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Bandpass Signaling Principles and Circuits Chap. 4
A linear bandpass filter can cause variations in the phase modulation at the output,
u 2 (t) = lg 2 (t), as a function of the amplitude modulation on the input complex envelope,
R 1 (t) = |g 1 (t)|. This is called AM-to-PM conversion. Similarly, the filter can cause variations in
the amplitude modulation at the output, R 2 (t), because of the PM on the input, u 1 (t). This is
called PM-to-AM conversion.
Because h(t) represents a linear filter, g 2 (t) will be a linear filtered version of g 1 (t); however,
u 2 (t) and R 2 (t)—the PM and AM components, respectively, of g 2 (t)—will be a nonlinear
filtered version of g 1 (t), since u 2 (t) and R 2 (t) are nonlinear functions of g 2 (t). The analysis of
the nonlinear distortion is very complicated. Although many analysis techniques have been
published in the literature, none has been entirely satisfactory. Panter [1965] gives a threechapter
summary of some of these techniques, and a classical paper is also recommended
[Bedrosian and Rice, 1968]. Furthermore, nonlinearities that occur in a practical system will
also cause nonlinear distortion and AM-to-PM conversion effects. Nonlinear effects can be
analyzed by several techniques, including power-series analysis; this is discussed in the
section on amplifiers that follows later in this chapter. If a nonlinear effect in a bandpass system
is to be analyzed, a Fourier series technique that uses the Chebyshev transform has been
found to be useful [Spilker, 1977].
Linear Distortion
In Sec. 2–6, the general conditions were found for distortionless transmission. For linear
bandpass filters (channels), a less restrictive set of conditions will now be shown to be satisfactory.
For distortionless transmission of bandpass signals, the channel transfer function,
H( f) = |H( f)|e ju( f ) , needs to satisfy the following requirements:
• The amplitude response is constant. That is,
|H(f)| = A
(4–27a)
where A is a positive (real) constant.
• The derivative of the phase response is a constant. That is,
- 1 du(f)
= T (4–27b)
2p df g
where T g is a constant called the complex envelope delay or, more concisely, the group
delay and u(f) = lH(f).
This is illustrated in Fig. 4–4. Note that Eq. (4–27a) is identical to the general requirement
of Eq. (2–150a), but Eq. (4–27b) is less restrictive than Eq. (2–150b). That is, if Eq. (2–150b)
is satisfied, Eq. (4–27b) is satisfied, where T d = T g ; however, if Eq. (4–27b) is satisfied,
Eq. (2–150b) is not necessarily satisfied, because the integral of Eq. (4–27b) is
u(f) = -2pfT g + u 0
(4–28)
where u 0 is a phase-shift constant, as shown in Fig. 4–4b. If u 0 happens to be nonzero,
Eq. (2–150b) is not satisfied.