1.4 Representation Using Basic Signals 85
The power of a sum of sinusoids,
x(t)=
∑
k
Akcos(kt)=
∑
k
xk(t) (1.15)
with harmonically or nonharmonically related frequencies{k}, is the sum of the power of each of the
sinusoidal components,
Px=
∑
k
Pxk (1.16)
1.4 Representation Using Basic Signals.......................................................
A fundamental idea in signal processing is to attempt to represent signals in terms of basic signals,
which we know how to process. In this section we consider some of these basic signals (complex
exponentials, sinusoids, impulse, unit-step, and ramp) that will be used to represent signals and for
which we will obtain their responses in a simple way in the next chapter.
1.4.1 Complex Exponentials
A complex exponential is a signal of the form
x(t)=Aeat
=|A|ert[cos( 0 t+θ)+jsin( 0 t+θ)] −∞<t<∞ (1.17)
whereA=|A|ejθ, anda=r+j 0 are complex numbers.
Using Euler’s identity,ejφ=cos(φ)+jsin(φ), and from the definitions ofA andaas complex
numbers, we have that
x(t)=|A|ejθe(r+j^0 )t=|A|erte(j^0 t+θ)
=|A|ert[cos( 0 t+θ)+jsin( 0 t+θ)]
We will see later that complex exponentials are fundamental in the Fourier representation of signals.
Remarks
n Suppose that A and a are real, then
x(t)=Aeat −∞<t<∞
is a decaying exponential if a< 0 , and a growing exponential if a> 0. See Figure 1.5.
