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Types of Skewness01:09

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If the frequency distribution of a data set is more inclined towards smaller or larger values, the distribution is said to be skewed. If data values are skewed to the right, then the distribution is called positively skewed. Conversely, if the plot is skewed to the left, the distribution is called negatively skewed.
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The measures of central tendency calculated from a data set may not reveal much about its intrinsic distribution. If a plot is made of the data set’s values, the mean and the median may not only differ, but also the plot may have more values on one side of the central tendencies. Such a data set is said to be skewed towards that side.
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In any LTI (Linear Time-Invariant) system, the convolution of two signals is denoted using a convolution operator, assuming all initial conditions are zero. The convolution integral can be divided into two parts: the zero-input or natural response and the zero-state or forced response, with t0 indicating the initial time.
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The important convolution properties include width, area, differentiation, and integration properties.
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Convolution computations can be simplified by utilizing their inherent properties.
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It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like addition, subtraction, and multiplication. There are two ways to circumvent this algebraic complexity. One way is to draw the vectors to scale, as in navigation, and read approximate vector lengths and angles (directions) from the graphs. The other way is to use the method of components.
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Skew Convolutional Codes.

Vladimir Sidorenko1, Wenhui Li2, Onur Günlü3

  • 1Institute for Communications Engineering, Technical University of Munich, 80333 München, Germany.

Entropy (Basel, Switzerland)
|December 5, 2020
PubMed
Summary
This summary is machine-generated.

A novel class of skew convolutional codes is introduced, extending fixed convolutional codes. These codes offer a compact description similar to fixed codes while being representable as periodic time-varying codes.

Keywords:
convolutional codesdual codesskew polynomialstime-varying codestrellises

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Area of Science:

  • Coding Theory
  • Information Theory
  • Digital Communications

Background:

  • Classical fixed convolutional codes are foundational in digital communication.
  • Existing codes may lack the desired balance between complexity and performance.
  • Time-varying codes offer flexibility but can be complex to describe.

Purpose of the Study:

  • To introduce a new class of convolutional codes, termed skew convolutional codes.
  • To demonstrate that skew convolutional codes provide a compact representation comparable to fixed codes.
  • To present the design principles for these new codes and their decoding.

Main Methods:

  • Development of generator and parity check matrices for skew convolutional codes.
  • Design of encoders and code trellises for the proposed codes.
  • Adaptation of decoding algorithms for memoryless channels.

Main Results:

  • Skew convolutional codes are shown to be a valid extension of fixed convolutional codes.
  • The proposed codes maintain a compact description, simplifying their implementation.
  • Standard decoding algorithms like Viterbi and BCJR can be applied.

Conclusions:

  • Skew convolutional codes represent a significant advancement in coding theory.
  • These codes offer a practical and efficient solution for error correction in communication systems.
  • The findings pave the way for new applications in data transmission.