MY-ASSIGNMENTEXPERT™可以为您提供illinois.edu ECE515 Information theory信息论课程的代写代考和辅导服务！
ECE515课程简介
Official Description
Feedback control systems emphasizing state space techniques. Basic principles, modeling, analysis, stability, structural properties, optimization, and design to meet specifications. Course Information: Same as ME 540. Prerequisite: ECE 486.
Topics
System modeling and analysis: system design as a control problem – constraints, goals and performance specifications, input-output and state space models; linearization; review of linear algebra; fundamentals of state-space analysis of linear systems
Prerequisites
System structural properties: stability; introduction to Lyapunov methods; controllability, observability; canonical forms and minimal realizations. Modeling uncertainties; system sensitivity and robustness measures.
Feedback system design: basic properties of feedback; stabilization and eigenvalue placement by state and output feedback; disturbance rejection; observers for estimating states, and observer feedback systems
Optimum feedback control: dynamic programming and the Hamilton-Jacobi-Bellman equation; synthesis of optimum state regulator systems; numerical methods
Introduction to the minimum principle: calculus of variations and necessary conditions for optimal trajectories; minimum principle for bounded controls; time-optimal control of linear systems; numerical methods
ECE515 Information theory HELP（EXAM HELP， ONLINE TUTOR）
Problem 3 (EQUaLITY IN KRAFT’s INEQUALITY) An $f$ prefix code is called full if it loses its prefix property by adding any new codeword to it. A string $\mathbf{x}$ is called undecodable if it is impossible to construct a sequence of codewords such that $\mathbf{x}$ is a prefix of their concatenation. Show that the following three statements are equivalent.
$\sum_{i=1}^n s^{-l_i}=1$, where $s$ is the cardinality of the code alphabet, $l_i$ is the codeword length of the ith codeword, and $n$ is the number of codewords.
To show that the three statements are equivalent, we need to prove the following implications:
$\$$ (a) \Rightarrow (b) \Rightarrow (c) \Rightarrow (a)\$
$\$(a) \backslash R i g h t a r r o w$ (b)\$:
Suppose that there exists an undecodable string $\$ \backslash$ mathbf ${x} \$$ in the code. Since the code is a prefix code, the only way to generate $\$ \backslash$ mathbf ${x} \$$ is to use it as a prefix of some codeword. Let $\$ \backslash$ mathbf ${x} \$$ be a prefix of the codeword $\$ \backslash$ mathbf ${y} \$$. Then, any other codeword $\$ \backslash$ mathbf ${z} \$$ that starts with $\$ \backslash$ mathbf ${y} \$$ must be longer than $\$ \backslash$ mathbf{y}$}$. Therefore, the length of $\$ \backslash$ mathbf ${y} \$$ is the maximum among all codewords starting with $\$ \backslash$ mathbf ${x} \$$. Let $\$ 1 \$$ be the length of $\$ \backslash$ mathbf ${y} \$$. Then, there are $\$ s^{\wedge}{1-$ $\mid \backslash$ mathbf ${x} \mid} \$$ codewords that start with $\$ \backslash$ mathbf ${x} \$$ and have length $\$ \mid \$$. Since $\$ \backslash$ mathbf ${x} \$$ is undecodable, none of these codewords can appear in the code. Therefore, the number of codewords of length $\$ \$ \$$ or longer is at most $\$ s^{\wedge}{1-\mid \backslash$ mathbf ${x} \mid}-1 \$$. It follows that
$$
\sum_{i=1}^n s^{-l_i} \leq s^{-|\mathbf{x}|}+\sum_{l=l_{\max }+1}^{\infty}\left(s^{1-l}\right)\left(s^{l-|\mathbf{x}|}-1\right)
$$
where $\$ I _{\max } \$$ is the length of the longest codeword and the first term on the right-hand side corresponds to codewords starting with $\$ \backslash$ mathbf ${x} \$$.
Since $\$ s^{\wedge}{1-\mid \backslash$ mathbf ${x} \mid}-1 \backslash$ leq $s^{\wedge}{1-\mid \backslash$ mathbf ${x}|| \$$, we have
$$
\sum_{l=l_{\max }+1}^{\infty}\left(s^{1-l}\right)\left(s^{l-|\mathbf{x}|}-1\right) \leq \sum_{l=l_{\max }+1}^{\infty} s^{2-l} .
$$
The right-hand side is a geometric series with ratio $\$ S^{\wedge}{-1} \$$, which converges to $\$ s^{\wedge}{-1} /\left(1-s^{\wedge}{-1}\right)=1 / s \$$. Therefore,
$$
\sum_{i=1}^n s^{-l_i} \leq s^{-|\mathbf{x}|}+s^{-1}
$$
(b) there is no undecodable string with respect to $f$,
To prove that $\left|f\left(x_i\right)\right|<-\log p\left(x_i\right)+1$, we first observe that $w_i$ is the cumulative probability of the $i$ most probable symbols. Therefore, $w_i \leq 1$ for all $i$. Furthermore, we have $w_{i-1} \leq w_i$, since $p\left(x_i\right) \geq p\left(x_{i+1}\right)$. It follows that $w_i-w_{i-1} = p\left(x_i\right) \leq w_i$, and therefore the binary expansion of $w_i$ has at most one non-zero bit to the left of the binary point.
Let $k$ be the number of bits used to represent $w_i$. Then, we have $2^{k-1} \leq w_i < 2^k$, which implies $k \geq \lceil \log w_i \rceil$. Moreover, $w_i$ is the sum of $p\left(x_j\right)$ for $j<i$, which implies $w_i \geq p\left(x_1\right) + \cdots + p\left(x_{i-1}\right)$. Using the fact that $p\left(x_i\right) \geq p\left(x_j\right)$ for $j \geq i$, we obtain \begin{align*} w_i &\geq p\left(x_1\right) + \cdots + p\left(x_{i-1}\right) \ &= 1 – p\left(x_i\right) \ &\geq 1 – \sum_{j=1}^n p\left(x_j\right) \ &= 0, \end{align*} which implies $\log w_i \geq 0$. Therefore, we have
$$
\left|f\left(x_i\right)\right| \leq k \leq\left\lceil\log w_i\right\rceil \leq\left\lceil-\log p\left(x_i\right)\right\rceil \leq-\log p\left(x_i\right)+1
$$
To show that the expected codeword length is smaller than the entropy plus one, we have
$\backslash$ begin ${$ align $}$
\& $\backslash$ leq $\backslash$ sum_{i ${1}^{\wedge} n$ p $\backslash$ left(x_i\right) $\left(-\backslash \log p \backslash l e f t\left(x_{-} \backslash \backslash r i g h t\right)+1\right) \backslash$
$\&=\mathrm{H}(\mathrm{X})+1$,
lend{align $\left.{ }^{\star}\right}$
where $\$ H(X) \$$ is the entropy of the source. Therefore, the expected codeword length is smaller than the entropy plus one.
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