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Sec. 3–3 Pulse Code Modulation 147
signal. Consequently, the spectrum of the PCM signal is not directly related to the spectrum of
the input analog signal (as will be shown in Secs. 3–4 and 3–5).
The bandwidth of (serial) binary PCM waveforms depends on the bit rate and the waveform
pulse shape used to represent the data. From Fig. 3–8, the bit rate is
R = nf s
(3–14)
where n is the number of bits in the PCM word (M = 2 n ) and f s is the sampling rate. For no
aliasing, we require that f s Ú 2B, where B is the bandwidth of the analog signal (that is to be
converted to the PCM signal). In Sec. 3–4, we see that the dimensionality theorem shows that
the bandwidth of the binary encoded PCM waveform is bounded by
B PCM Ú 1 2 R = 1 2 nf s
(3–15a)
1
the minimum bandwidth of
2 R = 1 2 nf s is obtained only when (sin x) > x type pulse shape is
used to generate the PCM waveform. However, usually a more rectangular type of pulse
shape is used, and consequently, the bandwidth of the binary-encoded PCM waveform will
be larger than this minimum. Details of line codes and pulse shape selection will be studied
later in Sections 3–5 and 3–6. The exact bandwidth that is attained will depend on what
selection is used. For example, referring to Fig. 3–15, suppose that one selects the typical
case of a rectangular pulse shape and uses a unipolar NRZ, a polar NRZ, or a bipolar RZ
PCM waveform as shown in Figs. 3–15b, 3–15c, and 3–15e. These are typical of waveforms
generated by popular PCM integrated circuits. Then, as shown in Figs. 3–16a, 3–16b, and
3–16d, the null bandwidth will be the reciprocal of the pulse width, which is 1>
T b R for
these cases of binary signaling. Thus, for rectangular pulses, the first null bandwidth is
B PCM = R = nf s (first null bandwidth)
(3–15b)
Table 3–2 presents a tabulation of this result for the case of the minimum sampling rate,
f s 2B. Note that the dimensionality theorem of Eq. (3–15a) demonstrates that the bandwidth
of the PCM signal has a lower bound given by
B PCM Ú nB
(3–15c)
where f s Ú 2B and B is the bandwidth of the corresponding analog signal. Thus, for reasonable
values of n, the bandwidth of the serial PCM signal will be significantly larger than the
bandwidth of the corresponding analog signal that it represents. For the example shown in
Fig. 3–8 where n 3, the PCM signal bandwidth will be at least three times wider than that
of the corresponding analog signal. Furthermore, if the bandwidth of the PCM signal is
reduced by improper filtering or by passing the PCM signal through a system that has a poor
frequency response, the filtered pulses will be elongated (stretched in width), so that pulses
corresponding to any one bit will smear into adjacent bit slots. If this condition becomes too
serious, it will cause errors in the detected bits. This pulse-smearing effect is called
intersymbol interference (ISI). The filtering specifications for a signal without ISI are
discussed in Sec. 3–6.