"Jerry Avins" <jya@ieee.org> wrote in message news:HtudnW3re_w6pfTfRVn-qQ@rcn.net...> robert bristow-johnson wrote: > > in article 8oSdnXLQvvg0lPTfRVn-pg@rcn.net, Jerry Avins at jya@ieee.org wrote > > on 04/22/2005 10:26: > > > > > > > > input: x(t) > > > > output: y(t) = x(t) + sin(2*pi*(1 Hz)*t) > > > > i don't see how you could make that into a nonlinear system (with two > > inputs). and it ain't time-invariant. > > So steady 60 Hz hum makes an audio amplifier nonlinear? Then any > additive noise is a sign of nonlinearity! I guess there are no linear > systems, so we can all pack up and go home. :-)Well, of course no real-world system is linear. Every amplifier is limited by its power rails, so right there, you know it will fail the linearity test with large enough inputs. So if you want to be technical, I would say that real-world systems can often be well-approximated as linear systems within a specified operating range. That assumption/approximation prevents us from having to all pack up and go home!
Uniqueness of Sinusoidal waves
Started by ●April 19, 2005
Reply by ●April 22, 20052005-04-22
Reply by ●April 22, 20052005-04-22
robert bristow-johnson wrote:> in article HtudnW3re_w6pfTfRVn-qQ@rcn.net, Jerry Avins at jya@ieee.org wrote > on 04/22/2005 13:47: > > >>robert bristow-johnson wrote: >> >>>in article 8oSdnXLQvvg0lPTfRVn-pg@rcn.net, Jerry Avins at jya@ieee.org wrote >>>on 04/22/2005 10:26: >>> >>> >>> >>>>Mark wrote: >>>> >>>> >>>> >>>>>The case I was thinking of is an RF mixer (multiplier) with an RF >>>>>input, an LO "signal" and an IF output. >>>>> >>>>>If the LO "signal" is considered an external input signal to the >>>>>system, then the system is non-linear and time invariant. >>> >>> >>>yup. a two input nonlinear system. >>> >>> >>> >>>>>If the LO "signal" is considered as an inherent part of the system, >>>>>(draw the doted line around the LO and the mixer) then the system is >>>>>linear but time variant. >>>> >>>>Well, while we're there, can anyone describe a "linear, time varying" >>>>system that one _can't_ as well (and for my part, better) describe as >>>>nonlinear? >>> >>> >>> input: x(t) >>> >>>output: y(t) = x(t) + sin(2*pi*(60 Hz)*t) >>> >>>i don't see how you could make that into a nonlinear system (with two >>>inputs). and it ain't time-invariant. >> >>So steady 60 Hz hum makes an audio amplifier nonlinear? > > > NO! perhaps i misunderstood you. i thought your challenge was: to describe > a linear time-varying system that one cannot as well describe as nonlinear. > i described a linear, time-varying system that cannot be alternatively > described as a nonlinear system with two inputs. it *can* be described as a > linear system with two inputs. > > >>Then any additive noise is a sign of nonlinearity! > > > NO! not at all! somehow, at least one of us (could be me) are > misunderstanding the other.It's probably me. I took input: x(t) output: y(t) = x(t) + sin(2*pi*(60 Hz)*t) to mean that the output consists of two parts, the second being a sinusoid of 1 Hz (which you graciously changed to 60 Hmm!), and independent of the input. Where did I go wrong? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●April 22, 20052005-04-22
Fred Marshall wrote:> "Jerry Avins" <jya@ieee.org> wrote in message > news:HtudnW3re_w6pfTfRVn-qQ@rcn.net... > >>robert bristow-johnson wrote: >> >>>in article 8oSdnXLQvvg0lPTfRVn-pg@rcn.net, Jerry Avins at jya@ieee.org >>>wrote >>>on 04/22/2005 10:26: >>> >>> >>> >>>>Mark wrote: >>>> >>>> >>>> >>>>>The case I was thinking of is an RF mixer (multiplier) with an RF >>>>>input, an LO "signal" and an IF output. >>>>> >>>>>If the LO "signal" is considered an external input signal to the >>>>>system, then the system is non-linear and time invariant. >>> >>> >>>yup. a two input nonlinear system. >>> >>> >>> >>>>>If the LO "signal" is considered as an inherent part of the system, >>>>>(draw the doted line around the LO and the mixer) then the system is >>>>>linear but time variant. >>>> >>>>Well, while we're there, can anyone describe a "linear, time varying" >>>>system that one _can't_ as well (and for my part, better) describe as >>>>nonlinear? >>> >>> >>> input: x(t) >>> >>> output: y(t) = x(t) + sin(2*pi*(1 Hz)*t) >>> >>>i don't see how you could make that into a nonlinear system (with two >>>inputs). and it ain't time-invariant. >> >>So steady 60 Hz hum makes an audio amplifier nonlinear? Then any additive >>noise is a sign of nonlinearity! I guess there are no linear systems, so >>we can all pack up and go home. :-) >> >>Jerry >>-- > > > Jerry, > > r b-j didn't say it was nonlinear. He said the opposite and he said it was > time-varyiing.You're right. In my fevered zeal, I overlooked that. But there two parts to Robert's output, one a linear replica of the input, the other, a sinusoid of constant frequency. That's a fair description of the first audio amplifier I designed and built. (Tracking down and subduing the hum was how I learned my trade.) Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●April 22, 20052005-04-22
Jon Harris wrote:> "Jerry Avins" <jya@ieee.org> wrote in message > news:HtudnW3re_w6pfTfRVn-qQ@rcn.net... > >>robert bristow-johnson wrote: >> >>>in article 8oSdnXLQvvg0lPTfRVn-pg@rcn.net, Jerry Avins at jya@ieee.org wrote >>>on 04/22/2005 10:26: >>> >>> >>> >>> input: x(t) >>> >>> output: y(t) = x(t) + sin(2*pi*(1 Hz)*t) >>> >>>i don't see how you could make that into a nonlinear system (with two >>>inputs). and it ain't time-invariant. >> >>So steady 60 Hz hum makes an audio amplifier nonlinear? Then any >>additive noise is a sign of nonlinearity! I guess there are no linear >>systems, so we can all pack up and go home. :-) > > > Well, of course no real-world system is linear. Every amplifier is limited by > its power rails, so right there, you know it will fail the linearity test with > large enough inputs. So if you want to be technical, I would say that > real-world systems can often be well-approximated as linear systems within a > specified operating range. That assumption/approximation prevents us from > having to all pack up and go home!Now that I've had some coffee to calm my nerves, I'm quite relieved. It was the my false conclusion that the added sinusoid classified the system as non linear, extending that to added noise, and recognizing that noise, like the poor, will always be with us, that gave me a bad turn. I'm better now. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●April 22, 20052005-04-22
Fred Marshall wrote:> No way. I ain't gonna do it! :-)Me neither. But in my heart of hearts, I know that if a frequency comes out that wasn't put in, it's non linear, and that includes modulators. I've been definitioned out of repeating that notion in public, and I was looking for a way back to the comfort of my old ways. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●April 22, 20052005-04-22
"Jerry Avins" <jya@ieee.org> wrote in message news:aKidndS7YdXlz_TfRVn-gQ@rcn.net...> Fred Marshall wrote: > > > No way. I ain't gonna do it! :-) > > Me neither. But in my heart of hearts, I know that if a frequency comes > out that wasn't put in, it's non linear, and that includes modulators. > I've been definitioned out of repeating that notion in public, and I was > looking for a way back to the comfort of my old ways.It seems that non-linear and time-variant get lumped together and mixed up quite a bit in casual use. BTW, many common audio effects such as chorusing, flanging, phasing, and some of your more sophisticated reverbs are linear/time-variant.
Reply by ●April 22, 20052005-04-22
"Jerry Avins" <jya@ieee.org> wrote in message news:aKidndS7YdXlz_TfRVn-gQ@rcn.net...> Fred Marshall wrote: > > >> No way. I ain't gonna do it! :-) > > Me neither. But in my heart of hearts, I know that if a frequency comes > out that wasn't put in, it's non linear, and that includes modulators. > I've been definitioned out of repeating that notion in public, and I was > looking for a way back to the comfort of my old ways. > > Jerry > -- >Well, at least you a) remember and b) admit it. :-) I'm going to start another thread that goes into maybe what's bothering you. Be on the lookout. News at 6! Fred
Reply by ●April 23, 20052005-04-23
Jerry Avins wrote:> robert bristow-johnson wrote: > >> in article 8oSdnXLQvvg0lPTfRVn-pg@rcn.net, Jerry Avins at jya@ieee.org >> wrote >> on 04/22/2005 10:26: >> >> >>> Mark wrote: >>> >>> >>>> The case I was thinking of is an RF mixer (multiplier) with an RF >>>> input, an LO "signal" and an IF output. >>>> >>>> If the LO "signal" is considered an external input signal to the >>>> system, then the system is non-linear and time invariant. >> >> >> >> yup. a two input nonlinear system. >> >> >>>> If the LO "signal" is considered as an inherent part of the system, >>>> (draw the doted line around the LO and the mixer) then the system is >>>> linear but time variant. >>> >>> >>> Well, while we're there, can anyone describe a "linear, time varying" >>> system that one _can't_ as well (and for my part, better) describe as >>> nonlinear? >> >> >> >> input: x(t) >> >> output: y(t) = x(t) + sin(2*pi*(1 Hz)*t) >> >> i don't see how you could make that into a nonlinear system (with two >> inputs). and it ain't time-invariant. > > > So steady 60 Hz hum makes an audio amplifier nonlinear? Then any > additive noise is a sign of nonlinearity! I guess there are no linear > systems, so we can all pack up and go home. :-) > > JerryYup. Take up basket weaving or something. Or model the system as a linear one with an unaccessible input (not too far from the truth with a 60Hz hum), or one with a perfectly metastable pole (or pole pair) that started life with just the right state. There is a slightly more inclusive definition that revolves around the term "affine", but I can never keep it straight for very long; it does take this kind of thing into account. Keep in mind that there are _no_ linear systems in real life: everything is nonlinear, but the math is way hard. So we pretend with varying degrees of success that our systems are linear, and get models with varying degrees of fidelity. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●April 23, 20052005-04-23
Tim Wescott wrote: ...> Keep in mind that there are _no_ linear systems in real life: everything > is nonlinear, but the math is way hard. So we pretend with varying > degrees of success that our systems are linear, and get models with > varying degrees of fidelity.Yup again. Linear systems are constructed of rigid bodies. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●April 23, 20052005-04-23
On Fri, 22 Apr 2005 14:01:58 -0700, "Jon Harris" <jon_harrisTIGER@hotmail.com> wrote:>"Jerry Avins" <jya@ieee.org> wrote in message >news:aKidndS7YdXlz_TfRVn-gQ@rcn.net... >> Fred Marshall wrote: >> >> > No way. I ain't gonna do it! :-) >> >> Me neither. But in my heart of hearts, I know that if a frequency comes >> out that wasn't put in, it's non linear, and that includes modulators. >> I've been definitioned out of repeating that notion in public, and I was >> looking for a way back to the comfort of my old ways. > >It seems that non-linear and time-variant get lumped together and mixed up quite >a bit in casual use. > >BTW, many common audio effects such as chorusing, flanging, phasing, and some of >your more sophisticated reverbs are linear/time-variant.Hi, unless I'm mistaken, another example of a linear/time-variant process is decimation (so long as no aliasing errors occur). [-Rick-]
