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Mathematics of the DFT
    Sinusoids and Exponentials
       Sinusoids
          Example Sinusoids

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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.
\includegraphics[width=\textwidth]{eps/sine}

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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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