Mid-l.errrL

Pl The meari ergodicity theorem.

P2 Letra*domvariatrle{bedistributr:tlrrccortlingto: u,e(.rj :{ ''-n

if re [0, 1] ,*h".oAi"*[ 0 e]sewtrtre

real constant. Let random variable 4: {2.ia) Determine constant 3.(b) Compute and lepresent thc curnulative distribution function of (.(c) Compute the mean and variturce of {.(c1) Dr:Li:rrnine the probability densitv function alcl the rnean of r7.

P3 Let signal ((t) : /siri(2*) +4cos(2{), with / and 17 br:ing inclepenclent ranclorn variables r:{ 0rneart and variance equal to 4. Dcternine the rnean and the autocorrelation. functiori of thesiglral ((t). Deternriue the autocorrelation fr.urction of the signal outprt by an ideal higlipassfi.lter with cutting frcquency t.,ro : 3 fed with {(t) at t}ie input.

Final

Fl Linear, tim+'invarialt systems: definitiorr, relatirtrr betrveen input and output signal, rclationbctween statistical miaslrros (autocorrclation functirrns) uf irrput *url output siglals.

F2 Pararuetcr estirnation: prol,lem posing, determination of the optimal estimator i1 the solse ofunilclrn cost fitnctir:u.

F3 Tra;usmitterl sigrals on a binary transmission chaitr are zs(f) : s:ign(cos(2rf)) arrd z1(i) :sign(cos(lrt+n)) for,

[0,2] . T]rc noise that a,ffccts transmissiori is additive, white. signal-independelt witlr a Gaussian distribution oI tnit meal and unit variancc. Decision over thetrammittcd signal is basecl on tra,r: samplcs of the received signal, taken at rno11cnts i, : I 1and t,: f . Wc assrnr-re that 2P6: Pr, ccjral costs of for correct clecisions md ec1ual ctrsts forwrorrg decisions.

{a) Dr::termine the sirnplest fcirrn cf Baves clecision criteriou.(h) Wliat will be the dccision for recciv'ed vector r

- [-0.7500.?5]?(c) Determiue P(DolS:).