1.2. BASIC CONCEPTS 11
number of bitsmneeded to code each sample is related to the number of quantized
signal levelsMby the relation
M= 2 m or m=log 2 M. (1.2.2)The bit rate associated with the PCM digital signal is thus given by
B=mfs≥( 2 ∆f)log 2 M, (1.2.3)where the Nyquist criterion,fs≥ 2 ∆f, was used. By noting thatM>Amax/ANand
using Eq. (1.2.1) together with log 210 ≈ 3 .33,
B>(∆f/ 3 )SNR, (1.2.4)where the SNR is expressed in decibel (dB) units.
Equation (1.2.4) provides the minimum bit rate required for digital representation
of an analog signal of bandwidth∆fand a specific SNR. When SNR>30 dB, the
required bit rate exceeds 10(∆f), indicating a considerable increase in the bandwidth
requirements of digital signals. Despite this increase, the digital format is almost al-
ways used for optical communication systems. This choice is made because of the
superior performance of digital transmission systems. Lightwave systems offer such
an enormous increase in the system capacity (by a factor∼ 105 ) compared with mi-
crowave systems that some bandwidth can be traded for improved performance.
As an illustration of Eq. (1.2.4), consider the digital conversion of an audio signal
generated in a telephone. The analog audio signal contains frequencies in the range
0.3–3.4 kHz with a bandwidth∆f= 3 .1 kHz and has a SNR of about 30 dB. Equa-
tion (1.2.4) indicates thatB>31 kb/s. In practice, a digital audio channel operates at
64 kb/s. The analog signal is sampled at intervals of 125μs (sampling ratefs=8 kHz),
and each sample is represented by 8 bits. The required bit rate for a digital video signal
is higher by more than a factor of 1000. The analog television signal has a bandwidth
∼4 MHz with a SNR of about 50 dB. The minimum bit rate from Eq. (1.2.4) is 66 Mb/s.
In practice, a digital video signal requires a bit rate of 100 Mb/s or more unless it is
compressed by using a standard format (such as MPEG-2).
1.2.2 Channel Multiplexing
As seen in the preceding discussion, a digital voice channel operates at 64 kb/s. Most
fiber-optic communication systems are capable of transmitting at a rate of more than
1 Gb/s. To utilize the system capacity fully, it is necessary to transmit many channels
simultaneously through multiplexing. This can be accomplished throughtime-division
multiplexing (TDM) orfrequency-divisionmultiplexing (FDM). In the case of TDM,
bits associated with different channels are interleaved in the time domain to form a
composite bit stream. For example, the bit slot is about 15μs for a single voice channel
operating at 64 kb/s. Five such channels can be multiplexed through TDM if the bit
streams of successive channels are delayed by 3μs. Figure 1.8(a) shows the resulting
bit stream schematically at a composite bit rate of 320 kb/s.
In the case of FDM, the channels are spaced apart in the frequency domain. Each
channel is carried by its own carrier wave. The carrier frequencies are spaced more than