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.

-- 
Jim Thomas            Principal Applications Engineer  Bittware, Inc
jthomas@bittware.com  http://www.bittware.com          (703) 779-7770
To mess up a Linux box, you need to work at it; to mess up your Windows
box, you just need to work on it. - Scott Granneman

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

"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
http://home.earthlink.net/~yatescr

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.
�����������������������������������������������������������������������

"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

"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.

I'm not sure what the 'official' definition is, but here is some relevant info
from http://www.wordiq.com/definition/Nyquist_theorem:

[It has to be noted that even if the concept of "twice the highest frequency" is
the more commonly used idea, it is not absolute. In fact the theorem stands for
"twice the bandwidth", which is totally different. Bandwidth is related with the
range between the first frequency and the last frequency that represent the
signal. Bandwidth and highest frequency are identical only in baseband signals,
that is, those that go very nearly down to DC. This concept led to what is
called undersampling, that is very used in software-defined radio.]

> > [ If true, just point out some good keywords for Google search. ]
>
> Bandpass sampling. It's also in Rick's book.

See also the link above.

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. 

> [ If true, just point out some good keywords for Google search. ]

Bandpass sampling. It's also in Rick's book. 

An easy way to think about it is like this: You know that an analog
signal that is to be sampled is normally bandlimited between 0 and 
+Fs/2. The process of sampling replicates this frequency spectrum every
Fs cycles. 

This can be generalized to sampling between k*Fs and k*Fs + Fs/2, 
where k is an integer. You still have a bandwidth of Fs/2, and the
duplicates that come from sampling still won't overlap. 
-- 
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124

"Bhaskar Thiagarajan" <bhaskart@my-deja.com> writes:

> "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).
> > However, if the wave is any other type, it will look different.
> >
> > 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.
> >
> > Trivial, but something I'd never thought of before. Or
> > am I wrong?
> 
> I see nothing wrong here. If you extend your example to more than just
> tones, say a signal with some BW that is not symmetric around it's center,
> then your point becomes even more important. For designs where undersampling
> is used, it now becomes important to pick the right alias (not all aliases
> have the correct spectral orientation). 

Right. That's exactly my point (I think?!). The aliases of a frequency f 
at k*Fs - f have their spectra reversed; the ones at k*Fs + f do not. Both
alias to the same frequency.

> If you don't have the flexibility to
> pick the right alias, then you have to adjust for that in the final digital
> quadrature mixing stage (which isn't too bad, since it's just a sign
> inversion on the complex exponential).

I think I follow that. 

> I'm probably just restating what you already knew....

The term "know" can be slippery! ... Thanks for the verification, Bhaskar!
-- 
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124

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. ]

Randy Yates wrote:

   ...

> Does the problem you see go away with the sample clock synchronized
> to t = 0? If so, then we're not seeing the same thing.

No, but agreeing to sample always at t=0 will simplify what I want to 
say. When sampling a unity-amplitude cosine, the first sample will be +1 
and the next (if it isn't also +1) must be less. It's clear that the 
alias is a cosine too. (My daughter brought home straight-A report cards 
all through high school. I used to tell her, "This is terrible: the only 
way you can go is down.")

When sampling a sine, the first sample is zero. The sign of the second 
sample depends in a simple way on the ratio of the analog signal to the 
sampling frequency. If the second sample falls in the first (or any odd) 
half cycle of the signal, the alias will be a positive sine. If it falls 
in the second (or any even) half cycle, the alias will be a negative 
sine. Simple. Pat. Right?

Please excuse the unnecessary stuff I wrote before. The problem hadn't 
settled in yet. (I was always naive yesterday.)

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