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Bandpass Signaling Principles and Circuits Chap. 4
independent pieces of information. N is said to be the number of dimensions required to
specify the waveform.
Computer simulation is often used to analyze communication systems. The bandpass
dimensionality theorem tells us that a bandpass signal B T Hz wide can be represented over a
T 0 -second interval, provided that at least N = 2B T T 0 samples are used. More details about the
bandpass sampling theorem are discussed in study-aid Prob. SA4–5.
4–7 RECEIVED SIGNAL PLUS NOISE
Using the representation of bandpass signals and including the effects of channel filtering, we
can obtain a model for the received signal plus noise. Referring to Fig. 4–1, the signal out of
the transmitter is
s(t) = Re[g(t)e jv ct ]
where g(t) is the complex envelope for the particular type of modulation used. (See Table
4–1.) If the channel is linear and time invariant, the received signal plus noise is
r(t) = s(t) * h(t) + n(t)
(4–35)
where h(t) is the impulse response of the channel and n(t) is the noise at the receiver input.
Furthermore, if the channel is distortionless, its transfer function is given by Eq. (4–29), and
consequently, the signal plus noise at the receiver input is
r(t) = Re[Ag(t - T g )e j1v ct+u(f c )2 + n(t)]
(4–36)
where A is the gain of the channel (a positive number usually less than 1), T g is the channel
group delay, and u(f c ) is the carrier phase shift caused by the channel. In practice, the values for
T g and u(f c ) are often not known, so that if values for T g and u(f c ) are needed by the receiver to
detect the information that was transmitted, receiver circuits estimate the received carrier phase
u(f c ) and the group delay (e.g., a bit synchronizer in the case of digital signaling). We will
assume that the receiver circuits are designed to make errors due to these effects negligible;
therefore, we can consider the signal plus noise at the receiver input to be
r(t) = Re[g(t)e jv ct ] + n(t)
(4–37)
where the effects of channel filtering, if any, are included by some modification of the complex
envelope g(t) and the constant A c that is implicit within g(t) (see Table 4–1) is adjusted to
reflect the effect of channel attenuation. Details of this approach are worked out in Sec. 8–6.
4–8 CLASSIFICATION OF FILTERS AND AMPLIFIERS
Filters
Filters are devices that take an input waveshape and modify the frequency spectrum to
produce the output waveshape. Filters may be classified in several ways. One is by the type of
construction used, such as LC elements or quartz crystal elements. Another is by the type of