Tarea 1 Cristian Parra Alvial ELO352: Comunicaciones por Fibra Optica Profesor: Ricardo Olivares 14 de Septiembre de 2017
Problema 1.1 Calculate the carrier frequency for optical communication systems operating at 0.88, 1.3, and 1.55 µ 1.55 µm. m. What is the photon energy (in eV) in each case? c λ h = 6, 63 · 10−34[J/s] J/s] = h · V = h · E = h
h = 4, 13 · 10−15[EV · s] 88[µm]] i) 0, 88[µm f = 3, 40 · 1014[H z ] 1 , 40[EV 40[EV ]] E = 1, 3[µm]] ii) ii) 1, 3[µm f = 2, 3 · 1014[H z ] = 0, 953[EV 953[EV ]] E = 55[µm]] iii) iii) 1, 55[µm 1014[H z ] f = 1, 93 · 1014[H 0 , 79[EV 79[EV ]] E = 0,
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Problema1.2 Calculate the transmission distance over which the optical power will attenuate by a factor of 10 for three fibers with losses of 0.2, 20, and 2000 dB/km. Assuming that the optical power decreases as exp( −αL), calculate a (in cm −1) for the three fibers. P out = 0, 1 P in i) 0, 2[db/km] ii) 20[db/km] iii) 2000[db/km] P out = P in · e−α·L → 0, 1 = e −αL 10 P out ) α[db/km] = − · log10( L P in 10 1 ) · log10 ( 10 L L = 50[km]
i) 0, 2[db/km] = −
Entonces calculando α para el largo L obtenido, se tiene: α =
−
1 ln(0, 1) = 4, 6 · 10−7 [ ] 50[km] cm 10 1 ) · log10 ( 10 L L = 0, 5[km]
ii) 20[db/km] = −
Entonces calculando α para el largo L obtenido, se tiene: α =
−
1 ln(0, 1) = 4, 6 · 10−5 [ ] 0, 5[km] cm 10 1 ) · log10 ( 10 L L = 5 · 10−3 [km]
iii) 2000[db/km] = −
Entonces calculando α para el largo L obtenido, se tiene: α =
−
ln(0, 1) −3 1 = 4, 6 ] · 10 [ 5 · 10−3 [km] cm
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Problema 1.3 Assume that a digital communication system can be operated at a bit rate of up to 1 % of the carrier frequency. How many audio channels at 64 kb/s can be transmitted over a microwave carrier at 5 GHz and an optical carrier at 1.55 µm? i) λ = 1, 55[µm] c f = = 1, 935 · 1014[Hz ] λ ii) f = 5[GHz ] B = 1 %f # de canales =
B[bit/s] 64[kb/s]
1, 935 · 1014 · 0, 01[bit/s] = 30, 24 · 106 canales i) # de canales = 64[kb/s] ii) # de canales =
5[GHz ] · 0, 01[bit/s] = 781, 25 canales 64[kb/s]
Problema 1.4 1−hour lecture script is stored on the computer hard disk in the ASCII format. Estimate the total number of bits assuming a delivery rate of 200 words per minute and on average 5 letters per word. How long will it take to transmit the script at a bit rate of 1 Gb/s? Se sabe que en el codigo ASCII una letra contiene 8 bits Bits = 1[hora] · 60[
min palabras letras bit ] · 200[ ] · 5[ ] · 8[ ] = 4, 8 · 105 [bits] hora min palabra letra 4, 8 · 105 [bits] = 4, 8 · 10−4 [s] T = 9 1 · 10 [bit/s]
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Problema 1.5 A 1.55−µm digital communication system operating at 1 Gb/s receives an average power of −40 dBm at the detector. Assuming that 1 and 0 bits are equally likely to occur, calculate the number of photons received within each 1 bit. Gb ] s λ = 1, 55[µm] B = 1[
P = −40[dbm]
→
P = 1 · 10−4 [mW ]
E = P · ∆T ∆T =
1 1 = = 10 −9 [s] 9 bit B 1 · 10 [ s ]
E = 1 · 10−4 [mW ] · 10−9 [s] = 1 · 10−16[J ] N fotones =
E h · V
=
Eλ h·c
≈
780
Problema 1.6 An analog voice signal that can vary over the range 0−50 mA is digitized by sampling it at 8 kHz. The first four sample values are 10, 21, 36, and 16 mA. Write the corresponding digital signal (a string of 1 and 0 bits) by using a 4−bit representation for each sample. V alores de muestreo 10, 21, 36, 16 [mA] f S = 8[kH z ] 4 bits por muestreo kbits ] B = 32[ s Sea D la diferencia entre las amplitudes de corriente, D = 50[mA] − 0[mA] = 50 L = 2N = 24 = 16 D ∆= = 3, 125 L 10 = 3, 2 → 0011 ∆ 21 = 6, 72 → 0110 ∆ 36 = 11, 52 → 1011 ∆ 16 = 5, 12 → 0101 ∆ 4
Problema 1.7 Sketch the variation of optical power with time for a digital NRZ bit stream 010111101110 by assuming a bit rate of 2.5 Gb/s. What is the duration of the shortest and widest optical pulse? El pulso m´as corto dura 1 bit, y el m´as largo dura 4 bits consecutivos, por lo tanto se tiene para el pulso m´as corto: B = 2, 5[
1 Gbit ] ⇒ T = = 4 · 10−10[s] Gb s 2, 5[ s ]
Y para el pulso m´as largo se tiene: 4 · T = 1, 6 · 10−9 [s]
Problema 1.8 A 1.55−µm fiber−optic communication system is transmitting digital signals over 100 km at 2 Gb/s. The transmitter launches 2 mW of average power into the fiber cable, having a net loss of 0.3 dB/km. How many photons are incident on the receiver during a single 1 bit? Assume that 0 bits carry no power, while 1 bits are in the form of a rectangular pulse occupying the entire bit slot (NRZ format). λ = 1, 55[µm] L = 100[km] B = 2[
10−
αL 10
Gb db ] P = 2[mW ] α = 0, 3[ ] s km
J h = 6, 6 · 10−34[ ] s 1 1 ∆T = = = 5 · 10−10 9 2 · 10 B 10 P rx ) L = − log 10( α P αL P rx = log 10( ) − 10 P P rx − = = 2 · 10−3 [mW ] → P rx = P · 10 P P rx ∆T N fotones = ≈ 7804[fotones] hV αL 10
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Problema 1.9 A 0.8−µm optical receiver needs at least 1000 photons to detect the 1 bits accurately. What is the maximum possible length of the fiber link for a 100 −Mb/s optical communication system designed to transmit −10 dBm of average power? The fiber loss is 2 dB/km at 0.8 µm. Assume the NRZ format and a rectangular pulse shape. λ = 0, 8[µm] N fotones = 1000[fotones] db ] km Mb ] B = 100[ s P = −10[dbm] → P = 10−4 [W ] α = 0, 2[
N · hV N · h · c · B = = 2, 482 · 10−5 [mW ] ∆T λ 10 10 Prec P rec ) ⇒ L = − log 10( ) ≈ 18[Km] α = − log 10( L P α P P rec =
Problema 1.10 A 1.3−µm optical transmitter is used to obtain a digital bit stream at a bit rate of 2 Gb/s. Calculate the number of photons contained in a single 1 bit when the average power emitted by the transmitter is 4 mW. Assume that the 0 bits carry no energy. P = 4[mW ] Gb ] s λ = 1, 3[µm] B = 2[
J h = 6, 62 · 10−34 [ ] s 1 P · ∆T c N fotones = , ∆T = , V = hV B λ P · λ = 13, 09 · 106 [fotones] N fotones = h·c·B
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