Bit Error Rate in Digital Photoreceivers

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ĒΓαβριήλ Παπαφιλίππου
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1 Bit Eo Rat in Digital hotocivs In th pvious slids, w saw that th photociv aks a dcision as to whth th covd wavfo is abov ( o blow ( th thshold lvl. Whn nois is psnt, a wong dcision can b ad, i.. w hav a bit o. W will now ain tchniqus fo calculating th bit o pobability and hnc th BER. Γ. Έλληνας, Διάλεξη 3-4, ελ. 9 Digital hotociv Rcovd puls tain (output voltag ntic-hall Γ. Έλληνας, Διάλεξη 3-4, ελ. 3 ag 5

2 Eapl of a bit o Wily Bit os a a consqunc of th nois psnt on th civd signal. Sinc th nois is ando and pobabilistic, it can b dscibd using a ando vaiabl. Γ. Έλληνας, Διάλεξη 3-4, ελ. 3 Eapl of a bit o S S Only two typs of bit can b snt in a binay syst: s and s. Ths vnts a utually clusiv, so w hav (S + (S. S is th vnt was snt S is th vnt was snt D D Only two typs of dcision can b ad: th dtctd signal is abov o blow th thshold lvl, i.. ith a o a is dtctd. Ths vnts a utually clusiv, so w hav (D + (D. D is th vnt was dtctd D is th vnt was dtctd Γ. Έλληνας, Διάλεξη 3-4, ελ. 3 ag 6

3 Conditional pobabilitis S S D.S D.S D.S D.S D D A total of fou utually clusiv outcos a possibl in a binay counications syst Γ. Έλληνας, Διάλεξη 3-4, ελ. 33 Conditional pobabilitis D.S D.S Th shadd gions psnt vnts that giv a bit o: D. S a is dtctd and a was snt D.S D.S D.S a is dtctd and a was snt Ths two vnts a utually clusiv, hnc: ( bit o ( D. S + ( D. S Γ. Έλληνας, Διάλεξη 3-4, ελ. 34 ag 7

4 obability of bit o ( D / S ( D. S ( S Bay s foula Raanging givs: ( D. S ( S ( D / S Siilaly, w hav: ( D S ( S ( D /. S Thus th bit o pobability can b wittn as: ( bit o ( S ( D / S + ( S ( D / S Γ. Έλληνας, Διάλεξη 3-4, ελ. 35 obability of bit o Th pvious foula can b usd to calculat th bit o pobability povidd: w know what th pobabilitis of snding s and s a (oftn w hav (S (S.5 and w can obtain th conditional pobabilitis (D /S and (D /S. W can obtain (D /S and (D /S if w know what th DFs associatd with cption of th bits and in th psnc of nois a. Ths pocsss can b vy accuatly appoiatd by gaussian ando vaiabls; th gaussian DF is plottd on th nt slid. Γ. Έλληνας, Διάλεξη 3-4, ελ. 36 ag 8

5 p Gaussian DF ( ( π p( Γ. Έλληνας, Διάλεξη 3-4, ελ. 37 Gaussian DF Th gaussian DF occus vy widly in any applications (and fo that ason is also calld th Noal distibution. On ason fo this is th cntal liit tho. This tho tlls us that if w tak th su of a lag nub of indpndnt vaiabls X, X,... X n, and if ach of ths aks a sall contibution to th su X X + X X n, thn th DF of X will appoach a gaussian shap as n. Th poof is byond th scop of this cous, but th ida can b illustatd bst by an apl,.g. oll n dic and add thi valus. If this vnt is patd nough tis, you gt a gaussian distibution. Γ. Έλληνας, Διάλεξη 3-4, ελ. 38 ag 9

6 optis of th gaussian DF p( ( X ( X an: X.5 by syty is th standad dviation: whn p( is usd to dscib th pobability of dtcting a nois cunt (o voltag thn psnts th s valu of th nois cunt (o voltag. Γ. Έλληνας, Διάλεξη 3-4, ελ. 39 Obtaining pobabilitis fo th gaussian DF Whn calculating th bit o pobability lat on, w will hav to valuat pobabilitis such as: ( X p( d This pssion cannot b calculatd analytically, w ust us nuical tchniqus. W dfin: Q( k π This can b obtaind nuically and thn plottd: k y dy Γ. Έλληνας, Διάλεξη 3-4, ελ. 4 ag

7 ag Γ. Έλληνας, Διάλεξη 3-4, ελ. 4 Q(k Γ. Έλληνας, Διάλεξη 3-4, ελ. 4 To calculat: [ ] d X ( ( π Lt: y π Q X dy X y / ( ( Obtaining pobabilitis fo th gaussian DF

8 Obtaining pobabilitis fo th gaussian DF p( ( X ( X Q p ( d Γ. Έλληνας, Διάλεξη 3-4, ελ. 43 Towads BER... In th contt of ou digital photociv, w can say that output voltag v(t gnatd idiatly aft th aplifi stag in spons to th tansission of and will hav an valus of V and V fo ths two pulss. Th thshold lvl (V th will b st btwn ths two valus. Howv, nois (du.g. to thal and aplifi contibutions will b supiposd on ths an valus, and th distibutions will follow that of a gaussian DF. Hnc th civd voltags fo and hav DFs givn by p (v and p (v spctivly: Γ. Έλληνας, Διάλεξη 3-4, ελ. 44 ag

9 Towads BER... dtctd voltag, v p (v (D /S (D /S V V th V p (v Assu Γ. Έλληνας, Διάλεξη 3-4, ελ. 45 QUESTION W saw ali that th bit o pobability is: + ( S ( D / S ( S ( D / S If w assu that ons and zos a qually likly to b snt, thn (S (S.5 and: [ D / S ( D / ] ( S + By considing an NRZ wavfo with V, and picking a thshold idway btwn this and V, i.. V th V /, show that: Q V ( Γ. Έλληνας, Διάλεξη 3-4, ελ. 46 ag 3

10 Bit Eo Rat in Digital hotocivs In th pvious slids, w saw that th photociv aks an o whnv nois pushs th wavfo to th wong sid of th thshold lvl. W also saw that w could odl this pocss using th gaussian distibution. W will now finish ou tatnt by showing how BER is latd to SNR. Γ. Έλληνας, Διάλεξη 3-4, ελ. 47 Digital hotociv Bit os can b ad h; th nub dpnds on th SNR of th civd signal Rcovd puls tain (output voltag ntic-hall Γ. Έλληνας, Διάλεξη 3-4, ελ. 48 ag 4

11 W saw ali that th bit o pobability is: Towads BER... + ( S ( D / S ( S ( D / S If w assu that ons and zos a qually likly to b snt, thn (S (S.5 and: [ D / S ( D / ] ( S + W will consid a NRZ wavfo with V, and pick a thshold idway btwn this and V, i.. V th V /. W f to this as a unipola wavfo. Γ. Έλληνας, Διάλεξη 3-4, ελ. 49 Towads BER... ( D / S Vth V p (v p (v p v ( v π v ( D / S ( v V th p V th ( v dv Γ. Έλληνας, Διάλεξη 3-4, ελ. 5 ag 5

12 Using th lationship: w hav: Towads BER... ( X Q D / S ( v V ( th Vth Q Γ. Έλληνας, Διάλεξη 3-4, ελ. 5 Towads BER... ( v V ( D / S p ( v π V th V p (v p (v v V th ( D / S ( v V th p ( v dv Γ. Έλληνας, Διάλεξη 3-4, ελ. 5 ag 6

13 Towads BER... By syty, w hav: V th V 3V th p (v Gn aa black aa ( D / S p( v 3 V th v dv Γ. Έλληνας, Διάλεξη 3-4, ελ. 53 Towads BER... Using w hav: ( X Q ( D / S ( v 3V Q 3V th Vth Q th V Γ. Έλληνας, Διάλεξη 3-4, ελ. 54 ag 7

14 ag 8 Γ. Έλληνας, Διάλεξη 3-4, ελ. 55 [ ] + / ( / ( th V Q V Q S D S D Hnc: Now, b that is th s nois voltag, so: an squa nois pow Towads BER... Γ. Έλληνας, Διάλεξη 3-4, ελ. 56 Also, if ons and zos a qually likly, Hnc th SNR is: an squa signal pow [ ] V V V + V Copaing with th bit o pobability, SNR Q V Q Bit Eo obability

15 plot of Q function Fo plot of Q function, fo -9, nd to find Q(k -9, which givs k 6.. Γ. Έλληνας, Διάλεξη 3-4, ελ. 57 and SNR Hnc w hav fo -9 : Q SNR Fo th plot of Q(k vsus k, w hav k 6., 9 i..: SNR 6. SNR 7. In db, w hav SNR log ( db Γ. Έλληνας, Διάλεξη 3-4, ελ. 58 ag 9

16 BER vsus SNR fo unipola NRZ Bit o pobability SNR (db Γ. Έλληνας, Διάλεξη 3-4, ελ. 59 Coplntay o function Not that w hav usd Q(k in ths calculations; ost ttbooks ak us of th coplntay o function fc( dfind as: fc ( π It is staightfowad to show this is latd to Q(k as follows: Q( k fc u k du (MATLAB, fo apl, uss fc(, not Q( Γ. Έλληνας, Διάλεξη 3-4, ελ. 6 ag 3

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