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In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
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A servo system exemplifies a second-order system, featuring a proportional controller and load elements that ensure the output position aligns with the input position. The relationship between these components is described by a second-order differential equation. Applying the Laplace transform under zero initial conditions yields the transfer function, showing how inputs are converted to outputs in the system.
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Spatial and Directional Modulation Systems for Near-Field Secure Transmission.

Ji Liu1,2, Yuan Zhong3, Yong Wang1

  • 1School of Telecommunications Engineering, Xidian University, Xi'an 710126, China.

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|February 13, 2026
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Summary
This summary is machine-generated.

This study introduces a new secure communication framework for 6G near-field environments. The spatial and directional modulation with artificial noise (SDMN-AN) enhances security and spectral efficiency against eavesdropping.

Keywords:
artificial noise (AN)bit error rate (BER)near-fieldspatial and directional modulation (SDM)

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Area of Science:

  • Wireless communication and signal processing.
  • Information security and cryptography.

Background:

  • The 6G era, with its massive antenna arrays, intensifies near-field effects, posing significant security risks.
  • Existing communication systems struggle to address the unique security challenges in these complex near-field environments.

Purpose of the Study:

  • To propose a novel artificial noise-aided spatial and directional modulation (SDMN-AN) framework for secure near-field communications.
  • To enhance both spectral efficiency and communication security in 6G near-field environments.

Main Methods:

  • Integration of legitimate receiver indices, modulation symbols, and artificial noise (AN) into the SDMN-AN framework.
  • Confining artificial noise to the null space of legitimate channels to mask transmissions.
  • Investigating two precoding strategies: maximum-ratio transmission (MRT) and zero-forcing (ZF).

Main Results:

  • The SDMN-AN framework demonstrates superior performance in mitigating eavesdropping threats.
  • Significant improvements in spectral efficiency were observed.
  • Analytical derivations of bit error rate (BER) bounds align with simulation results, validating the framework's effectiveness.

Conclusions:

  • The proposed SDMN-AN framework offers a robust solution for secure near-field communications in 6G.
  • The trade-offs between MRT and ZF precoding strategies provide flexibility for different hardware and detection requirements.
  • This framework is a compelling advancement for next-generation secure wireless networks.