The most common form of mapping with this property is called Gray coding, which is discussed in more
detail in Section 5.3.
Signal space concepts are applicable to any modulation where bits are encoded as one of several
possible analog signals. This is true for linear modulation techniques but generally not for nonlinear
techniques.
5.2
Passband Modulation Principles
The basic principle of passband digital modulation is to encode an information bit stream into a carrier
signal which is then transmitted over a communications channel. Demodulation is the process of extract-
ing this information bit stream from the received signal. Corruption of the transmitted signal by the
channel can lead to bit errors in the demodulation process. The goal of modulation is to send bits at a
high data rate while minimizing the probability of data corruption.
In general, modulated carrier signals encode information in the amplitude A(t), frequency f (t), or
phase (t) of a carrier signal. Thus, the modulated signal can be represented as
s(t) = A(t) cos[2(f
c
+ f (t))t + (t)] = A(t) cos(2f
c
t + (t)),
(5.36)
where (t) = 2f (t)t + (t) which combines frequency and phase modulation into angle modulation.
We can rewrite the right-hand side of (5.36) in terms of its in-phase and quadrature components as:
s(t) = A(t) cos (t) cos(2f
c
t)
- A(t) sin (t) sin(2f
c
t) = s
I
(t) cos(2f
c
t)
- s
Q
(t) sin(2f
c
t),
(5.37)
where s
I
(t) = A(t) cos (t) is called the in-phase component of s(t) and s
Q
(t) = A(t) sin (t) is called
its quadrature component. The in-phase and quadrature signal components are baseband signals with
bandwidth B: the corresponding transmitted s(t) is a passband signal with center frequency f
c
and
passband bandwidth 2B. We can also write s(t) as
s(t) =
{u(t)e
j2f
c
t
},
(5.38)
where u(t) = s
I
(t)
-js
Q
(t). These different representations will be useful in our discussion of linear mod-
ulation techniques, where the in-phase and quadrature signal components are often processed separately.
5.3
Linear Modulation
In linear modulation the information bit stream is encoded in the amplitude and/or phase of the trans-
mitted signal, not the frequency. There are three main types of linear modulation:
· Pulse Amplitude Modulation (M-PAM): information encoded in amplitude only.
· Phase Shift Keying (M-PSK): information encoded in phase only.
· Quadrature Amplitude Modulation (M-QAM): information encoded in both amplitude and phase.
Let s(t) =
{u(t)e
j2f
c
t
} denote the transmitted signal. Then for all linear modulation schemes,
u(t) =
n
s
n
g(t
- nT
s
),
s
n
= a
n
+ jb
n
,
(5.39)
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