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Framework for the Heart Rate Variability Analysis. Modeling the Pedalling Frequency: Effect of the Jitter and the Lack of Synchronisation

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(1)Framework for the Heart Rate Variability Analysis. Modeling the Pedalling Frequency: Effect of the Jitter and the Lack of Synchronisation Olivier Meste. To cite this version: Olivier Meste. Framework for the Heart Rate Variability Analysis. Modeling the Pedalling Frequency: Effect of the Jitter and the Lack of Synchronisation. 2007, 6 p. �hal-00357730�. HAL Id: hal-00357730 https://hal.archives-ouvertes.fr/hal-00357730 Submitted on 19 Mar 2010. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) LABORATOIRE. INFORMATIQUE, SIGNAUX ET SYSTÈMES DE SOPHIA ANTIPOLIS UMR 6070. F RAMEWORK FOR THE H EART RATE VARIABILITY ANALYSIS . M ODELING THE PEDALING FREQUENCY: EFFECT OF THE JITTER AND THE LACK OF SYNCHRONISATION Olivier MESTE Projet BIOMED Rapport de recherche ISRN I3S/RR–2007-04–FR Février 2007. L ABORATOIRE I3S: Les Algorithmes / Euclide B – 2000 route des Lucioles – B.P. 121 – 06903 Sophia-Antipolis Cedex, France – Tél. (33) 492 942 701 – Télécopie : (33) 492 942 898 http://www.i3s.unice.fr/I3S/FR/.

(3) R ÉSUMÉ : Dans ce rapport, on propose un modèle de génération d’un processus stochastique a partir du mouvement des deux jambes lors d’un exercice de pédalage. On montre qu’il est stationnaire et que le rapport d’amplitude du fondamental par rapport à la premiere harmonique est, en autre, fonction d’un retard systématique entre les deux jambes.. M OTS CLÉS : processus stochastique, rythme cardiaque à l’effort, fréquence de pédalage. A BSTRACT: In this report, we investigate the relation between the movement of the two legs during a pedaling exercise and the stochastic process that is observed. We show that it is stationary and that the ratio of the magnitude of the fundamental frequency and its first harmonic is at least a function of a constant delay between the two legs.. K EY WORDS : stochastic process, heart rate variability during exercise, pedaling frequency.

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(11) —F‡ow#n+uŽrˆxoƒeŒ‰%uŽn‹psueoysw*~ d˜ w3 ϕ „w*pzrotEpsrˆx|w*v„w*rˆx|w*r ]†omˆ~qpzrˆt £¯¦e²¥ uŽrˆx £§¦’´Ž¥  wtw3 R¢ E[(6)] =. a 1 b1 a 1 b1 ˜ ˜ E[cos(2πf0 d(t))] cos(2πf0 τ ) + E[sin(2πf0 d(t))] sin(2πf0 τ ) 2 2. ·~qpsrot £”ª’¥ †oJwxow)‰ew*yzƒv‚ ‹‡ow qw*nqŒE~. ˜ cos(2πf0 d(t)). uerˆx. ˜ sin(2πf0 d(t)). †otpz‰`psrot¢. ˜ ¯ cos(2πf0 d(t) + π) − sin(2πf0 d) ¯ sin(2πf0 d(t) + π) cos(2πf0 d(t)) = cos(2πf0 d) ˜ ¯ cos(2πf0 d(t) + π) + cos(2πf0 d) ¯ sin(2πf0 d(t) + π) sin(2πf0 d(t)) = sin(2πf0 d). » ~‹~qmoŒ“pzrot" ‹‡ˆu% : ‹‡ow#‰%uŽn‹psuŽ qpsƒerˆ~JƒŽ. d(t). uŽn‹w#~€ŒEuŽysyw)rˆƒemot‡°†| wtw3 F q‡owRuevovon‹ƒ {|pzŒEu% ‹pzƒrˆ~'¢. £€¤*º¥ £c¤¤]¥ £c¤ ¦e¥. £€¤*¨¥ —F‡own‹uerˆx|ƒeŒ¼‰%uŽn‹psueoyzw d(t) „w*pzrˆt ½)w*nqƒ Œ"w]uŽr°†mˆ~qpzrotR q‡owvonqw*‰`pzƒmˆ~Suevovon‹ƒ {`psŒEu% ‹pzƒrˆ~= q‡oww3{|von‹w*~‹~€psƒer £€¤*º¥ -w*}3ƒŒ“w*~¢ cos(2πf0 d(t) + π) ≈ −1 et sin(2πf0 d(t) + π) ≈ −2πf0 d(t). a 1 b1 ¯ cos(2πf0 τ ) − a1 b1 sin(2πf0 d) ¯ sin(2πf0 τ ) cos(2πf0 d) 2 2 a 1 b1 a 1 b1 =− c1 cos(2πf0 τ ) − s1 sin(2πf0 τ ) 2 2 E[(6)] = −. —F‡ow' qw*nqŒ £¯´e¥ p~x|w*‰ew*yzƒv„w]x2u~¢ ¾ psrˆ}3w d˜ uŽrˆx. £€¤)ª’¥. (7) → ˜ + τ )) cos(2πf0 (t + τ ) + ϕ) cos(2πf0 t + ϕ) a1 b1 cos(2πf0 d(t ˜ + τ )) sin(2πf0 (t + τ ) + ϕ) cos(2πf0 t + ϕ) −a1 b1 sin(2πf0 d(t. ϕ. E[(7)] =. uŽn‹wpzr„x|w)v-w)rˆxow)r’ *†om„~€psrot £§¦²e¥ uŽrˆx £¯¦Ž³¥ Jwtew3 R¢. a 1 b1 ˜ + τ ))] cos(2πf0 τ ) − a1 b1 E[sin(2πf0 d(t ˜ + τ ))] sin(2πf0 τ ) E[cos(2πf0 d(t 2 2. ¿JŠ8mˆ~qpzrˆt ‹‡ow#n‹w*~qmoy˜ +~J”nqƒŒ £€¤e¤]¥ † £c¤ ¦e¥ uŽrˆx £€¤*¨¥ †| w'©ˆr„uŽysyzŠ8tw3 R¢ E[(7)] = −. —F‡ow ‹w)n‹Œ £§²¥ ps~:x|w*‰ew*yzƒv«u~¢. a 1 b1 a 1 b1 c1 cos(2πf0 τ ) + s1 sin(2πf0 τ ) 2 2. £€¤]²e¥ £€¤*³¥. (5) → ˜ ˜ + τ )) a21 [cos(2πf0 t + ϕ) cos(2πf0 d(t)) cos(2πf0 (t + τ ) + ϕ) cos(2πf0 d(t ˜ ˜ − cos(2πf0 t + ϕ) cos(2πf0 d(t)) sin(2πf0 (t + τ ) + ϕ) sin(2πf0 d(t + τ )) ˜ ˜ + τ )) − sin(2πf0 t + ϕ) sin(2πf0 d(t)) cos(2πf0 (t + τ ) + ϕ) cos(2πf0 d(t ˜ ˜ + τ ))] + sin(2πf0 t + ϕ) sin(2πf0 d(t)) sin(2πf0 (t + τ ) + ϕ) sin(2πf0 d(t. ¿JŠ8mˆ~qpzrˆt £¯¦e²¥ † £¯¦Ž³¥ † £§¦’´Ž¥ † £¯¦Ž¶’¥ † £€¤e¤]¥ † £c¤ ¦e¥ q‡ow#w){|v„w]}À ‹uŽ qpsƒer«ƒŽ £§²¥ „w]}3ƒeŒ“w]~¢ E((5)) → a21 [cos(2πf 0τ )E[(−c1 + s1 2πf0 d(t))(−c1 + s1 2πf0 d(t + τ ))] 2 − sin(2πf0 τ )E[(−c1 + s1 2πf0 d(t))(−s1 − c1 2πf0 d(t + τ ))] + sin(2πf0 τ )E[(−s1 − c1 2πf0 d(t))(−c1 + s1 2πf0 d(t + τ ))] + cos(2πf0 τ )E[(−s1 − c1 2πf0 d(t))(−s1 − c1 2πf0 d(t + τ ))]. ¦. £€¤ ´e¥.

(12) —F‡ˆuŽ ps~:~qpzŒ“voyspz©ˆw*xue~¢ :pz q‡. E[(5)] =. £€¤*¶¥. a21 (1 + (2πf0 )2 RDD (τ )) 2 d(t). q‡ˆwR}3ƒen‹n‹w)yu% qpsƒer‚”morˆ}3 qpsƒerƒŽB q‡ow#von‹ƒ|}3w]~q~ – —F‡oRw' qw*nq(τŒ ) E -w)psrot“ q‡ow#~qmoŒÁƒe £§¹¥ † £c¤)ª`¥ † £€¤*³’¥ † £c¤]¶¥ †| wtew) R¢ DD. 1. Rx1 x1 (τ ) = E1 =. —F‡ow' qw*nqŒ. E2. (a1 2πf0 )2 1 cos(2πf0 τ )RDD (τ ) + (a21 − 2a1 b1 c1 + b21 ) cos(2πf0 τ ) 2 2. p~:}3ƒeŒ“vomo qw*x~qpzŒ“psysuenqysŠ8omo mˆ~€psrot" q‡ˆwRuŽvovon‹ƒ {|pzŒEuŽ qpsƒer‘¢. £¯¦Žº¥. cos(2π2f0 d(t) + 2π) ≈ +1 et sin(2π2f0 d(t) + 2π) ≈ 2π2f0 d(t). —F‡ow*r«Jwtew) ¢. £€¤*¹¥. ¯£ ¦|¤]¥ —=uŽ•`pzrˆt2psr’ qƒ9ue}*}3ƒemˆr ' q‡ˆw8~queŒ"w"‡`Š`v„ƒe q‡ow]~€p~' q‡ˆuer‘”ƒn' q‡owEvon‹w)‰`psƒemˆ~}*uŽy}3moyu% ‹pzƒr@uŽr„x ‹‡ow“nqw]~€moyz ‹~”n‹ƒeŒ † † †  wR~q‡oƒ%L ‹‡ˆu% uŽrˆx –oœ@wR©ˆrˆuŽysysŠ‚ƒeo ‹uŽpsr« q‡ˆuŽ : q‡ow vonqƒ|})w*~‹~ x(t) p~ ~€£§ ‹¦u%²e ‹¥ pzƒrˆ£§¦euŽ³n‹¥ ŠE:£§¦pz´e q‡9¥ u“£¯¦Ž}3¶’ƒe¥ n‹n‹w)yu% qpsƒer‚”morˆ}3 qpsƒeEr R= 0(τ ) w*š’EmˆuŽyj= ‹ƒ90 ¢ Rx2 x2 (τ ) = E2 =. (a2 2π2f0 )2 1 cos(2π2f0 τ )RDD (τ ) + (a22 + 2a2 b2 c2 + b22 ) cos(2π2f0 τ ) 2 2 3. 4. xx. 1 2 1 (a1 − 2a1 b1 c1 + b21 ) cos(2πf0 τ ) + (a1 2πf0 )2 cos(2πf0 τ )RDD (τ ) 2 2 1 2 1 2 + (a2 − 2a2 b2 c2 + b2 ) cos(2π2f0 τ ) + (a2 2π2f0 )2 cos(2π2f0 τ )RDD (τ ) 2 2. Rxx (τ ) =. ¯£ ¦e¦e¥ œ0w » ~‹~qmoŒ“wE q‡ˆuŽ ps~u9yzƒ%Fµvˆu~q~©„y˜ ‹w)n‹w*xV:‡opz qw2n‹uerˆx|ƒeŒÃroƒps~qw8:pz q‡Äu9‰%uenqpuŽrˆ})w :‡ˆƒ~qwE©ˆyz qw)n ”n‹w*š’mow)r„}3Šn‹w*~qv„ƒrˆ~qd(t) w p~ H(f ) –­®r ‹nqƒ|x|m„}3psrot«u«n+u% ‹pzƒ α „w) c w*w)r@ q‡ow“w)™-ƒn€ #ƒŽ q‡ow“ cJƒ«yzσw*t~' w“tw3  q‡ow uŽr„x b = αa £ 0 < α < 1¥ –-±=psrˆuŽysysŠ2 ‹‡ow v-ƒ%Jw)n~qv„w]}À ‹n‹uey=x|w*rˆ~qp˜ cŠ X(f ) ƒŽ x(t) p~ n‹w)yu% qpsƒer„~ teps‰ew*r2`Š¢b = αa 2. 1. 1. X(f ) =. 2. 2. a21 (1 − 2αc1 + α2 )(δ(f − f0 ) + δ(f + f0 )) + (a1 σπf0 )2 (|H(f − f0 )|2 + |H(f + f0 )|2 ) 4. a22 (1 + 2αc2 + α2 )(δ(f − 2f0 ) + δ(f + 2f0 )) + (a2 σπ2f0 )2 (|H(f − 2f0 )|2 4 +|H(f + 2f0 )|2 ) +. £¯¦Ž¨’¥ :pz q‡ c = cos(2πf d)¯ w3 c = cos(2π2f d)¯ —F‡ow‘uerˆuŽysŠ|~€p~8ƒe# q‡op~2nqw]~€moyz «}*uŽrŁ-w@~qpzŒ“voyspz©ˆw*xQ’ŠÆrow*teysw*}À ‹pzrˆtÄpsrQuV©ˆn+~c 2 qpsŒ“w q‡ˆw0})ƒeŒ“v„ƒrow)r’ +~ von‹ƒ|x|mˆ}3w]xR`Š# ‹‡ow=¡cp˜ q qw)n £ p§– we– σ = 0¥ – ¾ ƒˆ†] q‡owFŒEuŽtropz qmˆx|wFn+u% ‹pzƒ R ƒŽ„ q‡owJ”morˆ}3 qpsƒerˆ~ δ u%  q‡owJ”mor„xouŽŒ“w)r’ +uŽy ”n‹w*š’mow)r„}3Š‚uerˆx‚pz ‹~‡ˆuŽn‹Œ“ƒerop}'ps~Ftpz‰w)r«’Š‘¢ 1. 0. 2. 0. R=β. s. £¯¦%ª`¥. 1 − 2αc1 + α2 1 + 2αc2 + α2. :‡ow*nqw β ps~J ‹‡ow#n+u% qpsƒ“ƒŽB q‡ow#ŒEuŽtropz qmˆx|w a uerˆx a – lw3©ˆropsrot †` q‡ow#v-w)n‹ŒEuŽrow*r :x|w*ysu ŠEw){`vˆnqw]~q~qw*x2ue~:uv„w*n‹})w)r’ ‹uetew'ƒŽ ‹‡owv„w]xouŽyspsrot“v„w*nqpsƒ`x † JwRtew) u8~qw3 dƒeS=}3mon‹‰ew]~Fd¯”morˆ}3 qpsƒerƒe d uerˆx α –-—F‡owR qw)n‹ŒE~ β p~~qpsŒ"vˆyzŠu8~‹})uŽyspsrotE¸ue}3 qƒn: qƒ8-wuevovoysTpzw]x q‡ˆw q‡ow#‰%uŽysmow]~Fvoyzƒe € ‹w*x«pzr«©ˆt„– £c¤ ¥ – œ0w“}*uŽr0x|w*x|m„}3w*x‘”n‹ƒeŒ ‹‡owE}3mon‹‰ew]~ ‹‡ˆu%  ‹‡ow"”morˆxoueŒ"w*r’ ‹uŽyX})ƒeŒ“v-ƒerow*r R}*uŽr‘-w“‡opste‡ow*n ‹‡ˆuŽr@ q‡ˆƒ~qw ”n‹ƒeŒÁpz ‹~‡ˆuŽn‹Œ“ƒerop}:‡ow*r q‡owRv-w*xˆuŽyspzrotEŒ“ƒ%‰w)Œ“w)r’ p~}3ysƒ~qw qƒ8u8vomon‹wR~€psr`mˆ~€ƒpsx°†op§– we–„pzr„}3n‹w*ue~qpsrot β †„uerˆx q‡„u% "u‘v„w*nqŒEuerow)r’ "xow)yu Š@p~ vˆnqw]~€w*r "ue~#¸uen"u~# ‹‡ow2w){|w)n+}3p~€wEyue~€ ‹~*†Bp¯– we–=pzrˆ})nqw]ue~qpzrˆt d – » xˆx|w*xV ‹ƒ ‹‡ops~ ¸ue}3 *† :‡ow)rÆ q‡ˆw9”ƒen+}3w9w){`w*n€ ‹w*xǁ`ŠÄ ‹‡ow9yzw*t~E-w*})ƒeŒ“w*~8mor`ˆuŽyuŽrˆ})w*xÇ q‡ˆw α ‰%uŽysmowx|w]}3n‹w*ue~qw*~E}*uŽmˆ~qpsrot?u n‹w)psr|”ƒen+}3w*Œ"w*r’ JƒeB ‹‡owŒ“ueteropz qm„x|w*~:x|pz™jw)n‹w)rˆ})we– » xoxop˜ ‹pzƒrˆuŽysyzŠ†|:‡ow)r a > 2a ‹‡op~:n‹uŽ qpsƒp~:psrˆ}3n‹w*u~€w]x‚u~ q‡ˆwF¡cp˜ q qw)nŒ“ueteropz qm„x|wps~:‡ˆpzt‡ow)n]†|p§– we–|psrˆ})nqw]ue~qpzrot σ pzr £§¦e¨¥ – 1. r. 2. 100 T0. 0. r. r. ¨. 1. 2.

(13) 2 α=1 α=0.9 α=0.8 α=0.7. 1.8. 1.6. 1.4. 1.2. R. 1. 0.8. 0.6. 0.4. 0.2. 0. 0. 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. dr. ÈFÉ”Ê Ë ¤ÌÍ mon‹‰ew]~. R. ƒe| +uŽpsrow*x‚”ƒen~€w*‰ew)n+uŽy-‰%uŽysmow*~Fƒe α uerˆx. dr. 20. delay. 15. 10. 5. 100. 200. 300. 400. 500. 600. 700. 400. 500. 600. 700. t (s). 5. magnitude ratio. 4 3 2 1 0. 100. 200. 300 t (s). ÈFɔÊSË —FpsŒ"w2‰%uŽn‹psuŽ qpsƒerVƒŽF q‡ˆw2x|w*ysu Š‘”morˆ}À ‹pzƒržƒŽF ‹pzŒ“w qƒv – » vovˆnqƒ {|psŒ“uŽ qw]x@n+u% ‹pzƒ ~qƒeyspx?yzpsrow uŽr„x9n‹uŽ¦2 qpsÌ ƒ8ƒŽ ‹‡ow"Œ“ueteropz qm„x|w £ xˆue~q‡|µxoƒŽ € ‹w*x9yspsrow ¥ ƒŽ ‹‡owR”mor„£ xouŽŒ“¥ w)r’ +uŽyuŽrˆx9‡„uŽn‹Œ"ƒrop}R}3ƒRŒ"v-£ ƒerˆw)r’ ‹~ £§¦¥¥ £ -ƒŽ q qƒŒ ¥ ª.

(14) —Bƒpsyzysmˆ~€ qn+u% ‹w ‹‡op~:nqw]~€mˆy˜ J q‡ˆuŽ ps~:uer«u ‰w)n+uŽtwƒerow†| wtpz‰wpsr‚©„tˆ– £§¦ ‹ƒev2 qn+ue}3w ¥ †ouer2w){|ueŒ“voyzwƒe q‡ˆw ”morˆ}3 qpsƒer d¯ + d(t) psr@v-w)n+}3w*r RƒeS ‹‡owEv„w]xouŽyspsrotv„w*nqpsƒ|x†°:pz q‡ d = 10% † uerˆx –­®r q‡ˆwR„ƒe € ‹ƒeŒ qn+ue}3w†oJwR}3ƒeŒ“vˆuenqw ‹‡owRn+u% ‹pzƒ“ƒŽ ‹‡owRŒEuŽtrop˜ ‹mˆx|wRƒe q‡ow#”mˆrˆxouŽαŒ“=w*r0.8 +uŽy”n‹w*š’moaw*rˆ/a }3Š«:=pz q‡5 q‡ˆw ‡ˆuenqŒ“ƒerˆps} uerˆxV ‹‡ow8‰%uŽysmow‚ƒe :‡ˆw)n‹wE q‡ow‚n‹uerˆx|ƒeŒ¡cpz € qw*np~RrˆƒŽ +uŽ•w)r?psr ‹ƒ‘ue}*}3ƒmor’ *–­µ "ps~}3ysw*uen q‡„u% morˆx|w*n£¯¦e~q¥ mˆ}+‡@nqw]uŽysps~€ qp} })ƒerˆx|pz qRpsƒer„~ ‹‡ow“}3ƒeŒ“v-ƒerow*r’ uŽ  ‹‡ow ”mˆrˆxouŽŒ“w*r +uŽy=”n‹w*š’mow)r„}3Š9p~‡ˆpzt‡ow)n q‡ˆuer pz ‹~:‡ˆuenqŒ“ƒrops}e– r. 1. 2. Î. Ï ÐÃЬÑLÒKÑ K uŽy}3mˆysuŽ qpsƒer‚ƒeB ‹‡owR}3ƒnqn‹w)yu% ‹pzƒr‚uŽr„x«})nqƒ’~q~€Â®}3ƒen‹n‹w)yu% qpsƒer"”mor„}À qpsƒer„~:mˆ~qw*x2pzr« q‡ˆw#Œ"ƒ|x|w*yzyspsrotˆ– ­®Í rV q‡ˆw2~qw*š’mow)y¯†= w8:psysy mˆ~qwE ‹‡ow‚von‹ƒev-w)nq cŠ@ q‡ˆuŽ ϕ ps~umorˆp˜”ƒnqŒÃn‹uerˆx|ƒŒ¬‰%uŽn‹puŽoysw8x|w3©„row*x?ƒr? q‡ˆw psr ‹w)n‹‰%uŽy [0, π] uerˆx (n, m) u“}3ƒmovoysw'ƒe=psr’ qw*tew)n+~ > 0 – E[cos(nωt + nϕ) cos(mω(t + τ ) + mϕ)] Z = cos(nωt + nϕ) cos(mωt + mτ ) + mϕ)pφ (ϕ)dϕ Z 2π Z 2π 1 = ( cos(ωt(n − m) − mωτ + ϕ(n − m))dϕ + cos(ωt(n + m) + mωτ + ϕ(n + m))dϕ) 4π 0 0. —F‡owR~qw*})ƒerˆx‚ qw*nqŒ

(15) -w)psrotEw]šm„uŽyj qƒ“½*w)n‹ƒ”ƒnuŽr`ŠE‰%uŽysmow*~ƒŽ n uŽr„x R1 : E[cos(nωt + nϕ) cos(mω(t + τ ) + mϕ)] =. . m. †oJw©„rˆuŽysyzŠ‚tew3 #¢. 0 pour n 6= m 1 2 cos(nωτ ) pour n = m. £¯¦e²¥. E[cos(nωt + nϕ) sin(mω(t + τ ) + mϕ)] Z = cos(nωt + nϕ) sin(mωt + mτ ) + mϕ)pφ (ϕ)dϕ Z 2π Z 2π 1 ( sin(ωt(m − n) + mωτ + ϕ(m − n))dϕ + sin(ωt(n + m) + mωτ + ϕ(n + m))dϕ) = 4π 0 0. —F‡owR~qw*})ƒerˆx‚ qw*nqŒ

(16) -w)psrotEw]šm„uŽyj qƒ“½*w)n‹ƒ”ƒnuŽr`ŠE‰%uŽysmow*~ƒŽ n uŽr„x R2 : E[cos(nωt + nϕ) sin(mω(t + τ ) + mϕ)] =. . m. †oJw©„rˆuŽysyzŠ‚tew3 #¢. 0 pour n 6= m 1 2 sin(nωτ ) pour n = m. £¯¦Ž³’¥. E[sin(nωt + nϕ) cos(mω(t + τ ) + mϕ)] Z = sin(nωt + nϕ) cos(mωt + mτ ) + mϕ)pφ (ϕ)dϕ Z 2π Z 2π 1 = ( sin(ωt(n − m) − mωτ + ϕ(n − m))dϕ + sin(ωt(n + m) + mωτ + ϕ(n + m))dϕ) 4π 0 0. —F‡owR~qw*})ƒerˆx‚ qw*nqŒ

(17) -w)psrotEw]šm„uŽyj qƒ“½*w)n‹ƒ”ƒnuŽr`ŠE‰%uŽysmow*~ƒŽ n uŽr„x R3 : E[sin(nωt + nϕ) cos(mω(t + τ ) + mϕ)] =. . m. †oJw©„rˆuŽysyzŠ‚tew3 #¢. 0 pour n 6= m − 12 sin(nωτ ) pour n = m. £¯¦´e¥. E[sin(nωt + nϕ) sin(mω(t + τ ) + mϕ)] Z = sin(nωt + nϕ) sin(mωt + mτ ) + mϕ)pφ (ϕ)dϕ Z 2π Z 2π 1 ( cos(ωt(n − m) − mωτ + ϕ(n − m))dϕ − cos(ωt(n + m) + mωτ + ϕ(n + m))dϕ) = 4π 0 0. ².

(18) —F‡owR~qw*})ƒerˆx‚ qw*nqŒ

(19) -w)psrotEw]šm„uŽyj qƒ“½*w)n‹ƒ”ƒnuŽr`ŠE‰%uŽysmow*~ƒŽ n uŽr„x R4 : E[sin(nωt + nϕ) sin(mω(t + τ ) + mϕ)] =. ³. . m. †oJw©„rˆuŽysyzŠ‚tew3 #¢. 0 pour n 6= m 1 2 cos(nωτ ) pour n = m. £¯¦Ž¶’¥.

(20)

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