2022-08-10 - Truxton's blog

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Table of Contents

本篇笔记的主要内容

本篇笔记的主要内容是：

令人绝望（且最终放弃了的）kalman filter标准公式的推导；

从另一个角度简单理解kalman filter和bayesian filtering.

Geometric Series, Geometric Distributions （等比级数 / 几何级数）

最开始在这里看到的：🔗 [论文阅读：《Real-Time Audio-to-Score Alignment of Music Performances Containing Errors and Arbitrary Repeats and Skips》 - Truxton's blog] https://truxton2blog.com/paper_reading_real_time_audio-to-score_alignment_of_music_performances_containing_errors_and_arbitrary_repeats_and_skips/

通过做这道例题判断自己是否遗忘了这方面的知识！

求

[mathjax-d]\sum_{n=1}^{\infty} \frac{2 n}{3^{n+1}}[/mathjax-d]

答案

[mathjax]\frac{1}{2}[/mathjax]，见：https://math.stackexchange.com/questions/30732/how-can-i-evaluate-sum-n-0-inftyn1xn

再做一道题：求：

[mathjax-d]\sum_{k=1}^{\infty} k p(1-p)^{(k-1)} p[/mathjax-d]

答案

答案就是[mathjax]1[/mathjax]

过程：

其他参考资料：

🔗 [probability - Why does $\sum_{k=1}^{\infty} kp (1-p)^{(k-1)} p$ converge to 1 for all probabilities p? - Mathematics Stack Exchange] https://math.stackexchange.com/questions/3681183/why-does-sum-k-1-infty-kp-1-pk-1-p-converge-to-1-for-all-probabil

更加一般化的公式推导：preview 🔗 [sequences and series - How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? - Mathematics Stack Exchange] https://math.stackexchange.com/questions/30732/how-can-i-evaluate-sum-n-0-inftyn1xn

其他常见公式总结：preview 🔗 [series.pdf] https://people.math.osu.edu/husen.1/teaching/530/series.pdf

微积分Lim x->0 [log(1+ax)+log(1+bx)]/x

🔗 [Lim x->0 [log(1+ax)+log(1+bx)]/x =? | Socratic] https://socratic.org/questions/lt-x-0-log-1-ax-log-1-bx-x

（几乎没有学进去）继续在kalmanfilter.net上面学习kalman filter

继续在kalmanfilter.net上面学习kalman filter

今天从这里开始：🔗 [State Extrapolation Equation] https://www.kalmanfilter.net/stateextrap.html

要学习STATE EXTRAPOLATION EQUATION，也就是这张图的这个方程：

有点眼熟，是不是和这个差不多？（对应的笔记）

这部分的内容看起来很多，但本质上还是没有引入新的公式，毕竟这是「和物理相关的内容」：

匀加速运动，预测下一时间的位置、速度、加速度。其中加速度保持不变。本质上就是把多个公式融合成了矩阵乘法。

https://www.kalmanfilter.net/stateextrap.html

在这个页面的末尾出现了一个新的概念需要学习：Linear Time-Invariant（LTI）（线性时不变）系统。

A time-invariant system is one where a time delay (or shift) in the input sequence causes an equivalent time delay in the system's output sequence.
https://www.kalmanfilter.net/stateextrap.html