Hi, We know about the uniqueness abt sinusoidal waves when applied to a linear system... But how about when the input to the frequency synthesizer is a sinusoidal wave, as the o/p frequency would not be same as the input frequency? Thanks Sandeep
Uniqueness of Sinusoidal waves
Started by ●April 19, 2005
Reply by ●April 19, 20052005-04-19
Sandeep Chikkerur wrote:> Hi, > > We know about the uniqueness abt sinusoidal waves when applied to a > linear system...What? I don't have the faintest clue what you mean.> But how about when the input to the frequency synthesizer is a > sinusoidal wave, as the o/p frequency would not be same as the input > frequency?What is a "frequency synthesizer"? How does it work? Why would it accept a sinusoidal signal as input and not, say, two numbers describing the amplitude and frequency of the sinusoidal? I don't understand your question. Rune
Reply by ●April 19, 20052005-04-19
"Rune Allnor" <allnor@tele.ntnu.no> wrote in message news:<1113896208.248439.233290@l41g2000cwc.googlegroups.com>...> Sandeep Chikkerur wrote: > > Hi, > > > > We know about the uniqueness abt sinusoidal waves when applied to a > > linear system... > > What? I don't have the faintest clue what you mean. >[CUT]> > I don't understand your question. >me too..... please give us more elements. Bye Jack
Reply by ●April 19, 20052005-04-19
Sandeep Chikkerur wrote:> We know about the uniqueness abt sinusoidal waves when applied to a > linear system...The uniqueness of sinusoids is that they are the solution to the appropriate differential equation for the system. Different systems have different differential equations. A circular drum, for example, has Bessel functions as its basis functions.> But how about when the input to the frequency synthesizer is a > sinusoidal wave, as the o/p frequency would not be same as the input > frequency?For a linear system, including a system that includes integrals and derivatives sinusoid in means sinusoid out. Can you be more specific about the frequency synthesizer? -- glen
Reply by ●April 19, 20052005-04-19
glen herrmannsfeldt wrote:> Sandeep Chikkerur wrote: > >> We know about the uniqueness abt sinusoidal waves when applied to a >> linear system... > > > The uniqueness of sinusoids is that they are the solution to the > appropriate differential equation for the system. Different systems > have different differential equations. > > A circular drum, for example, has Bessel functions as its > basis functions. > >> But how about when the input to the frequency synthesizer is a >> sinusoidal wave, as the o/p frequency would not be same as the input >> frequency? > > > For a linear system, including a system that includes integrals and > derivatives sinusoid in means sinusoid out. Can you be more specific > about the frequency synthesizer? > > -- glen >For a linear time-invariant system, that is. And it can include not only integrals and derivatives, but time delays of various sorts, if you're feeling that way. But yes, please do be more specific about the frequency synthesizer. Somehow I doubt that it's a linear system, or time-invariant. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●April 19, 20052005-04-19
Tim Wescott wrote:> glen herrmannsfeldt wrote: > >> Sandeep Chikkerur wrote: >> >>> We know about the uniqueness abt sinusoidal waves when applied to a >>> linear system... >> >> >> >> The uniqueness of sinusoids is that they are the solution to the >> appropriate differential equation for the system. Different systems >> have different differential equations. >> >> A circular drum, for example, has Bessel functions as its >> basis functions. >> >>> But how about when the input to the frequency synthesizer is a >>> sinusoidal wave, as the o/p frequency would not be same as the input >>> frequency? >> >> >> >> For a linear system, including a system that includes integrals and >> derivatives sinusoid in means sinusoid out. Can you be more specific >> about the frequency synthesizer? >> >> -- glen >> > For a linear time-invariant system, that is. And it can include not > only integrals and derivatives, but time delays of various sorts, if > you're feeling that way. > > But yes, please do be more specific about the frequency synthesizer. > Somehow I doubt that it's a linear system, or time-invariant. >Due to phrasing, I wondered if OP meant "VCO" when saying "frequency synthesizer".
Reply by ●April 19, 20052005-04-19
in article 99dc70b0.0504190641.270932c3@posting.google.com, Jack Ace at jack.ace@libero.it wrote on 04/19/2005 10:41:> "Rune Allnor" <allnor@tele.ntnu.no> wrote in message > news:<1113896208.248439.233290@l41g2000cwc.googlegroups.com>... >> Sandeep Chikkerur wrote: >>> Hi, >>> >>> We know about the uniqueness abt sinusoidal waves when applied to a >>> linear system... >> >> What? I don't have the faintest clue what you mean. >> > [CUT] >> >> I don't understand your question. >> > > me too..... > > please give us more elements.i don't understand the question as stated either. i thought it might be the same sorta philosophical question as to why sinusoids appear in the theory of linear systems instead of triangle waves or square waves or something else that could conceivably be a basis set for functions. if that is the question, the answer is that the exponential function (and sinusoids are part of that class of functions because of Euler) are what we call "eigenfunctions" of linear, time-invariant, systems. that is, if an exponential goes in, the same exponential comes out, scaled by a known constant that depends on the system and the "alpha" in the exponent. similarly, if a sinusoid goes into an LTI system, a sinusoid of the same frequency comes out, but scaled and phase shifted. that can't be said for triangle waves. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●April 19, 20052005-04-19
Tim Wescott wrote:> glen herrmannsfeldt wrote:>> Sandeep Chikkerur wrote:>>> We know about the uniqueness abt sinusoidal waves when applied to a >>> linear system...(snip)>> For a linear system, including a system that includes integrals and >> derivatives sinusoid in means sinusoid out. Can you be more specific >> about the frequency synthesizer?> For a linear time-invariant system, that is. And it can include not > only integrals and derivatives, but time delays of various sorts, if > you're feeling that way.I meant it in the sense of http://mathworld.wolfram.com/LinearOperator.html which I thought included time invariance, but now I am not so sure. Though f and g don't specify time dependence. If f(t)=g(t+T), for any T, and the system satisfies linearity, then doesn't it have to be time invariant? The way I was remembering it, you specify a weight function and domain, and find out which differential equation can satisfy those. For a constant weight function, that is, time and space independent, and infinite domain, it is y''=A y, with sinusoids and exponentials as solutions. Only sinusoids don't go to infinity at +/- infinity. An explanation of basis functions and the system that they belong to is: http://mathworld.wolfram.com/GeneralizedFourierSeries.html -- glen
Reply by ●April 19, 20052005-04-19
glen herrmannsfeldt wrote:> Tim Wescott wrote: > >> glen herrmannsfeldt wrote: >- snip -> >> For a linear time-invariant system, that is. And it can include not >> only integrals and derivatives, but time delays of various sorts, if >> you're feeling that way. > > > I meant it in the sense of > > http://mathworld.wolfram.com/LinearOperator.html > > which I thought included time invariance, but now I am not so sure. > Though f and g don't specify time dependence.Linearity and time dependence are orthogonal qualities, although they are often taken together because that's what you need to do Laplace analysis, and you need shift invariance to do z-domain analysis.> > If f(t)=g(t+T), for any T, and the system satisfies linearity, then > doesn't it have to be time invariant?I don't think you're expressing it quite right. If you have a system H and a signal x(t), and y(t + T) = H(x(t + T), t) then H is time invariant (and you don't need to specify H(x, t), only H(x)). For example indefinite integrals and differentiation are linear and time-invariant, multiplication by sin(w * t) is time varying and linear, and the squaring operator y(t) = x(t)^2 is time invariant and nonlinear.> > The way I was remembering it, you specify a weight function and > domain, and find out which differential equation can satisfy those. > For a constant weight function, that is, time and space independent, > and infinite domain, it is y''=A y, with sinusoids and exponentials > as solutions. Only sinusoids don't go to infinity at +/- infinity. > > An explanation of basis functions and the system that they belong to is: > > http://mathworld.wolfram.com/GeneralizedFourierSeries.html > > -- glen >-- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●April 20, 20052005-04-20
Tim Wescott <tim@wescottnospamdesign.com> wrote in message news:<116atbft1b00e3c@corp.supernews.com>...> glen herrmannsfeldt wrote: > > Tim Wescott wrote: > > > >> glen herrmannsfeldt wrote: > > > - snip - > > > >> For a linear time-invariant system, that is. And it can include not > >> only integrals and derivatives, but time delays of various sorts, if > >> you're feeling that way. > > > > > > I meant it in the sense of > > > > http://mathworld.wolfram.com/LinearOperator.html > > > > which I thought included time invariance, but now I am not so sure. > > Though f and g don't specify time dependence. > > Linearity and time dependence are orthogonal qualities, although they > are often taken together because that's what you need to do Laplace > analysis, and you need shift invariance to do z-domain analysis. > > > > If f(t)=g(t+T), for any T, and the system satisfies linearity, then > > doesn't it have to be time invariant? > > I don't think you're expressing it quite right. If you have a system H > and a signal x(t), and y(t + T) = H(x(t + T), t) then H is time > invariant (and you don't need to specify H(x, t), only H(x)). > > For example indefinite integrals and differentiation are linear and > time-invariant, multiplication by sin(w * t) is time varying and linear, > and the squaring operator y(t) = x(t)^2 is time invariant and nonlinear. > > > > The way I was remembering it, you specify a weight function and > > domain, and find out which differential equation can satisfy those. > > For a constant weight function, that is, time and space independent, > > and infinite domain, it is y''=A y, with sinusoids and exponentials > > as solutions. Only sinusoids don't go to infinity at +/- infinity. > > > > An explanation of basis functions and the system that they belong to is: > > > > http://mathworld.wolfram.com/GeneralizedFourierSeries.html > > > > -- glen > >Hi When I say uniqueness of sinusiodal waves, I mean the input frequency of a sinusoidal wave is same as the output frequency, when applied to a Linear system. So, maintaining its frequency is the uniqueness of sinusoidal waves. What is a frequency synthesizer ? The one whose output frequency is some factor of input frequency which will not be same as the input frequency. My query is, if we apply the sinusoidal wave to a frequency synthesizer, then the output of synthesizer will not be same as the input frequency. So, why the output frequency is different from input frequency for sinusoidal waves? Is it not linear ? I hope this time I am clear... Sandeep
