41 research outputs found
On Solving a Generalized Chinese Remainder Theorem in the Presence of Remainder Errors
In estimating frequencies given that the signal waveforms are undersampled
multiple times, Xia et. al. proposed to use a generalized version of Chinese
remainder Theorem (CRT), where the moduli are which are
not necessarily pairwise coprime. If the errors of the corrupted remainders are
within \tau=\sds \max_{1\le i\le k} \min_{\stackrel{1\le j\le k}{j\neq i}}
\frac{\gcd(M_i,M_j)}4, their schemes can be used to construct an approximation
of the solution to the generalized CRT with an error smaller than .
Accurately finding the quotients is a critical ingredient in their approach. In
this paper, we shall start with a faithful historical account of the
generalized CRT. We then present two treatments of the problem of solving
generalized CRT with erroneous remainders. The first treatment follows the
route of Wang and Xia to find the quotients, but with a simplified process. The
second treatment considers a simplified model of generalized CRT and takes a
different approach by working on the corrupted remainders directly. This
approach also reveals some useful information about the remainders by
inspecting extreme values of the erroneous remainders modulo . Both of
our treatments produce efficient algorithms with essentially optimal
performance. Finally, this paper constructs a counterexample to prove the
sharpness of the error bound
High Speed Under-Sampling Frequency Measurements on FPGA
A Sampling rate is less than Nyquist rate in some applications because of hardware limitations. Consequently, extensive researches have been conducted on frequency detection from sub-sampled signals. Previous studies on under-sampling frequency measurements have mostly discussed under-sampling frequency detection in theory and suggested possible methods for fast under-sampling frequencies detection. This study examined few suggested methods on Field Programmable Gate Array (FPGA) for fast under-sampling frequencies measurement. Implementation of the suggested methods on FPGA has issues that make them improper for fast data processing. This study tastes and discusses different methods for frequency detection including Least Squares (LS), Direct State Space (DSS), Goertzel filter, Sliding DFT, Phase changes of Fast Furrier Transform (FFT), peak amplitude of FFT to conclude which one from these methods are suitable for fast under-sampling frequencies detection on FPGA. Moreover, our proposed approach for sub-sampling detection from real waveform has less complextity than previous approaches from complex waveform