# 数学代写|信息论代写information theory代考|STRONGLY TYPICAL SEQUENCES AND RATE DISTORTION

#### Doug I. Jones

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## 数学代写|信息论代写information theory代考|STRONGLY TYPICAL SEQUENCES AND RATE DISTORTION

In Section 10.5 we proved the existence of a rate distortion code of rate $R(D)$ with average distortion close to $D$. In fact, not only is the average distortion close to $D$, but the total probability that the distortion is greater than $D+\delta$ is close to 0 . The proof of this is similar to the proof in Section 10.5; the main difference is that we will use strongly typical sequences rather than weakly typical sequences. This will enable us to give an upper bound to the probability that a typical source sequence is not well represented by a randomly chosen codeword in (10.94). We now outline an alternative proof based on strong typicality that will provide a stronger and more intuitive approach to the rate distortion theorem.

We begin by defining strong typicality and quoting a basic theorem bounding the probability that two sequences are jointly typical. The properties of strong typicality were introduced by Berger and were explored in detail in the book by Csiszár and Körner [149]. We will define strong typicality (as in Chapter 11) and state a fundamental lemma (Lemma 10.6.2).

Definition A sequence $x^n \in \mathcal{X}^n$ is said to be $\epsilon$-strongly typical with respect to a distribution $p(x)$ on $\mathcal{X}$ if:

1. For all $a \in \mathcal{X}$ with $p(a)>0$, we have
$$\left|\frac{1}{n} N\left(a \mid x^n\right)-p(a)\right|<\frac{\epsilon}{|\mathcal{X}|} .$$
2. For all $a \in \mathcal{X}$ with $p(a)=0, N\left(a \mid x^n\right)=0$.
$N\left(a \mid x^n\right)$ is the number of occurrences of the symbol $a$ in the sequence $x^n$.

The set of sequences $x^n \in \mathcal{X}^n$ such that $x^n$ is strongly typical is called the strongly typical set and is denoted $A_\epsilon^{(n)}(X)$ or $A_\epsilon^{(n)}$ when the random variable is understood from the context.

## 数学代写|信息论代写information theory代考|CHARACTERIZATION OF THE RATE DISTORTION FUNCTION

We have defined the information rate distortion function as
$$R(D)=\min {q(\hat{x} \mid x): \sum{(x, \hat{x})} p(x) q(\hat{x} \mid x) d(x, \hat{x}) \leq D} I(X ; \hat{X}),$$

where the minimization is over all conditional distributions $q(\hat{x} \mid x)$ for which the joint distribution $p(x) q(\hat{x} \mid x)$ satisfies the expected distortion constraint. This is a standard minimization problem of a convex function over the convex set of all $q(\hat{x} \mid x) \geq 0$ satisfying $\sum_{\hat{x}} q(\hat{x} \mid x)=1$ for all $x$ and $\sum q(\hat{x} \mid x) p(x) d(x, \hat{x}) \leq D$.

We can use the method of Lagrange multipliers to find the solution. We set up the functional
\begin{aligned} J(q)= & \sum_x \sum_{\hat{x}} p(x) q(\hat{x} \mid x) \log \frac{q(\hat{x} \mid x)}{\sum_x p(x) q(\hat{x} \mid x)} \ & +\lambda \sum_x \sum_{\hat{x}} p(x) q(\hat{x} \mid x) d(x, \hat{x}) \ & +\sum_x v(x) \sum_{\hat{x}} q(\hat{x} \mid x), \end{aligned}
where the last term corresponds to the constraint that $q(\hat{x} \mid x)$ is a conditional probability mass function. If we let $q(\hat{x})=\sum_x p(x) q(\hat{x} \mid x)$ be the distribution on $\hat{X}$ induced by $q(\hat{x} \mid x)$, we can rewrite $J(q)$ as
\begin{aligned} J(q)=\sum_x & \sum_{\hat{x}} p(x) q(\hat{x} \mid x) \log \frac{q(\hat{x} \mid x)}{q(\hat{x})} \ & +\lambda \sum_x \sum_{\hat{x}} p(x) q(\hat{x} \mid x) d(x, \hat{x}) \ & +\sum_x v(x) \sum_{\hat{x}} q(\hat{x} \mid x) . \end{aligned}

# 信息论代写

## 学代写|信息论代写information theory代考|STRONGLY TYPICAL SEQUENCES AND RATE DISTORTION

$$\left|\frac{1}{n} N\left(a \mid x^n\right)-p(a)\right|<\frac{\epsilon}{|\mathcal{X}|} .$$

$N\left(a \mid x^n\right)$是符号$a$在序列$x^n$中出现的次数。

## 数学代写|信息论代写information theory代考|CHARACTERIZATION OF THE RATE DISTORTION FUNCTION

$$R(D)=\min {q(\hat{x} \mid x): \sum{(x, \hat{x})} p(x) q(\hat{x} \mid x) d(x, \hat{x}) \leq D} I(X ; \hat{X}),$$

\begin{aligned} J(q)= & \sum_x \sum_{\hat{x}} p(x) q(\hat{x} \mid x) \log \frac{q(\hat{x} \mid x)}{\sum_x p(x) q(\hat{x} \mid x)} \ & +\lambda \sum_x \sum_{\hat{x}} p(x) q(\hat{x} \mid x) d(x, \hat{x}) \ & +\sum_x v(x) \sum_{\hat{x}} q(\hat{x} \mid x), \end{aligned}

\begin{aligned} J(q)=\sum_x & \sum_{\hat{x}} p(x) q(\hat{x} \mid x) \log \frac{q(\hat{x} \mid x)}{q(\hat{x})} \ & +\lambda \sum_x \sum_{\hat{x}} p(x) q(\hat{x} \mid x) d(x, \hat{x}) \ & +\sum_x v(x) \sum_{\hat{x}} q(\hat{x} \mid x) . \end{aligned}

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