Peter K. wrote:> 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 ..... :-)Oh. Does that mean we're now Donne with the silly stuff? :-) Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Is the system whose output y(n) is cos(x(n)) stable?
Started by ●February 11, 2006
Reply by ●February 12, 20062006-02-12
Reply by ●February 12, 20062006-02-12
Carlos Moreno wrote:> 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?I don't understand where you're coming from. Too much math, maybe. The impulse response of a system is how the system responds when it's excited by a single impulse, like the hammer blow in my example. When it's done doing its thing, it's done. In your concept, where it repeats, what is the period? Note that nonlinear systems respond to impulses too. Our saying that they don't have impulse responses has a very limited meaning. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 13, 20062006-02-13
Jerry Avins wrote:>> [...] >> 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? > > > I don't understand where you're coming from. Too much math, maybe.Yeah, because I'm going to believe that *you* are having a hard time understanding math... Nice try Jerry, except that I already know you! ;-)> impulse response of a system is how the system responds when it's > excited by a single impulse, like the hammer blow in my example. When > it's done doing its thing, it's done. In your concept, where it repeats, > what is the period?1 sample. I mean, a signal that goes like {1, 3, 2, 1, 2, 5, 7, 2 ...} is an infinite sum of shifted impulses. One impulse of weight 1 centered at n = 0, one impulse of weight 3 centered at n = 1, one impulse of weight 2 centered at n = 2, and so on. *Because the system is linear and also time-invariant*, the output of the system is the sum of the outputs of each "sub-signal" with the proper weight (linearity), and the output of each sub-signal is the impulse response shifted the right amount of samples (the time-invariant part of the system). The above is nothing more than the convolution (or the intuitive/ visual way to understand the convolution, at least). That's why, if you have a system with impulse response that does not converge to 0 as n->oo, then you feed that system with, say, a unit step (x[n] = 1 for all n >= 0), and the output goes to oo as n->oo. But then, coming back to the original post -- the OP was using the impulse response criterion for a non-linear system, which makes the criterion entirely meaningless. I hope I could make it clear this time!> Note that nonlinear systems respond to impulses too. Our saying that > they don't have impulse responses has a very limited meaning.But notice that no-one (well, not me, at least) said that they do not have impulse responses -- it's just that impulse response for a non-linear system is not nearly as meanigful as it is for linear systems (in any case, it does not have as many "nice" implications as it does for LTI systems); and in particular, for this thread, a condition on the impulse response that is valid for LTI systems only was being used for a non-linear system. Carlos --
Reply by ●February 13, 20062006-02-13
Carlos Moreno wrote:> Jerry Avins wrote: > >>> [...] >>> 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? >> >> >> >> I don't understand where you're coming from. Too much math, maybe. > > > Yeah, because I'm going to believe that *you* are having a hard > time understanding math... Nice try Jerry, except that I already > know you! ;-)I mean that /you/ are relying too heavily on math to the exclusion of physical thought. A system's impulse response is its response to the signal 1, 0, 0, 0, 0, 0, ....>> impulse response of a system is how the system responds when it's >> excited by a single impulse, like the hammer blow in my example. When >> it's done doing its thing, it's done. In your concept, where it >> repeats, what is the period? > > > 1 sample.Many impulse responses are longer than that. You know what IIR means.> I mean, a signal that goes like {1, 3, 2, 1, 2, 5, 7, 2 ...} is an > infinite sum of shifted impulses. One impulse of weight 1 centered > at n = 0, one impulse of weight 3 centered at n = 1, one impulse of > weight 2 centered at n = 2, and so on.Right. The system's response is the superposition of its response to those individual impulses. If you mean that the response will last at least as long as the stimulus, I agree. A stimulus that lasts longer than one sample is not an impulse and the system's response to it, while composed of superposed impulse responses, is not its impulse response. Impulse response is a property of a system. It does not depend on the signal.> *Because the system is linear and also time-invariant*, the output > of the system is the sum of the outputs of each "sub-signal" with > the proper weight (linearity), and the output of each sub-signal > is the impulse response shifted the right amount of samples (the > time-invariant part of the system).So?> The above is nothing more than the convolution (or the intuitive/ > visual way to understand the convolution, at least).Superposition is not convolution. If you understand convolution that way, that's too little math, not too much.> That's why, if you have a system with impulse response that does > not converge to 0 as n->oo, then you feed that system with, say, > a unit step (x[n] = 1 for all n >= 0), and the output goes to oo > as n->oo.What's why?> But then, coming back to the original post -- the OP was using the > impulse response criterion for a non-linear system, which makes the > criterion entirely meaningless.> I hope I could make it clear this time! > >> Note that nonlinear systems respond to impulses too. Our saying that >> they don't have impulse responses has a very limited meaning. > > > But notice that no-one (well, not me, at least) said that they do > not have impulse responses -- it's just that impulse response for > a non-linear system is not nearly as meanigful as it is for linear > systems (in any case, it does not have as many "nice" implications > as it does for LTI systems); and in particular, for this thread, > a condition on the impulse response that is valid for LTI systems > only was being used for a non-linear system.That's true. For your own piece of mind, review "impulse response". Recall that it is the derivative of step response, another system property. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 13, 20062006-02-13
Jerry Avins wrote:> I mean that /you/ are relying too heavily on math to the exclusion of > physical thought.Wait -- the OP was using a stability criterion based on the impulse response; the context of his question already puts aside the physical world.> A system's impulse response is its response to the > signal 1, 0, 0, 0, 0, 0, ....Yes, and no-one would deny that.>>> impulse response of a system is how the system responds when it's >>> excited by a single impulse, like the hammer blow in my example. When >>> it's done doing its thing, it's done. In your concept, where it >>> repeats, what is the period? >> >> 1 sample. > > Many impulse responses are longer than that. You know what IIR means.Oops -- sorry, I misinterpreted your question; I thought you were asking what's the period of the repetitive "hammer hitting" ... Still, my reply was more figurative than anything else. Scratch all that part.>> I mean, a signal that goes like {1, 3, 2, 1, 2, 5, 7, 2 ...} is an >> infinite sum of shifted impulses. One impulse of weight 1 centered >> at n = 0, one impulse of weight 3 centered at n = 1, one impulse of >> weight 2 centered at n = 2, and so on. > > Right. The system's response is the superposition of its response to > those individual impulses. If you mean that the response will last at > least as long as the stimulus, I agree.No, that'snot what I was saying -- and by the way, it's not necessarily true (think a sine wave at the exact frequency corresponding to a zero on the unit circle). But still, let's put that aside (you're right in the general case, and in the context where the discussion was)> Impulse response is a property of a system. It does not depend on the > signal.Yes, but the system's output to a given signal does. It depends on the signal and it also depends on the system's impulse response.>> *Because the system is linear and also time-invariant*, the output >> of the system is the sum of the outputs of each "sub-signal" with >> the proper weight (linearity), and the output of each sub-signal >> is the impulse response shifted the right amount of samples (the >> time-invariant part of the system). > > So? > >> The above is nothing more than the convolution (or the intuitive/ >> visual way to understand the convolution, at least). > > Superposition is not convolution. If you understand convolution that > way, that's too little math, not too much.The above is a proof that the output of a LTI system to a given signal is given by the convolution of the signal and the impulse response; why do you feel that that's too little math?>> That's why, if you have a system with impulse response that does >> not converge to 0 as n->oo, then you feed that system with, say, >> a unit step (x[n] = 1 for all n >= 0), and the output goes to oo >> as n->oo. > > What's why?Re-read the entire phrase, skipping the "parenthetic" part: that's why if you feed that system with a unit step, the output goes to oo as n goes to infinity. (The "that" refers to the fact that you're superposing an infinite amount of impulse responses for which at n->oo -- which is the same regardless of the fact that they're shifted -- and the impulse responses converge to something other than 0)>> But then, coming back to the original post -- the OP was using the >> impulse response criterion for a non-linear system, which makes the >> criterion entirely meaningless.At this point, I'd like to emphasize that I was simply trying to make it more explicit for the OP why is it that he was mistaken in applying a criterion that is only meaningful for LTI systems; I was explaining my way of visualizing why that criterion works the way it does. Carlos --
Reply by ●February 13, 20062006-02-13
Carlos Moreno wrote:> Jerry Avins wrote: > >> I mean that /you/ are relying too heavily on math to the exclusion of >> physical thought. > > > Wait -- the OP was using a stability criterion based on the impulse > response; the context of his question already puts aside the physical > world. > >> A system's impulse response is its response to the signal 1, 0, 0, 0, >> 0, 0, .... > > > Yes, and no-one would deny that. > >>>> impulse response of a system is how the system responds when it's >>>> excited by a single impulse, like the hammer blow in my example. >>>> When it's done doing its thing, it's done. In your concept, where it >>>> repeats, what is the period? >>> >>> >>> 1 sample. >> >> >> Many impulse responses are longer than that. You know what IIR means. > > > Oops -- sorry, I misinterpreted your question; I thought you were > asking what's the period of the repetitive "hammer hitting" ... Still, > my reply was more figurative than anything else. Scratch all that > part. > >>> I mean, a signal that goes like {1, 3, 2, 1, 2, 5, 7, 2 ...} is an >>> infinite sum of shifted impulses. One impulse of weight 1 centered >>> at n = 0, one impulse of weight 3 centered at n = 1, one impulse of >>> weight 2 centered at n = 2, and so on. >> >> >> Right. The system's response is the superposition of its response to >> those individual impulses. If you mean that the response will last at >> least as long as the stimulus, I agree. > > > No, that'snot what I was saying -- and by the way, it's not necessarily > true (think a sine wave at the exact frequency corresponding to a zero > on the unit circle). But still, let's put that aside (you're right > in the general case, and in the context where the discussion was)Let's keep it in play. A zero on the unit circle or anywhere else isn't the point. A pole on the unit circle doesn't have absolute stability because it has no damping.>> Impulse response is a property of a system. It does not depend on the >> signal. > > > Yes, but the system's output to a given signal does. It depends on > the signal and it also depends on the system's impulse response.How does that make an impulse response endlessly repeat? An impulse response is the response to an impulse, not the response to an arbitrary signal. How can I put that so you might better understand its consequences?>>> *Because the system is linear and also time-invariant*, the output >>> of the system is the sum of the outputs of each "sub-signal" with >>> the proper weight (linearity), and the output of each sub-signal >>> is the impulse response shifted the right amount of samples (the >>> time-invariant part of the system). >> >> >> So? >> >>> The above is nothing more than the convolution (or the intuitive/ >>> visual way to understand the convolution, at least). >> >> >> Superposition is not convolution. If you understand convolution that >> way, that's too little math, not too much. > > > The above is a proof that the output of a LTI system to a given signal > is given by the convolution of the signal and the impulse response; > why do you feel that that's too little math?If you think that it makes any impulse response infinite and cyclic, yes.>>> That's why, if you have a system with impulse response that does >>> not converge to 0 as n->oo, then you feed that system with, say, >>> a unit step (x[n] = 1 for all n >= 0), and the output goes to oo >>> as n->oo. >> >> >> What's why? > > > Re-read the entire phrase, skipping the "parenthetic" part: that's > why if you feed that system with a unit step, the output goes to > oo as n goes to infinity.I have a counterexample in mind from EE 101. I leave it for now as a puzzle.> (The "that" refers to the fact that you're superposing an infinite > amount of impulse responses for which at n->oo -- which is the same > regardless of the fact that they're shifted -- and the impulse > responses converge to something other than 0) > >>> But then, coming back to the original post -- the OP was using the >>> impulse response criterion for a non-linear system, which makes the >>> criterion entirely meaningless. > > > At this point, I'd like to emphasize that I was simply trying to > make it more explicit for the OP why is it that he was mistaken in > applying a criterion that is only meaningful for LTI systems; I > was explaining my way of visualizing why that criterion works the > way it does.You told him, "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!)." We took that to mean that you believe the impulse response is cyclic. I now see what you meant, but the OP probably doesn't. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
