Question

3. (40 points) A binary communication system transmits signals s (0) (i = 1, 2). The receiver samples the received signal r(t) = s(t)+ n(t) at T and obtain the decision statistic r =r(T) = S(T) + n(T) = a, un, where the signal component is either a = + A or a, = -A with A >0 and n is the noise component. Assume that s (6) and s(l) are equally likely to be transmitted and the decision threshold is chosen as zero. a) If the noise component n is AWGN with zero mean and PSD (power spectral density) N./2, derive the theoretical expression of bit-error rate (BER) of this system as a function of the parameters A and N.. b) If the noise component n is uniformly distributed over [-1, +1] and A=0.8, derive the expression of BER of this system.

Answer 1

solltion in probability do error calculations in nolt) With Inst) [ai (+) and noise The received signal a(t) = silt) +01+) the opp of the filter is consist of signal component component [nolt)] ::21+)= ailt) + nolt) . assume that filter olp is sampled at t= Tb - E(Tb) = ai (76) + no (To) . The ilp to the threshold comparater is z = aitho the input of the reciever case 1 in Assume that the signal is not present at [s: (+) = 0 CS The analpe to the threshold comparate [zone) amScanner

& "no is assumed as a "Gaussion random variable with mean 20" NEO E(z) , Elno) zo [since mean = o assume ) mio - 022m2 Liftz) Poze m=2² noise power. ) m o noise power) - to z case lii) at the elp of the reciever symbol '1' is present Assume that binary St+) 264 x 1 2 Eustatheus) Tc het) Da,(+) holit) et noles z=a,tno ECZ) = f (as) + E (no) a auto -9, In this case also à is a gaussion $(4) l a random variable but mean is as I (2-2,37/2012 Cage füi) present at the ilp ats the symbol o is reciever Assume that binary wonobor sz=a, tno lp of comparater E(Z) = E[az] + E [no] Scanned with 792 +0 CamScanner sa2

-(2-2, 12 e 282 502 daca =) be binary symmery ag channel Ilio the ilp to the threshold comparater is either a, t no or azt no. so the threshold voltage is aita2 229, +no x to with 24itary threshold comparater is when t is silt). the ilp to the n reciever the ilp to the 2=9, +no as oilers aror occurs when the ilp to Pinary simbol i will be recieved voltage. . the comparater is less than the threshold 1 terror occurs [zeuth] Per = P(zevth) 7 - thf) dz — O DS 07 error occurs (2> Vth ) Freo =P(Z >uth )] = 8/240)dz — @ 5th 1 0 . 7 1 Pe, 2034 PRO 2012 0 Here peitpeo effect of noise is not same on both symbols o practically atteet of noise is same both the symbols Peq=Peo] Then the channel is called as binary symmetric 0.9 Scanned with CamScanner channel.

practically the after of nois is came on both the symbols so the eorar probability will be also same for both the symbols In this case the channel is called as "Binary symmetric channel " (BSc). H2/0), UTTA with a л Peo prestare s-42-02)265202 -(2- QO2 a taz wa? a(z) = a 12-71 day Q(1) 1 let y = 2.92 » dy - dz pe a to su est day - t r 5 estray 4 Z-42 0 9²/2 def Pe a 282 A-92 a,.az 20 compare with ala) pe - Q =0 (2) Pe minimum : 0(a) : minimum x is maximium Tre Scanned with CamScanner

wid I - a fait son obainen - afd difference signal power YNO V ux noise power - S, (t) = A Sz(t)= -A Ed of teza)? dt =yaTb pe = a[/antonio BER = P(1) Re, + Plo) peo Af(Z) gif (zlo) 126) Nutz) 02 20.2 -0.2 201) *(21) ds = f* $(27.) Az = 3° 14d2.2011 Peo) = 1 **t-lo) dz = *llo) de ) *942 -01 BER= {(0,1)+ į Cool) [azume Flo7= P(1) - +] (BER = 0.17 CS Scanned with CamScanner