Intriguing story about undersampling
 

Look at this http://spazioscuola.altervista.org/U...samplingAR.htm
An excerpt:

A STORY ABOUT UNDERSAMPLING

by Angelo Ricotta - Rome, Italy



ITALIAN VERSION

In the article "Turning Nyquist upside down by undersampling" by Bonnie Baker, EDN 12 May 2005, are reported two formulae to compute an allowable sampling frequency for undersampling a bandpass signal. I was surprised by that because I have been using the undersampling technique since the late seventies and even wrote a report (Angelo Ricotta, "Some remarks on the sampling and processing of SODAR data", Technical Report, IFA-CNR, July 1983) where I gave two simple and practical formulae to compute all the allowable sampling frequencies for undersampling a given bandpass signal. The report was written in Italian and was known, at least, among the Italian community working on SODAR systems in which a few people and even students utilized my formulae in an unfair way because they did not mention the source. On 10 October and 7 December 1991, also to stop the above misuses, I sent two letters, containing my formulae for undersampling, to EDN Signals & Noise Editor but I never received an answer...

Gee, I hope Bonnie didn't claim they were original. Certainly it's not
difficult to figure out.

More interesting to me is recent work on letting a delta-sigma ADC
sample at a rate to serve as a direct IF sampler--up around 5GHz. The
practice of getting it to work really right at such frequencies is much
more interesting than the formulas for what frequencies work and avoid
aliasing.

Cheers,
Tom

Greetings all,

I'm new to the group and find many of the various threads fascinating.
I'm currently building a homebrew HF rig.

I think undersampling has probably been around nearly as long as
sampling itself. This is because samplers are a subset of (RF) mixers,
and early engineers fluent in RF mixer skills would have had
undersampling in their toolboxes already.

As mixers, samplers are modelled as a multipliers with the sampler
clock serving as the local oscillator. What makes samplers a subset of
mixers are the constraints placed on the LO--in the ideal sampler a
Dirac delta function is used. This waveform has the property that all
harmonics (including the zeroth or DC and even harmonics) have the same
amplitude. Our doubly-balanced RF mixers typically respond to only odd
harmonics. Real-World samplers approximate the ideal sampler fairly
well over a wide frequency range.

Because samplers have a DC response, the baseband signal Fin is passed
by the sampler, something RF mixers don't normally do (but can be made
to do simply by putting a DC component on the local oscillator). When
viewing a sampler as a mixer, aliasing is nothing more or less than the
mixing product (LO-Fin) overlapping the baseband signal Fin.
Undersampling is nothing more or less than prefiltering the desired
band of frequencies and mixing them to baseband with the appropriate
harmonic of the local oscillator.

I've not found the notion of samplers as mixers in the literature, but
probably could if I looked hard enough. Perhaps authors consider the
notion either obvious or not useful, but to this old ham viewing
samplers as mixers is useful.

Bonnie's article contains a fun blunder. She states Nyquist and
Shannon developed sampling theory in the 1920's, which would have put
Shannon in his teens. Shannon's seminal paper on communication theory
was actually published in 1948. I've read that the 'Nyquist rate'
should really be called the 'Shannon rate' as he was the first to
develop it. Anyone know more?

Regards,
Glenn Dixon, AC7ZN

That's absolutely correct.

Tektronix has produced sampling oscilloscopes from at least the early
60's which have a frequency response of from several to several tens of
GHz but sample at rates often below 1 M sample/sec. Aliasing? You
betcha! As you pointed out, the aliased signal is the desired one -- a
frequency (or time) converted signal. The only requirement is that the
signal be repetative, since it takes samples from many cycles to create
the time-scaled waveform. Single-shot events, often misunderstood or
disregarded by RF engineers, require Nyquist-dictated sampling rates.
This is done by most conventional digital oscilloscopes. The Nyquist
criterion assumes that you want to recreate a nearly exact replica of
the waveform and that it extends down to DC; if you'll settle for a time
scaled or frequency shifted one, or one with a limited bandwidth, other
options are available.

Roy Lewallen, W7EL

wrote:
Greetings all,

I'm new to the group and find many of the various threads fascinating.
I'm currently building a homebrew HF rig.

I think undersampling has probably been around nearly as long as
sampling itself. This is because samplers are a subset of (RF) mixers,
and early engineers fluent in RF mixer skills would have had
undersampling in their toolboxes already.

As mixers, samplers are modelled as a multipliers with the sampler
clock serving as the local oscillator. What makes samplers a subset of
mixers are the constraints placed on the LO--in the ideal sampler a
Dirac delta function is used. This waveform has the property that all
harmonics (including the zeroth or DC and even harmonics) have the same
amplitude. Our doubly-balanced RF mixers typically respond to only odd
harmonics. Real-World samplers approximate the ideal sampler fairly
well over a wide frequency range.

Because samplers have a DC response, the baseband signal Fin is passed
by the sampler, something RF mixers don't normally do (but can be made
to do simply by putting a DC component on the local oscillator). When
viewing a sampler as a mixer, aliasing is nothing more or less than the
mixing product (LO-Fin) overlapping the baseband signal Fin.
Undersampling is nothing more or less than prefiltering the desired
band of frequencies and mixing them to baseband with the appropriate
harmonic of the local oscillator.

I've not found the notion of samplers as mixers in the literature, but
probably could if I looked hard enough. Perhaps authors consider the
notion either obvious or not useful, but to this old ham viewing
samplers as mixers is useful.

Bonnie's article contains a fun blunder. She states Nyquist and
Shannon developed sampling theory in the 1920's, which would have put
Shannon in his teens. Shannon's seminal paper on communication theory
was actually published in 1948. I've read that the 'Nyquist rate'
should really be called the 'Shannon rate' as he was the first to
develop it. Anyone know more?

Regards,
Glenn Dixon, AC7ZN

I read Angelo's full article (shoulda done that before I pontificated
the first time) and it answered the Nyquist vs. Shannon question I had.
Nyquist had the concept, Shannon proved it mathematically. Thanks,
Angelo.

Regards,
Glenn Dixon, AC7ZN