knock-down drag-out with the post-tribulationist
pre-millenialists.  those of us not in that group might throw up our hands
and ask "who gives a rats-ass?"

if you say that this mathematical operation:


    +inf
    SUM{ x[i]*h[n-i] }
   i=-inf


is _defined_ to be the "convolution operation of x[n] and h[n]", i have no
problem with that.  we gotta give that thing a name, and "convolution" is
that name.  in addition, if you define a symbol for that operation like:


    +inf
    SUM{ x[i]*h[n-i] }  =def  x[n] (*) h[n]  or  (x*h)[n]
   i=-inf


that's fine.  let's assign it a symbol and *that* is a definition.  (when i
type "=def", that's the same as the equal sign with the little triangle on
top.)

when you say that for LTI systems or filters, their input-output properties
are defined as:


    LTI{ x1[n] + x2[n] }  =def   LTI{ x1[n] } + LTI{ x2[n] }


and if y[n] =def LTI{ x[n] }  then


    LTI{ x[n-n0] }  =def  y[n-n0]


if you say that, i say "fine".  the first is literally the definition of
linearity, called "superposition" (the scaling property of linear systems
can be derived from the superposition property for any rational scaling
constant), and the second is literally the definition of time-invariancy.

if you say that the impulse function is defined as:


                   {1  for n = 0
    delta[n] =def  {
                   {0  for n <> 0


then i say "fine", it a discrete, well-defined function for every integer n
and we gotta give it a name and "impulse" and "kroenecker delta" are
perfectly good names to give it.

if you define the impulse response of an LTI system as:


    h[n]  =def  LTI{ delta[n] }


i say "fine".  no one is stopping us from hypothesizing putting an impulse
into an LTI system and looking at what comes out.  giving that output a name
"impulse response" and "h[n]" is fine and that is a definition.

NOW, when you roll all those previous definitions up and you discover (or
derive, i disagree with you that i "cannot derive it") that for any general
input x[n], that the output,


    y[n] =def LTI{ x[n] }


is:

            +inf
    y[n] =  SUM{ x[i]*h[n-i] }
           i=-inf


that equality is NOT a definition.  it is a result.  using the "=def" symbol
is not correct.  that's not just a semantic difference.

now here is why the semantics of "definition" matter:  it matters to how one
both performs a mathematical or logical (or philosophical) proof and how
another reviews and accepts or rejects such a proof.  when one sets out to
prove some point, that person needs to first lay out what the axioms are.
the axioms are facts that people accept at the outset as true (or at least
"tentatively" true), and then from those axioms (of which the rules of logic
are also) another ostensibly different point is discovered or derived.  if
the reviewers of the proof do not like the result, then they have to EITHER
pick apart the argument or procedure that the first person used to move from
the axioms to the result (they have to find a flaw in the proof) OR they
have to back-pedal on their acceptance of one or more axiom that they
started with.  if they do not accept the result nor do they find a flaw in
the argument nor do they reject any axiom, then they suffer what i would
call "cognitive dissonance".

how it applies here is that no EE student (or anyone else) should accept
that for a discrete-time LTI system, that

            +inf
    y[n] =  SUM{ x[i]*h[n-i] }
           i=-inf

is true by definition.  it is not the definition of an LTI system.  it is a
true result of applying the definitions of an LTI system and what h[n] is.

i mean maybe it's true that:

            +inf
    y[n] =  SUM{ (x[i]*h[n-i])^2 }
           i=-inf

but i would certainly have trouble deriving that from the definitions of
LTI.  so then it isn't true and the equal sign is in error.  the equal sign
for

            +inf
    y[n] =  SUM{ x[i]*h[n-i] }
           i=-inf

is not in error, but it is NOT an "=def" sign because if it was, i would
telling the reader or reviewer to not bother checking the veracity of that
equation but to just accept it as true because i define it as such.  if
students or reviewers went along with that, i might try "selling them some
land in Florida".

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

"robert bristow-johnson" <rbj@audioimagination.com> wrote in message 
news:BE385676.488D%rbj@audioimagination.com...
> crap!  had to fix a couple of typos.  can you find them?
>
> in article _4BQd.30988$by5.20298@newssvr19.news.prodigy.com, Brad Griffis 
> at
> bradgriffis@hotmail.com wrote on 02/16/2005 00:44:
>
>> "robert bristow-johnson" <rbj@audioimagination.com> wrote in message
>> news:BE37A0C2.482A%rbj@audioimagination.com...
>>
>>> 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
>
> i generally agree with Julius on most things and this is another.  he is
> *deriving* the convolution summation from two or three given facts:
>
>
> 1. that the input x[n] can be expressed as a summation of "constant"
> coefficients and the delta function:
>
>
>    x[n] = SUM{ x[i] * delta[n-i] }   =def  (x(*)delta)[n]
>            i
>
> the first equal sign is true, but not by definition.  you have to show 
> that
> it is from the definition of the kroeneker delta.  but it's not hard.
>
> 2.  h[n]  =def  LTI{ delta[n] }
>
> 3. then, using the linearity and time-invariancy properties of LTI systems
> that *are* there by definition, he derives that
>
>    y[n]  =def   LTI{x[n]}
>
>          =     LTI{ SUM{ x[i] * delta[n-i] } }
>                      i
>
>          =     SUM{ x[i] * LTI{ delta[n-i] } }
>                 i
>
>          =     SUM{ x[i] * h[n-i] }
>                 i
>
>          =def  (x(*)h)[n]
>
>
> the derivation makes use of the defined properties of LTI systems, but the
> fact that LTI systems convolute their input against their impulse response
> is derived from those defined properties.  if you understand that the
> equality symbol, with or without the little triangle (or whatever other
> version such as two little squiggles meaning "approximately equal"), 
> relates
> the two neighboring expressions, not necessarily the bottom expression to
> the top, then you understand although it is true that:
>
>    y[n] =  (x(*)h)[n]
>
> but not every equality that connects those are defined equalities.  some 
> of
> those equalities in between you have to work and show they're true.
>
>> Calculating the output of an LTI filter using convolution is a 
>> consequence
>> of linearity and time-invariance.
>
> i think i said that, too.  we have a disagreement about what is true by
> axiom (which is what any definition is, though not all axioms are
> definitions) and what is true by derivation.
>
>> When convolution was originally defined it was not the whim of a madman.
>
> i never implied that it was.
>
>> It was created methodically based on calculating the output of an LTI 
>> filter.
>
> absolutely true.  but it is a *derived* result, not one that is true by
> definition of what an LTI filter is.
>
>> Hence my statement that its DEFINITION was based on LTI filters.  It's 
>> the
>> physical correspondence that is a consequence.
>
> but what you mean by "DEFINITION" is not the meaning of the word.
> convolution in LTI systems is a true and real property of LTI systems that
> flows from the definition of LTI and the definition of what delta[n] and
> h[n] are, but it's not the definition.  it's a result.  more results are
> that the Z-transforms of h[n] and x[n] of LTI systems multiply to get you
> the Z-transform of the output y[n].  we don't say that is true by 
> definition
> (at least i don't).  but it *is* true.
>

This is just a semantics battle.  You can leave out all the equations as we 
all agree on them.

The triangle above the equal sign means "is defined to equal".  That's the 
difference between the "triangle equals" and the "normal equals".  (BTW the 
"three line equals" also means "defined equal to" and the "squiggly equals" 
means "approximately" as you stated.)

I think that web page actually illustrates my point well.  The convolution 
operation is a DEFINITION (as supported by the _definition_ of the triangle 
equals, ha ha!).  If you go further down the page you see that defnition 
applied to LTI systems and find that it's a perfect fit.  Of course it is a 
perfect fit since that's what is was defined for.

Here's maybe another way to say it.  You can go through all the math and 
derive the output of an LTI system.  In doing that you end up with the 
familiar convolution equation.  However, this does not mean that you have 
derived convolution.  You cannot derive it.  It is a definition.  You have 
derived the formula for the output to an LTI filter, which of couse is 
inextricably linked to convolution.

Here's yet another!  The output of an LTI filter can always be given using 
convolution.  The flip of that is not true though.  Convolution does not 
always result in giving the output of a LTI filter (i.e. in the case where 
the signals you're convolving are not an input and impulse response to an 
LTI filter but rather just two signals).  This is because convolution is a 
DEFINITION and because it is a definition it can be applied to any signals.

These semantics battles are tiring!  I'm dropping off the thread after this 
post, RBJ.  You've worn me down!  I hope you realize I agree with the meat 
of what you're saying and it is in fact almost exactly what I was trying to 
get across to the original person posting.  Just a tiny discrepancy in 
wording as to what is a definition and what is a consequence.

Cheers,

Brad

crap!  had to fix a couple of typos.  can you find them?

in article _4BQd.30988$by5.20298@newssvr19.news.prodigy.com, Brad Griffis at
bradgriffis@hotmail.com wrote on 02/16/2005 00:44:

> "robert bristow-johnson" <rbj@audioimagination.com> wrote in message
> news:BE37A0C2.482A%rbj@audioimagination.com...
> 
>> 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

i generally agree with Julius on most things and this is another.  he is
*deriving* the convolution summation from two or three given facts:


1. that the input x[n] can be expressed as a summation of "constant"
coefficients and the delta function:


    x[n] = SUM{ x[i] * delta[n-i] }   =def  (x(*)delta)[n]
            i

the first equal sign is true, but not by definition.  you have to show that
it is from the definition of the kroeneker delta.  but it's not hard.

2.  h[n]  =def  LTI{ delta[n] }

3. then, using the linearity and time-invariancy properties of LTI systems
that *are* there by definition, he derives that

    y[n]  =def   LTI{x[n]}

          =     LTI{ SUM{ x[i] * delta[n-i] } }
                      i

          =     SUM{ x[i] * LTI{ delta[n-i] } }
                 i

          =     SUM{ x[i] * h[n-i] }
                 i

          =def  (x(*)h)[n]


the derivation makes use of the defined properties of LTI systems, but the
fact that LTI systems convolute their input against their impulse response
is derived from those defined properties.  if you understand that the
equality symbol, with or without the little triangle (or whatever other
version such as two little squiggles meaning "approximately equal"), relates
the two neighboring expressions, not necessarily the bottom expression to
the top, then you understand although it is true that:

    y[n] =  (x(*)h)[n]

but not every equality that connects those are defined equalities.  some of
those equalities in between you have to work and show they're true.

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

i think i said that, too.  we have a disagreement about what is true by
axiom (which is what any definition is, though not all axioms are
definitions) and what is true by derivation.

> When convolution was originally defined it was not the whim of a madman.

i never implied that it was.

> It was created methodically based on calculating the output of an LTI filter.

absolutely true.  but it is a *derived* result, not one that is true by
definition of what an LTI filter is.

> Hence my statement that its DEFINITION was based on LTI filters.  It's the
> physical correspondence that is a consequence.

but what you mean by "DEFINITION" is not the meaning of the word.
convolution in LTI systems is a true and real property of LTI systems that
flows from the definition of LTI and the definition of what delta[n] and
h[n] are, but it's not the definition.  it's a result.  more results are
that the Z-transforms of h[n] and x[n] of LTI systems multiply to get you
the Z-transform of the output y[n].  we don't say that is true by definition
(at least i don't).  but it *is* true.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

in article _4BQd.30988$by5.20298@newssvr19.news.prodigy.com, Brad Griffis at
bradgriffis@hotmail.com wrote on 02/16/2005 00:44:

> "robert bristow-johnson" <rbj@audioimagination.com> wrote in message
> news:BE37A0C2.482A%rbj@audioimagination.com...
> 
>> 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

i generally agree with Julius on most things and this is another.  he is
*deriving* the convolution summation from two or three given facts:


1. that the input x[n] can be expressed as a summation of "constant"
coefficients and the delta function:


    x[n] = SUM{ x[i] * delta[n-i] }   =def  (x(*)delta)[i]
            i

the first equal sign is true, but not by definition.  you have to show that
it is from the definition of the kroeneker delta.  but it's not hard.

2.  h[n]  =def  LTI{ delta[n] }

3. then, using the linearity and time-invariancy properties of LTI systems
that *are* there by definition, he derives that

    y[n]  =def   LTI{x[n]}

          =     LTI{ SUM{ x[i] * delta[n-i] } }
                      i

          =     SUM{ x[i] * LTI{ delta[n-i] } }
                 i

          =     SUM{ x[i] * h[n-i] }
                 i

          =def  (x(*)h)[i]


the derivation makes use of the defined properties of LTI systems, but the
fact that LTI systems convolute their input against their impulse response
is derived from those defined properties.  if you understand that the
equality symbol, with or without the little triangle (or whatever other
version such as two little squiggles meaning "approximately equal"), relates
the two neighboring expressions, not necessarily the bottom expression to
the top.  it is true that:

    y[n] =  (x(*)h)[i]

but not every equality that connects those are defined equalities.  some of
those equalities in between you have to work and show they're true.

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

i think i said that, too.  we have a disagreement about what is true by
axiom (which is what any definition is, though not all axioms are
definitions) and what is true by derivation.

> When convolution was originally defined it was not the whim of a madman.

i never implied that it was.

> It was created methodically based on calculating the output of an LTI filter.

absolutely true.  but it is a *derived* result, not one that is true by
definition.

> Hence my statement that its DEFINITION was based on LTI filters.  It's the
> physical correspondence that is a consequence.

but what you mean by "DEFINITION" is not the meaning of the word.
convolution in LTI systems is a true and real property of LTI systems that
flows from the definition of LTI and the definition of what delta[n] and
h[n] are, but it's not the definition.  it's a result.  more results are
that the Z-transforms of h[n] and x[n] of LTI systems multiply to get you
the Z-transform of the output y[n].  we don't say that is true by definition
(at least i don't).  but it *is* true.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

"robert bristow-johnson" <rbj@audioimagination.com> wrote in message 
news:BE37A0C2.482A%rbj@audioimagination.com...
> in article AUmQd.5744$NO3.572@newssvr31.news.prodigy.com, Brad Griffis at
> bradgriffis@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

"Abdul" <abdul_araheem@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

in article 1114g3h2hle7s9d@corp.supernews.com, Tim Wescott at
tim@wescottnospamdesign.com wrote on 02/15/2005 13:33:

> robert bristow-johnson wrote:
>> in article AUmQd.5744$NO3.572@newssvr31.news.prodigy.com, Brad Griffis at
>> bradgriffis@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                  rbj@audioimagination.com

"Imagination is more important than knowledge."

Tim Wescott wrote:
> robert bristow-johnson wrote:
> 
>> in article AUmQd.5744$NO3.572@newssvr31.news.prodigy.com, Brad Griffis at
>> bradgriffis@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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robert bristow-johnson wrote:
> in article AUmQd.5744$NO3.572@newssvr31.news.prodigy.com, Brad Griffis at
> bradgriffis@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

in article AUmQd.5744$NO3.572@newssvr31.news.prodigy.com, Brad Griffis at
bradgriffis@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                  rbj@audioimagination.com

"Imagination is more important than knowledge."