Instrumentation & Measurement Magazine 23-2 - 101

be input) feed five sampled values into a set of summer circuits that compress the signal to four samples. These signals
are sampled by our microcontroller (and converted to digital values by its A/D) at a frequency lower than the Nyquist
frequency associated with the frequencies we are measuring,
and because the compressed measurements contain all of the
information about the original signal, we can use a computer
running MATLAB to determine frequency of the signal that
was input into our system. Our system works well and provides a simple demonstration of sub-Nyquist sampling.

References
[1]	 X. Chen et al., "A sub-Nyquist rate compressive sensing data

Fig. 5. The signal stored in the analog shift register. (This is actually the
signal as seen at the output of the first master-slave unit.)

acquisition front-end," IEEE J. Emerging and Selected Topics in
Circuits and Syst., vol. 2, no. 3, pp. 542-551, Sep. 2012.
[2]	 S. Engelberg, "Compressive sensing," IEEE Instrum. Meas. Mag.,
vol. 15, no. 1, pp. 42-46, Feb. 2012.
[3]	 B. K. Natarajan. "Sparse approximate solutions to linear
systems," SIAM J. Computing, vol. 24, pp. 227-234, Apr. 1995.
[4]	 N. Ailon, "Advanced Topics in Algorithms: Lecture 7 - December
2, 2010," Faculty of Computer Science, Technion-Israel Institute
of Technology, Haifa, Israel (accessed 7/5/19). [Online].
Available: http://www.cs.technion.ac.il/∼nailon/236620_
group10_fall2010/downloads/lecture7.pdf.
[5]	 S. Zhuang, W. Zhao, R. Wang, Q. Wang and S. Huang, "New
measurement algorithm for supraharmonics based on multiple
measurement vectors model and orthogonal matching pursuit,"
IEEE Trans. Instrum. Meas., vol. 68, no. 6, pp. 1671-1679, Jun. 2019.

Fig. 6. The signal as reconstructed by MATLAB from the four "compressed
samples."

[6]	 M. Simonnard, Linear Programming. Englewood Cliffs, NJ, USA:
Prentice-Hall, Inc., 1966.
[7]	 S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge,
UK: Cambridge University Press, 2004.
[8]	 S. Shalev-Shwartz. "Compressed sensing: basic results and
self contained proofs," Jan. 2009 (accessed 7/5/19). [Online].
Available: https://www.cs.huji.ac.il/∼shais/compressedSensing.
pdf.
[9]	 "Condition Number," Wikipedia (accessed 7/5/19). [Online].
Available: https://en.wikipedia.org/wiki/Condition_number.
[10]	"ADuC841," Analog Devices (accessed 7/5/19). [Online].
Available: https://www.analog.com/en/products/aduc841.
html.

Fig. 7. The final system. In the photo, you see a couple of power supplies
(at the left); a 'scope (with a built-in signal generator), a breadboard, and a
microcontroller (in the center); and a PC (at the right).

Conclusions
The system described above allows you to sample signals at
frequencies that are lower than the Nyquist sampling theorem
seems to allow. As we have seen, it works by having an analog
shift register (which samples at a frequency that is more than
twice the frequency of the highest frequency signal that may
April 2020	

Roi Arie (roiarie52@gmail.com) is currently completing his
bachelor's degree in electrical and electronics engineering at
the Jerusalem College of Technology, Israel. He currently holds
a position as an analog engineer, and his interests include analog and digital electronics.
Avraham Brand (ambrand613@gmail.com) is currently completing his bachelor's degree in electrical and electronics
engineering at the Jerusalem College of Technology, Israel. His research interests include signal processing and
communications.
Shlomo Engelberg (shlomoe@jct.ac.il) is the Dean of the School
of Engineering and Computer Science at the Jerusalem College
continued on page 104

IEEE Instrumentation & Measurement Magazine	101


https://www.cs.huji.ac.il/~shais/compressedSensing.pdf https://www.cs.huji.ac.il/~shais/compressedSensing.pdf http://www.cs.technion.ac.il/~nailon/236620_group10_fall2010/downloads/lecture7.pdf http://www.cs.technion.ac.il/~nailon/236620_group10_fall2010/downloads/lecture7.pdf https://en.wikipedia.org/wiki/Condition_number https://www.analog.com/en/products/aduc841.html https://www.analog.com/en/products/aduc841.html

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