Page 53 - Fundamentals of Communications Systems

Signals and Systems Review 2.5

EXAMPLE 2.7
Examples of aperiodic signals are
t 1
x(t) = e − τ x(t) = (2.12)
t

2.1.3 Real versus Complex Signals
Complex signals arise often in communication systems analysis and design.
The most common example is Fourier transforms which is discussed later in
this chapter. Another important communication application of complex signals
is in the representation of bandpass signals and Chapter 4 discusses this in
more detail. For this review, we will consider some simple characteristics of
complex signals. Define a complex signal and a complex exponential to be

z(t) = x(t) + jy(t) e j θ = cos(θ) + j sin(θ) (2.13)
where x(t) and y(t) are both real signals. Note this definition is often known as
Euler’s rule. A magnitude (α(t)) and phase (θ(t)) representation of a complex
signal is also commonly used, i.e.,

z(t) = α(t)e j θ(t) (2.14)
where

−1
2
2
α(t) =|z(t)|= x (t) + y (t) θ(t) = arg(z(t)) = tan (y(t), x(t)) (2.15)
It should be noted that the inverse tangent function in Eq. (2.15) uses two
arguments and returns a result in [0, 2π]. The complex conjugate operation is
defined as
∗
z (t) = x(t) − jy(t) = α(t)e − j θ(t) (2.16)
Some important formulas for analyzing complex signals are
2
2
2
2
2
2
|z(t)| = α(t) = z(t)z (t) = x (t) + y (t) cos(θ) + sin(θ) = 1
∗
1
1
∗

[z(t)] = x(t) = α(t) cos(θ(t)) = [z(t) + z (t)] cos(θ) = [e j θ + e − j θ ] (2.17)
2 2
1 1 j θ − j θ
∗
[z(t)] = y(t) = α(t) sin(θ(t)) = [z(t) − z (t)] sin(θ) = [e − e ]
2 j 2 j
EXAMPLE 2.8
The most common complex signal in communication engineering is the complex expo-
nential, i.e.,
exp[ j 2π f m t] = cos(2π f m t) + j sin(2π f m t)
This signal will be the basis of the frequency domain understanding of signals.