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Bandpass Signaling Principles and Circuits Chap. 4
Using Eq. (4–28), we obtain
v 2 (t) = Ax(t - T g ) cos[v c t + u(f c )] - Ay(t - T g ) sin[v c t + u(f c )]
where, by the use of Eq. (2–150b) evaluated at f = f c ,
u(f c ) = -v c T g + u 0 = -2pf c T d
Thus, the output bandpass signal can be described by
v 2 (t) = Ax(t - T g ) cos[v c (t - T d )] - Ay(t - T g ) sin[v c (t - T d )]
(4–30)
where the modulation on the carrier (i.e., the x and y components) has been delayed by the
group time delay, T g , and the carrier has been delayed by the carrier time delay, T d . Because
u(f c ) = -2pf c T d , where u(f c ) is the carrier phase shift, T d is also called the phase delay.
Equation (4–30) demonstrates that the bandpass filter delays the input complex envelope
(i.e., the input information) by T g , whereas the carrier is delayed by T d . This is distortionless
transmission, which is obtained when Eqs. (4–27a) and (4–27b) are satisfied. Note that T g will
differ from T d , unless u 0 happens to be zero.
In summary, the general requirements for distortionless transmission of either
baseband or bandpass signals are given by Eqs. (2–150a) and (2–150b). However, for the
bandpass case, Eq. (2–150b) is overly restrictive and may be replaced by Eq. (4–27b). In
this case, T d Z T g unless u 0 = 0 where T d is the carrier or phase delay and T g is the
envelope or group delay. For distortionless bandpass transmission, it is only necessary to
have a transfer function with a constant amplitude and a constant phase derivative over the
bandwidth of the signal.
Example 4–4 GROUP DELAY FOR A RC LOW-PASS FILTER
Using Eq. (4–27b), calculate and plot the group delay for a RC low-pass filter. Compare this result
for the group delay with that obtained in Example 2–18 for the time delay of a RC low-pass filter.
See Example4_04.m for the solution.
4–6 BANDPASS SAMPLING THEOREM
Sampling is used in software radios and for simulation of communication systems. If the
sampling is carried out at the Nyquist rate or larger ( f s Ú 2B, where B is the highest frequency
involved in the spectrum of the RF signal), the sampling rate can be ridiculous. For example,
consider a satellite communication system with a carrier frequency of f c = 6 GHz. The
sampling rate required can be at least 12 GHz. Fortunately, for signals of this type (bandpass
signals), it can be shown that the sampling rate depends only on the bandwidth of the signal,
not on the absolute frequencies involved. This is equivalent to saying that we can reproduce
the signal from samples of the complex envelope.