Spectrum of a Sinusoid
A sinusoid is any signal of the form
![]() |
(6.1) |
where
![$ A$](http://www.dsprelated.com/josimages_new/sasp2/img584.png)
![$ \phi\in[-\pi,\pi)$](http://www.dsprelated.com/josimages_new/sasp2/img889.png)
![$ \omega_0$](http://www.dsprelated.com/josimages_new/sasp2/img248.png)
![$ t$](http://www.dsprelated.com/josimages_new/sasp2/img344.png)
![$ (A,\omega_0,\phi)$](http://www.dsprelated.com/josimages_new/sasp2/img890.png)
![$ \omega$](http://www.dsprelated.com/josimages_new/sasp2/img89.png)
![$ \omega =
2\pi f$](http://www.dsprelated.com/josimages_new/sasp2/img254.png)
![$ f$](http://www.dsprelated.com/josimages_new/sasp2/img84.png)
By Euler's identity,
, we can write
![\begin{eqnarray*}
x(t) &=& A \frac{e^{j(\omega_0 t + \phi)} + e^{-j(\omega_0 t + \phi)}}{2}\\
&=& \left(\frac{A}{2}e^{j \phi}\right) e^{j\omega_0 t}
+ \left(\frac{A}{2}e^{-j \phi}\right) e^{-j\omega_0 t}\\
&\isdef & a e^{j\omega_0 t} + \overline{a} e^{-j\omega_0 t}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img892.png)
where
denotes the complex conjugate of
.
Thus, we can build a real sinusoid
as a linear combination of
positive- and negative-frequency complex sinusoidal components:
where
![]() |
(6.3) |
The spectrum of
is given by its
Fourier transform (see §2.2):
![\begin{eqnarray*}
X(\omega) &\isdef & \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt\nonumber \\ [5pt]
&=& \int_{-\infty}^{\infty} \left[a s_{\omega_0}(t) + \overline{a} s_{-\omega_0}(t)
\right] e^{-j\omega t} dt.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img896.png)
In this case,
is given by (5.2) and we have
We see that, since the Fourier transform is a linear operator, we need only work with the unit-amplitude, zero-phase, positive-frequency sinusoid
![$ s_{\omega_0}(t)\isdef e^{j\omega_0t}$](http://www.dsprelated.com/josimages_new/sasp2/img898.png)
![$ \omega_0>0$](http://www.dsprelated.com/josimages_new/sasp2/img899.png)
![$ as_{\omega_0}(t)$](http://www.dsprelated.com/josimages_new/sasp2/img900.png)
![$ x(t)$](http://www.dsprelated.com/josimages_new/sasp2/img109.png)
It remains to find the Fourier transform of
:
![\begin{eqnarray*}
S_{\omega_0}(\omega)
&=& \int_{-\infty}^{\infty} s_{\omega_0}(t) e^{-j\omega t} dt\\ [5pt]
&\isdef & \int_{-\infty}^{\infty} e^{j\omega_0 t} e^{-j\omega t} dt\\ [5pt]
&=& \int_{-\infty}^{\infty} e^{j(\omega_0-\omega) t} dt\\ [5pt]
&=& \left.\frac{1}{j(\omega_0-\omega)} e^{j(\omega_0-\omega) t}
\right\vert _{-\infty}^\infty\\ [5pt]
&=& \lim_{\Delta\to\infty} 2\frac{\sin[(\omega_0-\omega)\Delta]}{\omega_0-\omega}\\ [5pt]
&=& 2\pi\delta(\omega_0-\omega) = \delta(f_0-f),
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img902.png)
where
is the delta function or impulse
at frequency
(see Fig.5.4 for a plot, and
§B.10 for a mathematical introduction).
Since the delta function is even (
),
we can also write
. It is shown in §B.13 that the
sinc
limit
above approaches a delta function
.
However, we will only use the Discrete Fourier Transform (DFT)
in any practical applications, and in that case, the result is easy to
show [264].
The inverse Fourier transform is easy to evaluate by the sifting property6.3of delta functions:
![]() |
(6.6) |
Substituting into (5.4), the spectrum of our original sinusoid
is given by
![]() |
(6.7) |
which is a pair of impulses, one at frequency
![$ \omega=\omega_0$](http://www.dsprelated.com/josimages_new/sasp2/img911.png)
![$ 2\pi a = A \pi e^{j\phi}$](http://www.dsprelated.com/josimages_new/sasp2/img912.png)
![$ \omega=-\omega_0$](http://www.dsprelated.com/josimages_new/sasp2/img913.png)
![$ 2\pi\overline{a} = A\pi
e^{-j\phi}$](http://www.dsprelated.com/josimages_new/sasp2/img914.png)
Next Section:
Spectrum of Sampled Complex Sinusoid
Previous Section:
Optimal FIR Digital Filter Design