LTI vs non-LTI

On Tue, 30 Apr 2013 12:17:45 -0700, robert bristow-johnson wrote:

> i've been refraining from this thread, sorta.
> 
> On Apr 28, 5:36&nbsp;am, "manishp" <58525@dsprelated> wrote:
>>
>> In order for me to appreciate a little better, can I get some very
>> simple examples of,
>>
>> 1) a non-LTI system that is linear but is time variant (that is, it is
>> non time invariant)
> 
> your audio sound system where you ware wiggling the volume or tone
> control.
> 
>> 2) a non-LTI system that is time invariaint but non linear
> 
> a distortion circuit with a diode in it or a transistor driven into
> saturation or similar.
> 
> 
> now, to glen and Tim and Rick and Eric, about this mixer business...
> 
> i think we can derive, purely from the base definitions of "Linear" and
> "Time-Invariant", an input-output characteristic that follows the
> convolution integral.  if the mixer cannot be described by a
> convolution, then it ain't LTI.
> 
> we know that for LTI, that the only frequencies that come out are
> frequencies that go in (but possibly with their amplitudes and phases
> adjusted).
> 
> i don't see how heterodyning can be described as LTI.

Convolution kernals are only usually described as time invariant.  You 
can fully describe a linear time-varying system with a time-varying 
convolution kernel, i.e. instead of describing system y = h(x) with the 
convolution kernel

h(tau),

you describe system y = h(x, t) with the convolution kernel

h(tau, t)

where t is the time of the convolution and tau is the delay.

This isn't necessarily a _useful_ bit of information to know for anything 
other than analysis, but it does follow from superposition.

-- 
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?

Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com

Tim Wescott <tim@seemywebsite.com> wrote:
(snip)

> Diplomacy?  What's that?
 
> I guess it depends on whether you're viewing a mixer as a two-input one 
> output device, in which case it is indeed nonlinear, or if you're viewing 
> a mixer as a one-input device that autonomously acts in a time-varying 
> way (i.e., if you're viewing it as a mixer with attached LO, or just 
> modeling it with one line of math).  In the former case it's nonlinear, 
> in the latter it is linear but time varying.

Yes, that seems to me the problem. 

In the early days of radio, toys for kids, and some microwave mixers, 
the mixer is a single diode, where the non-linearity is what does the 
mixing.

As far as I know, a doubly balanced mixer, such as used for FM stereo,
can't be built from a single diode. For FM stereo, linearity in the
signal going in to the subcarrier coming out is important. You don't
want to generate cross terms within the audio range.

Yet both devices are called mixers.
 
> In either case one should specify what you mean -- as such, 
> perhaps we both erred.
 
> As for the statement on mixers being nonlinear, etc., well, here's my 
> guiding light on this (paraphrasing intentional, but meant to educate 
> rather than offend): "Any physical system, device, or whatever is 
> nonlinear.  But it is possible to operate many such systems in such a way 
> that the effects of nonlinearity can be kept acceptably small".

(snip)

Yes, but for the single diode mixer, it is the non-linearity that
causes the mixing. Hopefully all the unwanted terms can be filtered
out, and all works as desired. But you still want the right kind of
linearity such that the desired signal comes out right.

-- glen

Tim Wescott <tim@seemywebsite.com> wrote:
>> Tim Wescott <tim@seemywebsite.com> wrote:

(snip, I wrote)
>>>> Well, assuming that the signal has the appropriate band limiting.
  
>>> I certainly hope you're off on a tangent, and not saying anything about
>>> the signal needing to be bandlimited for my comments to be valid.  I'm
>>> talking about _systems_, not the signals that may be running through
>>> them.
 
>> OK, consider system f(x), a 44.1kHz sampler, and system g(y) an
>> anti-aliasing filter that cuts off before 44.1kHz.
  
>>>> Consider starting with a continuous time signal that is perfectly band
>>>> limited. Now sample it at different sampling rates above the Nyquist
>>>> rate, with an ideal sampler. All then contain the exact same
>>>> information, which is that in the original signal.
  
>>> So?  What does that have to do with a discussion about whether a system
>>> is time varying or not?
 
>> Are systems f(x) and g(y) time invariant? How about g(f(x))?
 
> I'm going to take your statement as meaning z = f(g(y)), as that puts the 
> anti-alias filter before the sampling -- I think that must be what you 
> meant.  (Either that or you're failing to differentiate reconstruction 
> from aliasing, which I very much doubt).

Seems to me that at least some people call the reconstruction filter
an anti-aliasing filter, though the name probably better describes what
comes before sampling.  Anyway...

> Given that:
 
> g(y) is (or darn well should be!) time invariant.
 
> f(x) is time varying.
 
> Even without knowing that f(x) is time varying, you know that g(f(x)) is 
> time varying because its input is in continuous-time and its output is in 
> discrete time.  The action of the sampler loses the detail of what's 
> going on between the sampling instants of x.  It happens that in your 
> example that detail is redundant -- but f(x) still throws that detail 
> away, changing it from redundant to nonexistent.
 
> Had you really wanted to flummox me you'd then follow this by system h
> (z), which is a perfect reconstructor such that the action of
 
> x_p = h(f(g(y))) 
 
> is indistinguishable but for delay from x = g(y).  At this point I would 
> say that h(f(g(y))) certainly _acts like_ a linear time invariant system, 
> and in many cases should be treated as such -- but with a great deal of 
> caution, always keeping in mind that there's some pretty oddball stuff 
> going on in the middle of that there seemingly-innocent system.

I think you have covered what I was trying to say.

An important, and not so obvious, point, is that a time variant system
followed by a time invariant system together can be time invariant.

Previously, I might have considered an inverse system to undo time
variance, but not this way. For reasons like that, I wasn't sure how
to describe the time variance of resampling. Now, it isn't at all easy
to build ideal resampling systems, but often close enough.

-- glen

On Tue, 30 Apr 2013 22:48:20 +0000, glen herrmannsfeldt wrote:

> Tim Wescott <tim@seemywebsite.com> wrote: (snip)
> 
>> Diplomacy?  What's that?
>  
>> I guess it depends on whether you're viewing a mixer as a two-input one
>> output device, in which case it is indeed nonlinear, or if you're
>> viewing a mixer as a one-input device that autonomously acts in a
>> time-varying way (i.e., if you're viewing it as a mixer with attached
>> LO, or just modeling it with one line of math).  In the former case
>> it's nonlinear, in the latter it is linear but time varying.
> 
> Yes, that seems to me the problem.
> 
> In the early days of radio, toys for kids, and some microwave mixers,
> the mixer is a single diode, where the non-linearity is what does the
> mixing.
> 
> As far as I know, a doubly balanced mixer, such as used for FM stereo,
> can't be built from a single diode. For FM stereo, linearity in the
> signal going in to the subcarrier coming out is important. You don't
> want to generate cross terms within the audio range.
> 
> Yet both devices are called mixers.
>  
>> In either case one should specify what you mean -- as such, perhaps we
>> both erred.
>  
>> As for the statement on mixers being nonlinear, etc., well, here's my
>> guiding light on this (paraphrasing intentional, but meant to educate
>> rather than offend): "Any physical system, device, or whatever is
>> nonlinear.  But it is possible to operate many such systems in such a
>> way that the effects of nonlinearity can be kept acceptably small".
> 
> (snip)
> 
> Yes, but for the single diode mixer, it is the non-linearity that causes
> the mixing. Hopefully all the unwanted terms can be filtered out, and
> all works as desired. But you still want the right kind of linearity
> such that the desired signal comes out right.

When you get into the guts of just about any mixer, there are 
nonlinearities that are making the mixing happen.

Actually a single-diode mixer can be made to be pretty good, but if 
you're a circuit designer with BOM money to spend on making the mixer 
good the first thing you're going to do (unless you're designing 
microwave equipment) is to use something other than a single-diode mixer.

The _intended_ behavior of a single-diode mixer+LO is to be linear, but 
when you're using it because you're trying for the World's Cheapest Radio 
you're going to freely sacrifice a lot of that to get the cheapness.

If you _really_ want a poor-performing, cheap mixer, then look to single-
transistor "converters", that just fed the RF into a selected point in an 
oscillator circuit, and picked off the IF from some other (or the same) 
point.  That's pretty much guaranteed to be worse than a diode mixer with 
a separate LO.

-- 
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?

Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com

Tim Wescott <tim@seemywebsite.com> wrote:

(snip, I wrote)
>> OK, but we usually use sampling in the context of reconstruction. We
>> want to be able to sample a continuous signal, and then reconstruct the
>> original, or something close to it.
 
> Ah ha!  This is why you and I are out of step in this whole discussion.
 
> When you're doing control systems work, you use sampling in the context 
> of doing a good-enough job of control.  Anti-aliasing and reconstruction 
> are actually your enemy in this case, because they only make signal 
> fidelity better if you ignore delay, whereas in a control system signal 
> fidelity must take delay into account -- which means that an aliased, but 
> unfiltered, signal is better than one that's had the snot filtered out of 
> it and 720 degrees of delay added at the edges of the anti-alias filter.
 
>> Consider passing the two downsampled signals through reconstruction
>> (anti-aliasing) filters?
 
>> If you consider the sampled signal as a container holding a previously
>> band-limited continuous signal, or if you consider that aliases don't
>> add any information to a signal, then it seems to me that the result is
>> different.
 
> I think you're allowing yourself to confuse your signals with your 
> systems -- for the whole screed see my response to your other post vis-a-
> vis anti-alias filters followed by sampling.
 
>>> What this means is that if a downsampling operation is in cascade with
>>> other operations, we are not permitted to swap the order of any of
>>> those operations and the downsampling process without modifying those
>>> operations in some way.
 
>> Yes, you have to be careful with the order of operations.
 
> And careful consideration of which signal processing stages are linear 
> and/or time invariant is a big help in doing this.  It's one of the 
> reasons you want to keep track.

OK, I think I am mostly satisfied now. 

And yes, I was not at all thinking about control systems work.

-- glen

robert bristow-johnson <rbj@audioimagination.com> wrote:
 
> i've been refraining from this thread, sorta.
 
> On Apr 28, 5:36&#4294967295;am, "manishp" <58525@dsprelated> wrote:

>> In order for me to appreciate a little better, can I get some very simple
>> examples of,

>> 1) a non-LTI system that is linear but is time variant (that is, 
>> it is non time invariant)
 
> your audio sound system where you ware wiggling the volume or tone
> control.

That is a nice example that we can all recognize.
 
>> 2) a non-LTI system that is time invariaint but non linear
 
> a distortion circuit with a diode in it or a transistor driven into
> saturation or similar.

Hmm, see below.
 
> now, to glen and Tim and Rick and Eric, about this mixer business...
 
> i think we can derive, purely from the base definitions of "Linear"
> and "Time-Invariant", an input-output characteristic that follows the
> convolution integral.  if the mixer cannot be described by a
> convolution, then it ain't LTI.
 
> we know that for LTI, that the only frequencies that come out are
> frequencies that go in (but possibly with their amplitudes and phases
> adjusted).
 
> i don't see how heterodyning can be described as LTI.

The single diode mixer is non-linear and time invariant, good.

Consider two ideal double balanced mixers f(x,y) and g(x,y). 

First, we expect f(x,y)=g(x,y) so it isn't hard to show that

   f(x,y)+f(x,y)=2 f(x,y) 

Also, it should be true that f(Ax, y) = A f(x,y) = f(x, A y).
for constant A.

So, in a certain sense it is linear, but

   f(Ax, Ay) is not A f(x,y) so in a different sense it isn't.

-- glen