### 数学代写|信息论作业代写information theory代考|INFM130

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|信息论作业代写information theory代考|The Number of Degrees of Freedom

The physics of propagation dictate that any observed electromagnetic field is an essentially bandlimited function. This basic property allows us to define the size of the signals’ space in terms of the number of degrees of freedom. Consider a one-dimensional, real, scalar waveform $f$ of a single scalar variable $t$. We assume that $f$ is square-integrable, and
$$\int_{-\infty}^{\infty} f^{2}(t) d t \leq E$$
This ensures that the waveform can be expanded in a series of, possibly complex, orthonormal basis functions $\left{\psi_{n}\right}$,
$$f(t)=\sum_{n=1}^{\infty} a_{n} \psi_{n}(t),$$
where
$$a_{n}=\int_{-\infty}^{\infty} f(t) \psi_{n}^{*}(t) d t .$$
The equality in (1.2) is intended in the “energy” sense:
$$\lim {N \rightarrow \infty} \int{-\infty}^{\infty}\left[f(t)-f_{N}(t)\right]^{2} d t=0,$$
where
$$f_{N}(t)=\sum_{n=1}^{N} a_{n} \psi_{n}(t) .$$
In the language of mathematics, $f$ is in $L^{2}(-\infty, \infty)$, and it can be viewed as a point in an infinite-dimensional space of coordinates given by the coefficients $\left{a_{n}\right}$ in (1.3). By varying the values of these coefficients, we can create distinct waveforms and use them to communicate information. If the orthonormal set of basis functions $\left{\psi_{n}\right}$ is complete, then using (1.2) we can construct any element in the space of signals defined by (1.1). By associating a waveform in this space with a given message that the transmitter wishes to communicate, the correct selection of the same waveform at the receiver implies that a certain amount of information is transferred between the two. One may reasonably expect that only a finite number of coefficients is in practice needed to specify the waveform up to any given accuracy, while using a larger number does not significantly improve the resolution at the receiver. It turns out that the question of what the smallest $N$ is beyond which varying higher-order coefficients does not change the form of the waveform significantly has a remarkably precise answer.

## 数学代写|信息论作业代写information theory代考|Space–Time Fields

The electromagnetic field is in general a function of four scalar variables: three spatial and one temporal. It follows that in order to appreciate the total field’s informational content in terms of degrees of freedom, we need to extend the treatment above to higher dimensions.

Let us first consider the canonical case of a two-dimensional domain of cylindrical symmetry, in which an electromagnetic field is radiated by current sources located inside a circular domain of radius $r$, and oriented perpendicular to the domain. The sources can also be induced by multiple scattering inside the domain. In any case, the radiated field away from the sources is completely determined by the field on the cut-set boundary surrounding the sources and through which it propagates – see Figure 1.7. On this boundary, we can refer to a scalar field $f(\phi, t)$ that is a function of only two scalar variables: one angular and one temporal. The corresponding four representations, linked by Fourier transforms, are depicted in Figure $1.8$, where $\omega$ indicates the transformed coordinate of the time variable $t$ and $w$ indicates the wavenumber that is the transformed coordinate of the angular variable $\phi$.

Letting $\Omega$ be the angular frequency bandwidth and $W$ be the wavenumber bandwidth, we now wish to determine the total number of degrees of freedom of the space-time field $f(\phi, t)$. To visualize the phase transition, we fix the bandwidth $\Omega$ and the size of the angular observation interval $S=2 \pi$, and scale the time support where the signal is observed $T \rightarrow \infty$ and the wavenumber bandwidth $W \rightarrow \infty$. Using the results of the monodimensional case, we have that as $T \rightarrow \infty$ the number of time-frequency degrees of freedom is of the order of
$$N_{0}=\frac{\Omega T}{\pi} .$$

## 数学代写|信息论作业代写information theory代考|The Number of Degrees of Freedom

$$\int_{-\infty}^{\infty} f^{2}(t) d t \leq E$$

$$f(t)=\sum_{n=1}^{\infty} a_{n} \psi_{n}(t)$$

$$a_{n}=\int_{-\infty}^{\infty} f(t) \psi_{n}^{*}(t) d t$$
(1.2) 中的等式意在“能量”意义上:
$$\lim N \rightarrow \infty \int-\infty^{\infty}\left[f(t)-f_{N}(t)\right]^{2} d t=0$$

$$f_{N}(t)=\sum_{n=1}^{N} a_{n} \psi_{n}(t)$$

## 数学代写|信息论作业代写information theory代考|Space–Time Fields

$$N_{0}=\frac{\Omega T}{\pi}$$

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## MATLAB代写

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