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Related Concept Videos

Inverse z-Transform by Partial Fraction Expansion01:20

Inverse z-Transform by Partial Fraction Expansion

The inverse z-transform is a crucial technique for converting a function from its z-domain representation back to the time domain. One effective method for finding the inverse z-transform is the Partial Fraction Method, which involves decomposing a function into simpler fractions with distinct coefficients. These fractions correspond to known z-transform pairs, facilitating the inverse transformation process.
To begin the process, the poles of the function are identified and the function is...
Definition of z-Transform01:26

Definition of z-Transform

The z-transform is a powerful mathematical tool used in the analysis of discrete-time signals and systems. It is an essential analytical tool, analogous to the Laplace transform used in continuous-time systems. It plays a crucial role in the analysis of signals and systems, complementing the discrete-time Fourier transform. Both the z-transform and the Laplace transform convert differential or difference equations into algebraic equations, simplifying the process of solving complex problems.
Difference Equation Solution using z-Transform01:24

Difference Equation Solution using z-Transform

The z-transform is a powerful tool for analyzing practical discrete-time systems, often represented by linear difference equations. Solving a higher-order difference equation requires knowledge of the input signal and the initial conditions up to one term less than the order of the equation.
The z-transform facilitates handling delayed signals by shifting the signal in the z-domain, which corresponds to delaying the signal in the time domain, and advancing signals by similarly shifting in the...
Relation of DFT to z-Transform01:20

Relation of DFT to z-Transform

The Discrete Fourier Transform (DFT) is a crucial tool for analyzing the frequency content of discrete-time signals. It converts a sequence of N samples from the time domain into its corresponding sequence in the frequency domain, where each sample represents a specific frequency component.
To understand how the DFT works, it's helpful to consider the z-transform, which is a method for representing discrete sequences in the complex frequency domain. The z-transform involves summing the terms of...
Properties of the z-Transform I01:17

Properties of the z-Transform I

The z-transform is a fundamental tool in digital signal processing, enabling the analysis of discrete-time systems through its various properties. It is an invaluable tool for analyzing discrete-time systems, offering a range of properties that simplify complex signal manipulations. One fundamental property is linearity. For any two discrete-time signals, the z-transform of their linear combination equals the same linear combination of their individual z-transforms. This property is essential...
Properties of the z-Transform II01:16

Properties of the z-Transform II

The property of Accumulation in signal processing is derived by analyzing the accumulated sum of a discrete-time signal and using the time-shifting property to determine its z-transform. This principle reveals that the z-transform of the summed signal is related to the z-transform of the original signal by a multiplicative factor.
Moreover, the convolution property indicates that the convolution of two signals in the time domain corresponds to the product of their z-transforms in the frequency...

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Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture
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Published on: February 23, 2018

Direct and inverse discrete Zernike transform.

Rafael Navarro1, Justo Arines, Ricardo Rivera

  • 1ICMA, Universidad de Zaragoza and Consejo Superior de Investigaciones Científicas, Facultad de Ciencias, Pedro Cerbuna 12, 50009 Zaragoza, Spain. rafaelnb@unizar.es

Optics Express
|January 7, 2010
PubMed
Summary
This summary is machine-generated.

A new invertible discrete Zernike transform (DZT) offers improved performance and stability. Non-redundant sampling ensures completeness for accurate Zernike expansions and inversions, beneficial for various applications.

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Area of Science:

  • Optics and Photonics
  • Image Processing
  • Numerical Analysis

Background:

  • Standard Zernike expansions can suffer from numerical instability and redundancy.
  • Efficient and accurate discrete representations of optical wavefronts are crucial.

Purpose of the Study:

  • To propose and implement an invertible discrete Zernike transform (DZT).
  • To investigate the impact of non-redundant sampling on Zernike expansions.
  • To demonstrate the DZT's superior performance compared to standard methods.

Main Methods:

  • Development of an invertible discrete Zernike transform (DZT).
  • Implementation using three non-redundant sampling strategies: random, hybrid, and deterministic (spiral).
  • Analysis of Zernike polynomial expansion completeness and orthonormality.

Main Results:

  • Non-redundant sampling (random, hybrid, spiral) guarantees completeness for Zernike expansions.
  • Completeness enables inversion via simple matrix transposition.
  • The DZT demonstrates enhanced performance, numerical stability, and robustness in simulations.

Conclusions:

  • The proposed invertible DZT provides a stable and efficient method for Zernike analysis.
  • Non-redundant sampling is key to achieving a complete and orthonormal basis.
  • The DZT is a valuable tool for applications requiring accurate Zernike decomposition.