Signals and Systems - Electrical Engineering

1.4 Representation Using Basic Signals 87

Sinusoids

Sinusoids are of the general form

Acos( 0 t+θ)=Asin( 0 t+θ+π/ 2 ) −∞<t<∞ (1.18)

whereAis the amplitude of the sinusoid, 0 = 2 πf 0 (rad/sec)is the frequency, andθis a phase shift. The

frequency and time variables are inversely related, as follows:

 0 = 2 πf 0 =

2 π

T 0

The cosine and the sine signals, as indicated above, are out of phase byπ/2 radians. The frequency
can also be expressed in hertz or 1/sec units, and in that case 0 = 2 πf 0 , and the period is found by
the relationf 0 = 1 /T 0 (it is important to point out the inverse relation between time and frequency
shown here, which will be important in the representation of signals later on).

Recall from Chapter 0, that the Euler’s identity provides the relation of the sinusoids with the complex
exponential

ej^0 t=cos( 0 t)+jsin( 0 t) (1.19)

that will allow us to represent in terms of sines and cosines any signal that is represented in terms of
complex exponentials. Likewise, the Euler’s identity also permits us to represent sines and cosines in
terms of complex exponentials, since

cos( 0 t)=

1

2

(

ej^0 t+e−j^0 t

)

(1.20)

sin( 0 t)=

1

2 j

(

ej^0 t−e−j^0 t

)

(1.21)

RemarksA sinusoid is characterized by its amplitude, frequency, and phase. When we allow these three
parameters to be functions of time, or

A(t)cos((t)t+θ(t))

the following different types of modulation systems in communications are obtained:

n Amplitude modulation (AM)—The amplitude A(t)changes according to the message, while the
frequency and the phase are constant,
n Frequency modulation (FM)—The frequency(t)changes according to the message, while the
amplitude and phase are constant,
n Phase modulation (PM)—The phaseθ(t)varies according to the message and the other parameters are
kept constant.