41 research outputs found

    On Solving a Generalized Chinese Remainder Theorem in the Presence of Remainder Errors

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    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 M1,M2,⋯ ,MkM_1, M_2, \cdots, M_k 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 τ\tau. 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 4τ4\tau. 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 τ\tau

    High Speed Under-Sampling Frequency Measurements on FPGA

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    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
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