B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

162 AMPLITUDE MODULATIONS AND DEMODULATIONS
From the frequency-shifting property, the inverse transform of this equation yields
<pusB (t) = m(t) cos W e t - mh(t) sin W e t
(4.20a)
Similarly, we can show that
!pLSB (t) = m(t) cos W e t + mh(t) sin W e t
(4.20b)
Hence, a general SSB signal <pssB (t) can be expressed as
!pSSB (t) = m(t) COS W e t =f m1, (t) sin W e t
(4.20c)
where the minus sign applies to USB and the plus sign applies to LSB.
Given the time domain expression of SSB-SC signals, we can now confirm analytically
(instead of graphically) that SSB-SC signals can be coherently demodulated:
<pssB (t) COS W e t = [m(t) COS W e t =f mh(t) sin W e t] 2 COS W e t
= m(t)[l + cos 2w e t] =f mh(t) sin 2w e t
= m(t) + [m(t) cos 2w e t =f m1,(t) sin 2w e t]
SSB-SC signal with carrier 2w e
Thus, the product <pssB (t) • 2 cos W e t yields the baseband signal and another SSB signal
with a carrier 2w e , The spectrum in Fig. 4.13e shows precisely this result. A low-pass filter
will suppress the unwanted SSB terms, giving the desired baseband signal m(t). Hence, the
demodulator is identical to the synchronous demodulator used for DSB-SC. Thus, any one of
the synchronous DSB-SC demodulators discussed earlier in Sec. 4.2 can be used to demodulate
an SSB-SC signal.
Example 4.6 Tone Modulation: SSB
Find <pssB (t) for a simple case of a tone modulation, that is, when the modulating signal is a
sinusoid m(t) = cos w m t. Also demonstrate the coherent demodulation of this SSB signal.
Recall that the Hilbert transform delays the phase of each spectral component by n /2.
In the present case, there is only one spectral component of frequency W m , Delaying the
phase of m(t) by n /2 yields
Hence, from Eq. (4.20c),
Thus,
<pssB (t) = cos W m t cos W e t =f sin W m t sin W e t
= cos (w e ± W m )t
<pusB (t) = COS (W e + W m )t and !pLSB (t) = COS (W e - Wm)t
To verify these results, consider the spectrum of m(t) (Fig. 4.16a) and its DSB-SC
(Fig. 4.16b ), USB (Fig. 4.16c ), and LSB (Fig. 4.16d) spectra. It is evident that the spectra
in Fig. 4.16c and d do indeed correspond to the <pusB (t) and <pLsB (t) derived earlier.