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

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.

A regularized two-dimensional sampling algorithm

In 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.

Research on Fabric Image Acquisition Based on Compressed Sensing

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.

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

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.

Reconstructing Digital Signals Using 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.