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.htmli 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."
LTI system and Non LTI system
Started by ●February 15, 2005
Reply by ●February 16, 20052005-02-16
Reply by ●February 16, 20052005-02-16
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.htmli 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."
Reply by ●February 16, 20052005-02-16
"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
Reply by ●February 16, 20052005-02-16
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."
