はじめに
信号処理や制御工学では実用上、ディジタル計算機で実現するために連続時間信号をAD変換して離散領域で演算した後、DA変換して連続系である制御対象に入力する。よって手を加える前の物理系に於いて入力と制御対象の間に0次ホールド回路と演算回路が挟まった形になる。本記事では0次ホールド回路を通過した正弦波のスペクトラムについて考察する。
“０次ホールド機構の周波数特性” の続きを読むLDL分解のrank-one更新の導出
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% inverse trigonometric functions
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% convolution
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% Laplace transform
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% Discrete Fourier Transform
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% Z transform
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% linear algebra
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% vector
% unit vector
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% probability theory
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\]
はじめに
$n\times n$行列$A$のLDL分解の計算量は$O(n^3)$であるが、$A$の分解が既に得られているとき、$A+\bm{x}\bm{x}^*$の分解を$O(n^2)$の計算量で求めることができる。本記事ではこの方法を導出する。
“LDL分解のrank-one更新の導出” の続きを読むCholesky分解のrank-one updateの導出
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% mathematics
% general purpose
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% inverse trigonometric functions
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% convolution
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% derivative
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% complex analysis
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% Laplace transform
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% Discrete Fourier Transform
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% Z transform
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% vector
% unit vector
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% probability theory
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% graph theory
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% programming
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\]
はじめに
$n\times n$行列$A$のCholesky分解の計算量は$O(n^3)$であるが、$A$の分解が既に得られているとき、$A+\bm{x}\bm{x}^*$の分解を$O(n^2)$の計算量で求めることができる。本記事ではこの方法を導出する。
“Cholesky分解のrank-one updateの導出” の続きを読む過去サンプルのDFTの再帰的計算
\[
% general purpose
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% mathematics
% general purpose
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% set theory
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% number theory
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% analysis
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% complex analysis
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% Laplace transform
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% Discrete Fourier Transform
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% Z transform
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% linear algebra
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% vector
% unit vector
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% probability theory
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\newcommand{\cind}[2]{\ind{#1\left| #2\right.}} % conditional indicator function
\providecommand{\Pr}{}\renewcommand{\Pr}[1]{\operatorname{Pr}\parens*{#1}}
\DeclarePairedDelimiterX{\cPrParens}[2]{(}{)}{#1\,\delimsize\vert\,#2}
\newcommand{\cPr}[2]{\operatorname{Pr}\cPrParens{#1}{#2}}
\newcommand{\E}[2]{\operatorname{E}_{#1}\bracks*{#2}}
\newcommand{\cE}[3]{\E{#1}{\left.#2\right|#3}}
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% graph theory
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% programming
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\]
はじめに
信号処理に於いて、時刻 $n$ 毎に直近 $N$ 個のサンプルの(時間的に逆向きの)DFTを計算したい状況がある。例えばDCTによる近似的な直交変換 LMS (Least Mean Square) アルゴリズムがそうである。その際に毎度 $N$ 点FFTを実行すると毎時刻 $O(N\log N)$ の計算量となる。しかし実は漸化式を用いてこれを回避し、毎度 $O(N)$ の計算量とする方法がある([1] p65)。本記事ではこれについて述べ、Mathematicaによる数値例を載せる。
“過去サンプルのDFTの再帰的計算” の続きを読むGauss-Seidel法の収束の十分条件
はじめに
Gauss-Seidel法が厳密解へ収束する主要な十分条件の一つに、係数行列が狭義優対角であることが知られている。腕試しに証明を試みたらできたので書き残しておく。
“Gauss-Seidel法の収束の十分条件” の続きを読む