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.

Randomized Trees for Time Series Representation and Similarity

2021 ◽  
pp. 108097
Author(s):  
Berk Görgülü ◽  
Mustafa Gökçe Baydoğan
Keyword(s):  
Time Series ◽  
Series Representation

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 Time series analysis of radiant heat using 75 hours VIIRS satellite day and night band nightfire data

2020 ◽  
Vol 2020 (1) ◽  
pp. 98-117
Author(s):  
Jyoti U. Devkota
Keyword(s):  
Time Series ◽  
Infrared Imaging ◽  
Time Series Data ◽  
Spline Interpolation ◽  
Radiant Heat ◽  
Series Data ◽  
Time Interval ◽  
Industrial Sites ◽  
Steel Mills ◽  
Gas Flares

Abstract The nightfires illuminated on the earth surface are caught by the satellite. These are emitted by various sources such as gas flares, biomass burning, volcanoes, and industrial sites such as steel mills. Amount of nightfires in an area is a proxy indicator of fuel consumption and CO2 emission. In this paper the behavior of radiant heat (RH) data produced by nightfire is minutely analyzed over a period of 75 hour; the geographical coordinates of energy sources generating these values are not considered. Visible Infrared Imaging Radiometer Suite Day/Night Band (VIIRS DNB) satellite earth observation nightfire data were used. These 75 hours and 28252 observations time series RH (unit W) data is from 2 September 2018 to 6 September 2018. The dynamics of change in the overall behavior these data and with respect to time and irrespective of its geographical occurrence is studied and presented here. Different statistical methodologies are also used to identify hidden groups and patterns which are not obvious by remote sensing. Underlying groups and clusters are formed using Cluster Analysis and Discriminant Analysis. The behavior of RH for three consecutive days is studied with the technique Analysis of Variance. Cubic Spline Interpolation and merging has been done to create a time series data occurring at equal minute time interval. The time series data is decomposed to study the effect of various components. The behavior of this data is also analyzed in frequency domain by study of period, amplitude and the spectrum.

Thoughts on Women Entrepreneurship: An Application of Market Basket Analysis with Google Trends Data

2021 ◽  
Author(s):  
Anıl Boz Semerci ◽  
Ayşe Abbasoğlu Özgören ◽  
Duygu İçen
Keyword(s):  
Time Series ◽  
Social Justice ◽  
Geographic Area ◽  
Google Trends ◽  
Digital Data ◽  
Specific Topic ◽  
Market Basket Analysis ◽  
Women Entrepreneurship ◽  
Market Basket ◽  
Regional Setting

Abstract This paper focuses on the popularity and awareness of keywords in Google Trends data related to entrepreneurship of women in a global and cross-regional setting by using market basket analysis. Google Trends is one of the digital data platforms that provides a time series index of the volume of queries users enter into Google in a given geographic area. It is the most popular tool for gathering any information, and it has been used in several topics. Market basket analysis indicates items that appear/used together and the frequency of these appearances. Such technique is appropriate in finding hidden associations between items, which is also crucial in assessing individuals’ thoughts on a specific topic. This study contributes to the literature in terms of being the first study to use market basket analysis on Google trends data in the context of women’s entrepreneurship finding hidden associations between items, which is crucial in assessing individuals’ thoughts on a specific topic. The results of the analysis are interpreted through the lens of genderresponsive strategies, equality, efficiency and social justice in different country and region contexts.

scholarly journals Edge4TSC: Binary Distribution Tree-Enabled Time Series Classification in Edge Environment

Sensors ◽  
2020 ◽  
Vol 20 (7) ◽  
pp. 1908
Author(s):  
Chao Ma ◽  
Xiaochuan Shi ◽  
Wei Li ◽  
Weiping Zhu
Keyword(s):  
Time Series ◽  
Deep Learning ◽  
Classification Accuracy ◽  
Time Series Data ◽  
Series Representation ◽  
Series Data ◽  
Feature Engineering ◽  
Time Series Classification ◽  
Binary Distribution ◽  
New Time

In the past decade, time series data have been generated from various fields at a rapid speed, which offers a huge opportunity for mining valuable knowledge. As a typical task of time series mining, Time Series Classification (TSC) has attracted lots of attention from both researchers and domain experts due to its broad applications ranging from human activity recognition to smart city governance. Specifically, there is an increasing requirement for performing classification tasks on diverse types of time series data in a timely manner without costly hand-crafting feature engineering. Therefore, in this paper, we propose a framework named Edge4TSC that allows time series to be processed in the edge environment, so that the classification results can be instantly returned to the end-users. Meanwhile, to get rid of the costly hand-crafting feature engineering process, deep learning techniques are applied for automatic feature extraction, which shows competitive or even superior performance compared to state-of-the-art TSC solutions. However, because time series presents complex patterns, even deep learning models are not capable of achieving satisfactory classification accuracy, which motivated us to explore new time series representation methods to help classifiers further improve the classification accuracy. In the proposed framework Edge4TSC, by building the binary distribution tree, a new time series representation method was designed for addressing the classification accuracy concern in TSC tasks. By conducting comprehensive experiments on six challenging time series datasets in the edge environment, the potential of the proposed framework for its generalization ability and classification accuracy improvement is firmly validated with a number of helpful insights.

scholarly journals Machine Tool Component Health Identification with Unsupervised Learning

2020 ◽  
Vol 4 (3) ◽  
pp. 86
Author(s):  
Thomas Gittler ◽  
Stephan Scholze ◽  
Alisa Rupenyan ◽  
Konrad Wegener
Keyword(s):  
Time Series ◽  
Machine Tool ◽  
Statistical Significance ◽  
Clustering Algorithms ◽  
Series Representation ◽  
Feature Representation ◽  
Machine Component ◽  
Test Cycle ◽  
Clustering Model ◽  
Online Measurements

Unforeseen machine tool component failures cause considerable losses. This study presents a new approach to unsupervised machine component condition identification. It uses test cycle data of machine components in healthy and various faulty conditions for modelling. The novelty in the approach consists of the time series representation as features, the filtering of the features for statistical significance, and the use of this feature representation to train a clustering model. The benefit in the proposed approach is its small engineering effort, the potential for automation, the small amount of data necessary for training and updating the model, and the potential to distinguish between multiple known and unknown conditions. Online measurements on machines in unknown conditions are performed to predict the component condition with the aid of the trained model. The approach was exemplarily tested and verified on different healthy and faulty states of a grinding machine axis. For the accurate classification of the component condition, different clustering algorithms were evaluated and compared. The proposed solution demonstrated encouraging results as it accurately classified the component condition. It requires little data, is straightforward to implement and update, and is able to precisely differentiate minor differences of faults in test cycle time series.

Making Image Retrieval and Classification More Accurate Using Time Series and Learned Constraints

2011 ◽  
pp. 145-170 ◽  
Author(s):  
Chotirat “Ann” Ratanamahatana ◽  
Eamonn Keogh ◽  
Vit Niennattrakul
Keyword(s):  
Time Series ◽  
Image Retrieval ◽  
Data Storage ◽  
Distance Measure ◽  
Series Representation ◽  
Feedback System ◽  
Multimedia Data ◽  
Video Frames ◽  
Computationally Intensive ◽  
Dynamic Time

After the generation of multimedia data turning digital, an explosion of interest in their data storage, retrieval, and processing, has drastically increased in the database and data mining community. This includes videos, images, and handwriting, where we now have higher expectations in exploiting these data at hand. We argue however, that much of this work’s narrow focus on efficiency and scalability has come at the cost of usability and effectiveness. Typical manipulations are in some forms of video/image processing, which require fairly large amounts for storage and are computationally intensive. In this work, we will demonstrate how these multimedia data can be reduced to a more compact form, that is, time series representation, while preserving the features of interest, and can then be efficiently exploited in Content-Based Image Retrieval. We also introduce a general framework that learns a distance measure with arbitrary constraints on the warping path of the Dynamic Time Warping calculation. We demonstrate utilities of our approach on both classification and query retrieval tasks for time series and other types of multimedia data including images, video frames, and handwriting archives. In addition, we show that incorporating this framework into the relevance feedback system, a query refinement can be used to further improve the precision/recall by a wide margin.

Sign in / Sign up

Export Citation Format

Share Document