Angelo Ricotta –

In the article “Turning Nyquist upside down by
undersampling” by Bonnie Baker, EDN 12 May 2005, are reported the two formulae _{} and _{} to compute an allowable sampling frequency for
undersampling (bandpass sampling, subsampling, downsampling, sub-Nyquist, super-Nyquist, harmonic sampling) a bandpass signal. I was surprised by that because I have been using the undersampling technique since the beginning of the eighties 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 the undersampling, to EDN Signals &
Noise Editor but I never received an answer. On 25 March 1994 I attended a
Burr-Brown’s Applications Seminar in **in **,

Let _{} be the lowest and _{} the highest frequency of a band-pass
signal to be sampled.

Based on the “accordion-pleated” paper model we
should have _{} and _{}, with _{}integer, to
avoid the folding of the spectrum on itself. Simple manipulations give _{} in which _{}. For
example, put _{}, _{}. Applying
the second of the above formulae we obtain _{}, i.e. _{}, and then from
the first we have **all **the allowable
sampling frequencies: _{}: _{}, _{}: _{} and, of course, _{}: _{}. Based on
the “accordion-pleated” paper model the order of the harmonics of the aliased
spectrum of the bandpass signal could be reversed or not depending on the
position of the original signal to respect to the chosen _{}: if the
corresponding _{} is odd the order is preserved, if it is even
the order is reversed. These calculations permit the adjustement of the
sampling frequency depending on the specific application. If the data at our disposal
are the bandwidth _{}, and the
carrier _{} of the signal, we may put _{} and _{}, so that _{} and _{} and we can carry on the computation as above. Instead,
the formulae reported in the cited article give the single value _{}, being _{} and _{}. The
formulae _{} and _{} can be easily deduced from _{}. By
definition _{} therefore _{} is always satisfied: note that you cannot use
every _{} for undersampling because you have to satisfy
the other constraint. To deduce _{}, assume _{}, the arithmetic mean of the two bounds of the fundamental
formula.

It is _{}. Putting _{} and substituting _{} we obtain
_{}. The substitution _{} produces an _{} lower than the arithmetic mean assumed before.
It is _{} and then _{} with _{}. A more direct way to obtain _{} is to notice that with
this particular sampling frequency the aliases of the spectrum of the signal
are **centred** on the pages _{} and its multiples (Fig.3, p.5, Ref.4).

It has to
be _{} from which _{}.

REFERENCES

1. “Reference Data For Radio
Engineers, Fifth Edition”, Howard W. Sams & Co., Inc., ITT, 1970, (p.21-14).

2. Julius S. Bendat, Allan G.
Piersol, “Random Data: Analysis and Measurement Procedures”,
Wiley-Interscience, 1971, (p.230).

3. E. Oran Brigham, “The Fast
Fourier Transform”, Prentice-Hall, Inc., 1974, (p.87).

5. Angelo Ricotta, “Development of
an acoustic radar and applications to the planetary boundary layer dynamics
studies”, Physics Thesis,

6. E. J. Owens, NOAA MARK VII
Acoustic Echo Sounder, NOAA Tech. Mem.,