Περιεχόμενα διάλεξης

3-4η Διάλεξη Bit Eo Rate Receive Sensitivity Γ. Έλληνας, Διάλεξη 3-4, σελ. Περιεχόμενα διάλεξης BER in Digital hotoeceives obability Theoy obability of Bit Eo SNR Quantu Liit on Optical Receive Sensitivity Digital Counications Analog Counications Γ. Έλληνας, Διάλεξη 3-4, σελ. age

BER in Digital hotoeceives Γ. Έλληνας, Διάλεξη 3-4, σελ. 3 Bit Eo Rate in Digital hotoeceives Although SNR is a vey useful figue of eit, in a digital syste we ae concened with accuately detecting only two types of signal level: binay zeos and ones. The photoeceive theefoe akes a decision as to whethe the ecoveed wavefo is above ( ) o below ( ) the theshold level. (This decision is usually ade halfway though the bit duation T B ). When noise is pesent, it can lead to a wong decision, i.e. a bit eo. Γ. Έλληνας, Διάλεξη 3-4, σελ. 4 age

Effect of noise analog IM wavefo Digital IM wavefo Γ. Έλληνας, Διάλεξη 3-4, σελ. 5 Wiley obability of Eo The bit eo ate (BER) is obtained by dividing the nube of eos (N e ) occuing ove a tie inteval t by the nube of pulses (ones and zeos) tansitted duing this inteval (N t ): BER N N e t N e B t T B T is the bit ate (bits/sec) and is equivalent to /T B whee T B is the bit duation. We assue that a is high and is low fo the duation of T B, but othe line coding schees ae possible. Γ. Έλληνας, Διάλεξη 3-4, σελ. 6 age 3

Coon data (line coding) foats Siple, but with no inheent Eo onitoing o coecting Capabilities and no tiing featues oble with long stings of bits -Loss of tiing synchonization N.B. We will assue NRZ coding hee. -Unipola code Tansition at the cente of each bit inteval. Negative Tansition and positive a. Siple to geneate and decode. Double the BW of an NRZ code, no inheent Eo onitoing o coecting Capabilities Γ. Έλληνας, Διάλεξη 3-4, σελ. 7 BER Given that ost of the noise is added at the photoeceive itself, it is desiable to axiize the signal level (at the input to the eceive) so that the pobability of bit eos can be educed. Howeve, this conflicts with the need to axiize tansission distances. The ability to calculate BER fo a given SNR is useful in calculating the eceive sensitivity. This equies use of pobability theoy. Γ. Έλληνας, Διάλεξη 3-4, σελ. 8 age 4

Γ. Έλληνας, Διάλεξη 3-4, σελ. 9 Soe obability Theoy Suppose we have an expeient that gives ise to an event A. If the expeient is epeated N ties and A occus n A ties, then the pobability of A occuing is given by: ( A) li N n N A It follows that: ( A) Exaple: Expeient: Tossing a coin Event A: Appeaance of tails (A).5 Γ. Έλληνας, Διάλεξη 3-4, σελ. age 5

Soe obability Theoy In an expeient giving ise to events A, A,.., A n, The event (A OR A... OR A n ) can be witten as A + A... + A n The event (A AND A... AND A n ) can be witten as A. A.... A n Conside a Venn diaga: Aea is (A ) A A Whole event space E (E) Γ. Έλληνας, Διάλεξη 3-4, σελ. Soe obability Theoy A A A OR A A + A A A A A A AND A A. A A A Γ. Έλληνας, Διάλεξη 3-4, σελ. age 6

Soe obability Theoy Exaple: Expeient: Tossing a coin Event A : Appeaance of heads; (A ).5 Event A : Appeaance of tails; (A ).5 A A A A ( A + A ) A A ( A. A ) In this case, A and A ae utually exclusive events; thei aeas do not intesect one anothe. Γ. Έλληνας, Διάλεξη 3-4, σελ. 3 Mutually exclusive events e.g. event A today is Sunday, event A today is Monday; cannot have the joint event A. A ( today is Sunday AND today is Monday) A 3 A A A n By inspection of the Venn diaga, fo utually exclusive events we have: ( A + A +... + A n ) (A ) + (A ) +... + (A n ) ( A. A.... A n ) Γ. Έλληνας, Διάλεξη 3-4, σελ. 4 age 7

Joint pobability A A A A (A ) (A ) A A A A (A. A ) (A + A ) ( ) ( ) ( ) ( A + A ) A + A A. A Γ. Έλληνας, Διάλεξη 3-4, σελ. 5 Conditional pobability A conditional pobability is one such as: (A /A ) pobability that A occus given that A has occued If A has occued, the event space is effectively educed to the aea shown in the Venn diaga below: A A (A ) new event space Γ. Έλληνας, Διάλεξη 3-4, σελ. 6 age 8

Conditional pobability A A (A ) new event space A A (A. A ) (A /A ) pobability that A occus given that A has occued puple aea/gold aea ( A / A ) ( A. A ) ( A ) Baye s foula Γ. Έλληνας, Διάλεξη 3-4, σελ. 7 Rando Vaiables A ando vaiable is a function X whose values depend on events Exaple: Toss a coin; if a head appeas, let X(head), if a tail appeas, let X(tail). In othe wods, a ando vaiable assigns nueical values (eal nubes) to events (such as the appeaance of heads, tails) which theselves ay o ay not be nueical in natue. The above exaple is a discete ando vaiable; X takes on a discete nube (in this case two) of values. Γ. Έλληνας, Διάλεξη 3-4, σελ. 8 age 9

Continuous Rando Vaiables A continuous ando vaiable is one whee X takes on a continuous ange of (eal nube) values. Exaple: Expeient: switch on a sinusoidal oscillato and easue its phase θ at an instant in tie. The phase can take on any value in the ange -π θ π If we choose the ando vaiable X(θ) cos θ, then evey value in the ange - X is a possible outcoe of this expeient. Altenatively, we could siply choose X (θ) θ, in which case evey value in the ange - π X πis a possible outcoe of this expeient. Γ. Έλληνας, Διάλεξη 3-4, σελ. 9 Cuulative Distibution Function (CDF) A continuous ando vaiable, by its vey natue, will have an uncountable nube of possible values, so that the pobability of obseving a specific value of X is vanishingly sall. Howeve, the pobability that X is between soe ange does exist; if we take the above exaple, with a ando vaiable X θ, then the pobability (X ).5. In geneal, we ae inteested in the pobability (X x). This is known as the cuulative distibution function (CDF) and is denoted by F(x). Γ. Έλληνας, Διάλεξη 3-4, σελ. age

opeties of the CDF F(x) F(- ) (X - ) F( ) (X ) (X x) fo all x, i.e. F(x) fo all x F(x) does not decease as x inceases If x < x, then: (x < X x ) F(x ) - F(x ) Γ. Έλληνας, Διάλεξη 3-4, σελ. - Last popety follows fo the utual exclusivity x < X x EVENT C x x X x EVENT B X x EVENT A EVENT B + EVENT C EVENT B and EVENT C ae utually exclusive Hence (A) (B) + (C) (C) (A) - (B) EVENT A (x < X x ) (X x ) - (X x ) F(x ) - F(x ) Γ. Έλληνας, Διάλεξη 3-4, σελ. age

obability Density Function (DF) Fo basic calculus, we have: x x df ( x) dx dx x [ F ( x) ] F ( x ) F ( x ) x The deivative df(x)/dx is given the sybol p(x) and is known as the pobability density function (DF). x x p ( x ) dx F ( x ) F ( x ) ( x X x ) Γ. Έλληνας, Διάλεξη 3-4, σελ. 3 opeties of the obability Density Function Fo the popeties fo F(x), we have: F(- ), F( ) p( x) dx F ( ) F ( ) Since F(x) neve deceases with inceasing x, it follows that its slope df(x)/dx, hence p(x). Γ. Έλληνας, Διάλεξη 3-4, σελ. 4 age

Exaple Conside the sinusoidal oscillato entioned ealie. The output will be of the fo: v( t) V cos ( ω t + φ ) V cos θ ( t) π/ Hence the phase will vay with tie as: θ(t ) π t π θ ( t) + φ ads T - π θ(t ): value of phase θ easued at tie t t - π/ Γ. Έλληνας, Διάλεξη 3-4, σελ. 5 Exaple π -π φ θ tie Because the phase vaies linealy with tie, thee is an equal likelihood of easuing the phase to be any value in the ange -π θ π. So, fo exaple, if X(θ) is the ando vaiable whose value is given by X(θ) θ, then: (θ a < X θ a + Δθ) (θ b < X θ b + Δθ) θ a and θ b ae any two abitay values in -π θ π Γ. Έλληνας, Διάλεξη 3-4, σελ. 6 age 3

Exaple So p(θ) will have a constant value ove -π θ π, and will be zeo elsewhee. Since we ust have p( θ ) dθ it follows that p(θ) will be unifoly distibuted as shown, i.e. this is an exaple of a unifo pobability distibution: p(θ) /π -π π θ Γ. Έλληνας, Διάλεξη 3-4, σελ. 7 Exaple p(θ) /π -π π F(θ) Unifo DF θ Coesponding CDF -π π θ Γ. Έλληνας, Διάλεξη 3-4, σελ. 8 age 4

Bit Eo Rate in Digital hotoeceives In the pevious slides, we saw that the photoeceive akes a decision as to whethe the ecoveed wavefo is above ( ) o below ( ) the theshold level. When noise is pesent, a wong decision can be ade, i.e. we have a bit eo. We will now exaine techniques fo calculating the bit eo pobability and hence the BER. Γ. Έλληνας, Διάλεξη 3-4, σελ. 9 Digital hotoeceive Recoveed pulse tain (output voltage) entice-hall Γ. Έλληνας, Διάλεξη 3-4, σελ. 3 age 5

Exaple of a bit eo Wiley Bit eos ae a consequence of the noise pesent on the eceived signal. Since the noise is ando and pobabilistic, it can be descibed using a ando vaiable. Γ. Έλληνας, Διάλεξη 3-4, σελ. 3 Exaple of a bit eo S S Only two types of bit can be sent in a binay syste: s and s. These events ae utually exclusive, so we have (S ) + (S ). S is the event was sent S is the event was sent D D Only two types of decision can be ade: the detected signal is above o below the theshold level, i.e. eithe a o a is detected. These events ae utually exclusive, so we have (D ) + (D ). D is the event was detected D is the event was detected Γ. Έλληνας, Διάλεξη 3-4, σελ. 3 age 6

Conditional pobabilities S S D.S D.S D.S D.S D D A total of fou utually exclusive outcoes ae possible in a binay counications syste Γ. Έλληνας, Διάλεξη 3-4, σελ. 33 Conditional pobabilities D.S D.S The shaded egions epesent events that give a bit eo: D. S a is detected and a was sent D.S D.S D.S a is detected and a was sent These two events ae utually exclusive, hence: ( bit eo ) ( D. S ) + ( D. S) Γ. Έλληνας, Διάλεξη 3-4, σελ. 34 age 7

obability of bit eo ( D / S) ( D. S ) ( S ) Baye s foula Reaanging gives: ( D. S ) ( S) ( D / S) Siilaly, we have: ( D S ) ( S ) ( D / ). S Thus the bit eo pobability can be witten as: ( bit eo ) ( S ) ( D / S ) + ( S) ( D / S) Γ. Έλληνας, Διάλεξη 3-4, σελ. 35 obability of bit eo The pevious foula can be used to calculate the bit eo pobability povided: we know what the pobabilities of sending s and s ae (often we have (S ) (S ).5) and we can obtain the conditional pobabilities (D /S ) and (D /S ). We can obtain (D /S ) and (D /S ) if we know what the DFs associated with eception of the bits and in the pesence of noise ae. These pocesses can be vey accuately appoxiated by gaussian ando vaiables; the gaussian DF is plotted on the next slide. Γ. Έλληνας, Διάλεξη 3-4, σελ. 36 age 8

p Gaussian DF ( x ) σ ( x) e πσ p(x) x Γ. Έλληνας, Διάλεξη 3-4, σελ. 37 Gaussian DF The gaussian DF occus vey widely in any applications (and fo that eason is also called the Noal distibution). One eason fo this is the cental liit theoe. This theoe tells us that if we take the su of a lage nube of independent vaiables X, X,... X n, and if each of these akes a sall contibution to the su X X + X +... + X n, then the DF of X will appoach a gaussian shape as n. The poof is beyond the scope of this couse, but the idea can be illustated best by an exaple, e.g. oll n dice and add thei values. If this event is epeated enough ties, you get a gaussian distibution. www.uses.on.net/zhcchz/java/quincunx/quincunx.8.htl Γ. Έλληνας, Διάλεξη 3-4, σελ. 38 age 9

opeties of the gaussian DF p(x) ( X ) ( X ) ean: X.5 by syety σ is the standad deviation: when p(x) is used to descibe the pobability of detecting a noise cuent (o voltage) then σ epesents the s value of the noise cuent (o voltage). Γ. Έλληνας, Διάλεξη 3-4, σελ. 39 Obtaining pobabilities fo the gaussian DF When calculating the bit eo pobability late on, we will have to evaluate pobabilities such as: ( X x) x p( x) dx This expession cannot be calculated analytically, we ust use nueical techniques. We define: Q( k ) π This can be obtained nueically and then plotted: k e y dy Γ. Έλληνας, Διάλεξη 3-4, σελ. 4 age

age Γ. Έλληνας, Διάλεξη 3-4, σελ. 4 Q(k) Γ. Έλληνας, Διάλεξη 3-4, σελ. 4 To calculate: [ ] dx e x X x x ) ( ) ( σ πσ Let: y x σ σ π σ x Q x X dy e x X x y / ) ( ) ( Obtaining pobabilities fo the gaussian DF

Obtaining pobabilities fo the gaussian DF p(x) ( X ( X x ) x x ) Q p ( x ) dx x σ x Γ. Έλληνας, Διάλεξη 3-4, σελ. 43 Towads BER... In the context of ou digital photoeceive, we can say that output voltage v(t) geneated iediately afte the aplifie stage in esponse to the tansission of and will have ean values of V and V fo these two pulses. The theshold level (V th ) will be set between these two values. Howeve, noise (due e.g. to theal and aplifie contibutions) will be supeiposed on these ean values, and the distibutions will follow that of a gaussian DF. Hence the eceived voltages fo and have DFs given by p (v) and p (v) espectively: Γ. Έλληνας, Διάλεξη 3-4, σελ. 44 age

Towads BER... detected voltage, v p (v) (D /S ) (D /S ) V V th V p (v) Assue σ σ σ Γ. Έλληνας, Διάλεξη 3-4, σελ. 45 QUESTION We saw ealie that the bit eo pobability is: e + ( S ) ( D / S ) ( S) ( D / S) If we assue that ones and zeos ae equally likely to be sent, then (S ) (S ).5 and: [ D / S ) ( D / )] ( S e + By consideing an NRZ wavefo with V, and picking a theshold idway between this and V, i.e. V th V /, show that: Q e V ( σ ) Γ. Έλληνας, Διάλεξη 3-4, σελ. 46 age 3

Bit Eo Rate in Digital hotoeceives In the pevious slides, we saw that the photoeceive akes an eo wheneve noise pushes the wavefo to the wong side of the theshold level. We also saw that we could odel this pocess using the gaussian distibution. We will now finish ou teatent by showing how BER is elated to SNR. Γ. Έλληνας, Διάλεξη 3-4, σελ. 47 Digital hotoeceive Bit eos can be ade hee; the nube depends on the SNR of the eceived signal Recoveed pulse tain (output voltage) entice-hall Γ. Έλληνας, Διάλεξη 3-4, σελ. 48 age 4

We saw ealie that the bit eo pobability is: Towads BER... e + ( S ) ( D / S ) ( S) ( D / S) If we assue that ones and zeos ae equally likely to be sent, then (S ) (S ).5 and: [ D / S ) ( D / )] ( S e + We will conside a NRZ wavefo with V, and pick a theshold idway between this and V, i.e. V th V /. We efe to this as a unipola wavefo. Γ. Έλληνας, Διάλεξη 3-4, σελ. 49 Towads BER... ( D / S ) Vth V p (v) p (v) p v σ ( v ) e πσ v ( D / S ) ( v V th ) p V th ( v ) dv Γ. Έλληνας, Διάλεξη 3-4, σελ. 5 age 5

Using the elationship: we have: Towads BER... x ( X x ) Q σ D / S ) ( v V ) ( th Vth Q σ Γ. Έλληνας, Διάλεξη 3-4, σελ. 5 Towads BER... ( v V ) ( D / S ) σ p ( v ) e πσ V th V p (v) p (v) v V th ( D / S ) ( v V th ) p ( v ) dv Γ. Έλληνας, Διάλεξη 3-4, σελ. 5 age 6

Towads BER... By syety, we have: V th V 3V th p (v) Geen aea black aea ( D / S) p( v) 3 V th v dv Γ. Έλληνας, Διάλεξη 3-4, σελ. 53 Towads BER... Using we have: x ( X x ) Q σ ( D / S ) ( v 3V Q 3V th Vth Q σ th ) V σ Γ. Έλληνας, Διάλεξη 3-4, σελ. 54 age 7

age 8 Γ. Έλληνας, Διάλεξη 3-4, σελ. 55 [ ] + σ σ ) / ( ) / ( th e V Q V Q S D S D Hence: Now, eebe that σ is the s noise voltage, so: ean squae noise powe σ Towads BER... Γ. Έλληνας, Διάλεξη 3-4, σελ. 56 Also, if ones and zeos ae equally likely, Hence the SNR is: ean squae signal powe [ ] V V V + σ V Copaing with the bit eo pobability, SNR Q V Q e σ Bit Eo obability

plot of Q function Fo plot of Q function, fo e -9, need to find Q(k) -9, which gives k 6.. Γ. Έλληνας, Διάλεξη 3-4, σελ. 57 e and SNR Hence we have fo e -9 : e Q SNR Fo the plot of Q(k) vesus k, we have k 6., 9 i.e.: SNR 6. SNR 7. In db, we have SNR log (7.) 8.6 db Γ. Έλληνας, Διάλεξη 3-4, σελ. 58 age 9

BER vesus SNR fo unipola NRZ Bit eo pobability -5-5 - - -5-5 -5 - - -5 - -5 5 5 5 SNR (db) Γ. Έλληνας, Διάλεξη 3-4, σελ. 59 Copleentay eo function Note that we have used Q(k) in these calculations; ost textbooks ake use of the copleentay eo function efc(x) defined as: efc ( x) π It is staightfowad to show this is elated to Q(k) as follows: Q( k ) x efc e u k du (MATLAB, fo exaple, uses efc(x), not Q(x)) Γ. Έλληνας, Διάλεξη 3-4, σελ. 6 age 3

Bipola foat Also, note that this analysis was caied out fo a unipola foat, i.e. one in which the s poduced a zeo ean voltage, while the s have a positive ean voltage. Had a bipola schee been chosen (i.e. V - A and V A) with a decision theshold set to volts, then: e Q ( SNR ) This epesents a 3 db ipoveent in the SNR equied fo a given BER! Γ. Έλληνας, Διάλεξη 3-4, σελ. 6 Noise effects Finally... This teatent has taken into account the vaious souces of noise that we ay encounte and teated thei ipact as aking the distibutions fo and gaussian. This is accuate povided theal and aplifie noise doinate shot noise. In the next pat of this lectue, we will see what happens in an ideal eceive, whee these noise souces ae tuned off and the only souce of noise is the statistical natue of photon detection itself. Γ. Έλληνας, Διάλεξη 3-4, σελ. 6 age 3

Quantu Liit on Optical Receive Sensitivity Γ. Έλληνας, Διάλεξη 3-4, σελ. 63 Quantu Liit on Optical Receive Sensitivity at A: Digital Counications Iagine that you can obseve the aival of photons at a detecto. The detecto counts the nube of electon-hole pais that ae geneated in an inteval Δt. The following assuptions can be ade:. The pobability of one photon being detected in Δt is popotional to Δt when Δt is vey sall.. The pobability that oe than one photon is detected in Δt is negligible when Δt is vey sall. 3. The nube of photons detected in any one inteval is independent of the nube of photons detected in any othe sepaate inteval. Γ. Έλληνας, Διάλεξη 3-4, σελ. 64 age 3

Quantu Liit on Optical Receive Sensitivity at A: Digital Counications Unde these conditions, we can show that the pobability of detecting L photons in a tie peiod T obeys the oisson distibution: N ( L) L N e L! N N is the expected nube (i.e. ean nube) of detected photons (i.e. geneated electon-hole pais) in the tie peiod T. Γ. Έλληνας, Διάλεξη 3-4, σελ. 65 Quantu Liit on Optical Receive Sensitivity at A: Digital Counications Ideally, optical enegy would only be sensed (and e-h pais geneated) if a is sent. Theefoe, the ideal eceive would then be an electon-hole pai counte, and would ake a decision based on a theshold cuent. The ost efficient situation is when the theshold is set between no pais geneated (a was sent) and at least one pai geneated (a was sent). Theefoe no eos occu if a was sent (because no caies can be geneated). Γ. Έλληνας, Διάλεξη 3-4, σελ. 66 age 33

Quantu Liit on Optical Receive Sensitivity at A: Digital Counications Howeve, eos ay occu fo a if the incident optical powe fails to geneate any caie pais at all (when N could be expected). The pobability of this occuing is: N () N e! N e N Γ. Έλληνας, Διάλεξη 3-4, σελ. 67 Quantu Liit on Optical Receive Sensitivity at A: Digital Counications Since s ae eceived with no eos ( () ), and s and s ae equally likely, the oveall eo pobability is: e e N Now: opt N hf η T whee N ean no. of photons eceived in T seconds when an optical powe opt is incident, and opt is the optical powe eceived fo a stea of s. Γ. Έλληνας, Διάλεξη 3-4, σελ. 68 age 34

Quantu Liit on Optical Receive Sensitivity at A: Digital Counications NRZ (non-etun zeo foat): fo NRZ foat, the ean optical powe is: opt N hf η T Γ. Έλληνας, Διάλεξη 3-4, σελ. 69 Quantu Liit on Optical Receive Sensitivity at A: Digital Counications If we equie: e N whee e is the equied eo pobability, then: e N ln( e ) ln( e ) hf η T Γ. Έλληνας, Διάλεξη 3-4, σελ. 7 age 35

Quantu Liit on Optical Receive Sensitivity at A: Digital Counications Now, the bit ate is B T /T, and f c/λ. ln( e ) B ηλ T hc This sets the lowe liit on the optical powe needed to achieve binay tansission at B T bits pe second with an eo ate of e, i.e. this is the eceive sensitivity R. Γ. Έλληνας, Διάλεξη 3-4, σελ. 7 Quantu Liit on Optical Receive Sensitivity at A: Digital Counications ln( R e ) B ηλ Note that eceive sensitivity wosens (i.e. becoes lage) with inceasing bit ate. In db, we have: R (db ) log log T hc (W ) R W (W ) R + 3 W Γ. Έλληνας, Διάλεξη 3-4, σελ. 7 age 36

Quantu Liit on Optical Receive Sensitivity at A: Digital Counications R (db ) ln( e ) hc log ηλ + log ( B ) T + 3 Slope of db pe decade incease in B T Γ. Έλληνας, Διάλεξη 3-4, σελ. 73 Quantu Liit on Optical Receive Sensitivity at A: Digital Counications Fo exaple: R (db) - - - B T (Gb/s) Γ. Έλληνας, Διάλεξη 3-4, σελ. 74 age 37

Typical eceive sensitivities vs bit ate Γ. Έλληνας, Διάλεξη 3-4, σελ. 75 Quantu Liit on Optical Receive Sensitivity at B: Analog Counications Recall SNR expession fo a pin photoeceive: SNR qb ( I + I ) i D p + 4kTBF R If we only conside the quantu noise, then thee will be no dak cuent, theal o aplifie noise. (Note: if we take an AD photoeceive, we would set M to iniize SNR fo the shot-noise liited case in any case). L n Γ. Έλληνας, Διάλεξη 3-4, σελ. 76 age 38

Quantu Liit on Optical Receive Sensitivity at B: Analog Counications Assuing that quantu noise is the only type of noise pesent, we have: SNR i p qbi Fo the definition fo quantu efficiency, we can substitute i p and I with the optical signal powe p(t) and the ean optical powe : I ηq hf i p q η hf p Γ. Έλληνας, Διάλεξη 3-4, σελ. 77 Hence: Quantu Liit on Optical Receive Sensitivity at B: Analog Counications SNR ηq p hf η qb ηq B hf hf SNR will be axiised if we have a signal coponent whose s value is. This then gives: SNR η B hf p Γ. Έλληνας, Διάλεξη 3-4, σελ. 78 age 39

Quantu Liit on Optical Receive Sensitivity at B: Analog Counications Reaanging gives the equied ean optical powe to aintain a given SNR fo analog signals: B hc ηλ SNR Copae this with the ean optical powe needed to aintain a given BER fo digital counications: ln( e ) B ηλ T hc Γ. Έλληνας, Διάλεξη 3-4, σελ. 79 Quantu Liit on Optical Receive Sensitivity at B: Analog Counications We can theefoe copae the fundaental (i.e. quantu) liit to eceive sensitivity fo both digital and analog counications: R digital R analogue ln( 4 SNR e ) BT B Noally, we choose the atio B T /B (bit ate to analog bandwidth) to be 6. Fo pevious lectue, we have an SNR of 5.6 db fo a bipola NRZ schee and a e of -9. Substituting, we get: R digital (db) - R analog (db) 3.43 db Γ. Έλληνας, Διάλεξη 3-4, σελ. 8 age 4