Stochastic Geometry¶

\[ \def\CC{\mathbb{C}} \def\RR{\mathbb{R}} \def\NN{\mathbb{N}} \DeclareMathOperator\expect{\mathbb{E}} \]

课程名称

本课完整名称如下。

概率、随机过程和随机几何及其应用
Probability, Random Process and Stochastic Geometry in Engineering

Random Variables¶

Transforms¶

2024年11月14日。

Probability generating function (PGF, \(G\)):

\(z \mapsto \expect z^\xi\), \(\xi \in \NN\).

Moment generating function (MGF, \(M\)):

\(s \mapsto \expect e^{s \xi}\), \(\xi \in \RR\). Note that it may not converge for all \(s \in \CC\).

Characteristic function (CF, \(\varphi\)):

\(\nu \mapsto \expect e^{j \nu \xi}\), \(\nu \in \RR\). Note that \(\abs{\expect e^{j \cdots}} \leq \expect \abs{e^{j \cdots}} = 1\).

Bounds of probabilities¶

2024年11月14日。

For a random variable \(\xi \in \RR\), there exist the following bounds of \(\Pr(\xi \geq x)\). They describe how fast \(\xi\), as a sum, converges to the central limit theorem.

Markov: \(\expect \xi \geq x \Pr(\xi \geq x)\), where \(x \in \RR^+\).
Generalized Markov: \(\expect \xi^r \geq x^r \Pr(\xi \geq x)\), where \(x,r \in \RR^+\).
Чебышёв: \(\expect \xi^2 \geq x^2 \Pr(\abs{\xi} \geq x)\), where \(x \in \RR^+\).
Chernoff: \(M(s) \coloneqq \expect e^{s \xi} \geq e^{s x} \Pr(\xi \geq x)\), where \(x\in \RR, s \in \RR^+\).

Filters¶

Matched filter¶

2024年10月16日。

“数字通信原理”同名小节。这里更一般。

模型：可能存在的已知信号为 \(v\)。接收到 \(v + x\) 或 \(x\)，其中 \(x\) 是平稳噪声。将它输入滤波器。
目标：最大化信噪比。（根据滤波器输出检测信号有无或判断信号种类）

设滤波器的传输函数为 \(h\)，则输出中

信号频谱密度：\(HV\)。（量纲：[幅度]/[频率]）
噪声功率谱密度：\(S_X \abs{H}^2\)。（量纲：[幅度平方]/[频率]）

瞬时信噪比

\[ \begin{split} r &= \frac {\eval{\abs{v_\text{output}}^2}_{t_0}} {\int S_X \abs{H}^2 \dd{f}} \\ &= \frac {\qty(H^*,\ V e^{j\omega t_0})^2} {\qty(H \sqrt{S_X}, H \sqrt{S_X})} \qq{(inverse Fourier transform)} \\ &= \frac {\qty(H^* \sqrt{S_X},\ V e^{j\omega t_0} / \sqrt{S_X})^2} {\qty(H^* \sqrt{S_X}, H^* \sqrt{S_X})} \\ &\leq \qty(\frac{V e^{j\omega t_0}}{\sqrt{S_X}},\ \frac{V e^{j\omega t_0}}{\sqrt{S_X}})^2 \qq{(Cauchy–Schwarz inequality)} \\ &= \qty(\frac{V}{\sqrt{S_X}},\ \frac{V}{\sqrt{S_X}})^2. \\ \end{split} \]

记号

\((x,y) = \int x^* y \dd{f}\) 是内积。

取等条件是 \(H^* \sqrt{S_X} \parallel V e^{j\omega t_0} / \sqrt{S_X}\)，即

\[ H \parallel \frac{V^* e^{-j\omega t_0}} {S_X}. \]

Wiener filter¶

2024年10月17日。

模型：存在未知平稳信号 \(V\)，接收到 \(U = V + X\)，其中 \(X\) 是平稳噪声。将它输入滤波器。
目标：最小化均方误差。（从滤波器输出估计原信号）

设最优滤波器输出 \(\hat{V}\)，把它与任意滤波器的输出 \(\tilde{V}\) 比较。以互相关为内积 \((\cdot, \cdot)\)，则目标等价于 \(\forall \tilde{V}, (\hat{V} - V, \hat{V} - V) \leq (\tilde{V} - V, \tilde{V} - V)\)。（这里 \(V, \hat{V}, \tilde{V}\) 是同一时刻的。）

任意时刻还是时间平均？

按原始定义，均方误差是时间的函数，并不能比大小，更不用说最小化了。

一般有两种解决方案，一是要求任意时刻都最小，不过这样未必存在最小的；二是用时间平均度量大小，最小化这一个数。

这里都处理平稳信号，均方误差是常函数，两种方案殊途同归。

从线性空间几何上考虑，若 \(\hat{V}\) 是 \(V\) 到子空间 \(\{\tilde{V}\}\) 的垂足，即 \(\forall \tilde{V}, (\hat{V} - V) \perp \tilde{V}\)，则满足要求。这称作 orthogonality principle。

注意 \(\tilde{V}\) 从 \(U\) 滤波而来，总是各时刻 \(U\) 的线性组合，所以其实可排除仅用于比较的任意滤波器，直接要求 \(\forall t', (\hat{V} - V) \perp U_{t'}\) 就够了。（这里 \(V, \hat{V}\) 的时刻仍相同，但未必与 \(U\) 一致。）回归互相关的形式，\((\hat{V} - V) \perp U_{t'} \iff (\hat{V}, U_{t'}) \equiv (V, U_{t'}) \iff R_{\hat{V} U} \equiv R_{V U} \iff S_{\hat{V} U} \equiv S_{V U}\)。

现在从中解出滤波器的传递函数 \(H\)。由 \(U \overset{h}{\rightarrow} \hat{V}\) 可知 \(S_{\hat{V} U} = H S_u\)，再结合 \(U = V + X\) 可得

\[ H = \frac{S_{V U}}{S_U} = \frac{S_V + S_{V X}}{S_V + S_X + 2 S_{V X}}. \]

Point Process¶

Random measure formalism of a point process¶

2024年11月5日，2024年11月18日。

In the space \(X\) (e.g. \(\RR^d\) where \(d \in \NN_+\)), for a specific set \(B \subset X\) (\(B\) stands for Borel), the number of points falling in \(B\), denoted as \(\Phi(B) \in \NN\), is a random variable. Moreover, \(\Phi\) is a measure with randomness, and we can use \(\Phi\) to describe the point process.

Measure

To say \(\Phi\) is a measure is to declare that \(\Phi\) is additive over countable disjoint sets.

形象地解释一下。我们用 \(B_1, B_2, \ldots\) 划分 \(X\)，数出 \(\Phi(B_1), \Phi(B_2), \ldots\) 这一组随机变量。考察这组随机变量的概率分布，就能分析其背后的点过程。

There are abundance tools for random variables, and we can generalize them to \(\Phi\). But the random measure \(\Phi\) includes infinite random variables. Therefore, many tools must be generalized into infinite-dimensional vectors, or equivalently, a function that maps from the index to the corresponding component in the vector.

Mean \(m \coloneqq \expect \xi\).

→ Intensity measure \(\Lambda(B) \coloneqq \expect \Phi(B)\).

Mean of a linear combination of variables is the linear combination of their means.

→ Campbell’s theorem for sums: Let \(S\) be the sum of \(f\) of points in a realization \(\sum_{x \in \Phi} f(x)\), denoted as \(\Phi(f)\) or \(S[f]\), then \(\expect S = \int_X f(x) \times \Lambda(\dd{x})\).

点过程面面观

\(\Phi(f) = \int f(x) \Phi(\dd{x})\) taking \(\Phi\) as a measure, and it also equals to \(\sum f(\Phi)\) taking \(\Phi\) as a set.

Probability generating function (PGF) \((z, w) \mapsto \expect z^\xi w^\eta\) and moment generating function (MGF) \((t,s) \mapsto \expect e^{t\xi + s\eta}\).

→ Probability generating functional (p.g.fl)

\[ v \mapsto \expect v^\Phi \coloneqq \expect \prod_{x\in\Phi} v(x) = \expect e^{\Phi(\log v)}, \]

where \(v: X \to (0, 1]\), and \(v^0 = 1\) (no point here), \(v^1 = v\) (here exists a point).

Example: For a Poisson variable \(\xi\) with parameter \(\lambda\),

\[ \expect z^\xi = \sum_{x \in \NN} \frac{(\lambda z)^x} {x!} e^{-\lambda} = e^{\lambda (z - 1)}. \]

For a Poisson point process with \(\Lambda(\dd{x}) = \lambda(x) \abs{\dd{x}}\),

\[ \expect v^\Phi = \prod_x e^{\Lambda(\dd{x}) (v(x) - 1)} = e^{\int \lambda(x) (v(x) - 1) \dd{x}}. \]

后备箱¶

Poisson过程未必齐次。
A sampling may be with or without replacement. 有放回（前者）的可能性更多。
一个网络中有若干点，每点的参数（如发射与否）可能统一控制（\(\expect e^\cdots\)），也可能各自独立决定（\(e^{\expect[\cdots]}\)）。
随机变量的函数可能一对一、多对一、无穷多对一，不过若只需数字特征，不求解函数的分布也可。