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