Page 128 - Fundamentals of Communications Systems

4.4 Chapter Four

A complex valued signal, denoted the complex envelope, is defined as
x z (t) = x I (t) + jx Q (t) = x A(t) exp[ jx P (t)]

The original bandpass signal can be obtained from the complex envelope by
√
x c (t) = 2
[x z (t) exp[ j 2π f c t]]. (4.5)
Since the complex exponential only determines the carrier frequency, the
complex signal x z (t) contains all the information in x c (t). Using this complex
baseband representation of bandpass signals greatly simplifies the notation for
communication system analysis. As you progress through the text hopefully
the additional simplicity provided by the complex envelope representation will
become evident.

EXAMPLE 4.1
Consider the bandpass signal
√ √
x c (t) = 2 cos(2π f m t) 2 cos(2π f c t) − sin(2π f m t) 2 sin(2π f c t)
where f m < f c . A plot of this bandpass signal is seen in Figure 4.2 with f c = 10 f m .
Obviously we have
x I (t) = 2 cos(2π f m t) x Q (t) = sin(2π f m t)

and
x z (t) = 2 cos(2π f m t) + j sin(2π f m t)

The amplitude and phase can be computed as
−1
2
x A (t) = 1 + 3 cos (2π f m t) x P (t) = tan [sin(2π f m t), 2 cos(2π f m t)].
A plot of the amplitude and phase of this signal is seen in Figure 4.3.

As an interesting historical note, the communications field did not always uti-
lize the complex baseband notation. Originally, communication theorist adopted
the “analytical signal” denoted

x an (t) = x z (t) exp[ j 2π f c t] (4.6)

as a way to understand communication signals. This complex analytical signal
was viewed as useful since it was more mathematically tractable and yet cap-
ture all the characteristics of communications signals. Slowly over time the field
came to realize that the important characteristics of a communication wave-
form are captured by the complex envelope, x z (t). An early tutorial paper on
the complex analytical signal is given in [Bed62]. Hence the complex envelope