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Sinusoid Magnitude Spectra

A sinusoid's frequency content may be graphed in the frequency domain as shown in Fig. 1.

Figure 1: Spectral magnitude representation of a unit-amplitude sinusoid at frequency $ 100$ Hz such as $ \cos(200\pi t)$ or $ \sin(200\pi t$). (Phase is not shown.)
\begin{figure}\input fig/sinefd.pstex_t

An example of a particular sinusoid graphed in Fig. 1 is given by

$\displaystyle x(t) = \cos(\omega_x t)
= \frac{1}{2}e^{j\omega_x t}
+ \frac{1}{2}e^{-j\omega_x t}


$\displaystyle \omega_x = 2\pi 100.

That is, this sinusoid has amplitude 1, frequency 100 Hz, and phase zero (or $ \pi/2$, if $ \sin(\omega_x t)$ is defined as the zero-phase case).

Figure 1 can be viewed as a graph of the magnitude spectrum of $ x(t)$, or its spectral magnitude representation [4]. Note that the spectrum consists of two components with amplitude $ 1/2$, one at frequency $ 100$ Hz and the other at frequency $ -100$ Hz.

Phase is not shown in Fig. 1 at all. The phase of the components could be written simply as labels next to the magnitude arrows, or the magnitude arrows can be rotated ``into or out of the page'' by the appropriate phase angle as illustrated in [7, Fig. 4.8 on p. 43].

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``Sinusoidal Modulation of Sinusoids'', by Julius O. Smith III, (Excerpt from ... ).
Copyright © 2005-12-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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