Encoder Lipschitz Integers: The Lipschitz integers that have the ”division with small remainder” property

Average Energy ◽

Division Algorithm ◽

Prime Integer ◽

Awgn Channel ◽

Abstract The residue class set of a Lipschitz integer is constructed by modulo function with primitive Lipschitz integer whose norm is a prime integer, i.e. prime Lipschitz integer. In this study, we consider primitive Lipschitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Lipschitz integer is less than the norm of the primitive Lipschitz integer used to construct the residue class set of the Lipschitz integer, then, the Euclid division algorithm works for this primitive Lipschitz integer. The Euclid division algorithm always works for prime Lipschitz integers. In other words, the prime Lipschitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Lipschitz residue class set that lies on primitive Lipschitz integers whose norm is not a prime integer. In this study, we solve this problem by defining Lipschitz integers that have the ”division with small remainder” property, namely, encoder Lipschitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Also, we investigate the performances of Lipschitz signal constellations (the left residue class set) obtained by modulo function with Lipschitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by agency of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Encoder Lipschitz Integers: The Lipschitz integers that have the ”division with small division” property

10.21203/rs.3.rs-1053517/v1 ◽

2021 ◽

Author(s):

Ramazan Duram ◽

Murat Güzeltepe

Keyword(s):

Gaussian Noise ◽

Figure Of Merit ◽

Residue Class ◽

Signal To Noise Ratio ◽

Average Energy ◽

Division Algorithm ◽

Prime Integer ◽

Awgn Channel ◽

Abstract The residue class set of a Lipschitz integer is constructed by modulo function with primitive Lipschitz integer whose norm is a prime integer, i.e. prime Lipschitz integer. In this study, we consider primitive Lipschitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Lipschitz integer is less than the norm of the primitive Lipschitz integer used to construct the residue class set of the Lipschitz integer, then, the Euclid division algorithm works for this primitive Lipschitz integer. The Euclid division algorithm always works for prime Lipschitz integers. In other words, the prime Lipschitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Lipschitz residue class set that lies on primitive Lipschitz integers whose norm is not a prime integer. In this study, we solve this problem by defining Lipschitz integers that have the ”division with small remainder” property, namely, encoder Lipschitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Also, we investigate the performances of Lipschitz signal constellations (the left residue class set) obtained by modulo function with Lipschitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by agency of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Encoder Hurwitz Integers: The Hurwitz integers that have the ”division with small division” property

10.21203/rs.3.rs-1053551/v1 ◽

2021 ◽

Author(s):

Ramazan Duram ◽

Murat Güzeltepe

Keyword(s):

Gaussian Noise ◽

Figure Of Merit ◽

Residue Class ◽

Signal To Noise Ratio ◽

Average Energy ◽

Division Algorithm ◽

Prime Integer ◽

Euclidean Metric ◽

Abstract The residue class set of a Hurwitz integer is constructed by modulo function with primitive Hurwitz integer whose norm is a prime integer, i.e. prime Hurwitz integer. In this study, we consider primitive Hurwitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Hurwitz integer is less than the norm of the primitive Hurwitz integer used to construct the residue class set of the Hurwitz integer, then, the Euclid division algorithm works for this primitive Hurwitz integer. The Euclid division algorithm always works for prime Hurwitz integers. In other words, the prime Hurwitz integers and halves-integer primitive Hurwitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Hurwitz residue class set that lies on primitive Hurwitz integers that their norms are not a prime integer and their components are in integers set. In this study, we solve this problem by defining Hurwitz integers that have the ”division with small remainder” property, namely, encoder Hurwitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Especially, Euclidean metric. Also, we investigate the performances of Hurwitz signal constellations (the left residue class set) obtained by modulo function with Hurwitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by means of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Encoder Hurwitz Integers: The Hurwitz integers that have the ”division with small remainder” property

10.21203/rs.3.rs-1053551/v4 ◽

2021 ◽

Author(s):

Ramazan Duran ◽

Murat Güzeltepe

Keyword(s):

Gaussian Noise ◽

Figure Of Merit ◽

Residue Class ◽

Signal To Noise Ratio ◽

Average Energy ◽

Division Algorithm ◽

Prime Integer ◽

Euclidean Metric ◽

Abstract The residue class set of a Hurwitz integer is constructed by modulo function with primitive Hurwitz integer whose norm is a prime integer, i.e. prime Hurwitz integer. In this study, we consider primitive Hurwitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Hurwitz integer is less than the norm of the primitive Hurwitz integer used to construct the residue class set of the Hurwitz integer, then, the Euclid division algorithm works for this primitive Hurwitz integer. The Euclid division algorithm always works for prime Hurwitz integers. In other words, the prime Hurwitz integers and halves-integer primitive Hurwitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Hurwitz residue class set that lies on primitive Hurwitz integers that their norms are not a prime integer and their components are in integers set. In this study, we solve this problem by defining Hurwitz integers that have the ”division with small remainder” property, namely, encoder Hurwitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Especially, Euclidean metric. Also, we investigate the performances of Hurwitz signal constellations (the left residue class set) obtained by modulo function with Hurwitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by means of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Encoder Hurwitz Integers: The Hurwitz integers that have the ”division with small division” property

10.21203/rs.3.rs-1053551/v2 ◽

2021 ◽

Author(s):

Ramazan Duram ◽

Murat Güzeltepe

Keyword(s):

Gaussian Noise ◽

Figure Of Merit ◽

Residue Class ◽

Signal To Noise Ratio ◽

Average Energy ◽

Division Algorithm ◽

Prime Integer ◽

Euclidean Metric ◽

Abstract The residue class set of a Hurwitz integer is constructed by modulo function with primitive Hurwitz integer whose norm is a prime integer, i.e. prime Hurwitz integer. In this study, we consider primitive Hurwitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Hurwitz integer is less than the norm of the primitive Hurwitz integer used to construct the residue class set of the Hurwitz integer, then, the Euclid division algorithm works for this primitive Hurwitz integer. The Euclid division algorithm always works for prime Hurwitz integers. In other words, the prime Hurwitz integers and halves-integer primitive Hurwitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Hurwitz residue class set that lies on primitive Hurwitz integers that their norms are not a prime integer and their components are in integers set. In this study, we solve this problem by defining Hurwitz integers that have the ”division with small remainder” property, namely, encoder Hurwitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Especially, Euclidean metric. Also, we investigate the performances of Hurwitz signal constellations (the left residue class set) obtained by modulo function with Hurwitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by means of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Encoder Hurwitz Integers: The Hurwitz integers that have the ”division with small remainder” property

10.21203/rs.3.rs-1053551/v3 ◽

2021 ◽

Author(s):

Ramazan Duram ◽

Murat Güzeltepe

Keyword(s):

Gaussian Noise ◽

Figure Of Merit ◽

Residue Class ◽

Signal To Noise Ratio ◽

Average Energy ◽

Division Algorithm ◽

Prime Integer ◽

Euclidean Metric ◽

Abstract The residue class set of a Hurwitz integer is constructed by modulo function with primitive Hurwitz integer whose norm is a prime integer, i.e. prime Hurwitz integer. In this study, we consider primitive Hurwitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Hurwitz integer is less than the norm of the primitive Hurwitz integer used to construct the residue class set of the Hurwitz integer, then, the Euclid division algorithm works for this primitive Hurwitz integer. The Euclid division algorithm always works for prime Hurwitz integers. In other words, the prime Hurwitz integers and halves-integer primitive Hurwitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Hurwitz residue class set that lies on primitive Hurwitz integers that their norms are not a prime integer and their components are in integers set. In this study, we solve this problem by defining Hurwitz integers that have the ”division with small remainder” property, namely, encoder Hurwitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Especially, Euclidean metric. Also, we investigate the performances of Hurwitz signal constellations (the left residue class set) obtained by modulo function with Hurwitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by means of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Threshold Computation for Spatially Coupled Turbo-Like Codes on the AWGN Channel

Entropy ◽

10.3390/e23020240 ◽

2021 ◽

Vol 23 (2) ◽

pp. 240

Author(s):

Muhammad Umar Farooq ◽

Alexandre Graell i Amat ◽

Michael Lentmaier

Keyword(s):

Gaussian Noise ◽

Belief Propagation ◽

Maximum A Posteriori ◽

Prediction Methods ◽

Threshold Analysis ◽

A Posteriori ◽

Awgn Channel ◽

Efficient Prediction ◽

Binary Erasure Channel

In this paper, we perform a belief propagation (BP) decoding threshold analysis of spatially coupled (SC) turbo-like codes (TCs) (SC-TCs) on the additive white Gaussian noise (AWGN) channel. We review Monte-Carlo density evolution (MC-DE) and efficient prediction methods, which determine the BP thresholds of SC-TCs over the AWGN channel. We demonstrate that instead of performing time-consuming MC-DE computations, the BP threshold of SC-TCs over the AWGN channel can be predicted very efficiently from their binary erasure channel (BEC) thresholds. From threshold results, we conjecture that the similarity of MC-DE and predicted thresholds is related to the threshold saturation capability as well as capacity-approaching maximum a posteriori (MAP) performance of an SC-TC ensemble.

Minimum Array Elements for Resolution of Several Direction of Arrival Estimation Methods in Various Noise-Level Environments

Journal of Telecommunications and Information Technology ◽

10.26636/jtit.2018.119417 ◽

2018 ◽

Vol 2 ◽

pp. 87-94

Author(s):

Ismail El Ouargui ◽

Said Safi ◽

Miloud Frikel

Keyword(s):

Gaussian Noise ◽

Noise Level ◽

Signal To Noise Ratio ◽

Direction Of Arrival ◽

Estimation Algorithm ◽

Doa Estimation ◽

Estimation Methods ◽

Minimum Number ◽

Informative Signals

The resolution of a Direction of Arrival (DOA) estimation algorithm is determined based on its capability to resolve two closely spaced signals. In this paper, authors present and discuss the minimum number of array elements needed for the resolution of nearby sources in several DOA estimation methods. In the real world, the informative signals are corrupted by Additive White Gaussian Noise (AWGN). Thus, a higher signal-to-noise ratio (SNR) oﬀers a better resolution. Therefore, we show the performance of each method by applying the algorithms in diﬀerent noise level environments.

BER Performance Analysis of Non-Coherent Q-Ary Pulse Position Modulation Receivers on AWGN Channel

Sensors ◽

10.3390/s21186102 ◽

2021 ◽

Vol 21 (18) ◽

pp. 6102

Author(s):

Xianhua Shi ◽

Yimao Sun ◽

Jie Tian ◽

Maolin Chen ◽

Youjiang Liu ◽

...

Keyword(s):

Gaussian Noise ◽

Theoretical Formula ◽

Pulse Position Modulation ◽

Error Performance ◽

Optimal Receiver ◽

Awgn Channel ◽

Simulation Results ◽

The Relationship ◽

Better Than

This paper introduces the structure of a Q-ary pulse position modulation (PPM) signal and presents a noncoherent suboptimal receiver and a noncoherent optimal receiver. Aiming at addressing the lack of an accurate theoretical formula of the bit error rate (BER) of a Q-ary PPM receiver in the additive white Gaussian noise (AWGN) channel in the existing literature, the theoretical formulas of the BER of a noncoherent suboptimal receiver and noncoherent optimal receiver are derived, respectively. The simulation results verify the correctness of the theoretical formulas. The theoretical formulas can be applied to a Q-ary PPM system including binary PPM. In addition, the analysis shows that the larger the Q, the better the error performance of the receiver and that the error performance of the optimal receiver is about 2 dB better than that of the suboptimal receiver. The relationship between the threshold coefficient of the suboptimal receiver and the error performance is also given.

A New Reliability Ratio Weighted Bit Flipping Algorithm for Decoding LDPC Codes

Wireless Communications and Mobile Computing ◽

10.1155/2021/6698602 ◽

2021 ◽

Vol 2021 ◽

pp. 1-11

Author(s):

Chakir Aqil ◽

Ismail Akharraz ◽

Abdelaziz Ahaitouf

Keyword(s):

Gaussian Noise ◽

Ldpc Codes ◽

Low Density ◽

Parity Check ◽

Coding Gain ◽

Decoding Complexity ◽

Low Density Parity Check ◽

Awgn Channel ◽

Infinite Loop

In this study, we propose a “New Reliability Ratio Weighted Bit Flipping” (NRRWBF) algorithm for Low-Density Parity-Check (LDPC) codes. This algorithm improves the “Reliability Ratio Weighted Bit Flipping” (RRWBF) algorithm by modifying the reliability ratio. It surpasses the RRWBF in performance, reaching a 0.6 dB coding gain at a Binary Error Rate (BER) of 10−4 over the Additive White Gaussian Noise (AWGN) channel, and presents a significant reduction in the decoding complexity. Furthermore, we improved NRRWBF using the sum of the syndromes as a criterion to avoid the infinite loop. This will enable the decoder to attain a more efficient and effective decoding performance.