and 280
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techniques have been applied to calculate the capacity limits of optical systems in [MS00]. The principles of signal detection are covered in the classic books by van Trees [vT68] and Wozencraft and Jacobs [WJ90]. For a derivation of shot noise statistics, see [Pap91]. The noise introduced by optical ampliers has been studied extensively in the literature. Amplier noise statistics have been derived using quantum mechanical approaches [Per73, Yam80, MYK82, Dan95] as well as semiclassical approaches [Ols89, RH90]. There was a great deal of effort devoted to realizing coherent receivers in the 1980s, but the advent of optical ampliers in the late 1980s and early 1990s provided a simpler alternative. See [BL90, KBW96] for a detailed treatment of coherent receivers. Equalization is treated extensively in many books on digital communication; see, for example, [LM93, Pro00]. The eld of error-correcting codes has developed rapidly since its founding by Hamming [Ham50] and Shannon [Sha48] more than a half-century ago. There are many textbooks on this topic; see, for example, [McE77, LC82]. A discussion of FEC techniques in submarine transmission systems appears in [Sab01].
Problems
4.1 A very simple line code used in early data networks is called bit stufng. The objective of this code is to prevent long runs of 1s or 0s but not necessarily achieve DC balance. The encoding works as follows. Suppose the maximum number of consecutive 1s that we are allowed in the bit stream is k . Then the encoder inserts a 0 bit whenever it sees k consecutive 1 bits in the input sequence. (a) Suppose the incoming data to be transmitted is 11111111111001000000 (read left to right). What is the encoded bit stream, assuming k = 5? (b) What is the algorithm used by the decoder to recover the data? Suppose the received bit stream is 0111110101111100011 (read left to right). What is the decoded bit stream? The SONET standard uses scrambling to prevent long runs of 1s and 0s from occurring in the transmitted bit stream. The scrambling is accomplished by a carefully designed feedback shift register shown in Figure 4.13. The shift register consists of ip-ops whose operation is controlled by a clock running at the bit rate and is reset at the beginning of each frame. (a) Suppose the incoming data to be transmitted is 11111111111001000000. Assume that the shift register contents are 1111111 at the beginning. What is the scrambled output? (b) Write a simulation program to compute the scrambled output as a function of the input. The input is a sequence of bits generated by a pseudo-random
4.2
Problems
281
sequence with equal probabilities for a 1 and a 0. Plot the longest run length of 1s and the longest run length of 0s observed as a function of the sequence length for sequences up to 10 million bits long. Again assume that the shift register contents are 1111111 at the beginning of the sequence. What do you observe? 4.3 Consider the optical duobinary modulation scheme we discussed in Section 4.3.1. If the data sequence is d(nT ) = 10101011010111100001, calculate (a) the differential encoding x(nT ) of d(nT ), and (b) the duobinary encoding y(nT ) of x(nT ). Recall that y(nT ) mod 2 = d(nT ). How can you compute the sequence y(nT ) directly from d(nT ) without going through the two-stage differential and duobinary encoding processes? 4.4 Consider the SNR of an APD receiver when both shot noise and thermal noise are present. Assuming that the excess noise factor of the APD is given by FA (Gm ) = Gx m opt for some x (0, 1), derive an expression for the optimum value Gm of the APD gain Gm that maximizes the SNR. 4.5 This problem deals with the noise gure of a chain of optical ampliers and placement of loss elements in the amplier. The loss element may be an optical add/drop multiplexer, or a gain-attening lter, or a dispersion compensation module used to compensate for accumulated dispersion along the link. The question is, where should this loss element be placedin front of the amplier, after the amplier, or inside the amplier? (a) Consider an optical amplier with noise gure F . Suppose we introduce a loss element in front of it, with loss 0 < 1 ( = 0 implies no loss, and = 1 implies 100% loss). Show that the noise gure of the combination is F /(1 ). Note that this loss element may also simply reect the coupling loss into the amplier. Observe that this combination has a poor noise gure.
Data in
D
D
D
D
D
D
D
+
Scrambled data out
+
Figure 4.13 The feedback shift register used for scrambling in SONET.
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Modulation and Demodulation
(b) Suppose the loss element is placed just after the amplier. Show that the noise gure of the combination is still F ; that is, placing a loss element after the amplier does not affect the noise gure. However, the price we pay in this case is a reduction in optical output power, since the amplier output is attenuated by the loss element placed after it. (c) Consider an optical amplier chain with two ampliers, with gains G1 and G2 , respectively, and noise gures F1 and F2 , respectively, with no loss be1, show that the noise gure of tween the two ampliers. Assuming G1 the combined amplier chain is F = F1 + F2 . G1
In other words, the noise gure of the chain is dominated by the noise gure of the rst amplier, provided its gain is reasonably large, which is usually the case. (d) consider Now the case where a loss element with loss is introduced between 1, and (1 )G1 G2 1, the rst and second amplier. Assuming G1 , G2 show that the resulting noise gure of the chain is given by F = F1 + F2 . (1 )G1
Observe that the loss element doesnt affect the noise gure of the cascade 1, which is usually the case. This is signicantly as long as (1 )G1 an important fact that is made use of in designing systems. The amplier is broken down into two stages, the rst stage having high gain and a low noise gure, and the loss element is inserted between the two stages. This setup has the advantage that there is no reduction in the noise gure or the output power. 4.6 Show that the BER for an OOK direct detection receiver is given by BER = Q 4.7 I1 I0 0 + 1 .
Consider a binary digital communication system with received signal levels m1 and 2 m0 for a 1 bit and 0 bit, respectively. Let 2 and 0 denote the noise variances for a 1 and 0 bit, respectively. Assume that the noise is Gaussian and that a 1 and 0 bit are equally likely. In this case, the bit error rate BER is given by BER = 1 m1 Td Q 2 1 1 Td m0 +Q 2 0 ,
Problems
283
where Td is the receivers decision threshold. Show that the value of Td that minimizes the bit error rate is given by Td =
2 2 m1 0 + m0 1 + 22 2 2 0 1 (m1 m0 )2 + 2(1 0 ) ln(1 /0 ) 2 2 1 0
.
(4.21)
For the case of high signal-to-noise ratios, it is reasonable to assume that (m1 m0 )2
2 2 2(1 0 ) ln(1 /0 ) 22 0 1
.
In this case, (4.21) can be simplied to Td = m0 1 + m1 0 . 1 + 0
With m1 = RP1 and m0 = RP0 , this is the same as (4.12). 4.8 Consider a pin direct detection receiver where the thermal noise is the main noise component and its variance has the value given by (4.17). What is the receiver sensitivity expressed in photons per 1 bit at a bit rate of 100 Mb/s and 1 Gb/s for a bit error rate of 1012? Assume that the operating wavelength is 1.55 m and the responsivity is 1.25 A/W. 4.9 Consider the receiver sensitivity, Prec (for an arbitrary BER, not necessarily 109 ), of an APD receiver when both shot noise and thermal noise are present but neglecting the dark current, for direct detection of on-offkeyed signals. Assume no power is transmitted for a 0 bit. (a) Derive an expression for Prec . opt (b) Find the optimum value Gm of the APD gain Gm that minimizes Prec . opt (c) For Gm = Gm , what is the (minimum) value of Prec ? 4.10 4.11 Derive (4.18). Plot the receiver sensitivity as a function of bit rate for an optically preamplied receiver for three different optical bandwidths: (a) the ideal case, Bo = 2Be , (b) Bo = 100 GHz, and (c) Bo = 30 THz, that is, an unltered receiver. Assume an amplier noise gure of 6 dB and the electrical bandwidth Be is half the bit rate, and use the thermal noise variance given by (4.17). What do you observe as the optical bandwidth is increased? You are doing an experiment to measure the BER of an optically preamplied receiver. The setup consists of an optical amplier followed by a variable attenuator to adjust the power going into the receiver, followed by a pin receiver. You plot the
4.12
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Modulation and Demodulation
BER versus the power going into the receiver over a wide range of received powers. Calculate and plot this function. What do you observe regarding the slope of this curve? Assume that Bo = 100 GHz, Be = 2 GHz, B = 2.5 Gb/s, a noise gure of 6 dB for the optical amplier, and a noise gure of 3 dB for the front-end amplier. 4.13 4.14 Derive (4.19). Another form of digital modulation that can used in conjunction with coherent be reception is phase-shift keying (PSK). Here 2P cos(2fc t) is received for a 1 bit and 2P cos(2fc t) is received for a 0 bit. Derive an expression for the bit error rate of a PSK homodyne coherent receiver. How many photons per bit are required to obtain a bit error rate of 109 ? A balanced coherent receiver is shown in Figure 4.14. The input signal and local oscillator are sent through a 3 dB coupler, and each output of the coupler is connected to a photodetector. This 3 dB coupler is different in that it introduces an additional phase shift of /2 at its second input and second output. The detected current is the difference between the currents generated by the two photodetectors. Show that this receiver structure avoids the 3 dB penalty associated with the receiver we discussed in Section 4.4.7. Use the transfer function for a 3 dB coupler given by (3.1). SONET and SDH systems use an 8-bit interleaved parity (BIP-8) check code with even parity to detect errors. The code works as follows. Let b0 , b1 , b2 , . . . denote the sequence of bits to be transmitted. The transmitter adds an 8-bit code sequence c0 , c1 , . . . , c7 , to the end of this sequence where ci = bi bi +8 bi +16 + . . . . Here denotes an "exclusive OR" operation (0 0 = 0, 0 1 = 1, 1 1 = 0). (a) Suppose the bits to be transmitted are 010111010111101111001110. What is the transmitted sequence with the additional parity check bits? (b) Suppose the received sequence (including the parity check bits at the end) is 010111010111101111001110. How many bits are in error? Assume that if a parity check indicates an error, it is caused by a single bit error in one of the bits over which the parity is computed.
4.15
4.16
Signal Local oscillator 3 dB coupler
i1
i = i1 - i2
i2
Figure 4.14 A balanced coherent receiver.