Uniqueness of Sinusoidal waves

Mark wrote:
> 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.
> 
>
There is a precise mathematical definition of linearity.

if L(x1)=y1
and L(x2)=y2

a system is linear if L( a1*x1 + a2*x2) = a1*y1 + a2*y2


if L(x1)=x1^2  and L(x2)=x2^2

and L(x1+x2)=x1^2+2*x1*x2+x2^2, L() is this case is nonlinear because it 
doesn't satisfy the linearity properties.  Multiplication can be shown 
to be linear, while squaring is not.

if L(x1)= f1*x1 , L(x2)=f1*x2, and L(x1+x1)=f1*x1+f1*x2, the system 
satisfies the linearity property.  if f1 is a function of time, it still 
satisfies the linearity property.

"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

in article d48lt0$mv9$1@newslocal.mitre.org, Stan Pawlukiewicz at
spam@spam.mitre.org wrote on 04/21/2005 12:54:

> Mark wrote:
>> 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.

absolutely.  a time-variant system has to be time-varying independently of
any non-null function of any input.

> There is a precise mathematical definition of linearity.
> 
>  if L{x1} = y1
> and L{x2} = y2
> 
> a system is linear if L{ a1*x1 + a2*x2 } = a1*y1 + a2*y2

it turns out that it's sufficient to show that if

     L{ x1 + x2 }  =  y1 + y2

it's still linear.  the scaling property can be derived from the additive
superposition property for any rational scaling factor.


-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

Mark wrote:

> 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
> 
You're nitpicking my nitpiks!  But you didn't include the expression, so 
I have to repeat it:

y(t) = sin(w*t)*x(t)

If both the sin(w*t) and x(t) are taken to be system inputs then the 
system is a two-input nonlinear system.  If, however, the sin(w*t) is 
taken to be an intrinsic property of the system and x(t) is taken as an 
input then the system is time varying.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Tim.

OK I'll agree with that.

If the two "signals" are taken to be "inputs" the system is non-linear
because it is changing in response to one (or more) of the inputs.

If one of the signals is taken to be intrinsic to the system itself and
not an "input" then the system is time variant and linear.

Agreed.

Thanks for the interesting discussion.

Mark

in article 1114137181.650701.13810@f14g2000cwb.googlegroups.com, Mark at
makolber@yahoo.com wrote on 04/21/2005 22:33:

> If the two "signals" are taken to be "inputs" the system is non-linear
> because it is changing in response to one (or more) of the inputs.
> 
> If one of the signals is taken to be intrinsic to the system itself and
> not an "input" then the system is time variant and linear.

just for the sake of semantics, i wouldn't call that "signal ... intrinsic
to the system" a signal at all.  i would call it a parameter.  if the
parameters of a system (linear or non-linear) changes, it is time-variant.
if the parameters defining a system do not change, it is time-invariant.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

"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.

On 19 Apr 2005 21:48:58 -0700, sandeep_mc81@yahoo.com (Sandeep
Chikkerur) wrote:

>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.

   By "linear system" I presume you mean something like antialias
filter->A/D converter->D/A converter->reconstruction filter[1]. If you
try to put a frequency in above 1/2 the sample rate, which would
normally be aliased or 'reflected' to something below 1/2 the sample
rate, but the antialias filter will keep it out. Thus any frequency
that goes in will come out.

>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 ?

   This is apparently what someone else described, a PLL with
frequency dividers and/or multipliers in the loop.
   Apparently, then, a frequency synthesizer is not a linear system.
One wouldn't necesarily expect it to be, as it's a different type of
system with a different purpose.

>
>I hope this time I am clear...

   Me too...

>Sandeep

1. It seems redundant to mention the filters, since in modern times
one usually thinks of the filters as integral (no pun...) parts of the
A/D and D/A, as they are flat out neccesary for most applications.

-----
http://mindspring.com/~benbradley

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...

thanks

Mark

Mark wrote:
> 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...

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
-- 
Engineering is the art of making what you want from things you can get.
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