8.6 Summary
359
fitting by P R = 9logB − 68.5 where B is the data rate in Mb/s. To find the
attenuation-limited transmission distance, include a 1-dB connector loss at
each end and a 6-dB system margin.
8.9 A 1550-nm single-mode digital fiber optic link needs to operate at 622 Mb/s
over 80 km without amplifiers. A single-mode InGaAsP laser launches an
average optical power of 3.0 dBm into the fiber. The fiber has a loss of
0.25 dB/km, and there is a splice with a loss of 0.1 dB every kilometer. The
coupling loss at the receiver is 0.5 dB, and the receiver uses an InGaAs APD
with a sensitivity of −39 dBm. Excess-noise penalties are predicted to be
1.5 dB. (a) Set up an optical power budget for this link and find the system
margin. (b) What is the system margin at 2.5 Gb/s with an APD sensitivity
of −31 dBm?
8.10 In the (7,4) linear Hamming code the first four bits of a code word are the
information bits b 1 , b 2 , b 3 , b 4 and the next three bits b 5 , b 6 , b 7 are the check
bits, which are given by b 5 = b 1 + b 3 + b 4 , b 6 = b 1 + b 2 + b 4 , and b 7 = b 2
+ b 3 + b 4 Make a table listing the sixteen possible 4-bit information words,
that is, 0000 through 1111, and the corresponding 7-bit code words.
8.11 (a) Find the binary equivalent of the polynomial x
8
+ x
7
+ x
3
+ x + 1. (b)
Find the polynomial equivalent of 10011011110110101.
8.12 Consider the 10-bit data unit 1010011110 and the divisor 1011. Use both
binary and algebraic division to show that the CRC remainder is 001.
8.13 Consider the generator polynomial x
3
+ x + 1.
(a) Show that the CRC for the data unit 1001 is
given by 110.
(b) If the resulting code word has an error in the first bit when it arrives at
the destination, show that the CRC calculated by the receiver is 101.
8.14 Why can the (255, 223) Reed-Solomon code correct up to 16 bytes, whereas
the (255, 239) is limited to correcting 8 byte errors? What is the overhead for
each of these two codes?
8.15 Verify the resulting expression in Eq. (8.34) for the intensity due to the
combined signal and local oscillator fields.
8.16 A homodyne ASK receiver has a 100-MHz bandwidth and contains a 1310nm pin photodiode with a responsivity of 0.6 A/W. It is shot noise limited
and needs a signal-to- noise ratio of 12 to achieve a 10
−9 BER. Find the
photocurrent that is generated if the local oscillator power is −3 dBm and the
phase error is 10°. Assume that both the signal and the local oscillator have
the same polarization.
8.17 For a Bit error rate of 10
−9 assume that the combined spectral width of
the signal carrier wave and the local oscillator should be 1 percent of the
transmitted bit rate.
(a) What spectral width is needed at 1310 nm for a 100-Mb/s data rate?
(b) What is the maximum allowed spectral width at 2.5 Gb/s?
8.18 (a) Verify that 10 photons per bit are required to get a bit error rate of 10
−9
for a direct-detection OOK system.
359
fitting by P R = 9logB − 68.5 where B is the data rate in Mb/s. To find the
attenuation-limited transmission distance, include a 1-dB connector loss at
each end and a 6-dB system margin.
8.9 A 1550-nm single-mode digital fiber optic link needs to operate at 622 Mb/s
over 80 km without amplifiers. A single-mode InGaAsP laser launches an
average optical power of 3.0 dBm into the fiber. The fiber has a loss of
0.25 dB/km, and there is a splice with a loss of 0.1 dB every kilometer. The
coupling loss at the receiver is 0.5 dB, and the receiver uses an InGaAs APD
with a sensitivity of −39 dBm. Excess-noise penalties are predicted to be
1.5 dB. (a) Set up an optical power budget for this link and find the system
margin. (b) What is the system margin at 2.5 Gb/s with an APD sensitivity
of −31 dBm?
8.10 In the (7,4) linear Hamming code the first four bits of a code word are the
information bits b 1 , b 2 , b 3 , b 4 and the next three bits b 5 , b 6 , b 7 are the check
bits, which are given by b 5 = b 1 + b 3 + b 4 , b 6 = b 1 + b 2 + b 4 , and b 7 = b 2
+ b 3 + b 4 Make a table listing the sixteen possible 4-bit information words,
that is, 0000 through 1111, and the corresponding 7-bit code words.
8.11 (a) Find the binary equivalent of the polynomial x
8
+ x
7
+ x
3
+ x + 1. (b)
Find the polynomial equivalent of 10011011110110101.
8.12 Consider the 10-bit data unit 1010011110 and the divisor 1011. Use both
binary and algebraic division to show that the CRC remainder is 001.
8.13 Consider the generator polynomial x
3
+ x + 1.
(a) Show that the CRC for the data unit 1001 is
given by 110.
(b) If the resulting code word has an error in the first bit when it arrives at
the destination, show that the CRC calculated by the receiver is 101.
8.14 Why can the (255, 223) Reed-Solomon code correct up to 16 bytes, whereas
the (255, 239) is limited to correcting 8 byte errors? What is the overhead for
each of these two codes?
8.15 Verify the resulting expression in Eq. (8.34) for the intensity due to the
combined signal and local oscillator fields.
8.16 A homodyne ASK receiver has a 100-MHz bandwidth and contains a 1310nm pin photodiode with a responsivity of 0.6 A/W. It is shot noise limited
and needs a signal-to- noise ratio of 12 to achieve a 10
−9 BER. Find the
photocurrent that is generated if the local oscillator power is −3 dBm and the
phase error is 10°. Assume that both the signal and the local oscillator have
the same polarization.
8.17 For a Bit error rate of 10
−9 assume that the combined spectral width of
the signal carrier wave and the local oscillator should be 1 percent of the
transmitted bit rate.
(a) What spectral width is needed at 1310 nm for a 100-Mb/s data rate?
(b) What is the maximum allowed spectral width at 2.5 Gb/s?
8.18 (a) Verify that 10 photons per bit are required to get a bit error rate of 10
−9
for a direct-detection OOK system.