Sandeep Chikkerur wrote: ...> Hi Fred and Jerry, > > My question is, why the unique property of sinusoidal waves is not > maintained for frequency synthesizers ? U can assume it to be a PLL.What you call "the unique property of sinusoidal waves" applies only to linear time-invariant systems. A frequency synthesizer of any kind is not such a system. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Uniqueness of Sinusoidal waves
Started by ●April 19, 2005
Reply by ●April 22, 20052005-04-22
Reply by ●April 22, 20052005-04-22
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. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●April 22, 20052005-04-22
I'll take the challenge. Here is an almost-equivalent nonlinear differential system: y''(t) = 4 pi^2 y(t) + x''(t) - 4 pi^2 x(t) subject to the constraint y'(t)^2 - 2 y'(t) x'(t) + x'(t)^2 + 4 pi^2 y(t)^2 - 8 pi^2 y(t)x(t) + 4 pi^2 x(t)^2 - 4 pi^2 = 0 Solutions to this are y(t) = x(t) + sin(2 pi t + initial_phase) where initial_phase is arbitrary. Usually if there is an algebraic differential equation for how a system varies with time then differential elimination can convert the system into a nonlinear time-invariant system that is equivalent up to an ambiguity in the origin of the time axis. I say "usually" only because I can't prove "always". The same works for algebraic difference equations. Another example: y'(t) = x(t) t is almost equivalent to y''(t) = x(t) + y'(t) x'(t) / x(t) -Frederick Umminger
Reply by ●April 22, 20052005-04-22
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 -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●April 22, 20052005-04-22
"Sandeep Chikkerur" <sandeep_mc81@yahoo.com> wrote in message news:d5d88eb5.0504212040.7e4dc8d4@posting.google.com...> "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message > news:<CtCdnTghfKUGbvrfRVn-1w@centurytel.net>... >> "Jerry Avins" <jya@ieee.org> wrote in message >> news:wNmdnXQgrssgXPrfRVn-pg@rcn.net... >> > 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 >> > -- >> > >> Jerry, >> >> I took one of "those things" and abstracted it to something only slightly >> different. >> Instead of a single, fixed frequency input - which I'm sure you'll agree >> is >> not necessary for what is in the block diagram to work - within reason - >> let's allow a variable frequency input instead. >> We need not worry about the reasons for using a crystal oscillator in >> other >> applications. >> >> Now, I was only answering the question about "frequency synthesizer". In >> the bigger context of this thread, the problem I see with answering "is >> this >> a linear system" is that the output amplitude is usually not a function >> of >> the input amplitude. So, even were the frequencies involved to meet the >> criterion for linearity (they don't), the amplitude doesn't. >> >> As Stan mentions: >> >> There is a precise mathematical definition of linearity. >> >> if L(x1)=y1 >> L(sin(w1t))=b1*sin(Mw1t) >> and L(x2)=y2 >> L(sin(w2t))=b1*sin(Mw2t) >> >> a system is linear if L( a1*x1 + a2*x2) = a1*y1 + a2*y2 >> >> L(a1*sin(w1t) + a2*sin(w2t)) does not apply because two inputs cannot be >> superimposed on a synthesizer. And scaling, a special case of >> superposition, also does not apply. >> >> If we multiply a sinusoid x by another sinusoid y we get sum and >> difference >> frequencies. >> Let y be the "carrier" input to a multiplier. >> Let x be the "input". >> We will highpass the output so that the carrier and difference >> frequencies >> are eliminated. >> So, all we see is the sum frequency component. >> If the frequency of x changes, the sum frequency at the output changes >> equally. >> The output is translated but it is linearly related to the input. >> >> L(sin(w1*t))=a1*sin(w0 + w1)*t >> L(sin(w2*t))=a1*sin(w0*t + w2*t) >> L(sin(w1*t) + sin(w2*t))=a1*sin(w0 + w1)*t +a1*sin(w0 + w2)*t >> So, it's linear. >> >> In the case of the synthesizer, if the frequency of x changes, the output >> frequency changes as a multiple of the frequency of x which is not a >> property of a linear system I believe. >> >> L(sin(w1*t))=a1*sin(M*w1*t) = a1*sin(w0 + w1)*t where w0=(M-1)w1 >> L(sin(w2*t))=a1*sin(M*w2*t) which is not equal to a1*sin((M-1)w1 + w2)*t >> So, ignoring superposition, even the frequency doesn't match what would >> be >> expected. >> >> There are a bunch of reasons why a frequency synthesizer (so defined) >> isn't >> linear. >> >> Fred > > Hi Fred and Jerry, > > My question is, why the unique property of sinusoidal waves is not > maintained for frequency synthesizers ? U can assume it to be a PLL.I think the question has been answered any number of times in this thread by now. Perhaps the answers went too far too fast. And, perhaps your strange terminology "the unique property of sinusoidal waves" isn't understood by anyone. Folks have asked what you mean by this and I don't see that you've not answered. In the mean time, we have tried to help by speculating on the meaning and have come up with answers like those below: Since we know that if one starts with a linear, time invariant (LTI) system: The system transfer function, as viewed from its impulse response will be a superposition (a summation) of damped sinusoids. This is fundamental. Also, and perhaps viewed as a resultl of the above, the steady-state system response to a single sinusoid will be a sinusoid at that same frequency. So, we presume that this is "the unique property of sinusoidal waves" that you refer to!! Further, since superposition applies in an LTI system, The steady-state system response to a sum of multiple sinusoids will be a sum of sinusoids at those same frequencies. Further, if the system is linear, time varying (LTV) then: The steady-state system response to a single sinusoid may include sinusoids at additional or different frequencies that are related in a simple way (frequency translation) to the original frequency. This is just an example. So, I say "may" because there are many cases and examples and you need to apply the tests for linearity that have been given many times here. So, we presume that an LTI system (and maybe an LTV system) is the only one that has "the unique property of sinusoidal waves" that you refer to. Thus, we focus on LTI systems in our answers to your questions. I just posted a long description of the many reasons why a frequency synthesizer is not a linear system. - there is generally no mechanism for superposition at the input - even if there were, amplitudes don't scale through the synthesizer - which is really another property of superposition. That doesn't mean one could not build one in principle where output amplitude would track the input amplitude but then that would not be the typical "frequency synthesizer" that you ask about. - frequencies at the output don't track what would be seen out of an LTV system so that aspect of the linearity test would fail. If our presumption is correct, that means it does not have "the unique property of sinusoidal waves" because we believe that only linear (LTI or LTV) systems have that property of yours. Please read the responses. Please keep asking questions. Our aim is to be helpful! Fred
Reply by ●April 22, 20052005-04-22
"Mark" <makolber@yahoo.com> wrote in message news:1114177064.964283.252540@f14g2000cwb.googlegroups.com...> Robert, > > 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. > > 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. > > Agreed, we are into the realm of semantics... >Mark, Well, you're close. It doesn't matter how you show the LO in the block diagram. What matters is that the LO isn't the "input" of interest for which the test for linearity will be done. Generally the system will be LTV for the IF input. Whether it is linear or not for the LO input is another matter. Take the case of a 4-quadrant multiplier which is as good a modulator as you can get. The "device" can be LTI, LTV or nonlinear as follows: Let us call "x" the input for which the system will be tested for linearity. That leaves "y" as the "other input" to the multiplier. If y is a constant, then, for x inputs, the sytem is linear time invariant (LTI). If y is a sinusoid, then, for x inputs, the system is linear time varying (LTV). (there are other periodic forms for y that yield the same result - maybe all periodic forms of y) If y is the same as the x input so that the output is x^2, the system is nonlinear. If y is some time varying, non-periodic signal then the system to x is most likely nonlinear because time stability of the sytem is implied by the linearity tests. That's why the periodic inputs work. Then, since it's a 4-quadrant multiplier, you can switch the signals on the inputs and run the same tests to conclude the sytem is LTV for the y input with x being a sinusoid. But, single-input, single-output systems are the most common of course and I have not addressed multiple-input linear systems here. Fred
Reply by ●April 22, 20052005-04-22
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. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●April 22, 20052005-04-22
"robert bristow-johnson" <rbj@audioimagination.com> wrote in message news:BE8E8EAE.674B%rbj@audioimagination.com...> 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. > > > -- > > r b-jr b-j, I agree with a tiny exception - it's not really a complete description. Input: x1(t) is whatever you like Input: x2(t) is whatever you like Output: y(t)=a*x1(t) + b*x2(t) And, I can assert that it's time invariant if I like. If x1(t)=0 x2(t)=sin(t) y(t)=b*sin(t) and x1(t)=sin(pi*(t) x2(t)=0 y(t)=a*sin(pi*t) and x1(t)=sin(pi*t) x2(t)=b*sin(t) y(t)=a*sin(pi*t) + b*sin(t) so superposition applies I think that means it's linear and time-invariant because it applies for all t. It looks like this where a and b are 1/3 and R is a resistor: +---------+ | | | | x1----->| R1=R2 | | | | | +----+----+ | |-----------+----> y | | +----+----+ +----+----+ | | | | | | | | x2----->| R2 | | R3=R2 | | | | | | | | | +---------+ +----+----+ | | | | GND / 0v y= (1/3)*x1 + (1/3)*x2 Fred
Reply by ●April 22, 20052005-04-22
"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. Fred
Reply by ●April 22, 20052005-04-22
"Jerry Avins" <jya@ieee.org> wrote in message news:8oSdnXLQvvg0lPTfRVn-pg@rcn.net...> Mark wrote: > > 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? > > Jerry > --Jerry, Sure! The tough part is in how you want to define LTV and how you want to define nonlinear I should think. So, it's up to you to give us a definition of both. Well, it would be repetitious to repeat a definition for LTV wouldn't it? So, how do you define "nonlinear" such that LTV is? Long ago I got hung up in such a discussion. I found that we were bringing in the idea of a "nonlinear element" like a diode, etc. But, I find no introduction of "nonlinear elements" in the definition of LTI or LTV - simply an input-output set of relationships. There are a bunch of ways that a system can fail the LTI or LTV tests so I guess we call all of them "nonlinear" If that's your definition too then all we have to do is say that a nonlinear system is a system fails the test for linearity. So, you want us to come up with an LTV and then prove that it's not linear by doing the tests for linearity (which it will pass) and then argue that it fails? No way. I ain't gonna do it! :-) Fred
