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Sec. 5–10 Multilevel Modulated Bandpass Signaling 377
also introduces AM on the transmitted signal. If the overall pulse shape satisfies the raised
cosine-rolloff filter characteristic, then, by the use of Eq. (3–74), the absolute bandwidth of
the M-level modulating signal is
B = 1 2 (1 + r)D
(5–105)
where D = R and r is the rolloff factor of the filter characteristic. Furthermore, from our
study of AM (and DSB-SC, to be specific), we know that the transmission bandwidth is
related to the modulation bandwidth by B T = 2B, so that the overall absolute transmission
bandwidth of the QAM signal with raised cosine filtered pulses is
B T = a 1 + r bR
/
(5–106)
This compares to an absolute bandwidth of infinity (and a null bandwidth of B t = 2R) for
the case of QAM with rectangular data pulses (as shown in Fig. 5–33).
Because M = 2 , which implies that = log 2 M = (ln M)(ln 2), the spectral efficiency
of QAM-type signaling with raised cosine filtering is
h = R ln M bit>s
=
(5–107)
B T (1 + r) ln 2 Hz
This result is important because it tells us how fast we can signal for a prescribed bandwidth.
The result also holds for MPSK, since it is a special case of QAM. Equation (5–107) is used
to generate Table 5–8, which illustrates the allowable bit rate per hertz of transmission bandwidth
for QAM signaling. For example, suppose that we want to signal over a
communications satellite that has an available bandwidth of 2.4 MHz. If we used BPSK
(M = 2) with a 50% rolloff factor, we could signal at a rate of B T × h = 2.4 × 0.677 = 1.60
Mbitss; but if we used QPSK (M = 4) with a 25% rolloff factor, we could signal at a rate of
2.4 × 1.6 = 3.84 Mbitss.
To conserve bandwidth, the number of levels M in Eq. (5–107) cannot be increased too
much, since, for a given peak envelope power (PEP), the spacing between the signal points in
the signal constellation will decrease and noise on the received signal will cause errors.
TABLE 5–8
SPECTRAL EFFICIENCY FOR QAM SIGNALING WITH RAISED COSINE-ROLLOFF PULSE
SHAPING (USE M = 4 for QPSK, OQPSK, and p/4 QPSK signaling)
Number of
Levels,
M (symbols)
Size of
DAC, ;
(bits)
h = R B T
a bit>s
Hz b
r = 0.0 r = 0.1 r = 0.25 r = 0.5 r = 0.75 r = 1.0
2 1 1.00 0.909 0.800 0.667 0.571 0.500
4 2 2.00 1.82 1.60 1.33 1.14 1.00
8 3 3.00 2.73 2.40 2.00 1.71 1.50
16 4 4.00 3.64 3.20 2.67 2.29 2.00
32 5 5.00 4.55 4.0 3.33 2.86 2.50