Efciency
251
allow 4.3
Spectral a small clipping probability (a few percent), which substantially reduces the power requirement while introducing only a small amount of signal distortion.
4.2.2
Applications of SCM
SCM is widely used by cable operators today for transmitting multiple analog video signals using a single optical transmitter. SCM is also being used in metropolitan-area networks to combine the signals from various users using electronic FDM followed by SCM. This reduces the cost of the network since each user does not require an optical transmitter/laser. We will study these applications further in Chapter 11. SCM is also used to combine a control data stream along with the actual data stream. For example, most WDM systems that are deployed carry some control information about each WDM channel along with the data that is being sent. This control information has a low rate and modulates a microwave carrier that lies above the data signal bandwidth. This modulated microwave carrier is called a pilot tone. We will discuss the use of pilot tones in Chapter 8. Often it is necessary to receive the pilot tones from all the WDM channels for monitoring purposes, but not the data. This can be easily done if the pilot tones use different microwave frequencies. If this is the case, and the combined WDM signal is photodetected, the detector output will contain an electronic FDM signal consisting of all the pilot tones from which the control information can be extracted. The information from all the data channels will overlap with one another and be lost.
4.3
Spectral Efciency
We saw in Chapter 2 that the ultimate bandwidth available in silica optical ber is about 400 nm from 1.2 m to 1.6 m, or about 50 THz. The natural question that arises is, therefore, what is the total capacity at which signals can be transmitted over optical ber? There are a few different ways to look at this question. The spectral efciency of a digital signal is dened as the ratio of the bit rate to the bandwidth used by the signal. The spectral efciency depends on the type of modulation and coding scheme used. Todays systems primarily use on-off keying of digital data and in theory can achieve a spectral efciency of 1 b/s/Hz. In practice, the spectral efciency of these systems is more like 0.4 b/s/Hz. Using this number, we see that the maximum capacity of optical ber is about 20 Tb/s. The spectral efciency can be improved by using more sophisticated modulation and coding schemes, leading to higher channel capacities than the number above. As spectral efciency becomes increasingly important, such new schemes are being invented, typically based on proven electrical counterparts.
252
Modulation and Demodulation
One such scheme that we discuss in the next section is optical duobinary modulation. It can increase the spectral efciency by a factor of about 1.5, typically, achieving a spectral efciency of 0.6 b/s/Hz.
4.3.1
Optical Duobinary Modulation
The fundamental idea of duobinary modulation (electrical or optical) is to deliberately introduce intersymbol interference (ISI) by overlapping data from adjacent bits. This is accomplished by adding a data sequence to a 1-bit delayed version of itself. For example, if the (input) data sequence is (0, 0, 1, 0, 1, 0, 0, 1, 1, 0), we would instead transmit the (output) data sequence (0, 0, 1, 0, 1, 0, 0, 1, 1, 0) + (, 0, 0, 1, 0, 1, 0, 0, 1, 1) = (0, 0, 1, 1, 1, 1, 0, 1, 2, 1). Here the denotes the initial value of the input sequence, which we assume to be zero. Note that while the input sequence is binary and consists of 0s and 1s, the output sequence is a ternary sequence consisting of 0s, 1s, and 2s. Mathematically, if we denote the input sequence by x(nT ) and the output sequence by y(nT ), duobinary modulation results if y(nT ) = x(nT ) + x(nT T ), where T is the bit period. In the example above, x(nT ) = (0, 0, 1, 0, 1, 0, 0, 1, 1, 0), 1 n 10, and y(nT ) = (0, 0, 1, 1, 1, 1, 0, 1, 2, 1), 1 n 10. Since the bits overlap with each other, how do we recover the input sequence x(nT ) at the receiver from y(nT )? This can be done by constructing the signal z(nT ) = y(nT ) z(nT T ) at the receiver. Note that here we subtract a delayed version of z(nT ) from y(nT ), and not a delayed version of y(nT ) itself. This operation recovers x(nT ) since z(nT ) = x(nT ), assuming we also initialize the sequence z(0) = 0. (For readers familiar with digital lters, y(nT ) is obtained from x(nT ) by a digital lter, and z(nT ) from y(nT ) by using the inverse of the same digital lter.) The reader should verify this by calculating z(nT ) for the example sequence above. To see that this holds generally, just calculate as follows: z(nT ) = = = = = = y(nT ) z(nT T ) y(nT ) y(nT T ) + z(nT 2T ) y(nT ) y(nT T ) + y(nT 2T ) z(nT 3T ) y(nT ) y(nT T ) + y(nT 2T ) . . . + (1)n1 y(T ) [x(nT ) + x(nT T )] [x(nT T ) x(nT 2T )] + . . . x(nT ) (4.1)
4.3
Spectral Efciency
253
There is one problem with this scheme, however; a single transmission error will cause all further bits to be in error, until another transmission error occurs to correct the rst one! This phenomenon is known as error propagation. To visualize error propagation, assume a transmission error occurs in some ternary digit in the example sequence y(nT ) above, and calculate the decoded sequence z(nT ). The solution to the error propagation problem is to encode the actual data to be transmitted, not by the absolute value of the input sequence x(nT ), but by changes in the sequence x(nT ). Thus the sequence x(nT ) = (0, 0, 1, 0, 1, 0, 0, 1, 1, 0) would correspond to the data sequence d(nT ) = (0, 0, 1, 1, 1, 1, 0, 1, 0, 1). A 1 in the sequence d(nT ) is encoded by changing the sequence x(nT ) from a 0 to a 1, or from a 1 to a 0. To see how differential encoding solves the problem, observe that if a sequence of consecutive bits are all in error, their differences will still be correct, modulo 2. Transmission of a ternary sequence using optical intensity modulation (the generalization of OOK for nonbinary sequences) will involve transmitting three different optical powers, say, 0, P , and 2P . Such a modulation scheme will also considerably complicate the demodulation process. We would like to retain the advantage of binary signaling while employing duobinary signaling to reduce the transmission bandwidth. To see how this can be done, compare y(nT ) = (0, 0, 1, 1, 1, 1, 0, 1, 2, 1) and d(nT ) = (0, 0, 1, 1, 1, 1, 0, 1, 0, 1) in our example, observe and that y(nT ) mod 2 = d(nT )! This result holds in general, and thus we may think that we could simply map the 2s in y(nT ) to 0s and transmit the resulting binary sequence, which could then be detected using the standard scheme. However, such an approach would eliminate the bandwidth advantage of duobinary signaling, as it should, because in such a scheme the differential encoding and the duobinary encoding have done nothing but cancel each others effects. The bandwidth advantage of duobinary signaling can only be exploited by using a ternary signaling scheme. A ternary signaling alternative to using three optical power levels is to use a combination of amplitude and phase modulation. Such a scheme is dubbed optical AM-PSK, and most studies of optical duobinary signaling today are based on AM-PSK. Conceptually, the carrier is a continuous wave signal, a sinusoid, which we can denote by a cos(t). The three levels of the ternary signal correspond to a cos(t) = a cos(t + ), 0 = 0 cos(t), and a cos(t), which we denote by 1, 0, and +1, respectively. The actual modulation is usually accomplished using an external modulator in the Mach-Zehnder arrangement (see Sections 3.3.7 and 3.5.4). These are the three signal levels corresponding to 0, 1, and 2, respectively, in y(nT ). This modulation scheme is clearly a combination of amplitude and phase modulation, hence the term AM-PSK. The AM-PSK signal retains the bandwidth advantage of duobinary signaling. However, for a direct detection receiver, the signals
254
Modulation and Demodulation
Baseband signal
DSB signal
Upper SSB signal
Lower SSB signal
0
B
wo - B
wo
wo + B
wo
wo + B
wo - B
wo
Figure 4.4 Spectrum of a baseband signal compared with the spectra of double sideband (DSB) and single sideband (SSB) modulated signals. The spectral width of the SSB signals is the same as that of the baseband signal, whereas the DSB signal has twice the spectral width of the baseband signal.
a cos(t) are indistinguishable so that the use of such a receiver merely identies 2 = 0 in y(nT ) naturally performing the mod 2 operation required to recover d(nT ) from y(nT ).
4.3.2
Optical Single Sideband Modulation
Another technique for increasing the spectral efciency is optical single sideband (SSB) modulation. Such a scheme can improve the spectral efciency by a factor of 2, if practical implementations capable of supporting transmission at 10 Gb/s and above can be found. Before we can dene what optical SSB modulation is, we need to understand the concept of sidebands in a digital signal. Consider a sinusoidal carrier signal cos(o t). Assume this is directly modulated o by a data signal that is also a sinusoid, cos(d t), for simplicity. Typically, d since o is an optical carrier frequency of the order of 200 THz and d is of the order of 10 GHz. Direct modulation amounts to forming the product cos(o t) cos(d t) = 0.5 cos((o + d )t) + 0.5 cos((o d )t). Thus the transmitted signal contains two sinusoids at o + d and o d for a data signal consisting of a single sinusoid at d . In general, for a digital signal with a (baseband) frequency spectrum extending from 0 to B Hz, the modulated signal has a spectrum covering the frequency range from o B Hz to o + B Hz, that is, a range of 2B Hz around the carrier frequency o . Each of the spectral bands of width B Hz on either side of the carrier frequency o is called a sideband, and such a signal is said to be a double sideband (DSB) signal. By appropriate ltering, we can eliminate one of these sidebands: either the lower or the upper one. The resulting signals are called single sideband (SSB) signals. DSB and SSB signals are illustrated in Figure 4.4. The difculty in implementing optical SSB modulation lies in designing the lters to eliminate one of the sidebandsthey have to be very sharp. Instead of ltering it entirely, allowing a small part, or vestige, of one of the sidebands to remain makes
4.3
Spectral Efciency
255
implementation easier. Such a scheme is called vestigial sideband (VSB) modulation. This is the modulation scheme used in television systems, and its use is currently being explored for optical systems, mainly for analog signal transmission. Optical SSB modulation is also being explored today either for analog signal transmission or, equivalently, for SCM systems, which are analog systems from the viewpoint of optical modulation.
4.3.3
Multilevel Modulation
The main technique used in digital communication to achieve spectral efciencies greater than 1 b/s/Hz is multilevel modulation. The simplest multilevel modulation scheme uses M > 2 amplitude levels of a sinusoidal carrier to represent M possible signal values. In such a scheme, each signal represents log2 M bits. However, the bandwidth occupied by a digital communication system transmitting R such symbols per second is nearly the same as that occupied by an R b/s digital system employing binary signals. Therefore, the bandwidth efciency of such a multilevel scheme is log2 M times higher, and about log2 M b/s/Hz. To date, such multilevel schemes have not been used in practical optical communication systems due to the complexities of detecting such signals at high bit rates. Another potential advantage of multilevel modulation is that the signaling rate on the channel is lower than the data rate. For example, a 16-level modulation scheme would be able to transmit at a date rate of 40 Gb/s but at a signaling rate of 10 Gbaud; that is, each signal occupies a period of 100 ps, and not 25 ps. This, in turn, helps mitigate the effects of dispersion and nonlinearities.
4.3.4
Capacity Limits of Optical Fiber
An upper limit on the spectral efciency and the channel capacity is given by Shannons theorem [Sha48]. Shannons theorem says that the channel capacity C for a binary linear channel with additive noise is given by S . C = B log2 1 + N Here B is the available bandwidth and S/N is the signal-to-noise ratio. A typical value of S/N is 100. Using this number yields a channel capacity of 350 Tb/s or an equivalent spectral efciency of 7 b/s/Hz. Clearly, such efciencies can only be achieved through the use of multilevel modulation schemes. In practice, todays long-haul systems operate at high power levels to overcome ber losses and noise introduced by optical ampliers. At these power levels, nonlinear effects come into play. These nonlinear effects can be thought of as adding additional noise, which increases as the transmitted power is increased. Therefore they in