S ( t )
r ( t )
A ( t )
n ( t )
Figure 2.1 Generic channel model of modulation of digital signals.
modulate one bit at a time; or we can have M
4 symbols (00, 01, 10, and 11) where we
modulate two bits at a time, and so on. An M -ary modulator takes as an input one of the M
symbols and produces one of M analog waveforms that can be transmitted over the physical
media. There are many functions needed to be incorporated in the physical layer in order
to make modulation work effectively to include frequency conversion, signal amplification,
Figure 2.1 presents a generic channel model which focuses on modulation and demod-
ulation of a digital signal. In this model, we start with M symbols and have a signaling
rate of T S . That is, every T S seconds, the modulator gets an input S(T ) which represents
one of the M symbols. At any given time t , the output of the modulator S(t) is one of
M waveforms. S(t) can be expressed as S(t) = S X n (t) . The waveform S(t) can be a set
of voltages or currents (over a wire), a set of optical signals (over an optical medium), a
set of electric field intensities (over the air), and so on. In many cases, S(t) is limited to
a time interval bounded by T S
seconds. 2 Simply, the modulator emits signals at a rate of
symbols per second. Note: M is a power of two in most cases. 3
R S =
1 /T S
If the symbol
rate of this channel model is R S , then the bit rate is mR S or m/T S bps.
In the model presented in Figure 2.1, the modulated signal is distorted by attenuation, fad-
ing, noise, and other interference. A(t) expresses attenuation and fading while n(t) expresses
added noise. The received signal, in most cases, is very different from the transmitted signal.
Communications references use models for attenuation and fading such as Rayleigh, Rician,
and log-normal models. One of the most common noise models implemented in the field
is the additive white Gaussian noise (AWGN) model. Reference  is known as a classic
textbook example of noise models. Although these models help engineers design excellent
demodulators, there are some caveats that accompany them. For starters, some of these
models are linear, while channels are not. For example, signal amplifiers are non-linear; this
is especially true for high power signals such as satellite communications where the channel
non-linearity characteristics can considerably deviate from the models used in the design.
Models could be time-invariant, while in reality channel gain is time variant. Another issue
in channel modeling is the difficulty in accounting for frequency dispersion in a channel
model. The AWGN model commonly used is often far from reality. With tactical radios,
we need to consider the fact that the enemy could use sophisticated jammers that introduce
noise patterns as far away from AWGN as possible; in such a case, the AWGN model seems
2 In some waveforms, spectrum management techniques call for pulse overlap - detection and estimation of mod-
ulated signals is an interesting field of study that you can reference. Simply remember that a pulse is not an
even-shaped pulse and the leading and tail ends of it can have low spectrum density. The tradeoff between pulse
overlap and pulse energy at the receiver is an interesting area of study.
3 For one bit per symbol, M = 2 (0 or 1). For two bits per symbol, M = 4 (00, 01, 10, 11). For three bits per
8 (000, 001, 010, 011, 100, 101, 110, 111), and so on. The T 1 example presented in this section
shows an unusual case where we have two digital symbols and three modulated signals.