Coprime Reconstruction of Super-Nyquist Periodic Signal and Sampling Moiré Effect

Abstract

Sampling moiré effect is well-known in signal processing. When a continuous periodic signal $x(t)$ is sampled using a sampling rate $f_{s}$ that does not respect the Nyquist condition, and the signal-frequency $f_{x}$ folds over and gives a special sequence in the sampled signal. The coprime reconstruction is made possible when the periodic signal frequency $f_{x}$ and the sampling frequency $f_{s}$ satisfy $f_{s}/f_{x}=N/M~$ with integer $M$, $N$, where $N/M$ is a reduced fraction. This article shows how to rearrange the N sample data into one entire period of the signal using the M periods of the signal. The reconstruction is equivalent to the decryption of the sample data using the method referred to the mathematic operator “multiplicative inverse modulo.” In the implementation to verify the coprime reconstruction, we used a commodity DSP board with 4.2 MHz sampling rate to reconstruct the 13.56 MHz signal acquired from the ADC circuit. The signal reconstruction result is comparable to the sampling result for the same signal sampled at 475 MHz sampling rate. The other simulation on the 234 kHz signal under 1 MHz sampling rate yields a very detail information on the signal with a finding that the multiplicative inverse modulo may also be used as the encryption of signal for network security use.