What is the relationship between the expected length of code words and the entropy of a random variable in variable length coding?
The relationship between the expected length of code words and the entropy of a random variable in variable length coding is a fundamental concept in information theory. In order to understand this relationship, it is important to first grasp the concept of entropy and its significance in classical entropy.
Entropy, in the context of classical entropy, is a measure of the uncertainty or randomness associated with a random variable. It quantifies the average amount of information required to specify an outcome of the random variable. The higher the entropy, the more uncertain or random the variable is.
Variable length coding is a technique used in data compression, where different symbols are encoded with different lengths of binary code words. The goal of variable length coding is to assign shorter code words to more frequent symbols and longer code words to less frequent symbols, in order to achieve a more efficient representation of the data.
The expected length of code words in variable length coding is the average length of the code words used to represent the symbols of the random variable. It is calculated by multiplying the probability of each symbol by the length of its corresponding code word, and summing up these values for all symbols.
Now, the relationship between the expected length of code words and the entropy of a random variable can be understood by considering the optimal variable length coding scheme. In an optimal coding scheme, the expected length of code words is minimized, resulting in the most efficient representation of the data.
Shannon's source coding theorem states that in an optimal coding scheme, the expected length of code words is equal to or greater than the entropy of the random variable. This means that the entropy of the random variable serves as a lower bound on the expected length of code words.
To illustrate this relationship, consider a simple example. Let's say we have a random variable with four symbols A, B, C, and D, and their respective probabilities are 0.4, 0.3, 0.2, and 0.1. The entropy of this random variable can be calculated as:
Entropy = – (0.4 * log2(0.4) + 0.3 * log2(0.3) + 0.2 * log2(0.2) + 0.1 * log2(0.1))
Once we have calculated the entropy, we can design a variable length coding scheme that assigns shorter code words to more frequent symbols and longer code words to less frequent symbols. Let's assume the following code words are assigned:
A: 0
B: 10
C: 110
D: 111
The expected length of code words can be calculated as:
Expected Length = 0.4 * 1 + 0.3 * 2 + 0.2 * 3 + 0.1 * 3
In this example, the entropy is approximately 1.8464, while the expected length of code words is 1.9. As we can see, the expected length of code words is greater than the entropy, which aligns with Shannon's source coding theorem.
The expected length of code words in variable length coding is related to the entropy of a random variable. The entropy serves as a lower bound on the expected length of code words, indicating that the more random or uncertain the random variable is, the longer the expected length of the code words will be. This relationship is fundamental in understanding the efficiency and effectiveness of variable length coding in data compression.
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