fourier anlysis of square wave...

Hi Everyone,
Just a short observation– This “digital aliasing” process is not
just a demon to be avoided; it is an extremely useful technique for moving
parts of the frequency spectrum around.  Rick Lyon’s recent blog is an
excellent example: 

http://www.dsprelated.com/showarticle/37.php

Regards,
Steve
Steve.Smith1@SpectrumSDI.com

Fred Marshall wrote:
> "Peter K." <p.kootsookos@remove.ieee.org> wrote in message 
> news:umysv2n3j.fsf@remove.ieee.org...
>> Steve Underwood <steveu@dis.org> writes:
>>
>>> This "once you are digitised you are safe" idea is often used like a
>>> mantra, for some reason. I'm puzzled where the idea even comes
>>> from. Has some popular book pushed the idea?
>> Not that I'm aware of, but I can see why people get the idea:
>>
>> If we define aliasing to be the effect of sampling at a frequency that
>> is too low for the frequency content of the sampled signal, then that
>> is _not_ the effect this thread is talking about. The thread is
>> (mostly) referring to the case where the signal is already sampled.
>>
>> Once a signal is sampled, the maximum frequency it can contain is pi
>> (or 1, depending on how you define "frequency").
>>
>> So is applying a nonlinearity to an already-sampled signal really
>> aliasing (as defined above)?
>>
>> I agree that it has similar aspects to aliasing, but I don't think
>> that the effect should necessarily go by the same name.
>>
>> Comments? Criticisms? Witticisms?
> 
> Peter,
> 
> After we sample we can talk about a Fourier Transform *or* a DFT.
> The DFT can be "unwound" into an infinite, discrete periodic sequence.  So 
> when you say:
> 
>> Once a signal is sampled, the maximum frequency it can contain is pi
>> (or 1, depending on how you define "frequency").
> 
> If there is subsampling of the temporal sequence it's pretty easy to 
> envision what happens in frequency.  If there's any overlap of nonzero 
> spectral components then that's what we normally call "aliasing".  It's the 
> overlap that appears to be "folding" but that's perhaps an unfortunate way 
> of looking at it.  It's more a matter of the negative frequencies at the 
> next higher sampling harmonic moving down to overlap the positive 
> frequencies at the next lower sampling harmonic.  When this happens, the 
> frequency of individual components is "translated" in a known way - but such 
> that the signal can't usually be reconstructed.
> 
> Similarly, if there's a nonlinear operation or even a linear time-varying 
> operation then there are new frequencies created.  If the new frequencies 
> cause overlap as above then that would be considered to be "aliasing".  With 
> the unwound DFT, that's pretty easy to see.
> 
> Any time a new frequency is created (including with sampling or subsampling) 
> then if the spectra mix with the original such that interpretation of 
> frequency becomes ambiguous, there's aliasing.  If there's no ambiguity then 
> there's no aliasing.  Examples of the latter are well known in down-shifting 
> schemes.

If I have a clean well managed signal, and I apply a non-linearity to 
achieve well defined higher frequencies from the original content, I can 
do all sorts of interesting things with the result. It really doesn't 
matter if those high frequencies have folded back into the sampling 
span. I know what they really are. However, this assumes I have total 
knowledge of the transform. The same transform inside a black box will 
produce the same answer, and yet the spectral mess would be considered 
aliased by most people - they have no idea what the signal they see 
really represents.

Put another way, you are contriving a lack of aliasing by simply knowing 
the answer.

Steve

SteveSmith wrote:
> Hi Everyone,
> Just a short observation– This “digital aliasing” process is not
> just a demon to be avoided; it is an extremely useful technique for moving
> parts of the frequency spectrum around.  Rick Lyon’s recent blog is an
> excellent example: 
> 
> http://www.dsprelated.com/showarticle/37.php

I think it's important to distinguish /images/ and /aliases/. Images are 
shifted and possibly inverted chunks of spectrum that stand alone in 
their new locations. Aliases overlay parts of the desired signal in ways 
that don't allow disentanglement. I think Ricks's article is about images.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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Steve Underwood wrote:

   ...

> Put another way, you are contriving a lack of aliasing by simply knowing 
> the answer.

I chose to read Fred's words another way: when the image doesn't overlap 
the signal (as in down-shifting schemes) it doesn't constitute an alias.
Will that do?

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

"Jerry Avins" <jya@ieee.org> wrote in message 
news:y8OdnXwMWtIlEMjanZ2dnUVZ_gednZ2d@rcn.net...
> Steve Underwood wrote:
>
>   ...
>
>> Put another way, you are contriving a lack of aliasing by simply knowing 
>> the answer.
>
> I chose to read Fred's words another way: when the image doesn't overlap 
> the signal (as in down-shifting schemes) it doesn't constitute an alias.
> Will that do?
>
> jerry
> -- 

Thanks Jerry.  Well, I don't accept the "contrived" part.  But I agree with 
Steve that there's some a priori knowledge necessary to make use of "doesn't 
overlap".  As you mention and I think Rick Lyons has done a good job of 
showing cases where downtranslation can take advantage of that.  I believe 
it's simply a matter of knowing the bandwidth and center frequency and aptly 
choosing the downtranslation frequency... something like that.

Yes, either you know what you're doing or you don't.  If you don't "know (at 
least part of) the answer" then the standard caveats would apply and some of 
these very valid methods couldn't be used - for lack of knowledge.  The fact 
that they exist and do work isn't comparable to contriving an obscure 
academic situation that happens to strangely work out.  It took some thought 
and careful construction to come up with these methods.

Just to state a very simple example:
Take a signal with bandwidth that's less than or equal to 0.8*fs/4 = 0.2*fs
If you know this much about the bandwidth (which you may or may not) then 
you also know that you can decimate by a factor of 2 without filtering 
first.  Doing this creates nonoverlapping images at fs/2 = fs'

Fred