Chapter 8: Response of Linear Systems to Random Inputs

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Caper 8: epone of Linear yem o anom Inpu 8- Inroucion 8- nalyi in e ime Domain 8- Mean an Variance Value of yem Oupu 8-4 uocorrelaion Funcion of yem Oupu 8-5 Crocorrelaion beeen Inpu an Oupu 8-6 ample of ime-domain yem nalyi 8-7 nalyi in e Frequency Domain 8-8 pecral Deniy a e yem Oupu 8-9 Cro-pecral Deniie beeen Inpu an Oupu 8- ample of Frequency-Domain nalyi 8- umerical Compuaion of yem Oupu Concep: Linear yem nalyi in e ime Domain Mean an Variance Value of yem Oupu uocorrelaion Funcion of yem Oupu Crocorrelaion beeen Inpu an Oupu ample of ime-domain yem nalyi nalyi in e Frequency Domain pecral Deniy a e yem Oupu Frequency Limie Filering Banlimie Wie oie IMO yem Cro-pecral Deniie beeen Inpu an Oupu ample of Frequency-Domain nalyi umerical Compuaion of yem Oupu imulaing ocaic yem epone oe an figure are bae on or aken from maerial in e coure ebook: Probabiliic Meo of ignal an yem nalyi (r e.) by George. Cooper an Clare D. McGillem; Ofor Pre, 999. IB: -9-554-9. B.J. Bazuin, pring 5 of 6 C 8

Caper 8: epone of Linear yem o anom Inpu Linear ranformaion of ignal: convoluion in e ime omain y yem y Linear ranformaion of ignal: muliplicaion in e Laplace omain e convoluion Inegral (applying a caual filer) y or y Commen: Dr. Bazuin ill ue caual filer, efine a,, an en oing o ypically prefer oing convoluion a y oe an figure are bae on or aken from maerial in e coure ebook: Probabiliic Meo of ignal an yem nalyi (r e.) by George. Cooper an Clare D. McGillem; Ofor Pre, 999. IB: -9-554-9. B.J. Bazuin, pring 5 of 6 C 8

more realiic yem D G C G P P C e Proce mu be oberve an conrolle. anom noie, inerference, or iurbance are preen in all inpu o e yem an meauremen of e oupu of e yem. e elecronic conrol mu eermine a i imporan an a in an conrol e yem in orer o operae i properly. We ue filer o eliminae noie in frequency ban no of inere. We ue correlaion an opimal ignal eecor o look for knon paern epece in e ignal. We ue a-priori informaion o eign e yem for all knon. We can collece aiic in e form of auo an cro-covariance o eermine curren operaion. Bae on operaing parameer e mu make eciion an ac upon em. oe an figure are bae on or aken from maerial in e coure ebook: Probabiliic Meo of ignal an yem nalyi (r e.) by George. Cooper an Clare D. McGillem; Ofor Pre, 999. IB: -9-554-9. B.J. Bazuin, pring 5 of 6 C 8

oe an figure are bae on or aken from maerial in e coure ebook: Probabiliic Meo of ignal an yem nalyi (r e.) by George. Cooper an Clare D. McGillem; Ofor Pre, 999. IB: -9-554-9. B.J. Bazuin, pring 5 4 of 6 C 8 e Mean Value a a yem Oupu fer efining e convoluion, e can ue e epece value operaor noe a proceing of Wie-ene aionary anom Procee i eire an uually implie. For a ie-ene aionary proce, i reul in e mean of e oupu i e mean of e inpu ime e coeren gain of e filer. e coeren gain of a filer i efine a: gain erefore, gain e Mean quare Value a a yem Oupu erefore, ake e convoluion of e auocorrelaion funcion an en um e filer-eige reul from o infiniy.

oe an figure are bae on or aken from maerial in e coure ebook: Probabiliic Meo of ignal an yem nalyi (r e.) by George. Cooper an Clare D. McGillem; Ofor Pre, 999. IB: -9-554-9. B.J. Bazuin, pring 5 5 of 6 C 8 e uocorrelaion a a yem Oupu e epece value of e prouc of o iinc convoluion: Ienifying an forming e inpu auocorrelaion or oice a fir, you convolve e inpu auocorrelaion funcion i e filer funcion an en you convolve e reul i a ime invere verion of e filer!

oe an figure are bae on or aken from maerial in e coure ebook: Probabiliic Meo of ignal an yem nalyi (r e.) by George. Cooper an Clare D. McGillem; Ofor Pre, 999. IB: -9-554-9. B.J. Bazuin, pring 5 6 of 6 C 8 e Poer pecral Deniy a a yem Oupu e poer pecral eniy i e Fourier ranform of e auocorrelaion: i ep i ep Were y For a W, ergoic proce, i i ere e ave goen before. o e nee o perform e Fourier ranform o fin e PD! i ep

aking e Poer pecral Deniy ep i Iolaing e ranform o funcion in ime (au) ep i ep j eparaing epenen an inepenen elemen bae on e inegraion ep ep j j j ep erefore j ep or elaion of pecral Deniy o e Crocorrelaion Funcion you mig epec, compuaion of e cro-pecral only a a ingle filer PD erm. an an oe an figure are bae on or aken from maerial in e coure ebook: Probabiliic Meo of ignal an yem nalyi (r e.) by George. Cooper an Clare D. McGillem; Ofor Pre, 999. IB: -9-554-9. B.J. Bazuin, pring 5 7 of 6 C 8

oe an figure are bae on or aken from maerial in e coure ebook: Probabiliic Meo of ignal an yem nalyi (r e.) by George. Cooper an Clare D. McGillem; Ofor Pre, 999. IB: -9-554-9. B.J. Bazuin, pring 5 8 of 6 C 8 8-6 ample of ime-domain nalyi: I i imporan o remember bo e ime omain an frequency omain relaionip in orer o pick e one a i eaier bae on e ignal ype. Wen e auocorrelaion a a imple form over a finie ime inerval, e ime omain can be eaier. Filer: u u anom Proce: (a) Fin e mean value of e oupu. (b) Fin e mean quare of e oupu. Fir: Coeren gain gain u u gain Oupu ignal erefore, gain

oe an figure are bae on or aken from maerial in e coure ebook: Probabiliic Meo of ignal an yem nalyi (r e.) by George. Cooper an Clare D. McGillem; Ofor Pre, 999. IB: -9-554-9. B.J. Bazuin, pring 5 9 of 6 C 8 o eliminae e abolue value, perform o inegral for eac par, n finally e auocorrelaion funcion (uing able) ep in j j ep in 4 4 in in in

oe an figure are bae on or aken from maerial in e coure ebook: Probabiliic Meo of ignal an yem nalyi (r e.) by George. Cooper an Clare D. McGillem; Ofor Pre, 999. IB: -9-554-9. B.J. Bazuin, pring 5 of 6 C 8 Poer pecral Deniy of a filere ie noie proce. Le aking e PD i ep i ep ep i ep i ep i

ime bae meo o collec yem ignal repone oe an figure are bae on or aken from maerial in e coure ebook: Probabiliic Meo of ignal an yem nalyi (r e.) by George. Cooper an Clare D. McGillem; Ofor Pre, 999. IB: -9-554-9. B.J. Bazuin, pring 5 of 6 C 8

ample: Wie oie Inpu Le For a ie noie proce, e mean quare (or n momen) i proporional o e filer poer. ignal-o-oie-aio (alay one for poer) P ignal i efine a Poie BQ i aume a e filer oe no cange e inpu ignal, bu ricly reuce e noie poer by e equivalen noie bani of e filer. e narroer e filer applie prior o ignal proceing, e greaer e of e ignal! erefore, alay apply an analog filer prior o proceing e ignal of inere! oe an figure are bae on or aken from maerial in e coure ebook: Probabiliic Meo of ignal an yem nalyi (r e.) by George. Cooper an Clare D. McGillem; Ofor Pre, 999. IB: -9-554-9. B.J. Bazuin, pring 5 of 6 C 8

oe an figure are bae on or aken from maerial in e coure ebook: Probabiliic Meo of ignal an yem nalyi (r e.) by George. Cooper an Clare D. McGillem; Ofor Pre, 999. IB: -9-554-9. B.J. Bazuin, pring 5 of 6 C 8 8- ample of Frequency-Domain nalyi oie in a linear feeback yem loop. Linear uperpoiion of o an o. ere are effecively o filer, one applie o an a econ apply o. an Generic efiniion of oupu Poer pecral Deniy:

Cange in e inpu o oupu ignal o noie raio. In Ou Ou B Q Were B Q If e noie i ae a e ignal inpu, e epece efiniion of noie equivalen bani reul. oe an figure are bae on or aken from maerial in e coure ebook: Probabiliic Meo of ignal an yem nalyi (r e.) by George. Cooper an Clare D. McGillem; Ofor Pre, 999. IB: -9-554-9. B.J. Bazuin, pring 5 4 of 6 C 8

eign noe: If i yem i in operaion, o mig you caracerize e ranfer funcion iou raically cange i normal operaion? Wa oul appen if you coul a a conrolle ie noie inpu? Given a you coul collec an compue e aiive inpu o oupu crocorrelaion, a mig you epec o fin? y a n a, for Perform e e i no noie an repea i noie an compare e reul. o oie, e epec for, Wi oie, e epec, for Can e fin e ifference an u e filer ep repone! ere are procee o caracerize running or operaing equipmen. ey valiae correc operaion! ey may provie an inicaion of impening problem or even failure! oe an figure are bae on or aken from maerial in e coure ebook: Probabiliic Meo of ignal an yem nalyi (r e.) by George. Cooper an Clare D. McGillem; Ofor Pre, 999. IB: -9-554-9. B.J. Bazuin, pring 5 5 of 6 C 8

8- umerical Compuaion of yem Oupu ee moifie CG for correlaion of earbea even an efining n even eecion reol. un e emonraion for M an FM. oie poer level an filering can grealy affec e reul of a yem. oe an figure are bae on or aken from maerial in e coure ebook: Probabiliic Meo of ignal an yem nalyi (r e.) by George. Cooper an Clare D. McGillem; Ofor Pre, 999. IB: -9-554-9. B.J. Bazuin, pring 5 6 of 6 C 8

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