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Sinusoid Magnitude Spectra

A sinusoid's frequency content may be graphed in the frequency domain as shown in Fig.4.6.

Figure 4.6: 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.)

An example of a particular sinusoid graphed in Fig.4.6 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 4.6 can be viewed as a graph of the magnitude spectrum of $ x(t)$ , or its spectral magnitude representation [46]. 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.4.6 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 Fig.4.16.

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
Copyright © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University