The most ubiquitous and important signal in electrical
engineering is the sinusoid.

st=Acos2πft+φ
or
Acosωt+φ
s
t
A
2
f
t
φ
or
A
ω
t
φ

(1)
AA is known as the sinusoid's

amplitude, and determines the sinusoid's size.
The amplitude conveys the sinusoid's physical units (volts,
lumens, etc). The

frequency
ff has units of Hz (Hertz) or

s-1
s
, and determines how rapidly the sinusoid oscillates
per unit time. The temporal variable

t
t always has units of seconds, and thus the frequency
determines how many oscillations/second the sinusoid has. AM
radio stations have carrier frequencies of about 1 MHz (one
mega-hertz or

106
10
6
Hz), while FM stations have carrier frequencies of
about 100 MHz. Frequency can also be expressed by the symbol

ωω, which has units of
radians/second. Clearly,

ω=2πf
ω
2
f
. In communications, we most often express frequency in
Hertz. Finally,

φφ
is the

phase, and determines the sine wave's behavior
at the origin (

t=0
t
0
). It has units of radians, but we can express it in
degrees, realizing that in computations we must convert from
degrees to radians. Note that if

φ=−π2
φ
2
, the sinusoid corresponds to a sine function, having a
zero value at the origin.

Asin2πft+φ=Acos2πft+φ−π2
A
2
f
t
φ
A
2
f
t
φ
2

(2)
Thus, the only difference between a sine and cosine signal is
the phase; we term either a sinusoid.

We can also define a discrete-time variant of the sinusoid:
Acos2πfn+φ
A
2
f
n
φ
. Here, the independent variable is
nn and represents the
integers. Frequency now has no dimensions, and takes on values
between 0 and 1.

Show that
cos2πfn=cos2π(f+1)n
2
f
n
2
f
1
n
, which means that a sinusoid having a frequency
larger than one corresponds to a sinusoid having a frequency
less than one.

As
cosα+β=cosαcosβ−sinαsinβ
α
β
α
β
α
β
,
cos2π(f+1)n=cos2πfncos2πn−sin2πfnsin2πn=cos2πfn
2
f
1
n
2
f
n
2
n
2
f
n
2
n
2
f
n
.

Notice that we shall call either sinusoid an analog
signal. Only when the discrete-time signal takes on a finite
set of values can it be considered a digital signal.

Can you think of a simple signal that has a finite number
of values but is defined in continuous time? Such a signal
is also an analog signal.

A square wave takes on the values
1
1
and
-1
-1
alternately. See the plot in the module Elemental Signals.

Comments:"Electrical Engineering Digital Processing Systems in Braille."