LTI vs non-LTI

On Sun, 28 Apr 2013 09:58:15 -0700, Rick Lyons wrote:

>
> I've always thought of mixers as being nonlinear because a mixer's
> output contains spectral components that did not exist at either of its
> two inputs.

"Have always thought"?  That's wishy-washy.

Most physical mixers are not linear from the LO port -- generally the 
local oscillator signal gets badly distorted.  But decent mixers _are_ 
linear going from the RF port to the IF port, and there's a lot of profit 
in treating them as such in analysis.

(Note, too, that mixers are often the first point of distortion in a 
radio circuit, because you _want_ them to be perfectly linear, but you 
have to _work_ to make them so.)

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

On Sun, 28 Apr 2013 17:12:07 +0000, Eric Jacobsen wrote:

<< snip >>

> Another good example was mentioned from different perspectives by Tim
> and Rick, namely a mixer.   Mixers, when looked at as a whole, are
> substantially non-linear and often implemented with non-linear devices. 
> However, if we constrain out view to a particular region of interest,
> say a sub-section of bandwidth, and design it properly, the mixer will
> behave linearly within that region of interest, which means we may be
> able to treat and analyze it like a linear function.
> 
> So, as is often the case, it depends on your definitions of "linear" and
> "time invariant".   ;)

It can be more than that.  I did not state it in my original response, 
but in my opinion the choice to approximate one's system as linear and 
time invariant should always be a conscious one.  That way, when your 
assumption breaks down and you need to make your approximation more 
accurate you'll be better equipped to know whether you should use a 
decorated linear approximation (i.e., treat a system as linear with 
quantization noise), or if you should throw your linear approximation out 
the window, or if there is some middle path.

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

About four years ago, there was a discussion in
this newsgroup with the subject "Is complex
conjugation an LTI operation?" Perhaps some of
the stuff there can be incorporated here too
leading to more fun arguments about what is meant
by a linear operation. After all, complex baseband
signals are everyday stuff for DSP folks.

Dilip Sarwate

Rick Lyons <R.Lyons@_bogus_ieee.org> wrote:

(snip)

> Hi manishp,
>   I may have to 'take back' my comment about 
> the absolute value' operation being nonlinear.
> I'll have to think more about that issue.

Seems to me that absolute value should be considered
non-linear. It is at least not an analytic function, which
makes it not so useful in some cases.

Least-squares is used not because it is the best solution
to the problem, but because the math is easier.

Statisticians like absolute value (and so least absolute
value for fits) better, but the math is much harder.

Also, median is supposed to be better than mean, but again
the math is easier for mean.

-- glen

>   Upsampling and downsampling are time variant 

Sir, can you please elaborate?
Essentially, you are saying that if I take a sequence, send it to 
downsampler at different times, I would see different output.

Isn't this an issue in practical applications

Thanks, manish

PS: thanks to all others for responding to my original question

On Mon, 29 Apr 2013 07:50:10 -0500, manishp wrote:

>>   Upsampling and downsampling are time variant
> 
> Sir, can you please elaborate?
> Essentially, you are saying that if I take a sequence, send it to
> downsampler at different times, I would see different output.
> 
> Isn't this an issue in practical applications
> 
> Thanks, manish
> 
> PS: thanks to all others for responding to my original question

Sampling is time varying because the input is ignored for all time except 
the sampling instant.  If you're willing to use impulse functionals in 
your analysis then sampling is just a form of mixing, in the linear, time-
varying sense that I was advocating earlier.

Downsampling in the discrete-time world is much like sampling from 
continuous-time to discrete time, and can be modeled similarly.  If 
you're willing to view continuous-time as the ultimate in discrete-time 
domains, then it's exactly the same (but taking that view is a hell of a 
leap, and not to be taken lightly without really thinking through the 
mathematical implications).

I seem to be disagreeing with Rick a lot these days, but upsampling by an 
integer amount (or, for that matter reconstructing to continuous time), 
when expressed the way I normally view it, is not time-varying.  Rather, 
it is just taking the "train-of-impulses" and running them through a hold 
filter.

If you view the discrete-time domain as entirely separate from the 
continuous-time domain, or if you view two different discrete-time 
domains as being disjoint and not related, then upsampling by an integer 
amount either is time varying, or the concept is invalid -- I'm not sure 
which.

Upsampling by a non-integer amount _is_ time varying.  The easiest way to 
determine this is by noting that upsampling by any rational but non-
integer ratio is tantamount to upsampling by an integer ratio, then 
downsampling by some other integer ratio -- and that downsampling step is 
time-varying, which "poisons" the whole process.

-- 
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

On 2013-04-29 04:14, glen herrmannsfeldt wrote:
[...]
> Also, median is supposed to be better than mean, but again
> the math is easier for mean.

Better for what? In which sense?

OF course, median returns (if odd) a value
belonging to the set, but this does not mean
this value is better. It depends.

bye,

-- 

piergiorgio

>I seem to be disagreeing with Rick a lot these days, but upsampling by an

>integer amount (or, for that matter reconstructing to continuous time), 
>when expressed the way I normally view it, is not time-varying.  Rather, 
>it is just taking the "train-of-impulses" and running them through a hold

>filter.
>
>If you view the discrete-time domain as entirely separate from the 
>continuous-time domain, or if you view two different discrete-time 
>domains as being disjoint and not related, then upsampling by an integer 
>amount either is time varying, or the concept is invalid -- I'm not sure 
>which.

In what way does an upsampling operation even change a signal?

Piergiorgio Sartor <piergiorgio.sartor.this.should.not.be.used@nexgo.removethis.de> wrote:
> On 2013-04-29 04:14, glen herrmannsfeldt wrote:
> [...]
>> Also, median is supposed to be better than mean, but again
>> the math is easier for mean.
 
> Better for what? In which sense?

Better in the statistical, or statistician sense.
 
> OF course, median returns (if odd) a value
> belonging to the set, but this does not mean
> this value is better. It depends.

If you know that the data samples a Guassian distribution,
then mean and standard deviation should have the appropriate
statistical properties. 

Often, though, they are used where the data is known not to be
Gaussian, but where they are easier to compute.

The distribution of home sale prices is not Gaussian, and you will 
more often see median used in economic reports on home prices.

-- glen

Tim Wescott <tim@seemywebsite.com> wrote:
> On Mon, 29 Apr 2013 07:50:10 -0500, manishp wrote:
 
>>>   Upsampling and downsampling are time variant
 
>> Sir, can you please elaborate?

(snip)
> Sampling is time varying because the input is ignored for all time except 
> the sampling instant.  If you're willing to use impulse functionals in 
> your analysis then sampling is just a form of mixing, in the linear, time-
> varying sense that I was advocating earlier.
 
> Downsampling in the discrete-time world is much like sampling from 
> continuous-time to discrete time, and can be modeled similarly.  If 
> you're willing to view continuous-time as the ultimate in discrete-time 
> domains, then it's exactly the same (but taking that view is a hell of a 
> leap, and not to be taken lightly without really thinking through the 
> mathematical implications).

Well, assuming that the signal has the appropriate band limiting.

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. 
 
> I seem to be disagreeing with Rick a lot these days, but upsampling by an 
> integer amount (or, for that matter reconstructing to continuous time), 
> when expressed the way I normally view it, is not time-varying.  Rather, 
> it is just taking the "train-of-impulses" and running them through a hold 
> filter.
 
> If you view the discrete-time domain as entirely separate from the 
> continuous-time domain, or if you view two different discrete-time 
> domains as being disjoint and not related, then upsampling by an integer 
> amount either is time varying, or the concept is invalid -- I'm not sure 
> which.
 
> Upsampling by a non-integer amount _is_ time varying.  The easiest way to 
> determine this is by noting that upsampling by any rational but non-
> integer ratio is tantamount to upsampling by an integer ratio, then 
> downsampling by some other integer ratio -- and that downsampling step is 
> time-varying, which "poisons" the whole process.

Maybe I don't understand "time varying" in the sense you use it here,
but up sampling by a rational factor should give the same signal you
would have seen sampling the original at the higher rate. (Assuming the
filters are all ideal, which they usually aren't, except in homework
problems.)

-- glen