Example Sinusoids

Figure 4.1 plots the sinusoid $A \sin(2\pi f t + \phi)$, for $A=10$, $f=2.5$, $\phi=\pi/4$, and $t\in[0,1]$. Study the plot to make sure you understand the effect of changing each parameter (amplitude, frequency, phase), and also note the definitions of ``peak-to-peak amplitude'' and ``zero crossings.''

Figure 4.1: An example sinusoid.

A ``tuning fork'' vibrates approximately sinusoidally. An ``A-440'' tuning fork oscillates at $440$ cycles per second. As a result, a tone recorded from an ideal A-440 tuning fork is a sinusoid at $f=440$ Hz. The amplitude $A$ determines how loud it is and depends on how hard we strike the tuning fork. The phase $\phi$ is set by exactly when we strike the tuning fork (and on our choice of when time 0 is). If we record an A-440 tuning fork on an analog tape recorder, the electrical signal recorded on tape is of the form

$$x(t) = A \sin(2\pi 440 t + \phi).$$

As another example, the sinusoid at amplitude $1$ and phase $\pi/2$ (90 degrees) is simply

$$x(t) = \sin(\omega t + \pi/2) = \cos(\omega t).$$

Thus, $\cos(\omega t)$ is a sinusoid at phase 90-degrees, while $\sin(\omega t)$ is a sinusoid at zero phase. Note, however, that we could just as well have defined $\cos(\omega t)$ to be the zero-phase sinusoid rather than $\sin(\omega t)$. It really doesn't matter, except to be consistent in any given usage. The concept of a ``sinusoidal signal'' is simply that it is equal to a sine or cosine function at some amplitude, frequency, and phase. It does not matter whether we choose $\sin()$ or $\cos()$ in the ``official'' definition of a sinusoid. You may encounter both definitions. Using $\sin()$ is nice since ``sinusoid'' naturally generalizes $\sin()$. However, using $\cos()$ is nicer when defining a sinusoid to be the real part of a complex sinusoid (which we'll talk about in §4.3.11).