scholarly journals Implementations of Shannon’s sampling theorem, a time-frequency approach

2005 ◽  
Vol 4 (1) ◽  
pp. 2-17
Author(s):  
Thomas Strohmer ◽  
Jared Tanner
Keyword(s):  
Sampling Theorem ◽  
Time Frequency ◽  
Shannon’S Sampling Theorem

scholarly journals Discrete sampling theorem to Shannon’s sampling theorem using the hyperreal numbers R

2021 ◽  
Vol 28 (2) ◽  
pp. 163-182
Author(s):  
José L. Simancas-García ◽  
Kemel George-González
Keyword(s):  
Computational Cost ◽  
Number System ◽  
Sampling Theorem ◽  
Discrete Sampling ◽  
Sampled Signal ◽  
Speed Up ◽  
Shannon’S Sampling Theorem ◽  
The Difference ◽  
Band Limited ◽  
Hyperreal Numbers

Shannon’s sampling theorem is one of the most important results of modern signal theory. It describes the reconstruction of any band-limited signal from a finite number of its samples. On the other hand, although less well known, there is the discrete sampling theorem, proved by Cooley while he was working on the development of an algorithm to speed up the calculations of the discrete Fourier transform. Cooley showed that a sampled signal can be resampled by selecting a smaller number of samples, which reduces computational cost. Then it is possible to reconstruct the original sampled signal using a reverse process. In principle, the two theorems are not related. However, in this paper we will show that in the context of Non Standard Mathematical Analysis (NSA) and Hyperreal Numerical System R, the two theorems are equivalent. The difference between them becomes a matter of scale. With the scale changes that the hyperreal number system allows, the discrete variables and functions become continuous, and Shannon’s sampling theorem emerges from the discrete sampling theorem.

Reconstructing Digital Signals Using Shannon's Sampling Theorem

1997 ◽  
Vol 13 (2) ◽  
pp. 226-238 ◽  
Author(s):  
Joseph Hamill ◽  
Graham E. Caldwell ◽  
Timothy R. Derrick
Keyword(s):  
Time Series ◽  
Critical Frequency ◽  
Spline Interpolation ◽  
Series Representation ◽  
Sampling Theorem ◽  
Frequency Characteristics ◽  
Digital Data ◽  
Original Signal ◽  
Frequency Content ◽  
Shannon’S Sampling Theorem

Researchers must be cognizant of the frequency content of analog signals that they are collecting. Knowing the frequency content allows the researcher to determine the minimum sampling frequency of the data (Nyquist critical frequency), ensuring that the digital data will have all of the frequency characteristics of the original signal. The Nyquist critical frequency is 2 times greater than the highest frequency in the signal. When sampled at a rate above the Nyquist, the digital data will contain all of the frequency characteristics of the original signal but may not present a correct time-series representation of the signal. In this paper, an algorithm known as Shannon's Sampling Theorem is presented that correctly reconstructs the time-series profile of any signal sampled above the Nyquist critical frequency. This method is superior to polynomial or spline interpolation techniques in that it can reconstruct peak values found in the original signal but missing from the sampled data time-series.

scholarly journals A Micro-Doppler Frequency Ambiguity Resolution Method Based on Complex-Valued U-Net

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Ming Long ◽  
Jun Yang ◽  
Saiqiang Xia ◽  
Mingjiu Lv ◽  
Xu Wei ◽  
...  
Keyword(s):  
Ambiguity Resolution ◽  
Pulse Repetition Frequency ◽  
Doppler Frequency ◽  
Sampling Theorem ◽  
Repetition Frequency ◽  
Pulse Repetition ◽  
Resolution Method ◽  
Time Frequency ◽  
Short Time ◽  
Complex Valued

In order to resolute the micro-Doppler frequency ambiguity caused by radar pulse repetition frequency not high enough (i.e., pulse dimension does not satisfy the requirement of Nyquist sampling theorem), this paper presents a micro-Doppler frequency ambiguity resolution method based on complex-valued U-net. The echo sequence is interpolated by zeros in the pulse dimension to increase the equivalent pulse repetition frequency, so that the echo sequence after zero interpolation contains the real micro-Doppler frequency; at the same time, some new frequency components are generated. The variation law of the echo sequence frequency after zero interpolation is analyzed. Then, the echo sequence in time domain after zero interpolation is transformed to the time-frequency domain by short-time Fourier transform (STFT). Finally, the time-frequency results can be segmented by the model, which is trained by complex-valued U-net to eliminate the redundant frequencies generated by zero interpolation; thus, the reconstruction of real micro-Doppler frequency is realized. Theoretical analysis and simulation results show that the proposed method can solve the problem of micro-Doppler frequency ambiguity. Compared with fully convolution network (FCN) and fully convolution residual network (FCRN), the proposed method has better performance and robustness.

Research on Fabric Image Acquisition Based on Compressed Sensing

2014 ◽  
Vol 989-994 ◽  
pp. 3698-3701
Author(s):  
Ji Cheng Dong ◽  
Sheng Qi Guan ◽  
Long Long Chen
Keyword(s):  
Theoretical Analysis ◽  
Compressed Sensing ◽  
Textile Industry ◽  
Image Acquisition ◽  
Sampling Rate ◽  
Image Data ◽  
Sampling Theorem ◽  
High Quality ◽  
Shannon’S Sampling Theorem

Amount of data in collecting data of fabric image in the textile industry put forward a new challenge to sensor end. Compressed Sensing (CS) breaks limit of conventional Shannon’s sampling theorem, so we can reconstruct a signal in Sub-sampling rate. In addition, theoretical analysis tells us that collecting the fabric image data by CS method have a better advantage than collecting the general image data. Having reconstructed three fabric images and one general image by CS method, we can easily find that the former have a high quality of reconstruction.

scholarly journals An MCM-Enhanced Compressive Sensing for Weak Fault Feature Extraction of Rolling Element Bearings under Variable Speeds

2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Ya He ◽  
Kun Feng ◽  
Minghui Hu ◽  
Jinmiao Cui
Keyword(s):  
Compressive Sensing ◽  
Circulant Matrix ◽  
Sampling Theorem ◽  
Rolling Element Bearings ◽  
Measurement Matrix ◽  
Rolling Bearings ◽  
Time Frequency ◽  
Rolling Element ◽  
Shannon Sampling Theorem ◽  
Weak Fault

The compressive sensing (CS) theory provides a new slight to the big-data problem led by the Shannon sampling theorem in rolling element bearings condition monitoring, where the measurement matrix of CS tends to be designed by the random matrix (RM) to preserve the integrity of signal roughly. However, when the signal to be analyzed is infected with strong noise, not only does the signal become insufficiently sparse, but the randomness of the measurement matrix will bring down the sensing efficiency, resulting in the loss of fault feature. Thus, a sensing-enhanced CS scheme based on a series of modes after VMD decomposition is proposed under this paper. The core of this scheme is as follows: (1) the principal mode of VMD with better sparsity replaces the raw signal for compressive sensing; (2) all these modes contain the time-frequency characteristics of the raw signal; (3) a new measurement matrix called mode-circulant matrix (MCM) is defined by circulating the mode matrix, and when the amount of samples is shrunk, the sensing efficiency can be enhanced greatly. Besides, considering the fault signal of rolling bearings under variable speed, there is a need to use order tracking to overcome the nonstationarity of the signal before applying CS theory. The analysis results of simulation and experiment prove that the VMD- and MCM-based CS can successfully extract the weak fault feature of rolling bearings with operating speed changing.

A regularized two-dimensional sampling algorithm

2018 ◽  
Vol 26 (1) ◽  
pp. 67-84
Author(s):  
Weidong Chen
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Sampling Theorem ◽  
Dimensional Case ◽  
Tikhonov Regularization Method ◽  
Two Dimensional ◽  
Sampling Algorithm ◽  
Shannon’S Sampling Theorem ◽  
Band Limited

AbstractIn this paper, a regularized sampling algorithm for band-limited signals is presented in the two-dimensional case. The convergence of the regularized sampling algorithm is studied and compared with Shannon’s sampling theorem and the Tikhonov regularization method by some examples.

scholarly journals Bandpass Sampling – An Opportunity To Stress The Importance Of In-Depth Understanding

2010 ◽  
Vol 1 (1) ◽  
pp. 43-46
Author(s):  
Harold P. E. Stern
Keyword(s):  
Mathematical Models ◽  
Sampling Theorem ◽  
Physical Processes ◽  
Complete Understanding ◽  
Bandpass Sampling ◽  
Nyquist Rate ◽  
Shannon’S Sampling Theorem ◽  
Bandpass Signals

Many bandpass signals can be sampled at rates lower than the Nyquist rate, allowing significant practical advantages. Illustrating this phenomenon after discussing (and proving) Shannon’s sampling theorem provides a valuable opportunity for an instructor to reinforce the principle that innovation is possible when students strive to have a complete understanding of physical processes and mathematical models. 

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