Framework for the Heart Rate Variability Analysis. Modeling the Pedalling Frequency: Effect of the Jitter and the Lack of Synchronisation
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(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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(12) Fu ps~:~qpzvoyspz©w*xue~¢ :pz q. E[(5)] =. £¤*¶¥. a21 (1 + (2πf0 )2 RDD (τ )) 2 d(t). qwR}3ennw)yu% qpsermor}3 qpserB qow#von|}3w]~q~ FoRw' qw*nq(τ ) E -w)psrot qow#~qmoÁe £§¹¥ £c¤)ª`¥ £¤*³¥ £c¤]¶¥ | wtew) R¢ DD. 1. Rx1 x1 (τ ) = E1 =. Fow' qw*nq. E2. (a1 2πf0 )2 1 cos(2πf0 τ )RDD (τ ) + (a21 − 2a1 b1 c1 + b21 ) cos(2πf0 τ ) 2 2. p~:}3evomo qw*x~qpzpsysuenqys8omo m~psrot" qwRuvovon {|pzEu qpser¢. £¯¦º¥. cos(2π2f0 d(t) + 2π) ≈ +1 et sin(2π2f0 d(t) + 2π) ≈ 2π2f0 d(t). Fow*r«Jwtew) ¢. £¤*¹¥. ¯£ ¦|¤]¥ =u`pzrt2psr q9ue}*}3emr ' qw8~que"w"``ve qow]~p~' quern' qowEvonw)`psem~}*uy}3moyu% pzr@urx ownqw]~moyz ~ne wR~qo%L u% urx o@wR©ruysyseo upsr« qu : qow vonq|})w*~~ x(t) p~ ~£§ ¦u%²e ¥ pzr£§¦eu³n¥ E:£§¦pz´e q9¥ u£¯¦}3¶e¥ nnw)yu% qpsermor}3 qpseEr R= 0(τ ) w*Emuyj= 90 ¢ 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¥ 0w » ~~qmowE qu ps~u9yz%Fµvu~q~©y w)nw*xV:opz qw2nuerx|eÃrops~qw8:pz qÄu9%uenqpur})w :~qwE©yz qw)n nw*mow)r}3nw*~qvr~qd(t) w p~ H(f ) ®r nq|x|m}3psrot«u«n+u% pz α w) c w*w)r@ qoww)-n # qow cJ«yzσw*t~' wtw3 qow urx b = αa £ 0 < α < 1¥ -±=psruysys2 ow v-%Jw)n~qvw]}À nuey=x|w*r~qp c X(f ) x(t) p~ nw)yu% qpser~ tepsew*r2`¢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 ) +. £¯¦¨¥ :pz q c = cos(2πf d)¯ w3 c = cos(2π2f d)¯ Fowueruys|~p~8e# qop~2nqw]~moyz «}*urÅ-w@~qpzvoyspz©w*xQÆrow*teysw*}À pzrtÄpsrQuV©n+~c 2 qpsw qw0})evrow)r +~ von|x|m}3w]xR`# ow=¡cp q qw)n £ p§ we σ = 0¥ ¾ ] qowFEutropz qmx|wFn+u% pz R qowJmor}3 qpser~ δ u% qowJmorxouw)r +uy nw*mow)r}3uerxpz ~unerop}'ps~Ftpzw)r«¢ 1. 0. 2. 0. R=β. s. £¯¦%ª`¥. 1 − 2αc1 + α2 1 + 2αc2 + α2. :ow*nqw β ps~J ow#n+u% qpsB qow#Eutropz qmx|w a uerx a lw3©ropsrot ` qow#v-w)nEurow*r :x|w*ysu Ew){`vnqw]~q~qw*x2ue~:uvw*n})w)r uetew' owvw]xouyspsrotvw*nqps`x JwRtew) u8~qw3 deS=}3monew]~Fd¯mor}3 qpsere d uerx α -FowR qw)nE~ β p~~qps"vyzu8~})uyspsrotE¸ue}3 qn: q8-wuevovoysTpzw]x qw qow#%uysmow]~Fvoyze w*x«pzr«©t £c¤ ¥ 0w}*ur0x|w*x|m}3w*xne owE}3monew]~ u% ow"morxoue"w*r uyX})ev-erow*r R}*ur-wopsteow*n ur@ q~qw neÁpz ~unerop}:ow*r qowRv-w*xuyspzrotE%w)w)r p~}3ys~qw q8u8vomonwR~psr`m~psx°op§ wepzr}3nw*ue~qpsrot β uerx qu% "uvw*nqEuerow)r "xow)yu @p~ vnqw]~w*r "ue~#¸uen"u~# ow2w){|w)n+}3p~wEyue~ ~*Bp¯ we=pzr})nqw]ue~qpzrt d » xx|w*xV ops~ ¸ue}3 * :ow)rÆ qw9en+}3w9w){`w*n w*xÇ`Ä ow9yzw*t~E-w*})ew*~8mor`uyur})w*xÇ qw α %uysmowx|w]}3nw*ue~qw*~E}*um~qpsrot?u nw)psr|en+}3w*"w*r JeB owueteropz qmx|w*~:x|pzjw)nw)r})we » xoxop pzruysyz|:ow)r a > 2a op~:nu qpsp~:psr}3nw*u~w]xu~ qwF¡cp q qw)nueteropz qmx|wps~:pztow)n]|p§ we|psr})nqw]ue~qpzrot σ pzr £§¦e¨¥ 1. r. 2. 100 T0. 0. r. r. ¨. 1. 2.
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(15) -w)psrotEw]muyj q½*w)nnur`E%uysmow*~ n urx R1 : E[cos(nωt + nϕ) cos(mω(t + τ ) + mϕ)] =. . m. oJw©ruysyztew3 #¢. 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. FowR~qw*})erx qw*nq
(16) -w)psrotEw]muyj q½*w)nnur`E%uysmow*~ n urx R2 : E[cos(nωt + nϕ) sin(mω(t + τ ) + mϕ)] =. . m. oJw©ruysyztew3 #¢. 0 pour n 6= m 1 2 sin(nωτ ) pour n = m. £¯¦³¥. 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. FowR~qw*})erx qw*nq
(17) -w)psrotEw]muyj q½*w)nnur`E%uysmow*~ n urx R3 : E[sin(nωt + nϕ) cos(mω(t + τ ) + mϕ)] =. . m. oJw©ruysyztew3 #¢. 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.
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