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# 数学代写|信息论代写Information Theory代考|KRAFT INEQUALITY

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## 数学代写|信息论代写Information Theory代考|KRAFT INEQUALITY

We wish to construct instantaneous codes of minimum expected length to describe a given source. It is clear that we cannot assign short codewords to all source symbols and still be prefix-free. The set of codeword lengths possible for instantaneous codes is limited by the following inequality.
Theorem 5.2.1 (Kraft inequality) For any instantaneous code (prefix code) over an alphabet of size $D$, the codeword lengths $l_1, l_2, \ldots, l_m$ must satisfy the inequality
$$\sum_i D^{-l_i} \leq 1$$
Conversely, given a set of codeword lengths that satisfy this inequality, there exists an instantaneous code with these word lengths.

Proof: Consider a $D$-ary tree in which each node has $D$ children. Let the branches of the tree represent the symbols of the codeword. For example, the $D$ branches arising from the root node represent the $D$ possible values of the first symbol of the codeword. Then each codeword is represented by a leaf on the tree. The path from the root traces out the symbols of the codeword. A binary example of such a tree is shown in Figure 5.2. The prefix condition on the codewords implies that no codeword is an ancestor of any other codeword on the tree. Hence, each codeword eliminates its descendants as possible codewords.

Let $l_{\max }$ be the length of the longest codeword of the set of codewords. Consider all nodes of the tree at level $l_{\max }$. Some of them are codewords, some are descendants of codewords, and some are neither. A codeword at level $l_i$ has $D^{l_{\max }-l_i}$ descendants at level $l_{\max }$. Each of these descendant sets must be disjoint. Also, the total number of nodes in these sets must be less than or equal to $D^{l_{\max }}$. Hence, summing over all the codewords, we have
$$\sum D^{l_{\max }-l_i} \leq D^{l_{\max }}$$
or
$$\sum D^{-l_i} \leq 1$$
which is the Kraft inequality.

## 数学代写|信息论代写Information Theory代考|OPTIMAL CODES

In Section 5.2 we proved that any codeword set that satisfies the prefix condition has to satisfy the Kraft inequality and that the Kraft inequality is a sufficient condition for the existence of a codeword set with the specified set of codeword lengths. We now consider the problem of finding the prefix code with the minimum expected length. From the results of Section 5.2 , this is equivalent to finding the set of lengths $l_1, l_2, \ldots, l_m$ satisfying the Kraft inequality and whose expected length $L=\sum p_i l_i$ is less than the expected length of any other prefix code. This is a standard optimization problem: Minimize
$$L=\sum p_i l_i$$
over all integers $l_1, l_2, \ldots, l_m$ satisfying
$$\sum D^{-l_i} \leq 1$$
A simple analysis by calculus suggests the form of the minimizing $l_i^*$. We neglect the integer constraint on $l_i$ and assume equality in the constraint. Hence, we can write the constrained minimization using Lagrange multipliers as the minimization of
$$J=\sum p_i l_i+\lambda\left(\sum D^{-l_i}\right)$$
Differentiating with respect to $l_i$, we obtain
$$\frac{\partial J}{\partial l_i}=p_i-\lambda D^{-l_i} \log _e D .$$
Setting the derivative to 0 , we obtain
$$D^{-l_i}=\frac{p_i}{\lambda \log _e D} .$$

## 数学代写|信息论代写Information Theory代考|KRAFT INEQUALITY

$$\sum_i D^{-l_i} \leq 1$$

$$\sum D^{l_{\max }-l_i} \leq D^{l_{\max }}$$

$$\sum D^{-l_i} \leq 1$$

## 数学代写|信息论代写Information Theory代考|OPTIMAL CODES

$$L=\sum p_i l_i$$

$$\sum D^{-l_i} \leq 1$$

$$J=\sum p_i l_i+\lambda\left(\sum D^{-l_i}\right)$$

$$\frac{\partial J}{\partial l_i}=p_i-\lambda D^{-l_i} \log _e D .$$

$$D^{-l_i}=\frac{p_i}{\lambda \log _e D} .$$

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