in article d5d88eb5.0504192048.4b1eae37@posting.google.com, Sandeep Chikkerur at sandeep_mc81@yahoo.com wrote on 04/20/2005 00:48:> 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....> So, [if] the output frequency is different from input > frequency for sinusoidal waves? Is it not linear ? > > I hope this time I am clear...i think more clear this time. yes, it is true that if a device outputs sinusoidal frequency components that did not exist in its input, that such a device must necessarily be non-linear. when a sinusoid is input to a linear time-invariant system, the output of such a system will be a sinusoid of exactly the same frequency, but scaled in amplitude and shifted in phase. this is a direct result of the fact that exponential functions are eigenfunctions of linear time-invariant systems. did you want to see such proven? (not sure that i'm willing to do it.) -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Uniqueness of Sinusoidal waves
Started by ●April 19, 2005
Reply by ●April 20, 20052005-04-20
Reply by ●April 20, 20052005-04-20
robert bristow-johnson wrote:> in article d5d88eb5.0504192048.4b1eae37@posting.google.com, Sandeep > Chikkerur at sandeep_mc81@yahoo.com wrote on 04/20/2005 00:48: > > >>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. > > ... > >>So, [if] the output frequency is different from input >>frequency for sinusoidal waves? Is it not linear ? >> >>I hope this time I am clear... > > > i think more clear this time. yes, it is true that if a device outputs > sinusoidal frequency components that did not exist in its input, that such a > device must necessarily be non-linear. when a sinusoid is input to a linear > time-invariant system, the output of such a system will be a sinusoid of > exactly the same frequency, but scaled in amplitude and shifted in phase. > this is a direct result of the fact that exponential functions are > eigenfunctions of linear time-invariant systems. > > did you want to see such proven? (not sure that i'm willing to do it.) >Oh I hate to nitpick. The system y(t) = sin(w*t)*x(t) is linear but time-varying. When x(t) is a sinusoid with frequency q y(t) will contain two frequency components at w+q and w-q. The property of only shifting a sinusoidal input's amplitude and phase belongs to linear time invariant systems, and is at least a necessary condition for a system to be such (I don't know if it's sufficient). -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●April 20, 20052005-04-20
Sandeep Chikkerur wrote:> 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... >Do you mean a device that takes a frequency command and generates a signal at that frequency, such that giving it a sine wave will result in frequency modulation, or do you mean a device that takes a sinusoidal input and gives an output at a fixed multiple? At any rate, if you have a device whose output frequency is different from its input frequency then it is at least linear time-varying. If it can be honestly described as a "frequency synthesizer" by either of my descriptions above then it is nonlinear. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●April 20, 20052005-04-20
"robert bristow-johnson" <rbj@audioimagination.com> wrote in message news:BE8B59F0.65A8%rbj@audioimagination.com...> in article d5d88eb5.0504192048.4b1eae37@posting.google.com, Sandeep > Chikkerur at sandeep_mc81@yahoo.com wrote on 04/20/2005 00:48: > >> 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. > ... >> So, [if] the output frequency is different from input >> frequency for sinusoidal waves? Is it not linear ? >> >> I hope this time I am clear... > > i think more clear this time. yes, it is true that if a device outputs > sinusoidal frequency components that did not exist in its input, that such > a > device must necessarily be non-linear. when a sinusoid is input to a > linear > time-invariant system, the output of such a system will be a sinusoid of > exactly the same frequency, but scaled in amplitude and shifted in phase. > this is a direct result of the fact that exponential functions are > eigenfunctions of linear time-invariant systems. > > did you want to see such proven? (not sure that i'm willing to do it.) > > -- > > r b-j rbj@audioimagination.com >Robert, There's a counter example to "must necessarily be nonlinear". Jerry and I and others had a long discussion about this some time back. I think we all learned something. I know that I did! You can have other sinousoids at the output of a linear system if the system is time-varying in a regular way (as in periodic). A good example is a multiplier with inputs x and y like this: Let x be the input leading to the output of interest. Let y be a sinusoid so that the system is time varying in a regular way. The system will meet all the linearity tests with regard to input x and the output. And, there are new frequencies at the output. y could be a square wave or ...... whatever ... as long as it's periodic. Fred
Reply by ●April 20, 20052005-04-20
Tim Wescott wrote:> Sandeep Chikkerur wrote:(snip)>> 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 ?(snip)> Do you mean a device that takes a frequency command and generates a > signal at that frequency, such that giving it a sine wave will result in > frequency modulation, or do you mean a device that takes a sinusoidal > input and gives an output at a fixed multiple?The one I think of for frequency synthesizer is: http://www.home.agilent.com/USeng/nav/-536894773.536881301/pd.html Which generates sines from 1mHz to 21MHz. That is millihertz to Megahertz. It has combinations of reference oscillators, phase locked oscillators, and mixers that depend on the desired frequency. I don't know if this relates to the OP question, though. -- glen
Reply by ●April 20, 20052005-04-20
in article 7-6dnd3GkPIjGPvfRVn-ug@centurytel.net, Fred Marshall at fmarshallx@remove_the_x.acm.org wrote on 04/20/2005 12:39:> There's a counter example to "must necessarily be nonlinear".looks like Tim is counter-exampling it below. in article 116d0d17telkg75@corp.supernews.com, Tim Wescott at tim@wescottnospamdesign.com wrote on 04/20/2005 12:21:> robert bristow-johnson wrote:>> it is true that if a device outputs >> sinusoidal frequency components that did not exist in its input, that such a >> device must necessarily be non-linear. when a sinusoid is input to a linear >> time-invariant system, the output of such a system will be a sinusoid of >> exactly the same frequency, but scaled in amplitude and shifted in phase. >> this is a direct result of the fact that exponential functions are >> eigenfunctions of linear time-invariant systems. >> >> did you want to see such proven? (not sure that i'm willing to do it.) >> > Oh I hate to nitpick. > > The system y(t) = sin(w*t)*x(t) is linear but time-varying.absolutely right. sometimes i remember to include "time-invariant" when i say "linear" and sometimes i forget.> The property of only shifting a sinusoidal input's amplitude and phase > belongs to linear time invariant systems, and is at least a necessary > condition for a system to be such (I don't know if it's sufficient).i think you can show it's sufficient (that sinusoidal in -> sinusoidal out of same frequency means the system is LTI) if the premise is true for all input frequencies. from that premise, you essentially get a frequency response and from that you get convolution and from that you get LTI. might not be causal, but it will be LTI. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●April 21, 20052005-04-21
Sandeep Chikkerur wrote: ...> 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.Can you give an example of such a thing? I never heard of one. (I know about tuned doublers and triplers, and about Western Electric's sub harmonic generator that used 60 Hz input to generate a 20 Hz ring tone, but I don't think you mean such devices.) ... Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●April 21, 20052005-04-21
"Jerry Avins" <jya@ieee.org> wrote in message news:_8udnVrBE_vGv_rfRVn-pw@rcn.net...> Sandeep Chikkerur wrote: > > ... > >> 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. > > Can you give an example of such a thing? I never heard of one. (I know > about tuned doublers and triplers, and about Western Electric's sub > harmonic generator that used 60 Hz input to generate a 20 Hz ring tone, > but I don't think you mean such devices.) > > ... > > JerryI think the term "frequency synthesizer" often applies to a PLL that has: - A reference clock input - most often from a crystal osciallator - thus a sinewave at relatively high frequency f0 compared to what might be synthesized. - A counter or frequency divider applied to the reference and fed into a PLL - PLL feedback has a frequency divider that serves to multiply the PLL output frequency. So, with the feedback divider factor F and the input divider I, the synthesizer output frequency is f0*I/F. There is lots of flexibility in this type of synthesizer. If the output of the PLL VCO is a sinewave then you have a sinewave in and a sinewave out. I think that's what the OP was referring to. Lots and lots of hits on Google. We used them for generating pixel, H and V clocks for video processing and we used them for generating phase-locked clocks for PC main boards when we were trying to lock 3 boards in sync for redundancy, etc. etc...... The ratio of in to out is often referred to as the "modulus". In video processing the modulus becomes the ratio of the number of pixels in a line to the H clock as I recall. Setting the modulus (in a display sync processor) can be important if one is trying to sync a discrete display (like an LCD) to an arbitrary source. Fred
Reply by ●April 21, 20052005-04-21
Fred Marshall wrote:> "Jerry Avins" <jya@ieee.org> wrote in message > news:_8udnVrBE_vGv_rfRVn-pw@rcn.net... > >>Sandeep Chikkerur wrote: >> >> ... >> >> >>>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. >> >>Can you give an example of such a thing? I never heard of one. ...> I think the term "frequency synthesizer" often applies to a PLL that has: > - A reference clock input - most often from a crystal osciallator - thus a > sinewave at relatively high frequency f0 compared to what might be > synthesized. > - A counter or frequency divider applied to the reference and fed into a PLL > - PLL feedback has a frequency divider that serves to multiply the PLL > output frequency.... Those things have a single, fixed frequency input, and the variable output is gotten with a combination of analog and digital circuits. They don't fit the description Sandeep gives above, at least as I read it. He doesn't seem to mean any of the various forms of clock synthesizer either in the lines above or in previous posts. That may be obtuseness on my part. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●April 21, 20052005-04-21
If the system is time varying in response to one of the input signals, I don't think that classifies as time variant, I think that classifies as non-linear. Take the case of a multiplier with the same signal applied to both inputs. It generates new output frequencies and is therefore a non-linear system. Of course the system is changing over time but it is changing in response to the input (which is changing with time) I consider a multiplier with two different input signals to be a non-linear time invariant system. The system is not changing with TIME so it is time invariant. It is changing in response to one or more input signals so it is non-linear. If the system changes in response to inputs that change over time, that is non-linearity. If the system changes with time on its own, that is time variant. Mark
