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Properties of the z-Transform I

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...

Inverse z-Transform by Partial Fraction Expansion

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...

Properties of the z-Transform II

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...

Definition of z-Transform

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.

Vector Representation of Complex Numbers

Vector Representation of Complex Numbers

Complex numbers, represented in Cartesian coordinates, can also be visualized as vectors. These vectors can be expressed in polar form, emphasizing their magnitude and angle. When a complex number is input into a function, the output is another complex number, highlighting the function's zero point from which the vector representation can originate.
Consider a function defined as the product of the complex factors in the numerator divided by the product of the complex factors in the...

Cartesian Vector Notation

Cartesian Vector Notation

Cartesian vector notation is a valuable tool in mechanical engineering for representing vectors in three-dimensional space, performing vector operations such as determining the gradient, divergence, and curl, and expressing physical quantities such as the displacement, velocity, acceleration, and force. By using Cartesian vector notation, engineers can more easily analyze and solve problems in various areas of mechanical engineering, including dynamics, kinematics, and fluid mechanics. This...

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Related Experiment Video

Updated: Feb 27, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs

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Published on: June 8, 2018

Two S-type Z-eigenvalue inclusion sets for tensors.

Yanan Wang¹, Gang Wang¹

¹School of Management Science, Qufu Normal University, Rizhao, Shandong 276826 P.R. China.

Journal of Inequalities and Applications

|July 7, 2017

Summary

This summary is machine-generated.

This study introduces new S-type Z-eigenvalue inclusion sets for tensors, offering tighter bounds than previous methods. These findings improve spectral radius estimations for weakly symmetric nonnegative tensors.

Keywords:

Z-eigenvalue inclusion sets largest Z-eigenvalue weakly symmetric nonnegative tensors

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Area of Science:

Numerical analysis
Tensor computations
Algebraic eigenvalue problems

Background:

Eigenvalue inclusion sets are crucial for analyzing tensor properties.
Existing methods for Z-eigenvalue inclusion sets have limitations in tightness.
Spectral radius estimation for tensors is an active research area.

Purpose of the Study:

To introduce novel S-type Z-eigenvalue inclusion sets for general tensors.
To demonstrate the improved tightness of these new sets compared to existing literature.
To derive sharper upper bounds for the spectral radius of weakly symmetric nonnegative tensors.

Main Methods:

Development of new S-type Z-eigenvalue inclusion sets based on tensor properties.
Comparative analysis of the proposed sets against established inclusion sets (Wang et al., 2017).
Derivation of upper bounds for the spectral radius using the new inclusion sets.

Main Results:

Two new S-type Z-eigenvalue inclusion sets for general tensors are presented.
The proposed inclusion sets are proven to be tighter than those by Wang et al.
Sharper upper bounds for the spectral radius of weakly symmetric nonnegative tensors are obtained.

Conclusions:

The new S-type Z-eigenvalue inclusion sets offer significant improvements in tensor analysis.
The derived spectral radius bounds enhance the understanding of weakly symmetric nonnegative tensors.
This work contributes to the advancement of tensor eigenvalue theory and computation.