A Mathematical Theory of Communication

A Mathematical Theory of Communication

2006 | Jin Woo Shin, Sang Joon Kim
This paper, "A Mathematical Theory of Communication," introduced the field of information theory and challenged the conventional belief that reducing data rates is necessary to minimize transmission errors. It demonstrated that achieving low error probabilities does not require reducing data rates and established an upper bound on data rates. The paper is divided into five parts, which are summarized into four main sections: Preliminary, Discrete Source and Discrete Channel, Discrete Source and Continuous Channel, and Continuous Source and Continuous Channel. The key concepts introduced include entropy, ergodicity, and capacity. Entropy measures the uncertainty of a source, ergodicity ensures that the source's statistics are consistent over time, and capacity defines the maximum information rate a channel can handle. The paper proved that if the source entropy is less than or equal to the channel capacity, error-free transmission is possible with arbitrarily small errors. If the source entropy exceeds the capacity, no encoding scheme can achieve a sufficiently low error rate. For continuous channels, the paper defined capacity and transmission rates, showing that the maximum number of binary digits that can be transmitted per second is limited by the channel's capacity. The concept of fidelity was introduced to measure the quality of signal recovery, and the transmission rate for a continuous source was defined in terms of fidelity. The paper also discussed practical aspects, such as the need for large block sizes to achieve high performance and the limitations of current encoding schemes like Turbo codes and LDPC codes. The assumption of ergodicity was explored, and the paper concluded with a discussion on the fundamental ideas of entropy and ergodicity, emphasizing their importance in information theory.This paper, "A Mathematical Theory of Communication," introduced the field of information theory and challenged the conventional belief that reducing data rates is necessary to minimize transmission errors. It demonstrated that achieving low error probabilities does not require reducing data rates and established an upper bound on data rates. The paper is divided into five parts, which are summarized into four main sections: Preliminary, Discrete Source and Discrete Channel, Discrete Source and Continuous Channel, and Continuous Source and Continuous Channel. The key concepts introduced include entropy, ergodicity, and capacity. Entropy measures the uncertainty of a source, ergodicity ensures that the source's statistics are consistent over time, and capacity defines the maximum information rate a channel can handle. The paper proved that if the source entropy is less than or equal to the channel capacity, error-free transmission is possible with arbitrarily small errors. If the source entropy exceeds the capacity, no encoding scheme can achieve a sufficiently low error rate. For continuous channels, the paper defined capacity and transmission rates, showing that the maximum number of binary digits that can be transmitted per second is limited by the channel's capacity. The concept of fidelity was introduced to measure the quality of signal recovery, and the transmission rate for a continuous source was defined in terms of fidelity. The paper also discussed practical aspects, such as the need for large block sizes to achieve high performance and the limitations of current encoding schemes like Turbo codes and LDPC codes. The assumption of ergodicity was explored, and the paper concluded with a discussion on the fundamental ideas of entropy and ergodicity, emphasizing their importance in information theory.
Reach us at info@study.space
[slides] A Mathematical Theory of Communication | StudySpace