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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

$\displaystyle 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

$\displaystyle 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).

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[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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