Homework 9 Electrical Engineering Department

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Calculate the error probabilities, P'Y ≠ a | X = a and P'Y ≠ b | X = b, and determine the channel capacity for the modified binary erasure channel (MBEC) with input density function Px ∈ {a, b}, where the sum of probabilities p + q + r = 1, and the transition probabilities are given by the channel's parameters. Additionally, analyze a binary erasure channel (BEC) with input density function Px ∈ {a, b}, to find the channel capacity for inputs X ∈ {a, b} and outputs Y ∈ {a, e, b}, with codewords Xc ∈ {0, 1} and e representing erasure events. Extend this analysis to include the probability of error, conditional probabilities, and channel capacity for the extended channel with multiple codewords. All work should be conducted on the provided pages, with additional pages added as necessary.

Paper For Above instruction

The study of binary erasure channels (BEC) and modified binary erasure channels (MBEC) plays a significant role in understanding the capacity and reliability of digital communication systems. This paper aims to analyze the error probabilities and channel capacities of such channels, providing comprehensive calculations rooted in information theory principles.

First, consider the modified binary erasure channel (MBEC) with input density function Px ∈ {a, b}, where the probabilities p, q, and r satisfy p + q + r = 1. The transition probabilities are given such that the inputs can be corrupted or erased during transmission. The parameters p and q typically denote specific error or erase probabilities, whereas r represents the probability that the transmitted bit is received correctly. The goal is to compute the error probabilities P'Y ≠ a | X = a and P'Y ≠ b | X = b, which reflect the likelihood of incorrect reception given a particular transmitted symbol. These probabilities are essential for understanding how often errors occur and are influenced by the channel's noise characteristics, especially in the presence of correlation parameters like p, q, and r.

The error probability calculations involve the conditional probabilities derived from the channel transition matrix. For example, the probability P'Y ≠ a | X = a may be expressed as the sum of transition probabilities where the received symbol differs from the transmitted symbol a, considering both erasures and errors. Similarly, P'Y ≠ b | X = b is calculated. These computations utilize the law of total probability, considering each channel state and the corresponding transition probabilities.

Beyond error probabilities, the channel capacity—a fundamental metric in information theory—measures the maximum achievable rate of reliable communication over the channel. For the simplified input case where X ∈ {a, b} and output Y ∈ {a, e, b}, the capacity is calculated by maximizing the mutual information I(X;Y) over the input distribution Px. The mutual information quantifies how much information about the input can be inferred from the output. For binary inputs, this often involves computing entropy terms and their differences, considering the effect of erasure probabilities.

Extending the analysis to a larger codebook with codewords Xc ∈ {000, 001, 010, 011, 100, 101, 110, 111}, the probability of error and channel capacity calculations become more complex. The probability of error encompasses the likelihood of the receiver incorrectly decoding a transmitted codeword due to channel noise. Conditional probabilities are derived for each codeword based on the transition matrix, including the likelihood of specific erasure patterns.

The channel capacity in this extended setting involves maximizing the mutual information over the joint input distribution of the codebook. This process considers the probability of each codeword and the corresponding output distribution, factoring in the channel's parameters. The calculation typically leverages the properties of the entropy function and the law of total probability, often requiring numerical optimization to determine the maximum mutual information achievable.

In conclusion, the analysis of the MBEC and extended binary erasure channels provides critical insights into their reliability and capacity limits. These calculations inform the design of coding schemes and error correction strategies that improve the robustness of digital communication systems. Precise evaluation of error probabilities and channel capacities is essential for optimizing performance and achieving efficient data transmission in noisy environments.

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