ECE 3800 Probabilistic Methods of Signal and System Analysis
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1 C 3800 Probablstc Methods o gal ad stem Aalss Course Topcs:. Probablt. adom varables 3. Multple radom varables 4. adom processes 5. Lear sstems Course Objectves: Ths course seeks to develop a mathematcal uderstadg o basc statstcal tools ad processes as appled to eretal statstcs ad statstcal sgal processg.. covert a glsh problem descrpto to a precse mathematcal probablstc statemet (a). use the geeral propertes o radom varables to solve a probablstc problem (a, e) 3. be able to use a set o stadard probablt dstrbuto uctos sutable or egeerg applcatos (a) 4. be able to calculate stadard statstcs rom mass, dstrbuto ad dest uctos (a) 5. calculate codece tervals or a populato mea (a) 6. recogze ad terpret a varet o determstc ad odetermstc radom processes that occur egeerg (a, b, e) 7. calculate the autocorrelato ad spectral dest o a arbtrar radom process (a) 8. uderstad stochastc pheomea such as whte, pk ad black ose (a) 9. relate the correlato o ad betwee put ad output o autocorrelato ad spectral dest (a) 0. uderstad the mathematcal characterstcs o stadard requec solato lters (a, e). uderstad the sgal-to-ose optmzato prcple as appled to lter desg (a, e, k). desg Weer ad matched ose lters (a, c, e) Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN:
2 . Itroducto to Probablt.. geerg Applcatos o Probablt.. adom permets ad vets Table o Cotets.3. Detos o Probablt permet Possble Outcomes Trals vet quall Lkel vets/outcomes Objects Attrbute ample pace Wth eplacemet ad Wthout eplacemet.4. The elatve-requec Approach Where A. 0 PrA Pr r N A N lm r A A N A Pr s deed as the probablt o evet A.. PrA PrB PrC 3. A mpossble evet, A, ca be represeted as Pr 0 4. A certa evet, A, ca be represeted as PrA..5. lemetar et Theor et ubset pace Null et or mpt et Ve Dagram qualt um or Uo Products or Itersecto Mutuall clusve or Dsjot ets Complemet Dereces Proos o et Algebra.6. The Aomatc Approach, or mutuall eclusve evets A. Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN:
3 .7. Codtoal Probablt Pr A Jot Probablt B PrA B PrB, or PrB 0 Pr A B Pr A B, or PrB 0 PrB Pr A, B PrA B PrA whe A ollows B A, B PrB, A PrA B PrB PrB A PrA Pr Margal Probabltes Total Probablt PrB PrB A PrA PrB A PrA PrB A PrA Baes Theorem PrB A PrA PrA B Pr B A Pr A Pr B A Pr A Pr B A Pr A.8. Idepedece Pr A, B PrB, A PrA PrB.9. Combed permets.0. Beroull Trals Pr.. Applcatos o Beroull Trals k k Aoccurg k tmes trals p k p q k Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN:
4 . adom Varables.. Cocept o a adom Varable.. Dstrbuto uctos Probablt Dstrbuto ucto (PD) 0, or 0 ad s o-decreasg as creases Pr or dscrete evets or cotuous evets.3. Dest uctos Probablt Dest ucto (pd) 0, or d u du Pr d Probablt Mass ucto (pm) 0, or u u u u Pr u uctos o radom varables u d d Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN:
5 Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN: Mea Values ad Momets st, geeral, th Momets d or Pr d g g or g g Pr d or Pr Cetral Momets d Pr Varace ad tadard Devato d Pr.5. The Gaussa adom Varable or, ep where s the mea ad s the varace dv v v ep Ut Normal (Apped D) du u u ep or
6 The Q-ucto s the complemet o the ormal ucto, : (Apped ) u Q ep du u.6. Dest uctos elated to Gaussa.7. Other Probablt Dest uctos poetal Dstrbuto T ep, or 0 M M 0, or 0 T ep, M 0, T T M T T M or 0 or 0 T T T M M M T Bomal Dstrbuto B k k p p k k k 0 B k k p p u k k k 0.8. Codtoal Probablt Dstrbuto ad Dest uctos PrA B PrA B PrB, or PrB 0 Pr A B Pr A B, or PrB 0 PrB Pr A B PrA, B Pr A B, or PrB 0 PrB PrB It ca be show that M s a vald probablt dstrbuto ucto wth all the epected characterstcs: 0 M, or M 0 ad M M s o-decreasg as creases Pr M M M Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN:
7 .9. amples ad Applcatos 3. everal adom Varables 3.. Two adom Varables Jot Probablt Dstrbuto ucto (PD), Pr, 0,, or ad,,, 0,, s o-decreasg as ether or creases, ad, Jot Probablt Dest ucto (pd),, 0, or ad, d d, u, v du dv, d ad, d,, Pr pected Values Correlato, g, g,, d d d d d d Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN:
8 Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN: Codtoal Probablt--evsted M M, Pr Pr,,,,, 3.3. tatstcal Idepedece, 3.4. Correlato betwee adom Varables d d, Covarace d d, Correlato coecet or ormalzed covarace, d d,
9 3.5. Dest ucto o the um o Two adom Varables Z Z Z, u v, z du dv z d d z z d z d 3.6. Probablt Dest ucto o a ucto o Two adom Varables 3.7. The Characterstc ucto u ep j u u epj u d The verse o the characterstc ucto s the deed as: u ep j u du Computg other momets s perormed smlarl, where: d u du u 0 d u du j epj u d j d j d j Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN:
10 Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN: lemets o tatstcs 4.. Itroducto 4.. amplg Theor--The ample Mea ample Mea, where are radom varables wth a pd. Varace o the sample mea Var N N Var 4.3. amplg Theor--The ample Varace N N ubased ~ ~ N N 4.4. amplg Dstrbutos ad Codece Itervals Gaussa Z tudet s t dstrbuto T ~ k k
11 Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN: Hpothess Testg Oe tal or two-tal testg 4.6. Curve ttg ad Lear egresso C a C C b 4.7. Correlato betwee Two ets o Data C C C
12 5. adom Processes 5.. Itroducto semble or eample, assume that there s a kow AM sgal trasmtted: st b At swt at a udetermed dstace the sgal s receved as t b At swt, 0 The receved sgal s med ad low pass ltered t h t t cosw t ht b At swt cosw t,0 t h t t cosw t h t b At 0.5s wt s,0 I the lter removes the wt term, we have b At t h t t cos w t s,0 Notce that based o the value o the radom varable, the output ca chage sgcatl! rom producg o output sgal, ( 0, ), to havg the output be postve or egatve ( 0to or to ). P.. Ths s ot how ou perorm o-coheret AM demodulato. To perorm coheret AM demodulato, all I eed to do s measured the value o the radom varable ad use t to sure that the output s a mamum (.e. m wth cos w t, where t. m m 5.. Cotuous ad Dscrete adom Processes 5.3. Determstc ad Nodetermstc adom Processes 5.4. tatoar ad Nostatoar adom Processes The requremet that all margal ad jot dest uctos be depedet o the choce o tme org s requetl more strget (tghter) tha s ecessar or sstem aalss. A more relaed requremet s called statoar the wde sese: where the mea value o a radom varable s depedet o the choce o tme, t, ad that the correlato o two radom varables depeds ol upo the tme derece betwee them. That s t ad t t t t 0 0 or t t ou wll tpcall deal wth Wde-ese tatoar gals. Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN:
13 5.5. rgodc ad Noergodc adom Processes A Process or Determg tatoart ad rgodct a) d the mea ad the d momet based o the probablt b) d the tme sample mea ad tme sample d momet based o tme averagg. c) I the meas or d momets are uctos o tme o-statoar d) I the tme average mea ad momets are ot equal to the probablstc mea ad momets or t s ot statoar, the t s o ergodc. or ergodc processes, all the statstcs ca be determed rom a sgle ucto o the process. Ths ma also be stated based o the tme averages. or a ergodc process, the tme averages (epected values) equal the esemble averages (epected values). That s to sa, d lm T T T T Note that ergodct caot est uless the process s statoar! t dt 5.6. Measuremet o Process Parameters 5.7. moothg Data wth a Movg Wdow Average A Process or Determg tatoart ad rgodct a) d the mea ad the d momet based o the probablt b) d the tme sample mea ad tme sample d momet based o tme averagg. c) I the meas or d momets are uctos o tme o-statoar d) I the tme average mea ad momets are ot equal to the probablstc mea ad momets or t s ot statoar, the t s o ergodc. Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN:
14 6. Correlato uctos 6.. Itroducto 6.. ample: Autocorrelato ucto o a Bar Process t, t d d, The above ucto s vald or all processes, statoar ad o-statoar. or W processes: t, t t t I the process s ergodc, the tme average s equvalet to the probablstc epectato, or ad lm t t dt t t T T T T 6.3. Propertes o Autocorrelato uctos 0 0 ) or 0 t ) 3) 4) I has a DC compoet, the has a costat actor. 5) I has a perodc compoet, the has a wll also have a perodc compoet o the same perod. 6) I s ergodc ad zero mea ad has o perodc compoet, the lm 0 7) Autocorrelato uctos ca ot have a arbtrar shape. Oe wa o specg shapes permssble s terms o the ourer trasorm o the autocorrelato ucto. That s, the the restrcto states that ep jwt 0 or all w dt 6.4. Measuremet o Autocorrelato uctos 6.5. amples o Autocorrelato uctos Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN:
15 6.6. Crosscorrelato uctos The cross-correlato s deed as: or jotl W processes: ad t, t d d, t, t d d, t, t t t t t t t, 6.7. Propertes o Cross-correlato uctos ) The propertes o the zoreth lag have o partcular sgcace ad do ot represet mea-square values. It s true that the ordered crosscorrelatos are equal at or 0 0 ) Crosscorrelato uctos are ot geerall eve uctos. There s a atsmmetr to the ordered crosscorrelatos: 3) The crosscorrelato does ot ecessarl have ts mamum at the zeroth lag. Ths makes sese ou are correlatg a sgal wth a tmed delaed verso o tsel. The crosscorrelato should be a mamum whe the lag equals the tme dela! 4) I ad are statstcall pedet, the the orderg s ot mportat t t t t ad 5) I s a statoar radom process ad d deretable wth respect to tme, the crosscorrelato o the sgal ad t s dervatve s gve b d d 6.8. amples ad Applcatos o Crosscorrelato uctos 6.9. Correlato Matrces or ampled uctos Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN:
16 7. pectral Dest 7.. Itroducto Thereore, we ca dee a power spectral dest or the esemble as: w ep w w w t w epwt dw d 7.. elato o pectral Dest to the ourer Trasorm w w w 7.3. Propertes o pectral Dest The power spectral dest as a ucto s alwas real, postve, ad a eve ucto w pectral Dest ad the Comple requec Plae 7.5. Mea-quare Values rom pectral Dest The mea squared value o a radom process s equal to the 0 th lag o the autocorrelato 0 w epw 0 dw w 0 ep 0 dw As a ote, sce the PD s real ad smmetrc, the tegral ca be perormed as d dw Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN:
17 w 0 0 dw 0 0 d 7.6. elato o pectral Dest to the Autocorrelato ucto The ourer Trasorm ep t ep t d d 7.7. Whte Nose As a result, we dee Whte Nose as w t 0 N Cross-pectral Dest The ourer Trasorm w w ep w d ad w ep w t w epwt dw Propertes o the uctos w coj ce the cross-correalto s real, the real porto o the spectrum s eve the magar porto o the spectrum s odd ad t w epwt w d dw Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN:
18 7.9. Autocorrelato ucto stmate o pectral Dest 7.0. Perodogram stmate o pectral Dest 7.. amples ad Applcatos o pectral Dest Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN:
19 8. espose o Lear stems to adom Iputs 8.. Itroducto 8.. Aalss the Tme Doma 8.3. Mea ad Mea-quare Value o stem Output 8,4. Autocorrelato ucto o stem Output 8.5. Crosscorrelato betwee Iput ad Output 8.6. ample o Tme-Doma stem Aalss 8.7. Aalss the requec Doma 8.8. pectral Dest at the stem Output 8.9. Cross-pectral Destes betwee Iput ad Output 8.0. amples o requec-doma Aalss 8.. Numercal Computato o stem Output 9. Optmum Lear stems 9.. Itroducto 9.. Crtera o Optmalt 9.3. estrctos o the Optmum stem 9.4. Optmzato b Parameter Adjustmet 9.5. stems That Mamze gal-to-nose ato 9.6. stems That Mmze Mea-quare rror Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN:
20 Appedces A. Mathematcal Tables A.. Trgoometrc Idettes A.. Idete Itegrals A.3. Dete Itegrals A.4. ourer Trasorm Operatos A.5. ourer Trasorms A.6. Oe-ded Laplace Trasorms B. requetl coutered Probablt Dstrbutos B.. Dscrete Probablt uctos B.. Cotuous Dstrbutos C. Bomal Coecets D. Normal Probablt Dstrbuto ucto. The Q-ucto. tudet's t Dstrbuto ucto G. Computer Computatos H. Table o Correlato ucto--pectral Dest Pars I. Cotour Itegrato Notes ad gures are based o or take rom materals the course tetbook: Probablstc Methods o gal ad stem Aalss (3rd ed.) b George. Cooper ad Clare D. McGllem; Oord Press, 999. IBN:
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