Mathematics of Electrical Noise

Transcription

1 class noes, M. odwell, copyrighed 0 ECE 45B / 8B, noes : Mahemaics o Elecrical Noise Mark odwell Uniersiy o Caliornia, ana Barbara rodwell@ece.ucsb.edu , a

2 raegy class noes, M. odwell, copyrighed 0 Thereis no imein hisclass odeelop hissubec in deail. raegy: gie backroundsuicien or correc calculaion o N, signal specraldensiies, correlaion uncions, correlaions, error raes. Moredeail can be oundin my noise class noeson heweb, or in helieraure. an der Zeil's book is comprehensie.

3 Topics class noes, M. odwell, copyrighed 0 Mah: disribuions, random ariables, epecaions, pairs o, oin disribuions, coariance and correlaions. andom processes, saionariy, ergodiciy, correlaion uncions, auocorrelaion uncion, power specral densiy. Noise models o deices: hermal and sho noise. Models o resisors, diodes, ransiors, anennas. Circui noise analysis: nework represenaion. oluion. Toal oupu noise. Toal inpu noise. generaor model. En/In model. Noise igure, noise emperaure. ignal / noise raio.

4 random ariables class noes, M. odwell, copyrighed 0

5 The irs sep: random ariables class noes, M. odwell, copyrighed 0 During an eperimen, a random ariable akes on a paricular The probabiliy ha lies beween and is P{ } d is he probabiliy disribuion uncion. alue.

6 Eample: The Gaussian Disribuion class noes, M. odwell, copyrighed 0 The Gaussian disribuion : ep We will deine shorly he mean and he sandard deiaion. Because o he* cenral limi heorem*, physicalrandom processes arising rom hesum o many small eecs hae probabiliy disribuions close o ha o he Gaussian. ~

7 Mean alues and epecaions class noes, M. odwell, copyrighed 0 Epecaion o a uncion g o he random ariable E g g d Mean alue o E d Epeced alue o E d

8 ariance class noes, M. odwell, copyrighed 0 The ariance rom is aerage o is alue is roo E mean - - square deiaion d The sandard deiaion hesquare roo o he ariance o is simply

9 eurning o he Gaussian Disribuion class noes, M. odwell, copyrighed 0 The noaion describing he Gaussian disribuion : should now be clear. ep ~

10 class noes, M. odwell, copyrighed 0 ariance s Epecaion o he quare he epecaion. he square o minus he square he epecaion o is The ariance

11 Pairs o andom ariables class noes, M. odwell, copyrighed 0 Toundersand random processes, we mus irs undersand pairs o random ariables. In an eperimen, a pair o random ariables akes on speciic paricular alues and y. and Y Their oin behaior is described by he oin disribuion Y, y P{ A B and C y D} D B C A Y, y ddy

12 Pairs o andom ariables class noes, M. odwell, copyrighed 0 Marginal disribuions mus also be deined P{ A B } B A B A Y, d y ddy and similarly or Y : P{ C y D } D C D C Y Y, y dy y ddy

13 aisical Independence class noes, M. odwell, copyrighed 0 In hecase where Y, y Y y, he ariables are said o be saisically independen. This is no generally epeced.

14 Epecaions o a pair o random ariables class noes, M. odwell, copyrighed 0 The epecaion o a uncion g, Y o he random ariables Y and Y is E g, y g, y Y, y ddy Epecaion o : E Y, y ddy d Epecaion o E Y, y ddy d...and similarly or Y and Y.

15 Correlaion beween random ariables class noes, M. odwell, copyrighed 0 The correlaion o and Y is Y E Y y Y, y ddy The coariance C Y E o and Y is Y y EY Y y y Y y Noe ha correlaion and coariance are hesame i eiher or Y hae zero mean alues.

16 Correlaion ersus Coariance class noes, M. odwell, copyrighed 0 When we are working wih olages and currens, we usually separae he mean alue DC bias rom he ime - arying componen. The random ariables hen hae zero mean. Correlaion is hen equal o coariance. I is hereore common in circui noise analysis o use he wo erms inerchangably. Bu, nonzero mean alues can reurn when we e.g. calculae condiional disribuions. Be careul.

17 Correlaion Coeicien class noes, M. odwell, copyrighed 0 The correlaion coeicien o Y CY / Y and Y is Noe hesandard conusion in erminology beween correlaion and coariance.

18 um o TWO andom ariables class noes, M. odwell, copyrighed 0 um o worandom ariables : Z Y E I E Z E Y E Y Y E E Y and Y boh hae zero means Z E E Y C Y Y This emphasizes he role o correlaion.

19 Pairs o Joinly Gaussian andom ariables class noes, M. odwell, copyrighed 0 I and Y are Joinly Gaussian : Y, y Y Y ep Y y Y y y y Y This deiniion can be eended oa larger # o ariables. In general, we can hae a Joinly Gaussian random ecor, means,, i n which is speciied by a se o, ariances E, and coariance s E i i i

20 Linear Operaions on JG's class noes, M. odwell, copyrighed 0 I and Y are Joinly Gaussian, and i we deine a by and W c dy Then and W are also Joinly Gaussian. This is saed wihou proo;he resul arises conoluion o Gaussian uncions produces a Gaussian uncion. because The resul holds or JGs o any number.

21 class noes, M. odwell, copyrighed 0 Probabiliy disribuion aer a Linear Operaion on JG's E C W W Ea by E E W a E b EY ab EY W c E d EY cd EY W E a by c dy E W E ac ace a by and W ad bc Y bdy W ad bc EY bd EY c W dy a a c by by c dy dy edious deails We can now calculae he oin disribuion o and W. W, w W W ep W w W w w w W

22 Why are JG's Imporan? class noes, M. odwell, copyrighed 0 The mah on helas slide was edious bu here is a clear conclusion : Wih JG's subeced o linear operaions, i is suicien o keep rack o means, correlaions, and ariances. Wih his inormaion, disribuion uncions can always be simply ound. This asly simpliies calculaio ns o noise propagaion in linear sysems linear circuis.

23 Uncorrelaed ariables. class noes, M. odwell, copyrighed 0 Uncorrelaed : C Y 0 aisically independen Y, y Y y : Independence implies zero correlaion. Zero correlaion does no imply independence. For JG's, uncorrelaed does imply independence

24 class noes, M. odwell, copyrighed 0 umming o Noise andom olages do add. he random noise olages o ime alues The insananeous - a correllai on erm mus be included. do no add - he wo random generaors The noise powerso P a random ariable he resisor is dissipaed in The power are applied o he resisor Two olages C P P E

25 ho Noise as a andom ariable class noes, M. odwell, copyrighed 0 The iber has ransmission probabiliy p. end one phoon,and call he # o receied phoonsn. E N N p and EN p so EN N N p p I we now send many phoons M o hem, ransmission o each is saisically independen,so- - - calling he # o receied phoonsn, E N M EN Mp and N M N M p p Now supposem, p, and Mp, N N The ariance o he coun approaches he mean alue o he coun.

26 Thermal Noise as a andom ariable class noes, M. odwell, copyrighed 0 A capacior C is conneced oa resisor. The resisor is in equilibriu m wih a "reseroir" a warm room a emperaure T can echange energy wih he room in he orm o hea. C can dissipae no power: i esablishes hermal equilibriu m wih he room ia he resisor. From hermodynamics, any independen degree o reedom o a sysem a emperaure T has mean energy kt/, hence E kt / C / kt / kt / C The noise olage has ariance kt/c.

27 random processes class noes, M. odwell, copyrighed 0

28 andom Processes class noes, M. odwell, copyrighed 0 Draw a se o graphs, on separae shees o o uncions o olage s. ime. paper, Pu hemino a garbage can. This garbage can is called he probabiliy sample space. Pick ou one shee a random. This is our random uncion o ime. The random process is. The paricular oucome is

29 Time Aerages s. ample pace Aerages class noes, M. odwell, copyrighed 0 ecall he deinion o he epecaion o a uncion g o a random ariable E g g d g g is he* aerage alue * o g, where he aerage is oer he sample space. Wih our random process deiniion, wecan deine an aerage oer he sample space a some paricular ime E A g g g g d d We can also deine an aerage o he uncion oer ime : :

30 Ergodic andom Processes class noes, M. odwell, copyrighed 0 An Ergodic random process has aerages oer ime equal o aerages oer hesaisical sample space E g Ag In some sense, we hae made "random ariaion wih ime" equialen o "random ariaion oer hesample space"

31 Time amples o andom Processes class noes, M. odwell, copyrighed 0 Wih ime samples a imes and he random process has alues and. and hae some oin probabiliy disribuion. They migh or migh no be oinly Gaussian.

32 andom Waeorms are andom ecors class noes, M. odwell, copyrighed 0 Using Nyquis' s sampling heorem, i a random signal and i is we pick regularly bandlimied, - spaced ime samples... n we coner our random process ino a random ecor., We can hus analyze random signals ecor analysis and geomery. using This is mosly beyond he scope o his class. n

33 class noes, M. odwell, copyrighed 0 lower orders,, order saionariy: lower orders..and,...,,,...,, order saionariy: N a saionary process do no ary wih ime. The saisicso nd h E E E E E E n n aionary andom Processes

34 class noes, M. odwell, copyrighed 0 esricions on he random processes we consider We will make ollowing resricions o make analysis racable: The process will be Ergodic. The process will be saionary o any order: all saisical properies are independen o ime. Many common processes are no saionary, including inegraed whie noise and / noise. The process will be Joinly Gaussian. This means ha i he alues o a random process are sampled a imes,, ec, o orm random ariables =, ec, hen,, ec. are a oinly Gaussian random ariable. In naure, many random processes resul rom he sum o a as number o small underlying random processses. From he cenral limi heorem, such processes can requenly be epeced o be Joinly Gaussian.

35 ariaion o a random process wih ime class noes, M. odwell, copyrighed 0 For he random process, look a = and =. E, d d To compue his we need o know he oin probabiliy disribuion. We hae assumed a Gaussian process. The aboe is called he Auocorrellaion uncion. IF he process is saionary, i is a uncion only o - =au, and hence E his is he auocorrellaion uncion. I describes how rapidly a random olage aries wih ime. PLEAE recall we are assuming zero-mean random processes DC bias subraced. Thus he auocorrellaion and he auo-coariance are he same

36 ariaion o a random process wih ime class noes, M. odwell, copyrighed 0 Noe ha 0 E gies he ariance o he random process. The auocorrelaion uncion gies us ariance o he random process and he correlaion beween is alues or wo momens in ime. I he process is Gaussian, his is enough o compleely describe he process. Narrow auocorrelaion: Fas ariaion Broad auocorrelaion low ariaion

37 class noes, M. odwell, copyrighed 0 Auocorrelaion is an Esimae o he ariaion wih Time I random ariables and Y are Joinly Gaussian, and hae zero mean, hen knowledge o he alue y o he ouome o Y resuls in a bes esimae o as ollows: E Y y Y y Y Y y "The epeced alue o he random ariable, gien ha he random ariable Y has alue y is..." Hence, he auocorrellaion uncion ells us he degree o which he signal a ime is relaed o he signal a ime A narrow auocorrelaion is indicaie o a quickly-arying random process

38 Power specral densiies class noes, M. odwell, copyrighed 0 The auocorrellaion uncion describes how a random process eoles wih ime. Find is Fourier ransorm: ep d This is called he power specral densiy o he signal. emembering he usual Fourier ransorm relaionships, i he power specrum is broad, he auocorrellaion uncion is narrow, and he signal aries rapidly--i has conen a high requencies, and he olages o any wo poins are srongly relaed only i he wo poins are close ogeher in ime. I he power specrum is narrow, he auocorrellaion uncion is broad, and he signal aries slowly--i has conen only a low requencies and he olages o any wo poins are srongly relaed unless i he wo poins are broadly separaed in ime.

39 Power specral densiies class noes, M. odwell, copyrighed 0 omega au omega au ime omega au omega au ime

40 Power pecral Densiies class noes, M. odwell, copyrighed 0 ecall he Fourier ransorm o speciically, 0 he power ha hepower specraldensiy The inerse ransorm holds, so ha o, i is called ep d ep d he power in specral densiy will he auocorrelaion uncion d he process, hen inegraing gie us is he power. This is he usiicaion or he erm,"power specral densiy"

41 Correlaed andom Processes class noes, M. odwell, copyrighed 0 Two processes can be saisically relaed. Consider wo random processes and Y. Deine Y he cross - correllaion uncion o E Y he processes They willhae a cross - specraldensiy as ollows : Y and hereore Y Y ep d ep d Y

42 ingle-ided Hz-based pecral Densiies class noes, M. odwell, copyrighed 0 Double - ided pecral Densiies E ep d ep d ingle - ided Hz - based pecral Densiies ~ E ep d ~ ep d

43 class noes, M. odwell, copyrighed 0 ingle-ided Hz-based pecral Densiies- Why? Why hisnoaion? The signal Power ~ per Hz o power in is low ~ high signal a requencie s lying close he bandwidh d d direcly he Was o bandwidh high low ~ low signal o herequency, high high low power. ~ d

44 class noes, M. odwell, copyrighed 0 ingle-ided Hz-based Cross pecral Densiies Double - ided Cross pecral Densiies ~ Y Y Y Y E Y Y E Y Y ep ep d ingle - ided Hz - based Cross pecral Densiies ~ Y Y d ep d ep d ~ Y d d is also oen wrien as Y

45 Eample: Cross pecral Densiies class noes, M. odwell, copyrighed 0 Y E Y Y YY Y Y Y P= / * YY Y Y e YY Y Or, in single ~ ~ - sided specral densiies ~ ~ e YY Y

46 Eample: Cross pecral Densiies class noes, M. odwell, copyrighed 0 The / / 0 E Power P / has epeced alue Y P= / And in he bandwidh beween P high low ~ d... low and Inegraing wih respec orequency oer whaeer bandwidh is gies he oalepeced power dissipaed in. high, relean Noe ha hecross - specral densiy is relean.

47 class noes, M. odwell, copyrighed 0 Our Noaion or pecral Densiies and Correlaions / ~ / ~ cross specral densiy crosscorrelaion uncion / ~ / ~ specral densiy power auocorrelaion uncion,, requency uncion o ime uncion o Oucome andom Process * * y y A Y E A E y y y y y Y Y Y Y y Y F F F F and processes For saionary ergodic * * y y Y. e wecan simply wri, or i clear wheher cone makes When

48 class noes, M. odwell, copyrighed 0 Eample: Noise passing hrough ilers & linear elecrical neworks I he iler has impulse responseh and ranser uncion h, hen or any in ou, ou h in o ou ou ou ou ou * ou h h h in in in in in h * * in ou ou ou in in * in h h h in in in in in * in in h ou I is riial ochange osingle - sided Hz - based specral densiies.