You are here

The Sinusoid

18 March, 2015 - 12:22

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

Sine Definition

s(t)=Acos(2\pi ft+\phi )orAcos(\omega t+\phi )

A 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 frequencyf has units of Hz (Hertz) or s-1, and determines how rapidly the sinusoid oscillates per unit time. The temporal variable 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 Hz), while FM stations have carrier frequencies of about 100 MHz. Frequency can also be expressed by the symbol ω, which has units of radians/second. Clearly, \omega =2\pi f. In communications, we most often express frequency in Hertz. Finally, φ is the phase, and determines the sine wave's behavior at the origin (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 \phi =-\frac{\pi}2{} , the sinusoid corresponds to a sine function, having a zero value at the origin.

Asin(2\pi ft+\phi)=Acos(2\pi ft+\phi=-\frac{\pi}{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: Acos(2\pi fn+\phi). Here, the independent variable is n and represents the integers. Frequency now has no dimensions, and takes on values between 0 and 1.

Exercise 1.4.1

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

Note: 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.

Exercise 1.4.2

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.