Spectrum of a Sinusoid

A sinusoid is any signal of the form

$\displaystyle x(t) = A\cos(\omega_0 t + \phi), \quad t\in{\bf R}$

where

is the amplitude (in arbitrary units), $\phi\in[-\pi,\pi)$ is the phase in radians, and $\omega _0$ is the frequency in radians per second. Time

is a real number that varies continuously from minus infinity to infinity in the ideal sinusoid. All three parameters $(A,\omega_0,\phi)$ are real numbers. In addition to radian frequency $\omega$ , it is useful to define $\omega = 2\pi f$ , where

is the frequency in Hertz (Hz).^2.1

By Euler's identity, $e^{j\theta} = \cos(\theta) + j\sin(\theta)$ , we can write

$\begin{eqnarray*} x(t) &=& A \frac{e^{j(\omega_0 t + \phi)} + e^{-j(\omega_0 t +... ...\ &\isdef & a e^{j\omega_0 t} + \overline{a} e^{-j\omega_0 t} \end{eqnarray*}$

where `` $\isdef$ '' means ``is defined as'', and $\overline{a}$ denotes the complex conjugate of . Thus, we can build a real sinusoid as a linear combination of positive- and negative-frequency complex sinusoidal components:

$\displaystyle x(t) = a s_{\omega_0}(t) + \overline{a} s_{-\omega_0}(t) \protect$

(2.1)

where

$\displaystyle s_{\omega_0}(t) \isdef e^{j\omega_0 t} \isdef e^{j2\pi f_0 t}, \qquad a\isdef \frac{A}{2}e^{j\phi}.$

The spectrum of is given by its Fourier transform (see §2.2):

$\begin{eqnarray*} X(\omega) &\isdef & \int_{-\infty}^{\infty} x(t) e^{-j\omega t... ...0}(t) + \overline{a} s_{-\omega_0}(t) \right] e^{-j\omega t} dt. \end{eqnarray*}$

In this case, is given by (1.1) and we have

$\displaystyle X(\omega) = a S_{\omega_0}(\omega) + \overline{a} S_{-\omega_0}(\omega). \protect$

(2.2)

We see that, since the Fourier transform is a linear operator, we need only work with the unit-amplitude, zero-phase, positive-frequency sinusoid $s_{\omega_0}(t)\isdef e^{j\omega_0t}$ . For $\omega_0>0$ , $as_{\omega_0}(t)$ may be called the analytic signal corresponding to

.^2.2

It remains to find the Fourier transform of $s_{\omega_0}(t)$ :

$\begin{eqnarray*} S_{\omega_0}(\omega) &=& \int_{-\infty}^{\infty} s_{\omega_0}... ...\omega}\\ [5pt] &=& 2\pi\delta(\omega_0-\omega) = \delta(f_0-f), \end{eqnarray*}$

where $\delta(\omega)$ is the delta function or impulse at frequency $\omega _0$ (see Eq. $\,$ (2.6)). Since the delta function is even ( $\delta(-\omega) = \delta(\omega)$ ), we can also write $S_{\omega_0}(\omega) = 2\pi\delta(\omega-\omega_0) = \delta(f-f_0)$ . It is shown in §2.4.12 that the sinc limit above approaches delta function $\delta(f_0-f)$ . However, we will only use the Discrete Fourier Transform (DFT) in any practical applications, and in that case, the result is easy to show [243].

The inverse Fourier transform is easy to evaluate by the sifting property^2.3of delta functions:

$\displaystyle s_{\omega_0}(t) = \frac{1}{2\pi}\int_{-\infty}^\infty S_{\omega_... ...\infty}^\infty \delta(\omega-\omega_0) e^{j\omega t} d\omega = e^{j\omega_0 t}$

Substituting into (1.2), the spectrum of our original sinusoid is given by

$\displaystyle X(\omega) = 2\pi\left[a \delta(\omega-\omega_0) + \overline{a}\delta(\omega+\omega_0)\right]$

which is a pair of impulses, one at frequency $\omega=\omega_0$ having complex amplitude $2\pi a = A \pi e^{j\phi}$ , summed with another at frequency $\omega=-\omega_0$ with complex amplitude $2\pi\overline{a} = A\pi e^{-j\phi}$ .

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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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Spectral Audio Signal Processing
Spectrum Analysis of Sinusoids
Spectrum of a Sinusoid