On Sun, 18 Oct 2009 17:18:42 -0700, Fully Half Baked wrote:> On Oct 18, 8:02 pm, Tim Wescott <t...@seemywebsite.com> wrote: >> On Sun, 18 Oct 2009 09:45:12 -0700, Fully Half Baked wrote: >> > On Oct 17, 8:05 pm, Tim Wescott <t...@seemywebsite.com> wrote: >> >> On Sat, 17 Oct 2009 12:30:43 -0500, fisico32 wrote: >> >> > A system is described by an equation that relates the output >> >> > function y(t) and the input function x(t). Both x(t) and y(t) are >> >> > functions of time. If the system is time invariant, it means that >> >> > the mechanisms of the equation are not time dependent (the >> >> > coefficients are constants). >> >> >> > Ex: y(t)=3*x(t)+ [x(t)]^2 >> >> >> > This means that y depends only "implicitly" on time, but not >> >> > explicitly: y(x)= y(x(t)). >> >> > y(t) can be written as a function of time only however: Ex: >> >> > x(t)=2*t, then y(t)=3*(2t)+ [2t]^2 >> >> >> > If the system is time variant, then the equation describing the >> >> > relation between input and output has time t variable appearing as >> >> > an explicit variable: >> >> >> > Ex: y(t)=5*t+x(t) >> >> >> > so y(t,x)=y(t, x(t)). >> >> > y(t) can be written only as a function of time t too. x(t)=2t, >> >> > then y(t)=5*t+2t >> >> >> > Q: If I was ONLY given the function y(t) and was asked if it is >> >> > the output of a time variant or invariant system, would I be able >> >> > to tell? >> >> >> > thanks >> >> > fisico32 >> >> >> Normal terminology in signals & systems is to call x(t) and y(t) >> >> _signals_. Yes, their values are functions of time, but you don't >> >> really care about the "functional" part nearly as much as you care >> >> about their behavior. Think of them as continuous vectors that >> >> "just are" more than as functions. >> >> >> A more general way to describe a time varying system is to define >> >> the system h as >> >> >> y(t) = h(x(t), t). >> >> >> In other words, h is some "thing" that acts on the input signal x(t) >> >> to generate the output signal y(t). The nice thing about the above >> >> definition is that you can immediately shift the input and output >> >> signals by some time t_s: >> >> >> y(t - t_s) =? h(x(t - t_s), t). >> >> >> If the above y and x _always_ match the non-shifted case for _all_ >> >> possible time shifts and _all_ possible input signals then the >> >> system is time invariant. >> >> >> Now, to answer your question: >> >> >> No. If you were given both the input and the output signals for all >> >> time, you could _sometimes_ determine that the system was either >> >> time varying or nonlinear. I don't think you could conclusively >> >> prove that the system was a linear time invariant system just from >> >> one sample x(t) and it's resulting sample y(t), however. >> >> >> --www.wescottdesign.com >> >> > Is it not true that if a system is nonlinear it's spectrum will have >> > changed which is easily measurable as long as the change isn't too >> > small to measure? >> >> Any system can _change_ the spectrum of the input. A linear system can >> only change the amplitude of energy that was already in the input, a >> time- varying system can only convolve the input spectrum with it's own >> "time- varying-ness" spectrum and change the amplitude of energy that's >> already in the input, and will do so in a way that obeys superposition. >> A nonlinear system can do any damn thing it pleases with the spectrum >> of the input signal _and_ won't obey superposition. >> >> But the OP is asking if he can look at the output signal _only_. If he >> really means what he says, that he's just been handed a signal and told >> "here, this is the output of a system" then he has no clue about the >> linearity or time invariance of the system. If he's given the input >> _and_ the output, then he may be able to say "that system isn't LTI", >> but with just one input and one output signal I don't think he can say >> _for sure_ that the system is indeed LTI, or if not if it is nonlinear >> or time varying. >> >> --www.wescottdesign.com > > Ok maybe my idea of what is linear or not is a bit too simplistic. What > I mean by change the spectrum is new frequencies are introduced not just > changes in amplitude or shifts in time. Having said that fisico32 has > said "not only nonlinear systems generate new frequencies in the output > spectrum" but I don't know how that can happen if a system is said to be > linear unless you deliberately generate new frequencies and add them to > the output but then the overall effect is still nonlinear.A time-varying system will add frequencies to the spectrum, because it multiplies the signal by a time-varying parameter. The effect on the spectrum is to convolve the signal's spectrum with the time-varying parameter's spectrum. So a time varying system can't generate frequencies from _nothing_, but it can certainly have more frequencies at the output than at the input. -- www.wescottdesign.com
time variant or invariant output?
Started by ●October 17, 2009
Reply by ●October 19, 20092009-10-19
Reply by ●October 19, 20092009-10-19
Tim Wescott wrote: ...> A time-varying system will add frequencies to the spectrum, because it > multiplies the signal by a time-varying parameter. The effect on the > spectrum is to convolve the signal's spectrum with the time-varying > parameter's spectrum. > > So a time varying system can't generate frequencies from _nothing_, but > it can certainly have more frequencies at the output than at the input.Can a nonlinear system generate frequencies from _nothing_? Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●October 19, 20092009-10-19
On Mon, 19 Oct 2009 11:17:09 -0400, Jerry Avins wrote:> Tim Wescott wrote: > > ... > >> A time-varying system will add frequencies to the spectrum, because it >> multiplies the signal by a time-varying parameter. The effect on the >> spectrum is to convolve the signal's spectrum with the time-varying >> parameter's spectrum. >> >> So a time varying system can't generate frequencies from _nothing_, but >> it can certainly have more frequencies at the output than at the input. > > Can a nonlinear system generate frequencies from _nothing_? >Define "nothing". A time-varying system cannot generate an output signal with spectral components that are unrelated to the spectrum of the input signal. A nonlinear system can -- define the system y = h(x, t) as y = sin(w*t) It's time-varying, it's nonlinear (it certainly doesn't obey superposition!), and it generates an output signal from as close to nothing as you can get. -- www.wescottdesign.com
Reply by ●October 19, 20092009-10-19
Tim Wescott wrote:> On Mon, 19 Oct 2009 11:17:09 -0400, Jerry Avins wrote: > >> Tim Wescott wrote: >> >> ... >> >>> A time-varying system will add frequencies to the spectrum, because it >>> multiplies the signal by a time-varying parameter. The effect on the >>> spectrum is to convolve the signal's spectrum with the time-varying >>> parameter's spectrum. >>> >>> So a time varying system can't generate frequencies from _nothing_, but >>> it can certainly have more frequencies at the output than at the input. >> Can a nonlinear system generate frequencies from _nothing_? >> > Define "nothing". > > A time-varying system cannot generate an output signal with spectral > components that are unrelated to the spectrum of the input signal. A > nonlinear system can -- define the system > > y = h(x, t) > > as y = sin(w*t) > > It's time-varying, it's nonlinear (it certainly doesn't obey > superposition!), and it generates an output signal from as close to > nothing as you can get.You lost me. (That's not hard.) As far as I can see, you're setting up sin(w*t) = h(x, t). Then the innards of h(x, t) are immaterial; y is given. Where's the non-linearity? Are you describing an oscillator with dummy input terminals? If so, we agree. A balanced diode bridge is non-linear. No x emerges at all, only its harmonics. It's so bad that the term "distortion" hardly applies. Nevertheless, when no signal is applied, none emerges. Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●October 19, 20092009-10-19
Jerry Avins <jya@ieee.org> wrote:> Tim Wescott wrote:(snip)>> So a time varying system can't generate frequencies from _nothing_, but >> it can certainly have more frequencies at the output than at the input.> Can a nonlinear system generate frequencies from _nothing_?Isn't that what we call an oscillator? There is in non-linear optics something called a phase conjugate mirror. It reflects a signal (light beam), reversed in time from the original. (It has to be large enough to do that.) One use for them is in fiber optics systems with dispersion: If you put a phase conjugate mirror in the middle of a long optical fiber, the second half will undo the dispersion for the first half. If you have a phase conjugate mirror with gain (they are active devices needing input power), it gets very interesting if you hold a shiny object nearby. (The reflected wavefront arrives back at the same phase and higher amplitude than when it left, just in time for a new reflection.) -- glen
Reply by ●October 19, 20092009-10-19
On 10/19/2009 10:26 AM, glen herrmannsfeldt wrote:> Jerry Avins<jya@ieee.org> wrote: >> Tim Wescott wrote: > (snip) > >>> So a time varying system can't generate frequencies from _nothing_, but >>> it can certainly have more frequencies at the output than at the input. > >> Can a nonlinear system generate frequencies from _nothing_? > > Isn't that what we call an oscillator? > > There is in non-linear optics something called a phase conjugate mirror. > > It reflects a signal (light beam), reversed in time from the original. > (It has to be large enough to do that.) One use for them is in > fiber optics systems with dispersion: If you put a phase conjugate > mirror in the middle of a long optical fiber, the second half will > undo the dispersion for the first half. > > If you have a phase conjugate mirror with gain (they are active > devices needing input power), it gets very interesting if you > hold a shiny object nearby. (The reflected wavefront arrives > back at the same phase and higher amplitude than when it left, > just in time for a new reflection.) > > -- glenThat sounds like it'd fit in the causal/non-causal thread really well. ;) -- Eric Jacobsen Minister of Algorithms Abineau Communications http://www.abineau.com
Reply by ●October 19, 20092009-10-19
On Mon, 19 Oct 2009 12:26:16 -0400, Jerry Avins wrote:> Tim Wescott wrote: >> On Mon, 19 Oct 2009 11:17:09 -0400, Jerry Avins wrote: >> >>> Tim Wescott wrote: >>> >>> ... >>> >>>> A time-varying system will add frequencies to the spectrum, because >>>> it multiplies the signal by a time-varying parameter. The effect on >>>> the spectrum is to convolve the signal's spectrum with the >>>> time-varying parameter's spectrum. >>>> >>>> So a time varying system can't generate frequencies from _nothing_, >>>> but it can certainly have more frequencies at the output than at the >>>> input. >>> Can a nonlinear system generate frequencies from _nothing_? >>> >> Define "nothing". >> >> A time-varying system cannot generate an output signal with spectral >> components that are unrelated to the spectrum of the input signal. A >> nonlinear system can -- define the system >> >> y = h(x, t) >> >> as y = sin(w*t) >> >> It's time-varying, it's nonlinear (it certainly doesn't obey >> superposition!), and it generates an output signal from as close to >> nothing as you can get. > > You lost me. (That's not hard.) As far as I can see, you're setting up > sin(w*t) = h(x, t). Then the innards of h(x, t) are immaterial; y is > given. Where's the non-linearity? Are you describing an oscillator with > dummy input terminals? If so, we agree.It was perhaps too simple an example. What's the simplest linear system? h(x) where y = 0. Not very interesting, but it _does_ obey superposition. So what's the simplest nonlinear system? h(x) where y = something nonzero, and unrelated to x. Like, ferinstance, y = sin(w*t). Could be interesting, may leave you scratching your head, but it certainly doesn't obey superposition! If it makes you feel better, let h(x,t) define y(t) = sin(w*t) when |x| > (some threshold), and 0 otherwise. In other words, it's an oscillator that's switched by the input signal. _Now_ it's palpably nonlinear, not only responds to x but has an average power output that is monotonically increasing with x _and_ is less trivial.> A balanced diode bridge is non-linear. No x emerges at all, only its > harmonics. It's so bad that the term "distortion" hardly applies. > Nevertheless, when no signal is applied, none emerges.I considered that as an example, except that someone would have come back with the plaint that for x a sine wave, you get x's harmonics out, and how's one set of spikes in the output spectrum (from doubling) distinguishable from another set (from mixing)? -- www.wescottdesign.com
Reply by ●October 19, 20092009-10-19
Eric Jacobsen <eric.jacobsen@ieee.org> wrote:> On 10/19/2009 10:26 AM, glen herrmannsfeldt wrote:>> There is in non-linear optics something called a phase conjugate mirror.>> It reflects a signal (light beam), reversed in time from the original. >> (It has to be large enough to do that.) One use for them is in >> fiber optics systems with dispersion: If you put a phase conjugate >> mirror in the middle of a long optical fiber, the second half will >> undo the dispersion for the first half.(snip)> That sounds like it'd fit in the causal/non-causal thread really well. ;)http://en.wikipedia.org/wiki/Nonlinear_optics#Optical_phase_conjugation The whole article on Nonlinear optics is pretty interesting. The link is to the section on phase conjugation. Now, how about DSP related audio phase conjugation systems? -- glen
Reply by ●October 21, 20092009-10-21
On Oct 19, 2:28�am, Jerry Avins <j...@ieee.org> wrote:> Fully Half Baked wrote: > > On Oct 18, 8:02 pm, Tim Wescott <t...@seemywebsite.com> wrote: > >> On Sun, 18 Oct 2009 09:45:12 -0700, Fully Half Baked wrote: > >>> On Oct 17, 8:05 pm, Tim Wescott <t...@seemywebsite.com> wrote: > >>>> On Sat, 17 Oct 2009 12:30:43 -0500, fisico32 wrote: > >>>>> A system is described by an equation that relates the output function > >>>>> y(t) and the input function x(t). Both x(t) and y(t) are functions of > >>>>> time. If the system is time invariant, it means that the mechanisms > >>>>> of the equation are not time dependent (the coefficients are > >>>>> constants). > >>>>> Ex: y(t)=3*x(t)+ [x(t)]^2 > >>>>> This means that y depends only "implicitly" on time, but not > >>>>> explicitly: y(x)= y(x(t)). > >>>>> y(t) can be written as a function of time only however: Ex: x(t)=2*t, > >>>>> then y(t)=3*(2t)+ [2t]^2 > >>>>> If the system is time variant, then the equation describing the > >>>>> relation between input and output has time t variable appearing as an > >>>>> explicit variable: > >>>>> Ex: y(t)=5*t+x(t) > >>>>> so y(t,x)=y(t, x(t)). > >>>>> y(t) can be written only as a function of time t too. x(t)=2t, then > >>>>> y(t)=5*t+2t > >>>>> Q: If I was ONLY given the function y(t) and was asked if it is the > >>>>> output of a time variant or invariant system, would I be able to > >>>>> tell? > >>>>> thanks > >>>>> fisico32 > >>>> Normal terminology in signals & systems is to call x(t) and y(t) > >>>> _signals_. �Yes, their values are functions of time, but you don't > >>>> really care about the "functional" part nearly as much as you care > >>>> about their behavior. �Think of them as continuous vectors that "just > >>>> are" more than as functions. > >>>> A more general way to describe a time varying system is to define the > >>>> system h as > >>>> y(t) = h(x(t), t). > >>>> In other words, h is some "thing" that acts on the input signal x(t) to > >>>> generate the output signal y(t). �The nice thing about the above > >>>> definition is that you can immediately shift the input and output > >>>> signals by some time t_s: > >>>> y(t - t_s) =? h(x(t - t_s), t). > >>>> If the above y and x _always_ match the non-shifted case for _all_ > >>>> possible time shifts and _all_ possible input signals then the system > >>>> is time invariant. > >>>> Now, to answer your question: > >>>> No. �If you were given both the input and the output signals for all > >>>> time, you could _sometimes_ determine that the system was either time > >>>> varying or nonlinear. �I don't think you could conclusively prove that > >>>> the system was a linear time invariant system just from one sample x(t) > >>>> and it's resulting sample y(t), however. > >>>> --www.wescottdesign.com > >>> Is it not true that if a system is nonlinear it's spectrum will have > >>> changed which is easily measurable as long as the change isn't too small > >>> to measure? > >> Any system can _change_ the spectrum of the input. �A linear system can > >> only change the amplitude of energy that was already in the input, a time- > >> varying system can only convolve the input spectrum with it's own "time- > >> varying-ness" spectrum and change the amplitude of energy that's already > >> in the input, and will do so in a way that obeys superposition. �A > >> nonlinear system can do any damn thing it pleases with the spectrum of > >> the input signal _and_ won't obey superposition. > > >> But the OP is asking if he can look at the output signal _only_. �If he > >> really means what he says, that he's just been handed a signal and told > >> "here, this is the output of a system" then he has no clue about the > >> linearity or time invariance of the system. �If he's given the input > >> _and_ the output, then he may be able to say "that system isn't LTI", but > >> with just one input and one output signal I don't think he can say _for > >> sure_ that the system is indeed LTI, or if not if it is nonlinear or time > >> varying. > > >> --www.wescottdesign.com > > > Ok maybe my idea of what is linear or not is a bit too simplistic. > > What I mean by change the spectrum is new frequencies are introduced > > not just changes in amplitude or shifts in time. > > Having said that fisico32 has said "not only nonlinear systems > > generate new frequencies in the output spectrum" but I don't know how > > that can happen if a system is said to be linear unless you > > deliberately generate new frequencies and add them to the output but > > then the overall effect is still nonlinear. > > To continue in a simplistic vein, imagine an amplifier with time-varying > gain; the input is a single sinusoid of 2KHz, and the gain varies > sinusoidally at a frequency of 50 Hz. What is the output spectrum? > > Jerry > -- > Engineering is the art of making what you want from things you can get. > �����������������������������������������������������������������������You got me there.
Reply by ●October 21, 20092009-10-21
On Oct 19, 5:17�am, Tim Wescott <t...@seemywebsite.com> wrote:> On Sun, 18 Oct 2009 17:18:42 -0700, Fully Half Baked wrote: > > On Oct 18, 8:02�pm, Tim Wescott <t...@seemywebsite.com> wrote: > >> On Sun, 18 Oct 2009 09:45:12 -0700, Fully Half Baked wrote: > >> > On Oct 17, 8:05�pm, Tim Wescott <t...@seemywebsite.com> wrote: > >> >> On Sat, 17 Oct 2009 12:30:43 -0500, fisico32 wrote: > >> >> > A system is described by an equation that relates the output > >> >> > function y(t) and the input function x(t). Both x(t) and y(t) are > >> >> > functions of time. If the system is time invariant, it means that > >> >> > the mechanisms of the equation are not time dependent (the > >> >> > coefficients are constants). > > >> >> > Ex: y(t)=3*x(t)+ [x(t)]^2 > > >> >> > This means that y depends only "implicitly" on time, but not > >> >> > explicitly: y(x)= y(x(t)). > >> >> > y(t) can be written as a function of time only however: Ex: > >> >> > x(t)=2*t, then y(t)=3*(2t)+ [2t]^2 > > >> >> > If the system is time variant, then the equation describing the > >> >> > relation between input and output has time t variable appearing as > >> >> > an explicit variable: > > >> >> > Ex: y(t)=5*t+x(t) > > >> >> > so y(t,x)=y(t, x(t)). > >> >> > y(t) can be written only as a function of time t too. x(t)=2t, > >> >> > then y(t)=5*t+2t > > >> >> > Q: If I was ONLY given the function y(t) and was asked if it is > >> >> > the output of a time variant or invariant system, would I be able > >> >> > to tell? > > >> >> > thanks > >> >> > fisico32 > > >> >> Normal terminology in signals & systems is to call x(t) and y(t) > >> >> _signals_. �Yes, their values are functions of time, but you don't > >> >> really care about the "functional" part nearly as much as you care > >> >> about their behavior. �Think of them as continuous vectors that > >> >> "just are" more than as functions. > > >> >> A more general way to describe a time varying system is to define > >> >> the system h as > > >> >> y(t) = h(x(t), t). > > >> >> In other words, h is some "thing" that acts on the input signal x(t) > >> >> to generate the output signal y(t). �The nice thing about the above > >> >> definition is that you can immediately shift the input and output > >> >> signals by some time t_s: > > >> >> y(t - t_s) =? h(x(t - t_s), t). > > >> >> If the above y and x _always_ match the non-shifted case for _all_ > >> >> possible time shifts and _all_ possible input signals then the > >> >> system is time invariant. > > >> >> Now, to answer your question: > > >> >> No. �If you were given both the input and the output signals for all > >> >> time, you could _sometimes_ determine that the system was either > >> >> time varying or nonlinear. �I don't think you could conclusively > >> >> prove that the system was a linear time invariant system just from > >> >> one sample x(t) and it's resulting sample y(t), however. > > >> >> --www.wescottdesign.com > > >> > Is it not true that if a system is nonlinear it's spectrum will have > >> > changed which is easily measurable as long as the change isn't too > >> > small to measure? > > >> Any system can _change_ the spectrum of the input. �A linear system can > >> only change the amplitude of energy that was already in the input, a > >> time- varying system can only convolve the input spectrum with it's own > >> "time- varying-ness" spectrum and change the amplitude of energy that's > >> already in the input, and will do so in a way that obeys superposition. > >> �A nonlinear system can do any damn thing it pleases with the spectrum > >> of the input signal _and_ won't obey superposition. > > >> But the OP is asking if he can look at the output signal _only_. �If he > >> really means what he says, that he's just been handed a signal and told > >> "here, this is the output of a system" then he has no clue about the > >> linearity or time invariance of the system. �If he's given the input > >> _and_ the output, then he may be able to say "that system isn't LTI", > >> but with just one input and one output signal I don't think he can say > >> _for sure_ that the system is indeed LTI, or if not if it is nonlinear > >> or time varying. > > >> --www.wescottdesign.com > > > Ok maybe my idea of what is linear or not is a bit too simplistic. What > > I mean by change the spectrum is new frequencies are introduced not just > > changes in amplitude or shifts in time. Having said that fisico32 has > > said "not only nonlinear systems generate new frequencies in the output > > spectrum" but I don't know how that can happen if a system is said to be > > linear unless you deliberately generate new frequencies and add them to > > the output but then the overall effect is still nonlinear. > > A time-varying system will add frequencies to the spectrum, because it > multiplies the signal by a time-varying parameter. �The effect on the > spectrum is to convolve the signal's spectrum with the time-varying > parameter's spectrum. > > So a time varying system can't generate frequencies from _nothing_, but > it can certainly have more frequencies at the output than at the input. > > --www.wescottdesign.comMaybe it's more like this http://en.wikipedia.org/wiki/Nonlinear_system
