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. 

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.

scholarly journals Quantitative Synaptic Biology: A Perspective on Techniques, Numbers and Expectations

2020 ◽  
Vol 21 (19) ◽  
pp. 7298
Author(s):  
Sofiia Reshetniak ◽  
Rubén Fernández-Busnadiego ◽  
Marcus Müller ◽  
Silvio O. Rizzoli ◽  
Christian Tetzlaff
Keyword(s):  
Mathematical Models ◽  
Super Resolution ◽  
Spatiotemporal Dynamics ◽  
Synaptic Function ◽  
Computational Studies ◽  
Complete Understanding ◽  
The Brain ◽  
Modelling Approaches

Synapses play a central role for the processing of information in the brain and have been analyzed in countless biochemical, electrophysiological, imaging, and computational studies. The functionality and plasticity of synapses are nevertheless still difficult to predict, and conflicting hypotheses have been proposed for many synaptic processes. In this review, we argue that the cause of these problems is a lack of understanding of the spatiotemporal dynamics of key synaptic components. Fortunately, a number of emerging imaging approaches, going beyond super-resolution, should be able to provide required protein positions in space at different points in time. Mathematical models can then integrate the resulting information to allow the prediction of the spatiotemporal dynamics. We argue that these models, to deal with the complexity of synaptic processes, need to be designed in a sufficiently abstract way. Taken together, we suggest that a well-designed combination of imaging and modelling approaches will result in a far more complete understanding of synaptic function than currently possible.

scholarly journals Ultrasonic Signal Reconstruction Using Compressed Sensing

2016 ◽  
Vol 855 ◽  
pp. 165-170
Author(s):  
Ren Jean Liou
Keyword(s):  
Compressed Sensing ◽  
Spline Interpolation ◽  
Signal To Noise Ratio ◽  
Signal Reconstruction ◽  
Sampling Theorem ◽  
Ultrasonic Signal ◽  
Ultrasonic Signals ◽  
Data Rates ◽  
Nyquist Rate ◽  
Simulation Results

Ultrasonic signal reconstruction for Structural Health Monitoring is a topic that has been discussed extensively. In this paper, we will apply the techniques of compressed sensing to reconstruct ultrasonic signals that are seriously damaged. To reconstruct the data, the application of conventional interpolation techniques is restricted under the criteria of Nyquist sampling theorem. The newly developed technique - compressed sensing breaks the limitations of Nyquist rate and provides effective results based upon sparse signal reconstruction. Sparse representation is constructed using Fourier transform basis. An l1-norm optimization is then applied for reconstruction. Signals with temperature characteristics were synthetically created. We seriously corrupted these signals and tested the efficacy of our approach under two different scenarios. Firstly, the signal is randomly sampled at very low rates. Secondly, selected intervals were completely blank out. Simulation results show that the signals are effectively reconstructed. It outperforms conventional Spline interpolation in signal-to-noise ratio (SNR) with low variation, especially under very low data rates. This research demonstrates very promising results of using compressed sensing for ultrasonic signal reconstruction.

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.

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