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Bandpass Signaling Principles and Circuits Chap. 4
and the PSD of the waveform is
where G( f) = [g(t)] and g (f) is the PSD of g(t).
Proof.
v ( f) = 1 4 [ g(f - f c ) + g (-f -f c )]
(4–13)
v(t) = Re{g(t)e jv ct } = 1 2 g(t)ejv ct + 1 2 g* (t)e -jv ct
Thus,
V(f) = [v(t)] = 1 2 [g(t)ejv ct ] + 1 2 [g *(t)e -jv ct ]
(4–14)
If we use [g * (t)] = G * (-f) from Table 2–1 and the frequency translation property of
Fourier transforms from Table 2–1, this equation becomes
which reduces to Eq. (4–12).
V(f) = 1 2 {G(f- f c) + G * [-(f + f c )]}
(4–15)
Example 4–2 SPECTRUM FOR A QUADRATURE MODULATED SIGNAL
Using the FFT, calculate and plot the magnitude spectrum for the QM signal that is described in
Example 4–1.
Solution
See Example4_02.m for the solution. Is the plot of the FFT results for the spectrum correct?
Using Eq. (4–15), we expect the sinusoid to produce delta functions at 9 Hz and at 11 Hz. Using
Eq. (4–15) and Fig. 2–6a, the rectangular pulse should produce a |Sa(x)| type spectrum that is
centered at the carrier frequency, f c = 10 Hz with spectral nulls spaced at 1 Hz intervals. Note that
the FFT approximation of the Fourier transform does not give accurate values for the weight of
the delta functions in the spectrum (as discussed in Section 2–8). However, the FFT can be used
to obtain accurate values for the weights of the delta functions by evaluating the Fourier series
coefficients. For example, see study-aid problem SA4–1.
The PSD for v(t) is obtained by first evaluating the autocorrelation for v(t):
R v (t) = 8v(t) v(t + t)9 = 8Re{g(t)e jvct } Re{g(t + t) e jv c(t + t) }9
Using the identity (see Prob. 2–74)
Re(c 2 ) Re(c 1 ) = 1 2 Re(c* 2 c 1) + 1 2 Re(c 2c 1 )
where c and c 1 = g(t + t)e jvc(t+t) 2 = g(t)e jv ct
, we get
R v (t) = 8 1 2 Re{g * (t)g(t + t) e -jv ct e jv c(t+t) }9 + 8 1 2 Re{g(t)g(t + t) ejv ct e jv c(t+t) }9