Initial Commit
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commit
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.gitignore
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.gitignore
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*.tex
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**/figs/*.pdf
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**/figs/*.svg
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**/figs/*.tex
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# Emacs
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auto/
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# Simulink Real Time
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*bio.m
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*pt.m
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*ref.m
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*ri.m
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*xcp.m
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*.mldatx
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*.slxc
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*.xml
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*_slrt_rtw/
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# data
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# data/
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# Windows default autosave extension
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*.asv
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# OSX / *nix default autosave extension
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*.m~
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# Compiled MEX binaries (all platforms)
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||||||
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*.mex*
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||||||
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||||||
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# Packaged app and toolbox files
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||||||
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*.mlappinstall
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*.mltbx
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# Generated helpsearch folders
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||||||
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helpsearch*/
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||||||
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# Simulink code generation folders
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||||||
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slprj/
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sccprj/
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||||||
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||||||
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# Matlab code generation folders
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||||||
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codegen/
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# Simulink autosave extension
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*.autosave
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# Octave session info
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||||||
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octave-workspace
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145
css/htmlize.css
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145
css/htmlize.css
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.org-bold { /* bold */ font-weight: bold; }
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||||||
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.org-bold-italic { /* bold-italic */ font-weight: bold; font-style: italic; }
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||||||
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.org-buffer-menu-buffer { /* buffer-menu-buffer */ font-weight: bold; }
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.org-builtin { /* font-lock-builtin-face */ color: #7a378b; }
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.org-button { /* button */ text-decoration: underline; }
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.org-calendar-today { /* calendar-today */ text-decoration: underline; }
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.org-change-log-acknowledgement { /* change-log-acknowledgement */ color: #b22222; }
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.org-change-log-conditionals { /* change-log-conditionals */ color: #a0522d; }
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.org-change-log-date { /* change-log-date */ color: #8b2252; }
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.org-change-log-email { /* change-log-email */ color: #a0522d; }
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.org-change-log-file { /* change-log-file */ color: #0000ff; }
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.org-change-log-function { /* change-log-function */ color: #a0522d; }
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||||||
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.org-change-log-list { /* change-log-list */ color: #a020f0; }
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||||||
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.org-change-log-name { /* change-log-name */ color: #008b8b; }
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||||||
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.org-comint-highlight-input { /* comint-highlight-input */ font-weight: bold; }
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||||||
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.org-comint-highlight-prompt { /* comint-highlight-prompt */ color: #00008b; }
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||||||
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.org-comment { /* font-lock-comment-face */ color: #999988; font-style: italic; }
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||||||
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.org-comment-delimiter { /* font-lock-comment-delimiter-face */ color: #999988; font-style: italic; }
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.org-completions-annotations { /* completions-annotations */ font-style: italic; }
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.org-completions-common-part { /* completions-common-part */ color: #000000; background-color: #ffffff; }
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.org-completions-first-difference { /* completions-first-difference */ font-weight: bold; }
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.org-constant { /* font-lock-constant-face */ color: #008b8b; }
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.org-diary { /* diary */ color: #ff0000; }
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||||||
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.org-diff-context { /* diff-context */ color: #7f7f7f; }
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||||||
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.org-diff-file-header { /* diff-file-header */ background-color: #b3b3b3; font-weight: bold; }
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||||||
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.org-diff-function { /* diff-function */ background-color: #cccccc; }
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||||||
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.org-diff-header { /* diff-header */ background-color: #cccccc; }
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||||||
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.org-diff-hunk-header { /* diff-hunk-header */ background-color: #cccccc; }
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.org-diff-index { /* diff-index */ background-color: #b3b3b3; font-weight: bold; }
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.org-diff-nonexistent { /* diff-nonexistent */ background-color: #b3b3b3; font-weight: bold; }
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.org-diff-refine-change { /* diff-refine-change */ background-color: #d9d9d9; }
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.org-dired-directory { /* dired-directory */ color: #0000ff; }
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||||||
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.org-dired-flagged { /* dired-flagged */ color: #ff0000; font-weight: bold; }
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||||||
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.org-dired-header { /* dired-header */ color: #228b22; }
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||||||
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.org-dired-ignored { /* dired-ignored */ color: #7f7f7f; }
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||||||
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.org-dired-mark { /* dired-mark */ color: #008b8b; }
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||||||
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.org-dired-marked { /* dired-marked */ color: #ff0000; font-weight: bold; }
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||||||
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.org-dired-perm-write { /* dired-perm-write */ color: #b22222; }
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||||||
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.org-dired-symlink { /* dired-symlink */ color: #a020f0; }
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.org-dired-warning { /* dired-warning */ color: #ff0000; font-weight: bold; }
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.org-doc { /* font-lock-doc-face */ color: #8b2252; }
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||||||
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.org-escape-glyph { /* escape-glyph */ color: #a52a2a; }
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||||||
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.org-file-name-shadow { /* file-name-shadow */ color: #7f7f7f; }
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||||||
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.org-flyspell-duplicate { /* flyspell-duplicate */ color: #cdad00; font-weight: bold; text-decoration: underline; }
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||||||
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.org-flyspell-incorrect { /* flyspell-incorrect */ color: #ff4500; font-weight: bold; text-decoration: underline; }
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||||||
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.org-fringe { /* fringe */ background-color: #f2f2f2; }
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||||||
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.org-function-name { /* font-lock-function-name-face */ color: teal; }
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||||||
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.org-header-line { /* header-line */ color: #333333; background-color: #e5e5e5; }
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||||||
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.org-help-argument-name { /* help-argument-name */ font-style: italic; }
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||||||
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.org-highlight { /* highlight */ background-color: #b4eeb4; }
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||||||
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.org-holiday { /* holiday */ background-color: #ffc0cb; }
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||||||
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.org-isearch { /* isearch */ color: #b0e2ff; background-color: #cd00cd; }
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||||||
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.org-isearch-fail { /* isearch-fail */ background-color: #ffc1c1; }
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||||||
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.org-italic { /* italic */ font-style: italic; }
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||||||
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.org-keyword { /* font-lock-keyword-face */ color: #0086b3; }
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||||||
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.org-lazy-highlight { /* lazy-highlight */ background-color: #afeeee; }
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||||||
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.org-link { /* link */ color: #0000ff; text-decoration: underline; }
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||||||
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.org-link-visited { /* link-visited */ color: #8b008b; text-decoration: underline; }
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||||||
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.org-log-edit-header { /* log-edit-header */ color: #a020f0; }
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||||||
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.org-log-edit-summary { /* log-edit-summary */ color: #0000ff; }
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||||||
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.org-log-edit-unknown-header { /* log-edit-unknown-header */ color: #b22222; }
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||||||
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.org-match { /* match */ background-color: #ffff00; }
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||||||
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.org-next-error { /* next-error */ background-color: #eedc82; }
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||||||
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.org-nobreak-space { /* nobreak-space */ color: #a52a2a; text-decoration: underline; }
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||||||
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.org-org-archived { /* org-archived */ color: #7f7f7f; }
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||||||
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.org-org-block { /* org-block */ color: #7f7f7f; }
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||||||
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.org-org-block-begin-line { /* org-block-begin-line */ color: #b22222; }
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||||||
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.org-org-block-end-line { /* org-block-end-line */ color: #b22222; }
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||||||
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.org-org-checkbox { /* org-checkbox */ font-weight: bold; }
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||||||
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.org-org-checkbox-statistics-done { /* org-checkbox-statistics-done */ color: #228b22; font-weight: bold; }
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||||||
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.org-org-checkbox-statistics-todo { /* org-checkbox-statistics-todo */ color: #ff0000; font-weight: bold; }
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||||||
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.org-org-clock-overlay { /* org-clock-overlay */ background-color: #ffff00; }
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||||||
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.org-org-code { /* org-code */ color: #7f7f7f; }
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||||||
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.org-org-column { /* org-column */ background-color: #e5e5e5; }
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||||||
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.org-org-column-title { /* org-column-title */ background-color: #e5e5e5; font-weight: bold; text-decoration: underline; }
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||||||
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.org-org-date { /* org-date */ color: #a020f0; text-decoration: underline; }
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||||||
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.org-org-document-info { /* org-document-info */ color: #191970; }
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||||||
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.org-org-document-info-keyword { /* org-document-info-keyword */ color: #7f7f7f; }
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||||||
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.org-org-document-title { /* org-document-title */ color: #191970; font-size: 144%; font-weight: bold; }
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||||||
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.org-org-done { /* org-done */ color: #228b22; font-weight: bold; }
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||||||
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.org-org-drawer { /* org-drawer */ color: #0000ff; }
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.org-org-ellipsis { /* org-ellipsis */ color: #b8860b; text-decoration: underline; }
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||||||
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.org-org-footnote { /* org-footnote */ color: #a020f0; text-decoration: underline; }
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||||||
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.org-org-formula { /* org-formula */ color: #b22222; }
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||||||
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.org-org-headline-done { /* org-headline-done */ color: #bc8f8f; }
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||||||
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.org-org-hide { /* org-hide */ color: #ffffff; }
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.org-org-latex-and-export-specials { /* org-latex-and-export-specials */ color: #8b4513; }
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.org-org-level-1 { /* org-level-1 */ color: #0000ff; }
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.org-org-level-2 { /* org-level-2 */ color: #a0522d; }
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.org-org-level-3 { /* org-level-3 */ color: #a020f0; }
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.org-org-level-4 { /* org-level-4 */ color: #b22222; }
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.org-org-level-5 { /* org-level-5 */ color: #228b22; }
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.org-org-level-6 { /* org-level-6 */ color: #008b8b; }
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.org-org-level-7 { /* org-level-7 */ color: #7a378b; }
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.org-org-level-8 { /* org-level-8 */ color: #8b2252; }
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.org-org-link { /* org-link */ color: #0000ff; text-decoration: underline; }
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.org-org-meta-line { /* org-meta-line */ color: #b22222; }
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.org-org-mode-line-clock { /* org-mode-line-clock */ color: #000000; background-color: #bfbfbf; }
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||||||
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.org-org-mode-line-clock-overrun { /* org-mode-line-clock-overrun */ color: #000000; background-color: #ff0000; }
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||||||
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.org-org-quote { /* org-quote */ color: #7f7f7f; }
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.org-org-scheduled { /* org-scheduled */ color: #006400; }
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.org-org-scheduled-previously { /* org-scheduled-previously */ color: #b22222; }
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.org-org-scheduled-today { /* org-scheduled-today */ color: #006400; }
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.org-org-sexp-date { /* org-sexp-date */ color: #a020f0; }
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||||||
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.org-org-special-keyword { /* org-special-keyword */ color: #a020f0; }
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.org-org-table { /* org-table */ color: #0000ff; }
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.org-org-tag { /* org-tag */ font-weight: bold; }
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.org-org-target { /* org-target */ text-decoration: underline; }
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||||||
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.org-org-time-grid { /* org-time-grid */ color: #b8860b; }
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.org-org-todo { /* org-todo */ color: #ff0000; font-weight: bold; }
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.org-org-upcoming-deadline { /* org-upcoming-deadline */ color: #b22222; }
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||||||
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.org-org-verbatim { /* org-verbatim */ color: #7f7f7f; }
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||||||
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.org-org-verse { /* org-verse */ color: #7f7f7f; }
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.org-org-warning { /* org-warning */ color: #ff0000; font-weight: bold; }
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.org-outline-1 { /* outline-1 */ color: #0000ff; }
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.org-outline-2 { /* outline-2 */ color: #a0522d; }
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.org-outline-3 { /* outline-3 */ color: #a020f0; }
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.org-outline-4 { /* outline-4 */ color: #b22222; }
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.org-outline-5 { /* outline-5 */ color: #228b22; }
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.org-outline-6 { /* outline-6 */ color: #008b8b; }
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.org-outline-7 { /* outline-7 */ color: #7a378b; }
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.org-outline-8 { /* outline-8 */ color: #8b2252; }
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.org-preprocessor { /* font-lock-preprocessor-face */ color: #7a378b; }
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.org-query-replace { /* query-replace */ color: #b0e2ff; background-color: #cd00cd; }
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.org-regexp-grouping-backslash { /* font-lock-regexp-grouping-backslash */ font-weight: bold; }
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.org-regexp-grouping-construct { /* font-lock-regexp-grouping-construct */ font-weight: bold; }
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.org-region { /* region */ background-color: #eedc82; }
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.org-secondary-selection { /* secondary-selection */ background-color: #ffff00; }
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||||||
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.org-shadow { /* shadow */ color: #7f7f7f; }
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.org-show-paren-match { /* show-paren-match */ background-color: #40e0d0; }
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.org-show-paren-mismatch { /* show-paren-mismatch */ color: #ffffff; background-color: #a020f0; }
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.org-string { /* font-lock-string-face */ color: #dd1144; }
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.org-tool-bar { /* tool-bar */ color: #000000; background-color: #bfbfbf; }
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||||||
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.org-tooltip { /* tooltip */ color: #000000; background-color: #ffffe0; }
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||||||
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.org-trailing-whitespace { /* trailing-whitespace */ background-color: #ff0000; }
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||||||
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.org-type { /* font-lock-type-face */ color: #228b22; }
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||||||
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.org-underline { /* underline */ text-decoration: underline; }
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||||||
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.org-variable-name { /* font-lock-variable-name-face */ color: teal; }
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||||||
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.org-warning { /* font-lock-warning-face */ color: #ff0000; font-weight: bold; }
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||||||
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.org-widget-button { /* widget-button */ font-weight: bold; }
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||||||
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.org-widget-button-pressed { /* widget-button-pressed */ color: #ff0000; }
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||||||
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.org-widget-documentation { /* widget-documentation */ color: #006400; }
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||||||
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.org-widget-field { /* widget-field */ background-color: #d9d9d9; }
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||||||
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.org-widget-inactive { /* widget-inactive */ color: #7f7f7f; }
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||||||
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.org-widget-single-line-field { /* widget-single-line-field */ background-color: #d9d9d9; }
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||||||
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css/readtheorg.css
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Load Diff
BIN
figs/velocity_time.png
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figs/velocity_time.png
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After Width: | Height: | Size: 44 KiB |
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figs/velocity_to_displacement_psd.png
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figs/velocity_to_displacement_psd.png
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After Width: | Height: | Size: 1.3 KiB |
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figs/velocity_to_voltage.png
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figs/velocity_to_voltage.png
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After Width: | Height: | Size: 7.4 KiB |
BIN
figs/velocity_to_voltage_psd.png
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figs/velocity_to_voltage_psd.png
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After Width: | Height: | Size: 2.4 KiB |
BIN
figs/voltage_to_velocity.png
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figs/voltage_to_velocity.png
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After Width: | Height: | Size: 6.2 KiB |
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<?xml version="1.0" encoding="utf-8"?>
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2019-08-15 jeu. 12:31 -->
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||||||
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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||||||
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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||||||
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<title>Compute Spectral Densities of signals with Matlab</title>
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||||||
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<meta name="generator" content="Org mode" />
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||||||
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<meta name="author" content="Dehaeze Thomas" />
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<style type="text/css">
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||||||
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<!--/*--><![CDATA[/*><!--*/
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.title { text-align: center;
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||||||
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margin-bottom: .2em; }
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.subtitle { text-align: center;
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font-size: medium;
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font-weight: bold;
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margin-top:0; }
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.todo { font-family: monospace; color: red; }
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.done { font-family: monospace; color: green; }
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.priority { font-family: monospace; color: orange; }
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||||||
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.tag { background-color: #eee; font-family: monospace;
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padding: 2px; font-size: 80%; font-weight: normal; }
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.timestamp { color: #bebebe; }
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.timestamp-kwd { color: #5f9ea0; }
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.org-right { margin-left: auto; margin-right: 0px; text-align: right; }
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||||||
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.org-left { margin-left: 0px; margin-right: auto; text-align: left; }
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.org-center { margin-left: auto; margin-right: auto; text-align: center; }
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||||||
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.underline { text-decoration: underline; }
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#postamble p, #preamble p { font-size: 90%; margin: .2em; }
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p.verse { margin-left: 3%; }
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pre {
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||||||
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border: 1px solid #ccc;
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box-shadow: 3px 3px 3px #eee;
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padding: 8pt;
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font-family: monospace;
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overflow: auto;
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margin: 1.2em;
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}
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pre.src {
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position: relative;
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overflow: visible;
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padding-top: 1.2em;
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}
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pre.src:before {
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display: none;
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||||||
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position: absolute;
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background-color: white;
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||||||
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<a accesskey="h" href="./index.html"> UP </a>
|
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<a accesskey="H" href="./index.html"> HOME </a>
|
||||||
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</div><div id="content">
|
||||||
|
<h1 class="title">Compute Spectral Densities of signals with Matlab</h1>
|
||||||
|
<div id="table-of-contents">
|
||||||
|
<h2>Table of Contents</h2>
|
||||||
|
<div id="text-table-of-contents">
|
||||||
|
<ul>
|
||||||
|
<li><a href="#org7cc41f2">1. Sensitivity of the instrumentation</a></li>
|
||||||
|
<li><a href="#orgac6792b">2. Convert the time domain from volts to velocity</a></li>
|
||||||
|
<li><a href="#org01a363c">3. Power Spectral Density and Amplitude Spectral Density</a></li>
|
||||||
|
<li><a href="#org90edcdd">4. Modification of a signal's Power Spectral Density when going through an LTI system</a></li>
|
||||||
|
<li><a href="#orgfb2734c">5. From PSD of the velocity to the PSD of the displacement</a></li>
|
||||||
|
</ul>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
This document presents the mathematics as well as the matlab scripts to do the spectral analysis of a measured signal.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Typically this signal is coming from an inertial sensor, a force sensor or any other sensor.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
We here take the example of a signal coming from a Geophone measurement the vertical velocity of the floor at the ESRF.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<div id="outline-container-org7cc41f2" class="outline-2">
|
||||||
|
<h2 id="org7cc41f2"><span class="section-number-2">1</span> Sensitivity of the instrumentation</h2>
|
||||||
|
<div class="outline-text-2" id="text-1">
|
||||||
|
<p>
|
||||||
|
The measured signal \(x\) by the ADC is in Volts.
|
||||||
|
The corresponding real velocity \(v\) in m/s.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
To obtain the real quantity as measured by the sensor, one have to know the sensitivity of the sensors and electronics used.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
|
||||||
|
<div id="org0a5ff56" class="figure">
|
||||||
|
<p><img src="figs/velocity_to_voltage.png" alt="velocity_to_voltage.png" />
|
||||||
|
</p>
|
||||||
|
<p><span class="figure-number">Figure 1: </span>Schematic of the instrumentation used for the measurement</p>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orgac6792b" class="outline-2">
|
||||||
|
<h2 id="orgac6792b"><span class="section-number-2">2</span> Convert the time domain from volts to velocity</h2>
|
||||||
|
<div class="outline-text-2" id="text-2">
|
||||||
|
<p>
|
||||||
|
Let's say, we know that the sensitivity of the geophone used is
|
||||||
|
\[ G_g(s) = G_0 \frac{\frac{s}{2\pi f_0}}{1 + \frac{s}{2\pi f_0}} \quad \left[\frac{V}{m/s}\right] \]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">G0 = <span class="org-highlight-numbers-number">88</span>; <span class="org-comment">% Sensitivity [V/(m/s)]</span>
|
||||||
|
f0 = <span class="org-highlight-numbers-number">2</span>; <span class="org-comment">% Cut-off frequency [Hz]</span>
|
||||||
|
|
||||||
|
Gg = G0<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>f0<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">+</span>s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>f0<span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
And the gain of the amplifier is 1000: \(G_m(s) = 1000\).
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">Gm = <span class="org-highlight-numbers-number">1000</span>;
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
If \({G_m(s)}^{-1} {G_g(s)}^{-1}\) is proper, we can simulate this dynamical system to go from the voltage to the velocity units (figure <a href="#orgd3dfdc8">2</a>).
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">data = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_028.mat', 'data'</span><span class="org-rainbow-delimiters-depth-1">)</span>; data = data.data;
|
||||||
|
|
||||||
|
t = data<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% [s]</span>
|
||||||
|
x = data<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">-</span>mean<span class="org-rainbow-delimiters-depth-1">(</span>data<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% The offset if removed (coming from the voltage amplifier) [v]</span>
|
||||||
|
|
||||||
|
dt = t<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">-</span>t<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>; Fs = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>dt;
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
|
||||||
|
<div id="orgd3dfdc8" class="figure">
|
||||||
|
<p><img src="figs/voltage_to_velocity.png" alt="voltage_to_velocity.png" />
|
||||||
|
</p>
|
||||||
|
<p><span class="figure-number">Figure 2: </span>Schematic of the instrumentation used for the measurement</p>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
We simulate this system with matlab:
|
||||||
|
</p>
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">v = lsim<span class="org-rainbow-delimiters-depth-1">(</span>inv<span class="org-rainbow-delimiters-depth-2">(</span>Gg<span class="org-type">*</span>Gm<span class="org-rainbow-delimiters-depth-2">)</span>, v, t<span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
And we plot the obtained velocity
|
||||||
|
</p>
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
||||||
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>t, v<span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||||
|
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">"Time [s]"</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">"Velocity [m/s]"</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
|
||||||
|
<div id="org4dd86c0" class="figure">
|
||||||
|
<p><img src="figs/velocity_time.png" alt="velocity_time.png" />
|
||||||
|
</p>
|
||||||
|
<p><span class="figure-number">Figure 3: </span>Measured Velocity</p>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org01a363c" class="outline-2">
|
||||||
|
<h2 id="org01a363c"><span class="section-number-2">3</span> Power Spectral Density and Amplitude Spectral Density</h2>
|
||||||
|
<div class="outline-text-2" id="text-3">
|
||||||
|
<p>
|
||||||
|
We now have the velocity in the time domain:
|
||||||
|
\[ v(t)\ [m/s] \]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
To compute the Power Spectral Density (PSD):
|
||||||
|
\[ S_v(f)\ \left[\frac{(m/s)^2}{Hz}\right] \]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
To compute that with matlab, we use the <code>pwelch</code> function.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
We first have to defined a window:
|
||||||
|
</p>
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">win = hanning<span class="org-rainbow-delimiters-depth-1">(</span>ceil<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">10</span><span class="org-type">*</span>Fs<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>; % <span class="org-highlight-numbers-number">10s</span> window
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>Sv, f<span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>v, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
||||||
|
loglog<span class="org-rainbow-delimiters-depth-1">(</span>f, Sv<span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||||
|
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||||
|
ylabel<span class="org-rainbow-delimiters-depth-1">(</span>'Power Spectral Density $<span class="org-type">\</span>left<span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">\</span>frac<span class="org-rainbow-delimiters-depth-3">{</span><span class="org-rainbow-delimiters-depth-4">(</span>m<span class="org-type">/</span>s<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">}{</span>Hz<span class="org-rainbow-delimiters-depth-3">}</span><span class="org-type">\</span>right<span class="org-rainbow-delimiters-depth-2">]</span>$'<span class="org-rainbow-delimiters-depth-1">)</span>
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
The Amplitude Spectral Density (ASD) is the square root of the Power Spectral Density:
|
||||||
|
</p>
|
||||||
|
\begin{equation}
|
||||||
|
\Gamma_{vv}(f) = \sqrt{S_{vv}(f)} \quad \left[ \frac{m/s}{\sqrt{Hz}} \right]
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
||||||
|
loglog<span class="org-rainbow-delimiters-depth-1">(</span>f, sqrt<span class="org-rainbow-delimiters-depth-2">(</span>Sv<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||||
|
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||||
|
ylabel<span class="org-rainbow-delimiters-depth-1">(</span>'Amplitude Spectral Density $<span class="org-type">\</span>left<span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">\</span>frac<span class="org-rainbow-delimiters-depth-3">{</span>m<span class="org-type">/</span>s<span class="org-rainbow-delimiters-depth-3">}{</span><span class="org-type">\</span>sqrt<span class="org-rainbow-delimiters-depth-4">{</span>Hz<span class="org-rainbow-delimiters-depth-4">}</span><span class="org-rainbow-delimiters-depth-3">}</span><span class="org-type">\</span>right<span class="org-rainbow-delimiters-depth-2">]</span>$'<span class="org-rainbow-delimiters-depth-1">)</span>
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org90edcdd" class="outline-2">
|
||||||
|
<h2 id="org90edcdd"><span class="section-number-2">4</span> Modification of a signal's Power Spectral Density when going through an LTI system</h2>
|
||||||
|
<div class="outline-text-2" id="text-4">
|
||||||
|
|
||||||
|
<div id="org9eca4b8" class="figure">
|
||||||
|
<p><img src="figs/velocity_to_voltage_psd.png" alt="velocity_to_voltage_psd.png" />
|
||||||
|
</p>
|
||||||
|
<p><span class="figure-number">Figure 4: </span>Schematic of the instrumentation used for the measurement</p>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
We can show that:
|
||||||
|
</p>
|
||||||
|
\begin{equation}
|
||||||
|
S_{yy}(\omega) = \left|G(j\omega)\right|^2 S_{xx}(\omega)
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
<p>
|
||||||
|
And we also have:
|
||||||
|
</p>
|
||||||
|
\begin{equation}
|
||||||
|
\Gamma_{yy}(\omega) = \left|G(j\omega)\right| \Gamma_{xx}(\omega)
|
||||||
|
\end{equation}
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orgfb2734c" class="outline-2">
|
||||||
|
<h2 id="orgfb2734c"><span class="section-number-2">5</span> From PSD of the velocity to the PSD of the displacement</h2>
|
||||||
|
<div class="outline-text-2" id="text-5">
|
||||||
|
|
||||||
|
<div id="orga1a0bc5" class="figure">
|
||||||
|
<p><img src="figs/velocity_to_displacement_psd.png" alt="velocity_to_displacement_psd.png" />
|
||||||
|
</p>
|
||||||
|
<p><span class="figure-number">Figure 5: </span>Schematic of the instrumentation used for the measurement</p>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
The displacement is the integral of the velocity.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
We then have that
|
||||||
|
</p>
|
||||||
|
\begin{equation}
|
||||||
|
S_{xx}(\omega) = \left|\frac{1}{j \omega}\right|^2 S_{vv}(\omega)
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Using a frequency variable in Hz:
|
||||||
|
</p>
|
||||||
|
\begin{equation}
|
||||||
|
S_{xx}(f) = \left| \frac{1}{j 2\pi f} \right|^2 S_{vv}(f)
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
<p>
|
||||||
|
For the Amplitude Spectral Density:
|
||||||
|
</p>
|
||||||
|
\begin{equation}
|
||||||
|
\Gamma_{xx}(f) = \frac{1}{2\pi f} \Gamma_{vv}(f)
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
<div class="note">
|
||||||
|
\begin{equation}
|
||||||
|
S_{xx}(\omega = 1) = S_{vv}(\omega = 1)
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
</div>
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Now if we want to obtain the Power Spectral Density of the Position or Acceleration:
|
||||||
|
For each frequency:
|
||||||
|
\[ \left| \frac{d sin(2 \pi f t)}{dt} \right| = | 2 \pi f | \times | \cos(2\pi f t) | \]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[ \left| \int_0^t sin(2 \pi f \tau) d\tau \right| = \left| \frac{1}{2 \pi f} \right| \times | \cos(2\pi f t) | \]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[ ASD_x(f) = \frac{1}{2\pi f} ASD_v(f) \ \left[\frac{m}{\sqrt{Hz}}\right] \]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[ ASD_a(f) = 2\pi f ASD_v(f) \ \left[\frac{m/s^2}{\sqrt{Hz}}\right] \]
|
||||||
|
And we have
|
||||||
|
\[ PSD_x(f) = {ASD_x(f)}^2 = \frac{1}{(2 \pi f)^2} {ASD_v(f)}^2 = \frac{1}{(2 \pi f)^2} PSD_v(f) \]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Note here that we always have
|
||||||
|
\[ PSD_x \left(f = \frac{1}{2\pi}\right) = PSD_v \left(f = \frac{1}{2\pi}\right) = PSD_a \left(f = \frac{1}{2\pi}\right), \quad \frac{1}{2\pi} \approx 0.16 [Hz] \]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
If we want to compute the Cumulative Power Spectrum:
|
||||||
|
\[ CPS_v(f) = \int_0^f PSD_v(\nu) d\nu \quad [(m/s)^2] \]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
We can also want to integrate from high frequency to low frequency:
|
||||||
|
\[ CPS_v(f) = \int_f^\infty PSD_v(\nu) d\nu \quad [(m/s)^2] \]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
The Cumulative Amplitude Spectrum is then the square root of the Cumulative Power Spectrum:
|
||||||
|
\[ CAS_v(f) = \sqrt{CPS_v(f)} = \sqrt{\int_f^\infty PSD_v(\nu) d\nu} \quad [m/s] \]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Then, we can obtain the Root Mean Square value of the velocity:
|
||||||
|
\[ v_{\text{rms}} = CAS_v(0) \quad [m/s \ \text{rms}] \]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<div id="postamble" class="status">
|
||||||
|
<p class="author">Author: Dehaeze Thomas</p>
|
||||||
|
<p class="date">Created: 2019-08-15 jeu. 12:31</p>
|
||||||
|
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||||||
|
</div>
|
||||||
|
</body>
|
||||||
|
</html>
|
||||||
305
index.org
Normal file
305
index.org
Normal file
@ -0,0 +1,305 @@
|
|||||||
|
#+TITLE: Compute Spectral Densities of signals with Matlab
|
||||||
|
:DRAWER:
|
||||||
|
#+LANGUAGE: en
|
||||||
|
#+EMAIL: dehaeze.thomas@gmail.com
|
||||||
|
#+AUTHOR: Dehaeze Thomas
|
||||||
|
|
||||||
|
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
|
||||||
|
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
|
||||||
|
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/zenburn.css"/>
|
||||||
|
#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.min.js"></script>
|
||||||
|
#+HTML_HEAD: <script type="text/javascript" src="./js/bootstrap.min.js"></script>
|
||||||
|
#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.stickytableheaders.min.js"></script>
|
||||||
|
#+HTML_HEAD: <script type="text/javascript" src="./js/readtheorg.js"></script>
|
||||||
|
|
||||||
|
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/MEGA/These/LaTeX/}{config.tex}")
|
||||||
|
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
|
||||||
|
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
|
||||||
|
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
|
||||||
|
#+PROPERTY: header-args:latex+ :results raw replace :buffer no
|
||||||
|
#+PROPERTY: header-args:latex+ :eval no-export
|
||||||
|
#+PROPERTY: header-args:latex+ :exports both
|
||||||
|
#+PROPERTY: header-args:latex+ :mkdirp yes
|
||||||
|
#+PROPERTY: header-args:latex+ :output-dir figs
|
||||||
|
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
|
||||||
|
|
||||||
|
#+PROPERTY: header-args:matlab :session *MATLAB*
|
||||||
|
#+PROPERTY: header-args:matlab+ :tangle filters.m
|
||||||
|
#+PROPERTY: header-args:matlab+ :comments org
|
||||||
|
#+PROPERTY: header-args:matlab+ :exports both
|
||||||
|
#+PROPERTY: header-args:matlab+ :results none
|
||||||
|
#+PROPERTY: header-args:matlab+ :eval no-export
|
||||||
|
#+PROPERTY: header-args:matlab+ :noweb yes
|
||||||
|
#+PROPERTY: header-args:matlab+ :mkdirp yes
|
||||||
|
#+PROPERTY: header-args:matlab+ :output-dir figs
|
||||||
|
:END:
|
||||||
|
|
||||||
|
This document presents the mathematics as well as the matlab scripts to do the spectral analysis of a measured signal.
|
||||||
|
|
||||||
|
Typically this signal is coming from an inertial sensor, a force sensor or any other sensor.
|
||||||
|
|
||||||
|
We here take the example of a signal coming from a Geophone measurement the vertical velocity of the floor at the ESRF.
|
||||||
|
|
||||||
|
* Matlab Init :noexport:ignore:
|
||||||
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||||
|
<<matlab-dir>>
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none :results silent :noweb yes
|
||||||
|
<<matlab-init>>
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
* Sensitivity of the instrumentation
|
||||||
|
The measured signal $x$ by the ADC is in Volts.
|
||||||
|
The corresponding real velocity $v$ in m/s.
|
||||||
|
|
||||||
|
To obtain the real quantity as measured by the sensor, one have to know the sensitivity of the sensors and electronics used.
|
||||||
|
|
||||||
|
#+begin_src latex :file velocity_to_voltage.pdf :post pdf2svg(file=*this*, ext="png") :exports results
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\node[block] (geophone) at (0, 0) {$G_g(s)$};
|
||||||
|
\node[above] at (geophone.north) {Geophone};
|
||||||
|
\node[block, right=1 of geophone] (ampli) {$G_a(s)$};
|
||||||
|
\node[above] at (ampli.north) {Amplifier};
|
||||||
|
\node[ADC, right=1 of ampli] (adc) {ADC};
|
||||||
|
|
||||||
|
\draw[double, <-] (geophone.west) -- node[midway, above]{$v$ [m/s]} ++(-1.4, 0);
|
||||||
|
\draw[->] (geophone.east) -- node[midway, above]{[V]} (ampli.west);
|
||||||
|
\draw[->] (ampli.east) -- node[midway, above]{[V]} (adc.west);
|
||||||
|
\draw[->] (adc.east) -- node[sloped]{$/$}node[midway, above]{$x$ [V]} ++(1.4, 0);
|
||||||
|
\end{tikzpicture}
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+NAME: fig:velocity_to_voltage
|
||||||
|
#+CAPTION: Schematic of the instrumentation used for the measurement
|
||||||
|
#+RESULTS: fig:velocity_to_voltage
|
||||||
|
[[file:figs/velocity_to_voltage.png]]
|
||||||
|
|
||||||
|
* Convert the time domain from volts to velocity
|
||||||
|
Let's say, we know that the sensitivity of the geophone used is
|
||||||
|
\[ G_g(s) = G_0 \frac{\frac{s}{2\pi f_0}}{1 + \frac{s}{2\pi f_0}} \quad \left[\frac{V}{m/s}\right] \]
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
G0 = 88; % Sensitivity [V/(m/s)]
|
||||||
|
f0 = 2; % Cut-off frequency [Hz]
|
||||||
|
|
||||||
|
Gg = G0*(s/2/pi/f0)/(1+s/2/pi/f0);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
And the gain of the amplifier is 1000: $G_m(s) = 1000$.
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
Gm = 1000;
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
If ${G_m(s)}^{-1} {G_g(s)}^{-1}$ is proper, we can simulate this dynamical system to go from the voltage to the velocity units (figure [[fig:voltage_to_velocity]]).
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
data = load('mat/data_028.mat', 'data'); data = data.data;
|
||||||
|
|
||||||
|
t = data(:, 3); % [s]
|
||||||
|
x = data(:, 1)-mean(data(:, 1)); % The offset if removed (coming from the voltage amplifier) [v]
|
||||||
|
|
||||||
|
dt = t(2)-t(1); Fs = 1/dt;
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src latex :file voltage_to_velocity.pdf :post pdf2svg(file=*this*, ext="png") :exports results
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\node[block] (ampli) at (0, 0) {${G_a(s)}^{-1}$};
|
||||||
|
\node[above] at (ampli.north) {Amplifier};
|
||||||
|
\node[block, right=1 of ampli] (geophone) {${G_g(s)}^{-1}$};
|
||||||
|
\node[above] at (geophone.north) {Geophone};
|
||||||
|
|
||||||
|
\draw[<-] (ampli.west) -- node[midway, above]{$x$ [V]}node[sloped]{$/$} ++(-1.4, 0);
|
||||||
|
\draw[->] (ampli.east) -- node[midway, above]{[V]}node[sloped]{$/$} (geophone.west);
|
||||||
|
\draw[->] (geophone.east) -- node[midway, above]{$v$ [m/s]}node[sloped]{$/$} ++(1.4, 0);
|
||||||
|
\end{tikzpicture}
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+NAME: fig:voltage_to_velocity
|
||||||
|
#+CAPTION: Schematic of the instrumentation used for the measurement
|
||||||
|
#+RESULTS: fig:voltage_to_velocity
|
||||||
|
[[file:figs/voltage_to_velocity.png]]
|
||||||
|
|
||||||
|
We simulate this system with matlab:
|
||||||
|
#+begin_src matlab
|
||||||
|
v = lsim(inv(Gg*Gm), v, t);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
And we plot the obtained velocity
|
||||||
|
#+begin_src matlab
|
||||||
|
figure;
|
||||||
|
plot(t, v);
|
||||||
|
xlabel("Time [s]"); ylabel("Velocity [m/s]");
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+NAME: fig:velocity_time
|
||||||
|
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
|
||||||
|
#+begin_src matlab :var filepath="figs/velocity_time.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
|
||||||
|
<<plt-matlab>>
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+NAME: fig:velocity_time
|
||||||
|
#+CAPTION: Measured Velocity
|
||||||
|
#+RESULTS: fig:velocity_time
|
||||||
|
[[file:figs/velocity_time.png]]
|
||||||
|
|
||||||
|
* Power Spectral Density and Amplitude Spectral Density
|
||||||
|
We now have the velocity in the time domain:
|
||||||
|
\[ v(t)\ [m/s] \]
|
||||||
|
|
||||||
|
To compute the Power Spectral Density (PSD):
|
||||||
|
\[ S_v(f)\ \left[\frac{(m/s)^2}{Hz}\right] \]
|
||||||
|
|
||||||
|
To compute that with matlab, we use the =pwelch= function.
|
||||||
|
|
||||||
|
We first have to defined a window:
|
||||||
|
#+begin_src matlab
|
||||||
|
win = hanning(ceil(10*Fs)); % 10s window
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
[Sv, f] = pwelch(v, win, [], [], Fs);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
figure;
|
||||||
|
loglog(f, Sv);
|
||||||
|
xlabel('Frequency [Hz]');
|
||||||
|
ylabel('Power Spectral Density $\left[\frac{(m/s)^2}{Hz}\right]$')
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+NAME: fig:psd_velocity
|
||||||
|
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
|
||||||
|
#+begin_src matlab :var filepath="figs/psd_velocity.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||||
|
<<plt-matlab>>
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+NAME: fig:psd_velocity
|
||||||
|
#+CAPTION: Power Spectral Density of the measured velocity
|
||||||
|
#+RESULTS: fig:psd_velocity
|
||||||
|
|
||||||
|
The Amplitude Spectral Density (ASD) is the square root of the Power Spectral Density:
|
||||||
|
\begin{equation}
|
||||||
|
\Gamma_{vv}(f) = \sqrt{S_{vv}(f)} \quad \left[ \frac{m/s}{\sqrt{Hz}} \right]
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
figure;
|
||||||
|
loglog(f, sqrt(Sv));
|
||||||
|
xlabel('Frequency [Hz]');
|
||||||
|
ylabel('Amplitude Spectral Density $\left[\frac{m/s}{\sqrt{Hz}}\right]$')
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+NAME: fig:asd_velocity
|
||||||
|
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
|
||||||
|
#+begin_src matlab :var filepath="figs/asd_velocity.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||||
|
<<plt-matlab>>
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+NAME: fig:asd_velocity
|
||||||
|
#+CAPTION: Power Spectral Density of the measured velocity
|
||||||
|
#+RESULTS: fig:asd_velocity
|
||||||
|
|
||||||
|
* Modification of a signal's Power Spectral Density when going through an LTI system
|
||||||
|
|
||||||
|
#+begin_src latex :file velocity_to_voltage_psd.pdf :post pdf2svg(file=*this*, ext="png") :exports results
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\node[block] (G) at (0, 0) {$G(s)$};
|
||||||
|
|
||||||
|
\draw[<-] (G.west) -- node[midway, above]{$x$} ++(-1.4, 0);
|
||||||
|
\draw[->] (G.east) -- node[midway, above]{$y$} ++(1.4, 0);
|
||||||
|
\end{tikzpicture}
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+NAME: fig:velocity_to_voltage_psd
|
||||||
|
#+CAPTION: Schematic of the instrumentation used for the measurement
|
||||||
|
#+RESULTS: fig:velocity_to_voltage_psd
|
||||||
|
[[file:figs/velocity_to_voltage_psd.png]]
|
||||||
|
|
||||||
|
We can show that:
|
||||||
|
\begin{equation}
|
||||||
|
S_{yy}(\omega) = \left|G(j\omega)\right|^2 S_{xx}(\omega)
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
And we also have:
|
||||||
|
\begin{equation}
|
||||||
|
\Gamma_{yy}(\omega) = \left|G(j\omega)\right| \Gamma_{xx}(\omega)
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
* From PSD of the velocity to the PSD of the displacement
|
||||||
|
|
||||||
|
#+begin_src latex :file velocity_to_displacement_psd.pdf :post pdf2svg(file=*this*, ext="png") :exports results
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\node[block] (G) at (0, 0) {$\frac{1}{s}$};
|
||||||
|
|
||||||
|
\draw[<-] (G.west) -- node[midway, above]{$v$} ++(-1, 0);
|
||||||
|
\draw[->] (G.east) -- node[midway, above]{$x$} ++(1, 0);
|
||||||
|
\end{tikzpicture}
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+NAME: fig:velocity_to_displacement_psd
|
||||||
|
#+CAPTION: Schematic of the instrumentation used for the measurement
|
||||||
|
#+RESULTS: fig:velocity_to_displacement_psd
|
||||||
|
[[file:figs/velocity_to_displacement_psd.png]]
|
||||||
|
|
||||||
|
The displacement is the integral of the velocity.
|
||||||
|
|
||||||
|
We then have that
|
||||||
|
\begin{equation}
|
||||||
|
S_{xx}(\omega) = \left|\frac{1}{j \omega}\right|^2 S_{vv}(\omega)
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
Using a frequency variable in Hz:
|
||||||
|
\begin{equation}
|
||||||
|
S_{xx}(f) = \left| \frac{1}{j 2\pi f} \right|^2 S_{vv}(f)
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
For the Amplitude Spectral Density:
|
||||||
|
\begin{equation}
|
||||||
|
\Gamma_{xx}(f) = \frac{1}{2\pi f} \Gamma_{vv}(f)
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
#+begin_note
|
||||||
|
\begin{equation}
|
||||||
|
S_{xx}(\omega = 1) = S_{vv}(\omega = 1)
|
||||||
|
\end{equation}
|
||||||
|
#+end_note
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
Now if we want to obtain the Power Spectral Density of the Position or Acceleration:
|
||||||
|
For each frequency:
|
||||||
|
\[ \left| \frac{d sin(2 \pi f t)}{dt} \right| = | 2 \pi f | \times | \cos(2\pi f t) | \]
|
||||||
|
|
||||||
|
\[ \left| \int_0^t sin(2 \pi f \tau) d\tau \right| = \left| \frac{1}{2 \pi f} \right| \times | \cos(2\pi f t) | \]
|
||||||
|
|
||||||
|
\[ ASD_x(f) = \frac{1}{2\pi f} ASD_v(f) \ \left[\frac{m}{\sqrt{Hz}}\right] \]
|
||||||
|
|
||||||
|
\[ ASD_a(f) = 2\pi f ASD_v(f) \ \left[\frac{m/s^2}{\sqrt{Hz}}\right] \]
|
||||||
|
And we have
|
||||||
|
\[ PSD_x(f) = {ASD_x(f)}^2 = \frac{1}{(2 \pi f)^2} {ASD_v(f)}^2 = \frac{1}{(2 \pi f)^2} PSD_v(f) \]
|
||||||
|
|
||||||
|
Note here that we always have
|
||||||
|
\[ PSD_x \left(f = \frac{1}{2\pi}\right) = PSD_v \left(f = \frac{1}{2\pi}\right) = PSD_a \left(f = \frac{1}{2\pi}\right), \quad \frac{1}{2\pi} \approx 0.16 [Hz] \]
|
||||||
|
|
||||||
|
If we want to compute the Cumulative Power Spectrum:
|
||||||
|
\[ CPS_v(f) = \int_0^f PSD_v(\nu) d\nu \quad [(m/s)^2] \]
|
||||||
|
|
||||||
|
We can also want to integrate from high frequency to low frequency:
|
||||||
|
\[ CPS_v(f) = \int_f^\infty PSD_v(\nu) d\nu \quad [(m/s)^2] \]
|
||||||
|
|
||||||
|
The Cumulative Amplitude Spectrum is then the square root of the Cumulative Power Spectrum:
|
||||||
|
\[ CAS_v(f) = \sqrt{CPS_v(f)} = \sqrt{\int_f^\infty PSD_v(\nu) d\nu} \quad [m/s] \]
|
||||||
|
|
||||||
|
Then, we can obtain the Root Mean Square value of the velocity:
|
||||||
|
\[ v_{\text{rms}} = CAS_v(0) \quad [m/s \ \text{rms}] \]
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
* Bibliography :ignore:
|
||||||
|
bibliographystyle:unsrt
|
||||||
|
bibliography:ref.bib
|
||||||
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|
|||||||
|
!function(a,b){"use strict";function c(c,g){var h=this;h.$el=a(c),h.el=c,h.id=e++,h.$window=a(b),h.$document=a(document),h.$el.bind("destroyed",a.proxy(h.teardown,h)),h.$clonedHeader=null,h.$originalHeader=null,h.isSticky=!1,h.hasBeenSticky=!1,h.leftOffset=null,h.topOffset=null,h.init=function(){h.$el.each(function(){var b=a(this);b.css("padding",0),h.$originalHeader=a("thead:first",this),h.$clonedHeader=h.$originalHeader.clone(),b.trigger("clonedHeader."+d,[h.$clonedHeader]),h.$clonedHeader.addClass("tableFloatingHeader"),h.$clonedHeader.css("display","none"),h.$originalHeader.addClass("tableFloatingHeaderOriginal"),h.$originalHeader.after(h.$clonedHeader),h.$printStyle=a('<style type="text/css" media="print">.tableFloatingHeader{display:none !important;}.tableFloatingHeaderOriginal{position:static !important;}</style>'),a("head").append(h.$printStyle)}),h.setOptions(g),h.updateWidth(),h.toggleHeaders(),h.bind()},h.destroy=function(){h.$el.unbind("destroyed",h.teardown),h.teardown()},h.teardown=function(){h.isSticky&&h.$originalHeader.css("position","static"),a.removeData(h.el,"plugin_"+d),h.unbind(),h.$clonedHeader.remove(),h.$originalHeader.removeClass("tableFloatingHeaderOriginal"),h.$originalHeader.css("visibility","visible"),h.$printStyle.remove(),h.el=null,h.$el=null},h.bind=function(){h.$scrollableArea.on("scroll."+d,h.toggleHeaders),h.isWindowScrolling||(h.$window.on("scroll."+d+h.id,h.setPositionValues),h.$window.on("resize."+d+h.id,h.toggleHeaders)),h.$scrollableArea.on("resize."+d,h.toggleHeaders),h.$scrollableArea.on("resize."+d,h.updateWidth)},h.unbind=function(){h.$scrollableArea.off("."+d,h.toggleHeaders),h.isWindowScrolling||(h.$window.off("."+d+h.id,h.setPositionValues),h.$window.off("."+d+h.id,h.toggleHeaders)),h.$scrollableArea.off("."+d,h.updateWidth)},h.toggleHeaders=function(){h.$el&&h.$el.each(function(){var b,c=a(this),d=h.isWindowScrolling?isNaN(h.options.fixedOffset)?h.options.fixedOffset.outerHeight():h.options.fixedOffset:h.$scrollableArea.offset().top+(isNaN(h.options.fixedOffset)?0:h.options.fixedOffset),e=c.offset(),f=h.$scrollableArea.scrollTop()+d,g=h.$scrollableArea.scrollLeft(),i=h.isWindowScrolling?f>e.top:d>e.top,j=(h.isWindowScrolling?f:0)<e.top+c.height()-h.$clonedHeader.height()-(h.isWindowScrolling?0:d);i&&j?(b=e.left-g+h.options.leftOffset,h.$originalHeader.css({position:"fixed","margin-top":h.options.marginTop,left:b,"z-index":3}),h.leftOffset=b,h.topOffset=d,h.$clonedHeader.css("display",""),h.isSticky||(h.isSticky=!0,h.updateWidth()),h.setPositionValues()):h.isSticky&&(h.$originalHeader.css("position","static"),h.$clonedHeader.css("display","none"),h.isSticky=!1,h.resetWidth(a("td,th",h.$clonedHeader),a("td,th",h.$originalHeader)))})},h.setPositionValues=function(){var a=h.$window.scrollTop(),b=h.$window.scrollLeft();!h.isSticky||0>a||a+h.$window.height()>h.$document.height()||0>b||b+h.$window.width()>h.$document.width()||h.$originalHeader.css({top:h.topOffset-(h.isWindowScrolling?0:a),left:h.leftOffset-(h.isWindowScrolling?0:b)})},h.updateWidth=function(){if(h.isSticky){h.$originalHeaderCells||(h.$originalHeaderCells=a("th,td",h.$originalHeader)),h.$clonedHeaderCells||(h.$clonedHeaderCells=a("th,td",h.$clonedHeader));var b=h.getWidth(h.$clonedHeaderCells);h.setWidth(b,h.$clonedHeaderCells,h.$originalHeaderCells),h.$originalHeader.css("width",h.$clonedHeader.width())}},h.getWidth=function(c){var d=[];return c.each(function(c){var e,f=a(this);if("border-box"===f.css("box-sizing"))e=f[0].getBoundingClientRect().width;else{var g=a("th",h.$originalHeader);if("collapse"===g.css("border-collapse"))if(b.getComputedStyle)e=parseFloat(b.getComputedStyle(this,null).width);else{var i=parseFloat(f.css("padding-left")),j=parseFloat(f.css("padding-right")),k=parseFloat(f.css("border-width"));e=f.outerWidth()-i-j-k}else e=f.width()}d[c]=e}),d},h.setWidth=function(a,b,c){b.each(function(b){var d=a[b];c.eq(b).css({"min-width":d,"max-width":d})})},h.resetWidth=function(b,c){b.each(function(b){var d=a(this);c.eq(b).css({"min-width":d.css("min-width"),"max-width":d.css("max-width")})})},h.setOptions=function(c){h.options=a.extend({},f,c),h.$scrollableArea=a(h.options.scrollableArea),h.isWindowScrolling=h.$scrollableArea[0]===b},h.updateOptions=function(a){h.setOptions(a),h.unbind(),h.bind(),h.updateWidth(),h.toggleHeaders()},h.init()}var d="stickyTableHeaders",e=0,f={fixedOffset:0,leftOffset:0,marginTop:0,scrollableArea:b};a.fn[d]=function(b){return this.each(function(){var e=a.data(this,"plugin_"+d);e?"string"==typeof b?e[b].apply(e):e.updateOptions(b):"destroy"!==b&&a.data(this,"plugin_"+d,new c(this,b))})}}(jQuery,window);
|
||||||
85
js/readtheorg.js
Normal file
85
js/readtheorg.js
Normal file
@ -0,0 +1,85 @@
|
|||||||
|
$(function() {
|
||||||
|
$('.note').before("<p class='admonition-title note'>Note</p>");
|
||||||
|
$('.seealso').before("<p class='admonition-title seealso'>See also</p>");
|
||||||
|
$('.warning').before("<p class='admonition-title warning'>Warning</p>");
|
||||||
|
$('.caution').before("<p class='admonition-title caution'>Caution</p>");
|
||||||
|
$('.attention').before("<p class='admonition-title attention'>Attention</p>");
|
||||||
|
$('.tip').before("<p class='admonition-title tip'>Tip</p>");
|
||||||
|
$('.important').before("<p class='admonition-title important'>Important</p>");
|
||||||
|
$('.hint').before("<p class='admonition-title hint'>Hint</p>");
|
||||||
|
$('.error').before("<p class='admonition-title error'>Error</p>");
|
||||||
|
$('.danger').before("<p class='admonition-title danger'>Danger</p>");
|
||||||
|
});
|
||||||
|
|
||||||
|
$( document ).ready(function() {
|
||||||
|
|
||||||
|
// Shift nav in mobile when clicking the menu.
|
||||||
|
$(document).on('click', "[data-toggle='wy-nav-top']", function() {
|
||||||
|
$("[data-toggle='wy-nav-shift']").toggleClass("shift");
|
||||||
|
$("[data-toggle='rst-versions']").toggleClass("shift");
|
||||||
|
});
|
||||||
|
// Close menu when you click a link.
|
||||||
|
$(document).on('click', ".wy-menu-vertical .current ul li a", function() {
|
||||||
|
$("[data-toggle='wy-nav-shift']").removeClass("shift");
|
||||||
|
$("[data-toggle='rst-versions']").toggleClass("shift");
|
||||||
|
});
|
||||||
|
$(document).on('click', "[data-toggle='rst-current-version']", function() {
|
||||||
|
$("[data-toggle='rst-versions']").toggleClass("shift-up");
|
||||||
|
});
|
||||||
|
// Make tables responsive
|
||||||
|
$("table.docutils:not(.field-list)").wrap("<div class='wy-table-responsive'></div>");
|
||||||
|
});
|
||||||
|
|
||||||
|
$( document ).ready(function() {
|
||||||
|
$('#text-table-of-contents ul').first().addClass('nav');
|
||||||
|
// ScrollSpy also requires that we use
|
||||||
|
// a Bootstrap nav component.
|
||||||
|
$('body').scrollspy({target: '#text-table-of-contents'});
|
||||||
|
|
||||||
|
// add sticky table headers
|
||||||
|
$('table').stickyTableHeaders();
|
||||||
|
|
||||||
|
// set the height of tableOfContents
|
||||||
|
var $postamble = $('#postamble');
|
||||||
|
var $tableOfContents = $('#table-of-contents');
|
||||||
|
$tableOfContents.css({paddingBottom: $postamble.outerHeight()});
|
||||||
|
|
||||||
|
// add TOC button
|
||||||
|
var toggleSidebar = $('<div id="toggle-sidebar"><a href="#table-of-contents"><h2>Table of Contents</h2></a></div>');
|
||||||
|
$('#content').prepend(toggleSidebar);
|
||||||
|
|
||||||
|
// add close button when sidebar showed in mobile screen
|
||||||
|
var closeBtn = $('<a class="close-sidebar" href="#">Close</a>');
|
||||||
|
var tocTitle = $('#table-of-contents').find('h2');
|
||||||
|
tocTitle.append(closeBtn);
|
||||||
|
});
|
||||||
|
|
||||||
|
window.SphinxRtdTheme = (function (jquery) {
|
||||||
|
var stickyNav = (function () {
|
||||||
|
var navBar,
|
||||||
|
win,
|
||||||
|
stickyNavCssClass = 'stickynav',
|
||||||
|
applyStickNav = function () {
|
||||||
|
if (navBar.height() <= win.height()) {
|
||||||
|
navBar.addClass(stickyNavCssClass);
|
||||||
|
} else {
|
||||||
|
navBar.removeClass(stickyNavCssClass);
|
||||||
|
}
|
||||||
|
},
|
||||||
|
enable = function () {
|
||||||
|
applyStickNav();
|
||||||
|
win.on('resize', applyStickNav);
|
||||||
|
},
|
||||||
|
init = function () {
|
||||||
|
navBar = jquery('nav.wy-nav-side:first');
|
||||||
|
win = jquery(window);
|
||||||
|
};
|
||||||
|
jquery(init);
|
||||||
|
return {
|
||||||
|
enable : enable
|
||||||
|
};
|
||||||
|
}());
|
||||||
|
return {
|
||||||
|
StickyNav : stickyNav
|
||||||
|
};
|
||||||
|
}($));
|
||||||
BIN
mat/data_028.mat
Normal file
BIN
mat/data_028.mat
Normal file
Binary file not shown.
Loading…
Reference in New Issue
Block a user