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AM, FM, and Digital Modulated Systems Chap. 5
(which contains the source information). To maximize the signaling efficiency (so that
there is a low probability of error), the power in the data term needs to be maximized.
This is accomplished by letting ∆u = D p = 90 = p2 radians, which corresponds to a
digital modulation index of h = 1. For this optimum case of h = 1, the BPSK signal
becomes
s(t) = - A c m(t) sin v c t
(5–77)
Throughout the text, we assume that ∆u = 90, h = 1, is used for BPSK signaling (unless
otherwise stated). Equation (5–77) shows that BPSK is equivalent to DSB-SC signaling with
a polar baseband data waveform. The complex envelope for this BPSK signal is
g(t) = jA c m(t) for BPSK
(5–78)
Using Eq. (3–41), we obtain the PSD for the complex envelope, viz.,
g
(f) = A 2 c T b a sin pfT 2
b
b (for BPSK)
(5–79)
pfT b
where m(t) has values of ;1, so that s(t) has an average normalized power of A 2 c >2. The PSD
for the corresponding BPSK signal is readily evaluated by translating the baseband spectrum
to the carrier frequency, as specified by substituting Eq. (5–79) into Eq. (5–2b). The resulting
BPSK spectrum is shown in Fig. 5–20b. The null-to-null bandwidth for BPSK is also 2R, the
same as that found for OOK.
To detect BPSK, synchronous detection must be used, as illustrated in Fig. 5–22a. Since
there is no discrete carrier term in the BPSK signal, a PLL may be used to extract the carrier
reference only if a low-level pilot carrier is transmitted together with the BPSK signal.
Otherwise, a Costas loop or a squaring loop (Fig. 5–3) may be used to synthesize the carrier
BPSK in
Low-pass
filter
Binary output
cos c t
(a) Detection of BPSK (Coherent Detection)
DPSK in
Low-pass
filter
Decoded binary output
One-bit
delay, T b
(b) Detection of DPSK (Partially Coherent Detection)
Figure 5–22
Detection of BPSK and DPSK.