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Bandpass Signaling Principles and Circuits Chap. 4
n=q
v(t) = Re ea2 a c n e j(nv 0-v c )t b e jvct f
n=1
so that Eq. (4–1a) follows, where
q
g(t) K 2 a c n e j(nv 0-v c )t
n=1
(4–7)
(4–8)
Because v(t) is a bandpass waveform with nonzero spectrum concentrated near f = f c , the
Fourier coefficients c n are nonzero only for values of n in the range ±nf 0 ≈ f c . Therefore,
from Eq. (4–8), g(t) has a spectrum that is concentrated near f = 0. That is, g(t) is a
baseband waveform.
The waveforms g(t), (and consequently) x(t), y(t), R(t), and u(t) are all baseband
waveforms, and, except for g(t), they are all real waveforms. R(t) is a nonnegative real
waveform. Equation (4–1) is a low-pass-to-bandpass transformation. The e jv ct
factor
in Eq. (4–1a) shifts (i.e., translates) the spectrum of the baseband signal g(t) from baseband
up to the carrier frequency f c . In communications terminology, the frequencies in
the baseband signal g(t) are said to be heterodyned up to f c . The complex envelope,
g(t), is usually a complex function of time, and it is the generalization of the phasor
concept. That is, if g(t) happens to be a complex constant, then v(t) is a pure sinusoidal
waveshape of frequency f c , and this complex constant is the phasor representing the
sinusoid. If g(t) is not a constant, then v(t) is not a pure sinusoid, because the amplitude
and phase of v(t) vary with time, caused by the variations in g(t).
Representing the complex envelope in terms of two real functions in Cartesian
coordinates, we have
g(x) K x(t) + jy(t)
where x(t) = Re{g(t)} and y(t) = Im{g(t)}. x(t) is said to be the in-phase modulation
associated with v(t), and y(t) is said to be the quadrature modulation associated with
v(t). Alternatively, the polar form of g(t), represented by R(t) and u(t), is given by
Eq. (4–2), where the identities between Cartesian and polar coordinates are given by
Eqs. (4–3) and (4–4). R(t) and u(t) are real waveforms, and in addition, R(t) is always
nonnegative. R(t) is said to be the amplitude modulation (AM) on v(t), u(t) is said to be
the phase modulation (PM) on v(t).
Example 4–1 IN-PHASE AND QUADRATURE MODULATED SIGNALING
Let x(t) = cos(2pt) and y(t) be a rectangular pulse described by
0, t 6 1
y(t) = c 1, 1 … t … 2
0, t 7 2
Using Eq. (4–1a), plot the resulting modulated signal over the time interval 0 6 t 6 4 sec.
Assume that the carrier frequency is 10 Hz. See Example4_01.m for the solution.