⚡ electrical engineering

Symbol Error Analysis of Linear Receivers in Terahertz Channels under Channel-Noise Dependence

This paper presents a comprehensive framework for analyzing the symbol error rate of linear receivers (ZF and MMSE) in terahertz channels under diverse fading models, explicitly accounting for channel-noise dependence caused by hardware impairments to quantify significant performance degradation and the trade-offs between detection methods.

Original authors: Almutasem Bellah Enad, Jihad Fahs, Hadi Sarieddeen, Hakim Jemaa, Tareq Y. Al-Naffouri

Published 2026-06-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Almutasem Bellah Enad, Jihad Fahs, Hadi Sarieddeen, Hakim Jemaa, Tareq Y. Al-Naffouri

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to send a secret message across a very crowded, noisy room using a high-powered flashlight. This is essentially what this paper is about, but instead of a room, it's the "Terahertz" (THz) band of the wireless spectrum—a super-fast, futuristic way to send data that could power the internet of the future.

Here is the breakdown of the paper's story, using simple analogies:

The Setting: The Noisy Room

In this scenario, the Terahertz signal is your flashlight beam. It's incredibly fast and carries a lot of information (like a video stream). However, the "room" (the air) is tricky. It has obstacles like fog, walls, and even the air molecules themselves that absorb the light. This makes the signal wobble and fade, a problem engineers call "fading."

Usually, engineers assume that the noise (static in the room) is random and has nothing to do with how the signal fades. They think, "The signal gets weak because of the wall, and the noise is just random static."

The Paper's Big Discovery:
This paper says, "Wait a minute!" In these high-tech systems, the noise and the signal fading are actually linked.

The Analogy: Imagine that every time your flashlight beam hits a patch of fog (signal fading), the fog itself starts shaking the air, creating more static noise right at that moment. The noise isn't random; it's reacting to the signal. The paper calls this "channel-noise dependence."

The Problem: The Two Receivers

To fix the wobbly signal, the receiver (the person catching the message) uses a "cleaner" or a filter. The paper tests two main types of cleaners:

The "Zero-Forcing" (ZF) Cleaner:
- How it works: This cleaner is like a stubborn math genius. It looks at the wobbly signal and says, "I will divide the signal by the exact amount of wobble to make it perfect again."
- The Flaw: If the signal gets very weak (like hitting a thick wall), this cleaner tries to divide by a tiny number. This causes the static noise to explode, turning a whisper into a deafening roar. It's great when things are clear, but it breaks down in bad conditions.
The "MMSE" Cleaner:
- How it works: This cleaner is more cautious. It knows the signal might be weak, so it doesn't try to fix it perfectly. Instead, it applies a "regularization" (a safety brake). It accepts a little bit of distortion to keep the noise from exploding.
- The Catch: Because it doesn't fix the signal perfectly, it introduces a slight "bias" or shift. It's like the cleaner slightly misaligns the letters of your message. For simple messages (like 4-QAM), this doesn't matter. But for complex, high-speed messages (like 16-QAM or 64-QAM), this slight misalignment causes errors.

The Experiment: Testing the Cleaners

The authors built a mathematical framework to test these two cleaners under three different "room" conditions:

Rayleigh Fading: A standard, generic noisy room.
Indoor THz (Alpha-Mu): A room with specific types of indoor obstacles (like furniture and walls).
Outdoor THz (Mixture-Gamma): A room with outdoor elements like fog and atmospheric absorption.

They also tested what happens when the noise and signal are linked (the "fog shaking the air" scenario) versus when they are unlinked (the old assumption).

The Results: What They Found

The "Linked" Noise Matters: When the noise and signal are linked (which happens in real hardware), the performance of the receivers changes significantly. If you ignore this link, you get the wrong answer. The paper found that high correlation between noise and signal can degrade performance by about 6.5 dB (a huge drop in quality).
The Trade-off:
- In good conditions: The "Zero-Forcing" (ZF) cleaner is usually better because it doesn't introduce that slight misalignment.
- In bad conditions: The "MMSE" cleaner is much safer. Even though it has that slight misalignment (bias), it prevents the noise from exploding. The paper found that while MMSE is slightly worse than ZF for complex messages (about 1 dB loss), it is much more stable when the signal is fading deeply.
The "Bias" Issue: The paper highlights a specific problem: MMSE shifts the decision boundaries. Imagine the cleaner is supposed to sort red balls from blue balls, but it slightly shifts the line between the red and blue buckets. For simple messages, the balls still land in the right bucket. For complex messages, some balls land in the wrong bucket, causing errors.

The Bottom Line

The paper provides a new, more accurate "rulebook" for engineers designing these super-fast THz systems. It proves that:

You cannot assume noise is random; it often depends on the signal.
The "Zero-Forcing" method is risky in bad weather because it amplifies noise.
The "MMSE" method is a bit "lazy" (it introduces a bias), but it's a safer, more robust choice for real-world, high-speed connections, especially when the signal is weak.

In short, the paper tells us that to build the future of ultra-fast wireless, we need to stop pretending the noise is random and start using receivers that are smart enough to handle the messy, linked reality of the real world.

Technical Summary: Symbol Error Analysis of Linear Receivers in Terahertz Channels under Channel-Noise Dependence

Problem Statement
The terahertz (THz) band (100 GHz to 10 THz) is critical for achieving terabit-per-second data rates in future wireless systems. However, the full potential of THz technology is hindered by propagation challenges, including severe attenuation, molecular absorption, and hardware impairments. A significant gap in existing performance analysis frameworks is the assumption of independence between the channel and additive noise. In practical THz systems, this assumption often breaks down due to channel-induced molecular absorption noise and hardware imperfections (e.g., transmitter and receiver distortions), which create a statistical dependence between the channel state and the noise. Traditional models that ignore this correlation fail to accurately predict the performance of linear detectors, particularly in deep-fading scenarios. Furthermore, while Zero-Forcing (ZF) and Minimum Mean-Square Error (MMSE) detectors are standard, the specific impact of channel-noise dependence on their Symbol Error Rate (SER), especially regarding the inherent bias introduced by MMSE in higher-order modulations, remains under-explored across diverse THz fading models.

Methodology
The paper develops a comprehensive analytical framework to evaluate the SER of linear detectors (ZF and MMSE) in Single-Input Single-Output (SISO) THz systems. The methodology involves the following key components:

System and Channel Modeling:
- Fading Models: The study considers three distinct fading environments: Rayleigh fading (reference), the $\alpha-\mu$ distribution for indoor THz scenarios, and the Mixture-Gamma (MG) distribution for outdoor THz scenarios.
- Noise Correlation: The framework explicitly models the dependence between the channel ( $h$ $h$ ) and noise ( $n$ $n$ ). Two approaches are utilized:
  - Direct Statistical Correlation: Using a complex correlation coefficient ( $\lambda$ ) for Gaussian channels.
  - Copula-Based Modeling: Employing Farlie-Gumbel-Morgenstern (FGM) and Frank copulas to capture complex dependency structures between non-Gaussian channel magnitudes and noise.
- Hardware Impairments: A specific THz system model incorporates distortion noise from transmitters and receivers, where the noise variance is proportional to the channel power ( $|h|^2$ ).
Analytical Derivation:
- ZF Detector: The effective noise is modeled as the ratio of dependent variables (noise over channel). The paper derives the Probability Density Function (PDF) of this ratio for MG-distributed channels using Meijer G-functions.
- MMSE Detector: The analysis derives the PDF of the post-filtered received signal. Crucially, it addresses the "mismatch" issue: MMSE introduces a scaling bias that shifts decision boundaries relative to the standard ZF regions. The paper derives SER expressions for MMSE operating under standard ZF decision regions, which is a common practical assumption.
- Asymptotic Analysis: High-SNR approximations are derived to characterize the functional relationship between SER and Signal-to-Noise Ratio (SNR), focusing on deep-fade events.
Validation:
- Theoretical expressions are validated against extensive Monte Carlo simulations ( $N = 5 \times 10^6$ trials) using MATLAB.
- Simulations cover various modulation schemes (4-QAM and 16-QAM) and correlation strengths.

Key Contributions
The paper makes the following technical contributions:

Extended Framework: It extends prior work on ZF detection in Rayleigh and indoor THz channels to include outdoor THz environments modeled by the MG distribution and to include MMSE detection.
Correlation Modeling: It provides a generalized solution for channel-noise dependence using both direct statistical correlation and copula-based methods (FGM and Frank), deriving the effective noise PDF for these scenarios.
MMSE Bias Analysis: It formally analyzes the performance degradation of MMSE detectors when using ZF-based decision regions. The paper derives that while MMSE offers numerical stability, the inherent bias causes a performance loss compared to ZF in higher-order modulations (e.g., 16-QAM) due to shifted decision boundaries.
Closed-Form and Semi-Analytical Expressions: The authors derive novel semi-analytical formulations, closed-form solutions, and accurate approximations for SER across Rayleigh, $\alpha-\mu$ , and MG channels.

Results
The simulation results demonstrate strong agreement with the derived theoretical expressions, validating the framework's accuracy. Key findings include:

Impact of Correlation: Channel-noise dependence significantly affects receiver performance. For a target SER of $10^{-3}$ , a 70% correlation results in approximately a 6.5 dB degradation in effective SNR compared to the independent case.
ZF vs. MMSE Performance:
- For 4-QAM, ZF and MMSE detectors exhibit identical performance, confirming theoretical predictions.
- For 16-QAM, the MMSE detector (operating with ZF decision regions) incurs a performance loss of approximately 1 dB compared to ZF at an SNR of 5 dB. This loss is attributed to the bias shifting the decision boundaries.
- However, MMSE demonstrates superior robustness in severe fading conditions where ZF suffers from unbounded noise amplification (channel inversion).
Outdoor THz: The framework successfully characterizes SER for outdoor scenarios using the MG distribution, showing that asymptotic expressions closely match theoretical curves in high-SNR regions.

Significance and Claims
The paper claims to provide a reliable analytical foundation for evaluating and optimizing linear detectors in future THz communication systems. Its significance lies in:

Realism: By incorporating channel-noise dependence and hardware impairments, the framework offers a more physically consistent analysis than traditional independent-noise models.
Practical Trade-offs: The analysis clarifies the trade-off between the numerical stability of MMSE (which prevents noise amplification in deep fades) and the decision-boundary mismatch that degrades performance in higher-order modulations.
Design Guidance: The results suggest that while ZF is optimal in high-SNR regimes for 4-QAM, MMSE remains a practical and robust alternative for higher-order modulations in realistic THz environments, provided the performance gap (approx. 1 dB) is acceptable for the application's reliability requirements.

The authors conclude that their framework is a valuable tool for the design of THz systems, particularly as it can be extended to other channel and noise models, and it highlights the necessity of accounting for channel-noise correlation in next-generation network design.

The Setting: The Noisy Room

The Problem: The Two Receivers

The Experiment: Testing the Cleaners

The Results: What They Found

The Bottom Line

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