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Bandpass Signaling Principles and Circuits Chap. 4
where the complex envelope of the input is denoted by
g(t) = R(t)e ju(t) = x(t) + jy(t)
and x(t) and y(t) are the quadrature components. [See Eq. (4–2).] Because the frequency of the
oscillator is the same as the carrier frequency of the incoming signal, the oscillator has been
frequency synchronized with the input signal. Furthermore, if, in addition, u 0 = 0, the oscillator
is said to be phase synchronized with the in-phase component, and the output becomes
v out (t) = 1 2 A 0x(t)
(4–77a)
If u 0 = 90°,
v out = 1 2 A 0y(t)
(4–77b)
Equation (4–76) also indicates that a product detector is sensitive to AM and PM. For
example, if the input contains no angle modulation, so that u(t) = 0, and if the reference phase
is set to zero (i.e., u 0 = 0), then
v out (t) = 1 2 A 0R(t)
(4–78a)
which implies that x(t) Ú 0, and the real envelope is obtained on the product detector
output, just as in the case of the envelope detector discussed previously. However, if an
angle-modulated signal A c cos [v c t + u(t)] is present at the input and u 0 = 90°, the product
detector output is
or
v out (t) = 1 2 A 0 Re{A ce j[u(t)-90°] }
v out (t) = 1 2 A 0A c sin u(t)
(4–78b)
In this case, the product detector acts like a phase detector with a sinusoidal characteristic,
because the output voltage is proportional to the sine of the phase difference between the
input signal and the oscillator signal. Phase detector circuits are also available that yield
triangle and sawtooth characteristics [Krauss, Bostian, and Raab, 1980]. Referring to
Eq. (4–78b) for the phase detector with a sinusoidal characteristic, and assuming that the
phase difference is small [i.e., |u(t)| p>2], we see that sin u(t) ≈ u(t) and
v out (t) L 1 2 A 0A c u(t)
(4–79)
which is a linear characteristic (for small angles). Thus, the output of this phase detector
is directly proportional to the phase differences when the difference angle is small. (See
Fig. 4–20a.)
The product detector acts as a linear time-varying device with respect to the input v in (t),
in contrast to the envelope detector, which is a nonlinear device. The property of being either
linear or nonlinear significantly affects the results when two or more components, such as a
signal plus noise, are applied to the input. This topic will be studied in Chapter 7.