A Basic Aliasing Question

"Richard Owlett" <rowlett@atlascomm.net> wrote in message
news:10ikcm0abkntoef@corp.supernews.com...
> Bhaskar Thiagarajan wrote:
> >[SNIP]  For designs where undersampling
> > is used, it now becomes important to pick the right alias...
>
> Does this mean that in some cases you can sample at less than Nyquist
> criterion?
> [ If true, just point out some good keywords for Google search. ]

Yes, that's true (do read Jon's response where he clarifies the Nyquist
criterion).
Both 'bandpass sampling' and 'undersampling' should give you plenty of hits
on Google.

Cheers
Bhaskar

Richard Owlett wrote:

> Bhaskar Thiagarajan wrote:
> 
>> [SNIP]  For designs where undersampling
>> is used, it now becomes important to pick the right alias...
> 
> 
> Does this mean that in some cases you can sample at less than Nyquist 
> criterion?
> [ If true, just point out some good keywords for Google search. ]

I think it means not sampling fast enough to accomplish what's needed.
Subsampling is something else, but I think it's what Bhaskar meant when
he wrote undersampling. Rick Lyons's books have the best treatment of
subsampling that I know of.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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"Jon Harris" <goldentully@hotmail.com> writes:

> "Randy Yates" <randy.yates@sonyericsson.com> wrote in message
> news:xxp3c2dsoig.fsf@usrts005.corpusers.net...
>> Richard Owlett <rowlett@atlascomm.net> writes:
>>
>> > Bhaskar Thiagarajan wrote:
>> > >[SNIP]  For designs where undersampling
>> > > is used, it now becomes important to pick the right alias...
>> >
>> > Does this mean that in some cases you can sample at less than Nyquist
>> > criterion?
>>
>> Yes.
>
> I guess that depends on how you define the "Nyquist criterion".  If you define
> it to require sampling at more than twice the highest frequency component in the
> original signal, then the answer is yes.  But if you define the Nyquist
> criterion to require sampling at more than twice the bandwidth of the original
> signal, than no.

Yo', Jon, dude: chill. We all know everything depends on everything
else, meaning depends on context, what "is" is is debatable, and
the answer to the universal question is "39" (or was it 37?). 
-- 
%  Randy Yates                  % "How's life on earth? 
%% Fuquay-Varina, NC            %  ... What is it worth?" 
%%% 919-577-9882                % 'Mission (A World Record)', 
%%%% <yates@ieee.org>           % *A New World Record*, ELO
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Randy Yates <randy.yates@sonyericsson.com> wrote in message news:<xxpsmaermef.fsf@usrts005.corpusers.net>...
> Here's something I don't remember about aliasing - someone please
> verify whether or not this is correct. Assume the sample rate is
> 2000 Hz for the sake of illustration.
> 
> Conventional wisdom tells us that a 1200 Hz signal will look
> like an 800 Hz signal due to aliasing. I say it may 
> look the same on a spectrum analyzer (i.e., its magnitude
> spectrum may look the same), but it may actually be different
> since the negative and positive frequency components of such
> a wave are swapped. Hence if the original wave was a cosine
> wave, the aliased wave will still be cosine (i.e., will look
> identical to the original) since the positive and negative
> frequency components are identical (cos x = (e^(i*x) + e^(-i*x))/2).

The "cosine" form is preserved, yes, but with an ambiguity, what 
the frequency is concerned.

> However, if the wave is any other type, it will look different. 

Yes. A base-band continuous pulse would look different from a band-pass 
continuous pulse that produces the same discrete set of samples.
But since both can be synthesized by sinusoidals, that doesn't help 
much, since the sinusoidals are ambiguous.

> Said perhaps more simply, xa(t) = -x(t). Since a cosine is 
> even-symmetric, xa(t) = -x(t) = x(t). Not true for a sine
> wave or sinusoids of other phases.

This is the sine form, the cosine form you found in the correction 
you posted afterwards.

> Trivial, but something I'd never thought of before. Or
> am I wrong?

You seem to be right in the two assertions (sampled sinusoidals 
being ambiguous wrt frequency, and the symmetry properties of 
sines/cosines) when viewing them individually. I don't see any 
connection between the ambiguity and the symmetry properties, 
though. I can't see how a cosine being even symmetrical around 
some time reference causes it to become ambiguous when sampled, 
if that's what you mean.

Rune

Randy Yates wrote:
> We all know everything depends on everything
> else, meaning depends on context, what "is" is is debatable, and
> the answer to the universal question is "39" (or was it 37?). 

It was 42.

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jthomas@bittware.com  http://www.bittware.com          (703) 779-7770
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