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数学代写|信息论代写Information Theory代考|EE381K Enhanced estimators for optimal decoding

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数学代写|信息论代写Information Theory代考|Enhanced estimators for optimal decoding

1. In the preceding sections decoding was performed according to the principle of the maximum likelihood function (7.1.6) or, equivalently, the minimum distance (7.1.8). It is of our great interest to study what an estimator of the error probability equals to if decoding is performed on the basis of ‘distance’ $D(\xi, \eta)$ defined somewhat differently. In the present paragraph we suppose that ‘distance’ $D(\xi, \eta)$ is some arbitrarily given function. Certainly, a transition to a new ‘distance’ cannot diminish the probability of decoding error but, in principle, it can decrease an upper bound for an estimator of the specified probability.

Theorem 7.4. Suppose that we have a channel $[P(\eta \mid \xi), P(\xi)]$ (just as in Theorem 7.1), which is an $n$-th power of channel $[P(y \mid x), P(x)]$. Let the decoding be performed on the basis of the minimum distance
$$D(\xi, \eta)=\sum_{j=1}^{n} d\left(x_{j}, y_{j}\right)$$
where $d(x, y)$ is a given function.
The amount $\ln M$ of transmitted information increases with $n$ according to the law
$$\ln M=\ln \left[e^{n R}\right] \leqslant n R$$

$\left(R<I_{x y}\right.$ is independent of $n$ ). Then there exists a sequence of codes having the probability of decoding error
$$P_{\mathrm{er}} \leqslant 2 e^{-n\left[s_{0} \gamma^{\prime}\left(s_{0}\right)-\gamma\left(s_{0}\right)\right]} .$$
Here $s_{0}$ is one of the roots $s_{0}, t_{0}, r_{0}$ of the following system of equations:
\begin{aligned} &\gamma\left(s_{0}\right)=\varphi_{r}^{\prime}\left(r_{0}, t_{0}\right) \ &\varphi_{t}^{\prime}\left(r_{0}, t_{0}\right)=0 \ &\left(s_{0}-r_{0}\right) \gamma\left(s_{0}\right)-\gamma\left(s_{0}\right)+\varphi\left(r_{0}, t_{0}\right)+R=0 . \end{aligned}
Also, $\gamma(s)$ and $\varphi(r, t)$ are the functions below:
\begin{aligned} \gamma(s) &=\ln \sum_{x, y} e^{s d(x, y)} P(x) P(y \mid x) \ \varphi(r, t) &=\ln \sum_{x, y, x^{\prime}} e^{(r-t) d(x, y)+t d\left(x^{\prime}, y\right)} P(x) P(y \mid x) P\left(x^{\prime}\right) . \end{aligned}
Besides, $\varphi_{r}^{\prime}=\frac{\partial \varphi}{\partial r}, \varphi_{t}^{\prime}=\frac{\partial \varphi}{\partial t}$.

数学代写|信息论代写Information Theory代考|Some general relations between entropies and mutual informations for encoding and decoding

1. The noisy channel in consideration is characterized by conditional probabilities $P(\eta \mid \xi)=\prod_{i} P\left(y_{i} \mid x_{i}\right)$. The probabilities $P(\xi)=\prod_{i} P\left(x_{i}\right)$ of the input variable $\xi$ determine the method of obtaining the random code $\xi_{1}, \ldots, \xi_{M}$. The transmitted message indexed by $k=1, \ldots, M$ is associated with $k$-th code point $\xi_{k}$. During decoding, the observed random variable $\eta$ defines the index $l(\eta)$ of the received message. As it as mentioned earlier, we select a code point $\xi_{l}$, which is the ‘closest’ to the observed point $\eta$ in terms of some ‘distance’.

The transformation $l=l(\eta)$ is degenerate. Therefore, due to inequality (6.3.9) we have
$$I_{k, l}\left(\mid \xi_{1}, \ldots, \xi_{M}\right) \leqslant I_{k, \eta}\left(\mid \xi_{1}, \ldots, \xi_{M}\right) .$$
Applying (6.3.9) we need to juxtapose $k$ with $y, l$ with $x_{1}$ and interpret $x_{2}$ as a random variable complementing $l$ to $\eta$ (in such a way that $\eta$ coincides with $x_{1}, x_{2}$ ).
The code $\xi_{1}, \ldots, \xi_{M}$ in (7.6.1) is assumed to be fixed. Averaging (7.6.1) over various codes with weight $P\left(\xi_{1}\right) \ldots P\left(\xi_{M}\right)$ and denoting the corresponding results as $I_{k l \mid} \mid \xi_{1} \ldots \xi_{M}, I_{k \eta \mid \xi_{1} \ldots \xi_{M}}$, we obtain
$$I_{k l \mid} \xi_{1}, \ldots, \xi_{M} \leqslant I_{k \eta \mid \xi_{1}, \ldots, \xi_{M}} .$$

数学代写|信息论代写Information Theory代考|Enhanced estimators for optimal decoding

$$D(\xi, \eta)=\sum_{j=1}^{n} d\left(x_{j}, y_{j}\right)$$

$$\ln M=\ln \left[e^{n R}\right] \leqslant n R$$
$\left(R<I_{x y}\right.$ 独立于 $\left.n\right)$ 。那么存在一个具有解码错吴概率的代码序列
$$P_{\mathrm{er}} \leqslant 2 e^{-n\left[s\left(s \gamma^{\prime}(s 0)-\gamma(s 0)\right]\right.} .$$

$$\gamma\left(s_{0}\right)=\varphi_{r}^{\prime}\left(r_{0}, t_{0}\right) \quad \varphi_{t}^{\prime}\left(r_{0}, t_{0}\right)=0\left(s_{0}-r_{0}\right) \gamma\left(s_{0}\right)-\gamma\left(s_{0}\right)+\varphi\left(r_{0}, t_{0}\right)+R=0 .$$

$$\gamma(s)=\ln \sum_{x, y} e^{s d(x, y)} P(x) P(y \mid x) \varphi(r, t) \quad=\ln \sum_{x, y, x^{\prime}} e^{(r-t) d(x, y)+t d\left(x^{\prime}, y\right)} P(x) P(y \mid x) P\left(x^{\prime}\right) .$$

数学代写|信息论代写Information Theory代考|Some general relations between entropies and mutual informations for encoding and decoding

$$I_{k, l}\left(\mid \xi_{1}, \ldots, \xi_{M}\right) \leqslant I_{k, \eta}\left(\mid \xi_{1}, \ldots, \xi_{M}\right)$$

$I_{k l \mid} \mid \xi_{1} \ldots \xi_{M}, I_{k \eta \mid \xi 1 \ldots \xi M ， ~}$ 我们获得
$$I_{k l} \mid \xi_{1}, \ldots, \xi_{M} \leqslant I_{k \eta \mid \xi_{1}, \ldots, \xi_{M}}$$

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