\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\newcommand {\mathllap }[2][]{{#1#2}}\)
\(\newcommand {\mathrlap }[2][]{{#1#2}}\)
\(\newcommand {\mathclap }[2][]{{#1#2}}\)
\(\newcommand {\mathmbox }[1]{#1}\)
\(\newcommand {\clap }[1]{#1}\)
\(\newcommand {\LWRmathmakebox }[2][]{#2}\)
\(\newcommand {\mathmakebox }[1][]{\LWRmathmakebox }\)
\(\newcommand {\cramped }[2][]{{#1#2}}\)
\(\newcommand {\crampedllap }[2][]{{#1#2}}\)
\(\newcommand {\crampedrlap }[2][]{{#1#2}}\)
\(\newcommand {\crampedclap }[2][]{{#1#2}}\)
\(\newenvironment {crampedsubarray}[1]{}{}\)
\(\newcommand {\crampedsubstack }{}\)
\(\newcommand {\smashoperator }[2][]{#2\limits }\)
\(\newcommand {\adjustlimits }{}\)
\(\newcommand {\SwapAboveDisplaySkip }{}\)
\(\require {extpfeil}\)
\(\Newextarrow \xleftrightarrow {10,10}{0x2194}\)
\(\Newextarrow \xLeftarrow {10,10}{0x21d0}\)
\(\Newextarrow \xhookleftarrow {10,10}{0x21a9}\)
\(\Newextarrow \xmapsto {10,10}{0x21a6}\)
\(\Newextarrow \xRightarrow {10,10}{0x21d2}\)
\(\Newextarrow \xLeftrightarrow {10,10}{0x21d4}\)
\(\Newextarrow \xhookrightarrow {10,10}{0x21aa}\)
\(\Newextarrow \xrightharpoondown {10,10}{0x21c1}\)
\(\Newextarrow \xleftharpoondown {10,10}{0x21bd}\)
\(\Newextarrow \xrightleftharpoons {10,10}{0x21cc}\)
\(\Newextarrow \xrightharpoonup {10,10}{0x21c0}\)
\(\Newextarrow \xleftharpoonup {10,10}{0x21bc}\)
\(\Newextarrow \xleftrightharpoons {10,10}{0x21cb}\)
\(\newcommand {\LWRdounderbracket }[3]{\underset {#3}{\underline {#1}}}\)
\(\newcommand {\LWRunderbracket }[2][]{\LWRdounderbracket {#2}}\)
\(\newcommand {\underbracket }[1][]{\LWRunderbracket }\)
\(\newcommand {\LWRdooverbracket }[3]{\overset {#3}{\overline {#1}}}\)
\(\newcommand {\LWRoverbracket }[2][]{\LWRdooverbracket {#2}}\)
\(\newcommand {\overbracket }[1][]{\LWRoverbracket }\)
\(\newcommand {\LATEXunderbrace }[1]{\underbrace {#1}}\)
\(\newcommand {\LATEXoverbrace }[1]{\overbrace {#1}}\)
\(\newenvironment {matrix*}[1][]{\begin {matrix}}{\end {matrix}}\)
\(\newenvironment {pmatrix*}[1][]{\begin {pmatrix}}{\end {pmatrix}}\)
\(\newenvironment {bmatrix*}[1][]{\begin {bmatrix}}{\end {bmatrix}}\)
\(\newenvironment {Bmatrix*}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\)
\(\newenvironment {vmatrix*}[1][]{\begin {vmatrix}}{\end {vmatrix}}\)
\(\newenvironment {Vmatrix*}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\)
\(\newenvironment {smallmatrix*}[1][]{\begin {matrix}}{\end {matrix}}\)
\(\newenvironment {psmallmatrix*}[1][]{\begin {pmatrix}}{\end {pmatrix}}\)
\(\newenvironment {bsmallmatrix*}[1][]{\begin {bmatrix}}{\end {bmatrix}}\)
\(\newenvironment {Bsmallmatrix*}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\)
\(\newenvironment {vsmallmatrix*}[1][]{\begin {vmatrix}}{\end {vmatrix}}\)
\(\newenvironment {Vsmallmatrix*}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\)
\(\newenvironment {psmallmatrix}[1][]{\begin {pmatrix}}{\end {pmatrix}}\)
\(\newenvironment {bsmallmatrix}[1][]{\begin {bmatrix}}{\end {bmatrix}}\)
\(\newenvironment {Bsmallmatrix}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\)
\(\newenvironment {vsmallmatrix}[1][]{\begin {vmatrix}}{\end {vmatrix}}\)
\(\newenvironment {Vsmallmatrix}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\)
\(\newcommand {\LWRmultlined }[1][]{\begin {multline*}}\)
\(\newenvironment {multlined}[1][]{\LWRmultlined }{\end {multline*}}\)
\(\let \LWRorigshoveleft \shoveleft \)
\(\renewcommand {\shoveleft }[1][]{\LWRorigshoveleft }\)
\(\let \LWRorigshoveright \shoveright \)
\(\renewcommand {\shoveright }[1][]{\LWRorigshoveright }\)
\(\newenvironment {dcases}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {dcases*}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {rcases}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {rcases*}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {drcases}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {drcases*}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {cases*}{\begin {cases}}{\end {cases}}\)
\(\newcommand {\MoveEqLeft }[1][]{}\)
\(\def \LWRAboxed #1!|!{\fbox {\(#1\)}&\fbox {\(#2\)}} \newcommand {\Aboxed }[1]{\LWRAboxed #1&&!|!} \)
\( \newcommand {\LWRABLines }[1][\Updownarrow ]{#1 \notag \\}\newcommand {\ArrowBetweenLines }{\ifstar \LWRABLines \LWRABLines } \)
\(\newcommand {\shortintertext }[1]{\text {#1}\notag \\}\)
\(\newcommand {\vdotswithin }[1]{\hspace {.5em}\vdots }\)
\(\newcommand {\LWRshortvdotswithinstar }[1]{\vdots \hspace {.5em} & \\}\)
\(\newcommand {\LWRshortvdotswithinnostar }[1]{& \hspace {.5em}\vdots \\}\)
\(\newcommand {\shortvdotswithin }{\ifstar \LWRshortvdotswithinstar \LWRshortvdotswithinnostar }\)
\(\newcommand {\MTFlushSpaceAbove }{}\)
\(\newcommand {\MTFlushSpaceBelow }{\\}\)
\(\newcommand \lparen {(}\)
\(\newcommand \rparen {)}\)
\(\newcommand {\ordinarycolon }{:}\)
\(\newcommand {\vcentcolon }{\mathrel {\mathop \ordinarycolon }}\)
\(\newcommand \dblcolon {\vcentcolon \vcentcolon }\)
\(\newcommand \coloneqq {\vcentcolon =}\)
\(\newcommand \Coloneqq {\dblcolon =}\)
\(\newcommand \coloneq {\vcentcolon {-}}\)
\(\newcommand \Coloneq {\dblcolon {-}}\)
\(\newcommand \eqqcolon {=\vcentcolon }\)
\(\newcommand \Eqqcolon {=\dblcolon }\)
\(\newcommand \eqcolon {\mathrel {-}\vcentcolon }\)
\(\newcommand \Eqcolon {\mathrel {-}\dblcolon }\)
\(\newcommand \colonapprox {\vcentcolon \approx }\)
\(\newcommand \Colonapprox {\dblcolon \approx }\)
\(\newcommand \colonsim {\vcentcolon \sim }\)
\(\newcommand \Colonsim {\dblcolon \sim }\)
\(\newcommand {\nuparrow }{\mathrel {\cancel {\uparrow }}}\)
\(\newcommand {\ndownarrow }{\mathrel {\cancel {\downarrow }}}\)
\(\newcommand {\bigtimes }{\mathop {\Large \times }\limits }\)
\(\newcommand {\prescript }[3]{{}^{#1}_{#2}#3}\)
\(\newenvironment {lgathered}{\begin {gathered}}{\end {gathered}}\)
\(\newenvironment {rgathered}{\begin {gathered}}{\end {gathered}}\)
\(\newcommand {\splitfrac }[2]{{}^{#1}_{#2}}\)
\(\let \splitdfrac \splitfrac \)
\(\newcommand {\LWRoverlaysymbols }[2]{\mathord {\smash {\mathop {#2\strut }\limits ^{\smash {\lower 3ex{#1}}}}\strut }}\)
\(\newcommand{\alphaup}{\unicode{x03B1}}\)
\(\newcommand{\betaup}{\unicode{x03B2}}\)
\(\newcommand{\gammaup}{\unicode{x03B3}}\)
\(\newcommand{\digammaup}{\unicode{x03DD}}\)
\(\newcommand{\deltaup}{\unicode{x03B4}}\)
\(\newcommand{\epsilonup}{\unicode{x03F5}}\)
\(\newcommand{\varepsilonup}{\unicode{x03B5}}\)
\(\newcommand{\zetaup}{\unicode{x03B6}}\)
\(\newcommand{\etaup}{\unicode{x03B7}}\)
\(\newcommand{\thetaup}{\unicode{x03B8}}\)
\(\newcommand{\varthetaup}{\unicode{x03D1}}\)
\(\newcommand{\iotaup}{\unicode{x03B9}}\)
\(\newcommand{\kappaup}{\unicode{x03BA}}\)
\(\newcommand{\varkappaup}{\unicode{x03F0}}\)
\(\newcommand{\lambdaup}{\unicode{x03BB}}\)
\(\newcommand{\muup}{\unicode{x03BC}}\)
\(\newcommand{\nuup}{\unicode{x03BD}}\)
\(\newcommand{\xiup}{\unicode{x03BE}}\)
\(\newcommand{\omicronup}{\unicode{x03BF}}\)
\(\newcommand{\piup}{\unicode{x03C0}}\)
\(\newcommand{\varpiup}{\unicode{x03D6}}\)
\(\newcommand{\rhoup}{\unicode{x03C1}}\)
\(\newcommand{\varrhoup}{\unicode{x03F1}}\)
\(\newcommand{\sigmaup}{\unicode{x03C3}}\)
\(\newcommand{\varsigmaup}{\unicode{x03C2}}\)
\(\newcommand{\tauup}{\unicode{x03C4}}\)
\(\newcommand{\upsilonup}{\unicode{x03C5}}\)
\(\newcommand{\phiup}{\unicode{x03D5}}\)
\(\newcommand{\varphiup}{\unicode{x03C6}}\)
\(\newcommand{\chiup}{\unicode{x03C7}}\)
\(\newcommand{\psiup}{\unicode{x03C8}}\)
\(\newcommand{\omegaup}{\unicode{x03C9}}\)
\(\newcommand{\Alphaup}{\unicode{x0391}}\)
\(\newcommand{\Betaup}{\unicode{x0392}}\)
\(\newcommand{\Gammaup}{\unicode{x0393}}\)
\(\newcommand{\Digammaup}{\unicode{x03DC}}\)
\(\newcommand{\Deltaup}{\unicode{x0394}}\)
\(\newcommand{\Epsilonup}{\unicode{x0395}}\)
\(\newcommand{\Zetaup}{\unicode{x0396}}\)
\(\newcommand{\Etaup}{\unicode{x0397}}\)
\(\newcommand{\Thetaup}{\unicode{x0398}}\)
\(\newcommand{\Varthetaup}{\unicode{x03F4}}\)
\(\newcommand{\Iotaup}{\unicode{x0399}}\)
\(\newcommand{\Kappaup}{\unicode{x039A}}\)
\(\newcommand{\Lambdaup}{\unicode{x039B}}\)
\(\newcommand{\Muup}{\unicode{x039C}}\)
\(\newcommand{\Nuup}{\unicode{x039D}}\)
\(\newcommand{\Xiup}{\unicode{x039E}}\)
\(\newcommand{\Omicronup}{\unicode{x039F}}\)
\(\newcommand{\Piup}{\unicode{x03A0}}\)
\(\newcommand{\Varpiup}{\unicode{x03D6}}\)
\(\newcommand{\Rhoup}{\unicode{x03A1}}\)
\(\newcommand{\Sigmaup}{\unicode{x03A3}}\)
\(\newcommand{\Tauup}{\unicode{x03A4}}\)
\(\newcommand{\Upsilonup}{\unicode{x03A5}}\)
\(\newcommand{\Phiup}{\unicode{x03A6}}\)
\(\newcommand{\Chiup}{\unicode{x03A7}}\)
\(\newcommand{\Psiup}{\unicode{x03A8}}\)
\(\newcommand{\Omegaup}{\unicode{x03A9}}\)
\(\newcommand{\alphait}{\unicode{x1D6FC}}\)
\(\newcommand{\betait}{\unicode{x1D6FD}}\)
\(\newcommand{\gammait}{\unicode{x1D6FE}}\)
\(\newcommand{\digammait}{\mathit{\unicode{x03DD}}}\)
\(\newcommand{\deltait}{\unicode{x1D6FF}}\)
\(\newcommand{\epsilonit}{\unicode{x1D716}}\)
\(\newcommand{\varepsilonit}{\unicode{x1D700}}\)
\(\newcommand{\zetait}{\unicode{x1D701}}\)
\(\newcommand{\etait}{\unicode{x1D702}}\)
\(\newcommand{\thetait}{\unicode{x1D703}}\)
\(\newcommand{\varthetait}{\unicode{x1D717}}\)
\(\newcommand{\iotait}{\unicode{x1D704}}\)
\(\newcommand{\kappait}{\unicode{x1D705}}\)
\(\newcommand{\varkappait}{\unicode{x1D718}}\)
\(\newcommand{\lambdait}{\unicode{x1D706}}\)
\(\newcommand{\muit}{\unicode{x1D707}}\)
\(\newcommand{\nuit}{\unicode{x1D708}}\)
\(\newcommand{\xiit}{\unicode{x1D709}}\)
\(\newcommand{\omicronit}{\unicode{x1D70A}}\)
\(\newcommand{\piit}{\unicode{x1D70B}}\)
\(\newcommand{\varpiit}{\unicode{x1D71B}}\)
\(\newcommand{\rhoit}{\unicode{x1D70C}}\)
\(\newcommand{\varrhoit}{\unicode{x1D71A}}\)
\(\newcommand{\sigmait}{\unicode{x1D70E}}\)
\(\newcommand{\varsigmait}{\unicode{x1D70D}}\)
\(\newcommand{\tauit}{\unicode{x1D70F}}\)
\(\newcommand{\upsilonit}{\unicode{x1D710}}\)
\(\newcommand{\phiit}{\unicode{x1D719}}\)
\(\newcommand{\varphiit}{\unicode{x1D711}}\)
\(\newcommand{\chiit}{\unicode{x1D712}}\)
\(\newcommand{\psiit}{\unicode{x1D713}}\)
\(\newcommand{\omegait}{\unicode{x1D714}}\)
\(\newcommand{\Alphait}{\unicode{x1D6E2}}\)
\(\newcommand{\Betait}{\unicode{x1D6E3}}\)
\(\newcommand{\Gammait}{\unicode{x1D6E4}}\)
\(\newcommand{\Digammait}{\mathit{\unicode{x03DC}}}\)
\(\newcommand{\Deltait}{\unicode{x1D6E5}}\)
\(\newcommand{\Epsilonit}{\unicode{x1D6E6}}\)
\(\newcommand{\Zetait}{\unicode{x1D6E7}}\)
\(\newcommand{\Etait}{\unicode{x1D6E8}}\)
\(\newcommand{\Thetait}{\unicode{x1D6E9}}\)
\(\newcommand{\Varthetait}{\unicode{x1D6F3}}\)
\(\newcommand{\Iotait}{\unicode{x1D6EA}}\)
\(\newcommand{\Kappait}{\unicode{x1D6EB}}\)
\(\newcommand{\Lambdait}{\unicode{x1D6EC}}\)
\(\newcommand{\Muit}{\unicode{x1D6ED}}\)
\(\newcommand{\Nuit}{\unicode{x1D6EE}}\)
\(\newcommand{\Xiit}{\unicode{x1D6EF}}\)
\(\newcommand{\Omicronit}{\unicode{x1D6F0}}\)
\(\newcommand{\Piit}{\unicode{x1D6F1}}\)
\(\newcommand{\Rhoit}{\unicode{x1D6F2}}\)
\(\newcommand{\Sigmait}{\unicode{x1D6F4}}\)
\(\newcommand{\Tauit}{\unicode{x1D6F5}}\)
\(\newcommand{\Upsilonit}{\unicode{x1D6F6}}\)
\(\newcommand{\Phiit}{\unicode{x1D6F7}}\)
\(\newcommand{\Chiit}{\unicode{x1D6F8}}\)
\(\newcommand{\Psiit}{\unicode{x1D6F9}}\)
\(\newcommand{\Omegait}{\unicode{x1D6FA}}\)
\(\let \digammaup \Digammaup \)
\(\renewcommand {\digammait }{\mathit {\digammaup }}\)
\(\newcommand {\smallin }{\unicode {x220A}}\)
\(\newcommand {\smallowns }{\unicode {x220D}}\)
\(\newcommand {\notsmallin }{\LWRoverlaysymbols {/}{\unicode {x220A}}}\)
\(\newcommand {\notsmallowns }{\LWRoverlaysymbols {/}{\unicode {x220D}}}\)
\(\newcommand {\rightangle }{\unicode {x221F}}\)
\(\newcommand {\intclockwise }{\unicode {x2231}}\)
\(\newcommand {\ointclockwise }{\unicode {x2232}}\)
\(\newcommand {\ointctrclockwise }{\unicode {x2233}}\)
\(\newcommand {\oiint }{\unicode {x222F}}\)
\(\newcommand {\oiiint }{\unicode {x2230}}\)
\(\newcommand {\ddag }{\unicode {x2021}}\)
\(\newcommand {\P }{\unicode {x00B6}}\)
\(\newcommand {\copyright }{\unicode {x00A9}}\)
\(\newcommand {\dag }{\unicode {x2020}}\)
\(\newcommand {\pounds }{\unicode {x00A3}}\)
\(\newcommand {\iddots }{\unicode {x22F0}}\)
\(\newcommand {\utimes }{\overline {\times }}\)
\(\newcommand {\dtimes }{\underline {\times }}\)
\(\newcommand {\udtimes }{\overline {\underline {\times }}}\)
\(\newcommand {\leftwave }{\left \{}\)
\(\newcommand {\rightwave }{\right \}}\)
\(\newcommand {\toprule }[1][]{\hline }\)
\(\let \midrule \toprule \)
\(\let \bottomrule \toprule \)
\(\newcommand {\cmidrule }[2][]{}\)
\(\newcommand {\morecmidrules }{}\)
\(\newcommand {\specialrule }[3]{\hline }\)
\(\newcommand {\addlinespace }[1][]{}\)
\(\newcommand {\LWRsubmultirow }[2][]{#2}\)
\(\newcommand {\LWRmultirow }[2][]{\LWRsubmultirow }\)
\(\newcommand {\multirow }[2][]{\LWRmultirow }\)
\(\newcommand {\mrowcell }{}\)
\(\newcommand {\mcolrowcell }{}\)
\(\newcommand {\STneed }[1]{}\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\newcommand {\tothe }[1]{^{#1}}\)
\(\newcommand {\raiseto }[2]{{#2}^{#1}}\)
\(\newcommand {\ang }[2][]{(\mathrm {#2})\degree }\)
\(\newcommand {\num }[2][]{\mathrm {#2}}\)
\(\newcommand {\si }[2][]{\mathrm {#2}}\)
\(\newcommand {\LWRSI }[2][]{\mathrm {#1\LWRSInumber \,#2}}\)
\(\newcommand {\SI }[2][]{\def \LWRSInumber {#2}\LWRSI }\)
\(\newcommand {\numlist }[2][]{\mathrm {#2}}\)
\(\newcommand {\numrange }[3][]{\mathrm {#2\,\unicode {x2013}\,#3}}\)
\(\newcommand {\SIlist }[3][]{\mathrm {#2\,#3}}\)
\(\newcommand {\SIrange }[4][]{\mathrm {#2\,#4\,\unicode {x2013}\,#3\,#4}}\)
\(\newcommand {\tablenum }[2][]{\mathrm {#2}}\)
\(\newcommand {\ampere }{\mathrm {A}}\)
\(\newcommand {\candela }{\mathrm {cd}}\)
\(\newcommand {\kelvin }{\mathrm {K}}\)
\(\newcommand {\kilogram }{\mathrm {kg}}\)
\(\newcommand {\metre }{\mathrm {m}}\)
\(\newcommand {\mole }{\mathrm {mol}}\)
\(\newcommand {\second }{\mathrm {s}}\)
\(\newcommand {\becquerel }{\mathrm {Bq}}\)
\(\newcommand {\degreeCelsius }{\unicode {x2103}}\)
\(\newcommand {\coulomb }{\mathrm {C}}\)
\(\newcommand {\farad }{\mathrm {F}}\)
\(\newcommand {\gray }{\mathrm {Gy}}\)
\(\newcommand {\hertz }{\mathrm {Hz}}\)
\(\newcommand {\henry }{\mathrm {H}}\)
\(\newcommand {\joule }{\mathrm {J}}\)
\(\newcommand {\katal }{\mathrm {kat}}\)
\(\newcommand {\lumen }{\mathrm {lm}}\)
\(\newcommand {\lux }{\mathrm {lx}}\)
\(\newcommand {\newton }{\mathrm {N}}\)
\(\newcommand {\ohm }{\mathrm {\Omega }}\)
\(\newcommand {\pascal }{\mathrm {Pa}}\)
\(\newcommand {\radian }{\mathrm {rad}}\)
\(\newcommand {\siemens }{\mathrm {S}}\)
\(\newcommand {\sievert }{\mathrm {Sv}}\)
\(\newcommand {\steradian }{\mathrm {sr}}\)
\(\newcommand {\tesla }{\mathrm {T}}\)
\(\newcommand {\volt }{\mathrm {V}}\)
\(\newcommand {\watt }{\mathrm {W}}\)
\(\newcommand {\weber }{\mathrm {Wb}}\)
\(\newcommand {\day }{\mathrm {d}}\)
\(\newcommand {\degree }{\mathrm {^\circ }}\)
\(\newcommand {\hectare }{\mathrm {ha}}\)
\(\newcommand {\hour }{\mathrm {h}}\)
\(\newcommand {\litre }{\mathrm {l}}\)
\(\newcommand {\liter }{\mathrm {L}}\)
\(\newcommand {\arcminute }{^\prime }\)
\(\newcommand {\minute }{\mathrm {min}}\)
\(\newcommand {\arcsecond }{^{\prime \prime }}\)
\(\newcommand {\tonne }{\mathrm {t}}\)
\(\newcommand {\astronomicalunit }{au}\)
\(\newcommand {\atomicmassunit }{u}\)
\(\newcommand {\bohr }{\mathit {a}_0}\)
\(\newcommand {\clight }{\mathit {c}_0}\)
\(\newcommand {\dalton }{\mathrm {D}_\mathrm {a}}\)
\(\newcommand {\electronmass }{\mathit {m}_{\mathrm {e}}}\)
\(\newcommand {\electronvolt }{\mathrm {eV}}\)
\(\newcommand {\elementarycharge }{\mathit {e}}\)
\(\newcommand {\hartree }{\mathit {E}_{\mathrm {h}}}\)
\(\newcommand {\planckbar }{\mathit {\unicode {x210F}}}\)
\(\newcommand {\angstrom }{\mathrm {\unicode {x212B}}}\)
\(\let \LWRorigbar \bar \)
\(\newcommand {\bar }{\mathrm {bar}}\)
\(\newcommand {\barn }{\mathrm {b}}\)
\(\newcommand {\bel }{\mathrm {B}}\)
\(\newcommand {\decibel }{\mathrm {dB}}\)
\(\newcommand {\knot }{\mathrm {kn}}\)
\(\newcommand {\mmHg }{\mathrm {mmHg}}\)
\(\newcommand {\nauticalmile }{\mathrm {M}}\)
\(\newcommand {\neper }{\mathrm {Np}}\)
\(\newcommand {\yocto }{\mathrm {y}}\)
\(\newcommand {\zepto }{\mathrm {z}}\)
\(\newcommand {\atto }{\mathrm {a}}\)
\(\newcommand {\femto }{\mathrm {f}}\)
\(\newcommand {\pico }{\mathrm {p}}\)
\(\newcommand {\nano }{\mathrm {n}}\)
\(\newcommand {\micro }{\mathrm {\unicode {x00B5}}}\)
\(\newcommand {\milli }{\mathrm {m}}\)
\(\newcommand {\centi }{\mathrm {c}}\)
\(\newcommand {\deci }{\mathrm {d}}\)
\(\newcommand {\deca }{\mathrm {da}}\)
\(\newcommand {\hecto }{\mathrm {h}}\)
\(\newcommand {\kilo }{\mathrm {k}}\)
\(\newcommand {\mega }{\mathrm {M}}\)
\(\newcommand {\giga }{\mathrm {G}}\)
\(\newcommand {\tera }{\mathrm {T}}\)
\(\newcommand {\peta }{\mathrm {P}}\)
\(\newcommand {\exa }{\mathrm {E}}\)
\(\newcommand {\zetta }{\mathrm {Z}}\)
\(\newcommand {\yotta }{\mathrm {Y}}\)
\(\newcommand {\percent }{\mathrm {\%}}\)
\(\newcommand {\meter }{\mathrm {m}}\)
\(\newcommand {\metre }{\mathrm {m}}\)
\(\newcommand {\gram }{\mathrm {g}}\)
\(\newcommand {\kg }{\kilo \gram }\)
\(\newcommand {\of }[1]{_{\mathrm {#1}}}\)
\(\newcommand {\squared }{^2}\)
\(\newcommand {\square }[1]{\mathrm {#1}^2}\)
\(\newcommand {\cubed }{^3}\)
\(\newcommand {\cubic }[1]{\mathrm {#1}^3}\)
\(\newcommand {\per }{/}\)
\(\newcommand {\celsius }{\unicode {x2103}}\)
\(\newcommand {\fg }{\femto \gram }\)
\(\newcommand {\pg }{\pico \gram }\)
\(\newcommand {\ng }{\nano \gram }\)
\(\newcommand {\ug }{\micro \gram }\)
\(\newcommand {\mg }{\milli \gram }\)
\(\newcommand {\g }{\gram }\)
\(\newcommand {\kg }{\kilo \gram }\)
\(\newcommand {\amu }{\mathrm {u}}\)
\(\newcommand {\nm }{\nano \metre }\)
\(\newcommand {\um }{\micro \metre }\)
\(\newcommand {\mm }{\milli \metre }\)
\(\newcommand {\cm }{\centi \metre }\)
\(\newcommand {\dm }{\deci \metre }\)
\(\newcommand {\m }{\metre }\)
\(\newcommand {\km }{\kilo \metre }\)
\(\newcommand {\as }{\atto \second }\)
\(\newcommand {\fs }{\femto \second }\)
\(\newcommand {\ps }{\pico \second }\)
\(\newcommand {\ns }{\nano \second }\)
\(\newcommand {\us }{\micro \second }\)
\(\newcommand {\ms }{\milli \second }\)
\(\newcommand {\s }{\second }\)
\(\newcommand {\fmol }{\femto \mol }\)
\(\newcommand {\pmol }{\pico \mol }\)
\(\newcommand {\nmol }{\nano \mol }\)
\(\newcommand {\umol }{\micro \mol }\)
\(\newcommand {\mmol }{\milli \mol }\)
\(\newcommand {\mol }{\mol }\)
\(\newcommand {\kmol }{\kilo \mol }\)
\(\newcommand {\pA }{\pico \ampere }\)
\(\newcommand {\nA }{\nano \ampere }\)
\(\newcommand {\uA }{\micro \ampere }\)
\(\newcommand {\mA }{\milli \ampere }\)
\(\newcommand {\A }{\ampere }\)
\(\newcommand {\kA }{\kilo \ampere }\)
\(\newcommand {\ul }{\micro \litre }\)
\(\newcommand {\ml }{\milli \litre }\)
\(\newcommand {\l }{\litre }\)
\(\newcommand {\hl }{\hecto \litre }\)
\(\newcommand {\uL }{\micro \liter }\)
\(\newcommand {\mL }{\milli \liter }\)
\(\newcommand {\L }{\liter }\)
\(\newcommand {\hL }{\hecto \liter }\)
\(\newcommand {\mHz }{\milli \hertz }\)
\(\newcommand {\Hz }{\hertz }\)
\(\newcommand {\kHz }{\kilo \hertz }\)
\(\newcommand {\MHz }{\mega \hertz }\)
\(\newcommand {\GHz }{\giga \hertz }\)
\(\newcommand {\THz }{\tera \hertz }\)
\(\newcommand {\mN }{\milli \newton }\)
\(\newcommand {\N }{\newton }\)
\(\newcommand {\kN }{\kilo \newton }\)
\(\newcommand {\MN }{\mega \newton }\)
\(\newcommand {\Pa }{\pascal }\)
\(\newcommand {\kPa }{\kilo \pascal }\)
\(\newcommand {\MPa }{\mega \pascal }\)
\(\newcommand {\GPa }{\giga \pascal }\)
\(\newcommand {\mohm }{\milli \ohm }\)
\(\newcommand {\kohm }{\kilo \ohm }\)
\(\newcommand {\Mohm }{\mega \ohm }\)
\(\newcommand {\pV }{\pico \volt }\)
\(\newcommand {\nV }{\nano \volt }\)
\(\newcommand {\uV }{\micro \volt }\)
\(\newcommand {\mV }{\milli \volt }\)
\(\newcommand {\V }{\volt }\)
\(\newcommand {\kV }{\kilo \volt }\)
\(\newcommand {\W }{\watt }\)
\(\newcommand {\uW }{\micro \watt }\)
\(\newcommand {\mW }{\milli \watt }\)
\(\newcommand {\kW }{\kilo \watt }\)
\(\newcommand {\MW }{\mega \watt }\)
\(\newcommand {\GW }{\giga \watt }\)
\(\newcommand {\J }{\joule }\)
\(\newcommand {\uJ }{\micro \joule }\)
\(\newcommand {\mJ }{\milli \joule }\)
\(\newcommand {\kJ }{\kilo \joule }\)
\(\newcommand {\eV }{\electronvolt }\)
\(\newcommand {\meV }{\milli \electronvolt }\)
\(\newcommand {\keV }{\kilo \electronvolt }\)
\(\newcommand {\MeV }{\mega \electronvolt }\)
\(\newcommand {\GeV }{\giga \electronvolt }\)
\(\newcommand {\TeV }{\tera \electronvolt }\)
\(\newcommand {\kWh }{\kilo \watt \hour }\)
\(\newcommand {\F }{\farad }\)
\(\newcommand {\fF }{\femto \farad }\)
\(\newcommand {\pF }{\pico \farad }\)
\(\newcommand {\K }{\mathrm {K}}\)
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Trigonometrische Approximation und Fourier-Reihen
trigonometrische Approximation: Gegeben sei eine \(2\pi \)-periodische, stückweise stetige Funktion \(f\colon \real \to \real \), d. h. \(\forall _{x \in \real }\; f(x + 2\pi ) = f(x)\),
sodass für alle \(x_0 \in \real \) die Grenzwerte
\(\lim _{h \to 0+0} f(x_0 - h) = y_0^-\) und \(\lim _{h \to 0+0} f(x_0 + h) = y_0^+\) existieren.
Gesucht ist \(g_n(x) := \frac {1}{2} a_0 + \sum _{k=1}^n (a_k\cos (kx) + b_k\sin (kx))\), sodass
\(F := \norm {g_n - f}_{L^2}^2 = \int _{-\pi }^\pi (g_n - f)^2 \dx \to \min \).
Durch \(\partial _{a_0} F = \partial _{a_j} F = \partial _{b_j} F \overset {!}{=} 0\) für \(j = 1, \dotsc , n\) erhält man folgende Formeln.
reelle Fourier-Reihe:
Seien \(a_k := \frac {1}{\pi } \int _{-\pi }^\pi f(x)\cos (kx) \dx \) für \(k \in \natural _0\) und \(b_k := \frac {1}{\pi } \int _{-\pi }^\pi f(x)\sin (kx) \dx \) für \(k \in
\natural \).
Das reelle Fourier-Polynom ist gegeben durch \(g_n(x) := \frac {1}{2} a_0 + \sum _{k=1}^n (a_k\cos (kx) + b_k\sin (kx))\).
Die reelle Fourier-Reihe ist gegeben durch \(g(x) := \frac {1}{2} a_0 + \sum _{k=1}^\infty (a_k\cos (kx) + b_k\sin (kx))\).
komplexe Fourier-Reihe: Durch Einsetzen von \(\cos (kx) = \frac {e^{\iu kx} + e^{-\iu kx}}{2}\) und \(\sin (kx) = \frac {e^{\iu kx} - e^{-\iu kx}}{2\iu }\) in die
reelle Fourier-Reihe erhält man die komplexe Fourier-Reihe \(g(x) = \sum _{k\in \integer } c_k e^{\iu kx}\) mit \(c_k :=
\frac {2}{\pi } \int _{-\pi }^\pi f(x) e^{-\iu kx} \dx \) bzw. \(c_0 := \frac {a_0}{2}\) für \(k = 0\), \(c_k := \frac {a_k - \iu b_k}{2}\) und \(c_{-k} := \frac {a_k + \iu b_k}{2}\)
für \(k \in \natural \).
Satz (Konvergenz der Fourier-Reihe): Gegeben sei eine \(2\pi \)-periodische, stückweise stetige Funktion \(f\colon \real \to \real \) mit stückweise stetiger
Ableitung.
Dann konvergiert die Fourier-Reihe \(g(x) := \frac {1}{2} a_0 + \sum _{k=1}^\infty (a_k\cos (kx) + b_k\sin (kx))\) in \(x_0 \in \real \)
gegen \(f(x_0)\), wenn \(f\) stetig in \(x_0\) ist, und
gegen \(\frac {y_0^- + y_0^+}{2}\), wenn \(x_0\) eine Sprungstelle von \(f\) ist.
Gibbs-Phänomen: Anhand der Rechteck-Funktion \(f := \chi _{[-\pi /2,\pi /2]}\) erkennt man, dass die Fourier-Reihe zwar punktweise f. ü.
konvergiert, die \(L^\infty \)-Norm der Differenz aber nicht konvergiert, weil \(g_n(x_0) \to 1.08949\) für \(n \to \infty \) und \(x_0 = x_0(n)\) der Maximalstelle von \(g_n\).
Fourier-Transformation: Sei \(f \in L^1(\real ) := L^1(\real , \complex )\).
Dann ist \((\F (f))(k) = F(k) := \int _\real f(x)e^{-\iu kx} \dx \) die Fourier-Transformation.
inverse Fourier-Transformation: Sei \(f \in L^1(\real )\) mit \(F = \F (f) \in L^1(\real )\).
Dann ist \(f(x) = \frac {1}{2\pi } \int _\real F(k) e^{\iu kx}\d k\) die inverse Fourier-Transformation.
Eigenschaften der Fourier-Transformation:
Eigenschaft |
Funktion |
Fourier-Transformierte |
Fourier-Transformation |
\(f(x)\) |
\(F(k)\) |
inverse Fourier-Transformation |
\(F(x)\) |
\(2\pi f(-k)\) |
Faltung |
\((f_1 \ast f_2)(x)\) |
\(F_1(k) F_2(k)\) |
Multiplikation |
\(f_1(x) f_2(x)\) |
\(\frac {1}{2\pi } (F_1 \ast F_2)(k)\) |
Translation |
\(f(x - a)\) |
\(e^{-\iu ak} F(k)\) |
Modulation |
\(e^{\iu ax} f(x)\) |
\(F(k - a)\) |
Skalierung |
\(f(x/a)\) |
\(|a| F(ak)\) |
Ableitung |
\(f^{(p)}(x)\) |
\((\iu k)^p F(k)\) |
Frequenzableitung |
\((-\iu x)^p f(x)\) |
\(F^{(p)}(k)\) |
komplexes Konjugat |
\(\overline {f(x)}\) |
\(\overline {F(-k)}\) |
hermitesche Symmetrie |
\(f(x) \in \real \) |
\(F(k) = \overline {F(-k)}\) |
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Diracsche Delta-Distribution
Dirac-Delta: Die diracsche Delta-Distribution \(\delta \) ist eine Distribution, die eine stetige Funktion
\(\phi \) im Punkt \(t = 0\) auswertet, d. h. \(\int _\real \phi (t) \delta (t) \dt := \phi (0)\) (keine Funktion, da \(\int _\real \delta (t) \dt = 1\)). Für die Auswertung in \(u \in \real
\) schreibt man \((\phi \ast \delta )(u) = \int _\real \phi (t) \delta (u - t) \dt := \phi (u)\). Die Ableitung ist gegeben durch \(\int _\real \phi (t) \delta ^{(n)}(t) \dt := (-1)^n \phi ^{(n)}(0)\).
Delta-Folgen: Delta-Folgen approximieren \(\delta \), z. B. die Glockenkurve \(\delta _\varepsilon (x) :=
\frac {1}{\sqrt {2\pi \varepsilon }} e^{-x^2/(2\varepsilon )}\) und die Lorentz-Kurve \(\delta _\varepsilon (x) := \frac
{1}{\pi } \cdot \frac {\varepsilon }{x^2 + \varepsilon ^2}\) (erfüllen jeweils \(\int _\real \delta _\varepsilon (x) \dx = 1\)).
Fourier-Transformation mit dem Dirac-Delta:
Funktion \(f(x)\)
|
Fourier-Transformierte \(F(k)\)
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\(\delta (x - a)\)
|
\(e^{-\iu ka}\)
|
\(c\)
|
\(c \delta (k)\)
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\(\cos (k_0 x)\)
|
\(\frac {1}{2} (\delta (k - k_0) + \delta (k + k_0))\)
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\(\sin (k_0 x)\)
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\(\frac {1}{2\iu } (\delta (k - k_0) - \delta (k + k_0))\)
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Sampling-Theorem
In diesem Abschnitt wird eine andere Definition für die Fourier-Transformation benutzt, nämlich \((\F (f))(k) = F(k) := \int _\real f(x) e^{-2\pi \iu kx} \dx \) (bzw. \(f(x) = \int _\real F(k)
e^{2\pi \iu kx} \d k\)).
Delta-Kamm: Die Transformation eines kontinuierlichen Signals \(f\) in ein diskretes Signal \(\widehat {f}\) kann man mit dem Delta-Kamm \(c(x) := \sum _{n \in
\integer } \delta (x - n\Delta x)\) darstellen durch
\(\widehat {f}(x) := f(x) c(x) = \sum _{n \in \integer } f(n\Delta x) \delta (x - n\Delta x)\), wobei \(\Delta x\) die Sampling-Gitterweite ist.
Dabei besitzt \(\widehat {f}(x)\) die Fourier-Transformierte \(\widehat {F}(k) = \frac {1}{\Delta x} \sum _{n \in \integer } F(k - \frac {n}{\Delta x})\).
Rekonstruktion des originalen Signals:
Um \(f(x)\) aus \(\widehat {F}(k)\) zu rekonstruieren, verfährt man wie folgt.
Multipliziere \(\widehat {F}(k)\) mit der Rechtecksfunktion \(R(k) := \Delta x \cdot \chi _{[-k_c,k_c]}(k)\)
(es gilt \(F(k) = \widehat {F}(k) R(k)\) genau dann, wenn \(f\) bandbegrenzt durch \(k_c\) ist).
Wende die inverse FT auf \(\widehat {F}(k) R(k)\) an, d. h.
\(\F ^{-1}(\widehat {F}(k) R(k)) = \sum _{n \in \integer } f(n\Delta x) 2k_c \Delta x \sinc (2k_c (x - n\Delta x))\) mit \(\sinc (x) := \frac {\sin (\pi x)}{\pi x}\).
Sampling-Theorem: Sei \(f(\cdot )\) eine Funktion. Gibt es eine Abschneidefrequenz \(k_c > 0\) mit
\(\forall _{|k| \ge k_c}\; F(k) = 0\), dann kann \(f\) aus der gesampelten Funktion \(\widehat {f}\) exakt rekonstruiert werden, wenn \(k_c \le \frac {1}{2 \Delta x} = \frac {k_s}{2}\) mit der Samplerate \(k_s := \frac {1}{\Delta x}\) und der Nyquist-Frequenz \(\frac {k_s}{2}\).
In diesem Fall gilt \(f(x) = \sum _{n \in \integer } f(n\Delta x) 2k_c \Delta x \sinc (2k_c (x - n\Delta x))\).
Aliasing: Wenn \(f(x)\) unterabgetastet wird, d. h. \(k_s \not \ge 2k_c\), dann besitzt die rekonstruierte Funktion Artefakte, sog. Aliasing-Effekte.
diskrete 1D-Fourier-Transformation:
Seien \(\vec {g} := (g_n)_{n=0}^{N-1} \in \complex ^N\) und \(\omega _N := e^{2\pi \iu /N}\) die \(N\)-te Einheitswurzel.
Dann heißt \(G_v := \sum _{n=0}^{N-1} \omega _N^{-vn} g_n\) für \(v = 0, \dotsc , N - 1\) diskrete Fourier-Transf. (DFT).
\(g_n = \frac {1}{N} \sum _{v=0}^{N-1} \omega _N^{vn} G_v\) für \(n = 0, \dotsc , N - 1\) heißt inverse DFT.
Es gilt \(G_v = \sqrt {N} \innerproduct {\vecs {b}{v}, \vec {g}}\) und \(g_n = \frac {1}{\sqrt {N}} \innerproduct {\vecs {b}{-n}, \vec {G}}\) mit den Basisvektoren
\(\vecs {b}{v} := \frac {1}{\sqrt {N}} (\omega _N^0, \omega _N^{v}, \dotsc , \omega _N^{(N-1)v})^\tp \), wobei \(\innerproduct {\vec {x}, \vec {y}} := \vec {x}^\ast \vec {y}\) mit \(\vec {x}^\ast
:= \overline {\vec {x}^\tp }\).
diskrete 2D-Fourier-Transformation: Sei \(g := (g_{m,n})_{m,n=0}^{M-1,N-1} \in \complex ^{M \times N}\).
Dann heißt \(G_{u,v} := \sum _{m=0}^{M-1} \sum _{n=0}^{N-1} \omega _M^{-um} \omega _N^{-vn} g_{m,n}\) für \(u = 0, \dotsc , M - 1\), \(v = 0, \dotsc , N - 1\) diskrete 2D-Fourier-Transf. (2D-DFT). \(g_{m,n} = \frac {1}{MN} \sum _{u=0}^{M-1} \sum _{v=0}^{N-1} \omega _M^{um} \omega _N^{vn} G_{u,v}\) heißt inverse 2D-DFT.
Es gilt \(G_{u,v} = \sqrt {MN} \innerproduct {B_{u,v}, g}\) und \(g_{m,n} = \frac {1}{\sqrt {MN}} \innerproduct {B_{-m,-n}, G}\) mit den Basismatrizen
\(B_{u,v} := \frac {1}{\sqrt {MN}} (\omega _M^0, \omega _M^u, \dotsc , \omega _M^{(M-1)u})^\tp (\omega _N^0, \omega _N^v, \dotsc , \omega _N^{(N-1)v})\),
wobei \(\innerproduct {G, H} := \sum _{m=0}^{M-1} \sum _{n=0}^{N-1} \overline {g_{m,n}} h_{m,n}\).
Eigenschaften:
1D-Periodizität: \(G_{v+\ell N} = G_v\), \(g_{n+\ell N} = g_n\) für alle \(\ell \in \integer \)
2D-Periodizität: \(G_{u+kM,v+\ell N} = G_{u,v}\), \(g_{m+kM,n+\ell N} = g_{m,n}\) für alle \(k, \ell \in \integer \)
1D-DFT einer reellen Folge ist hermitesch: \(g_n \in \real \implies \overline {G_v} = G_{-v} = G_{N-v}\)
1D-DFT einer hermiteschen Folge ist reell: \(g_{N-n} = g_{-n} = \overline {g_n} \implies G_v \in \real \)
2D-Faltungssatz: 2D-DFT von \(f_{m,n} = \sum _{m’=0}^{M-1} \sum _{n’=0}^{N-1} h_{m’,n’} g_{m-m’,n-n’}\) ist \(MN \cdot (H \cdot G)\)