## Spectrum of a Sinusoid

A *sinusoid* is any signal of the form

(6.1) |

where is the

*amplitude*(in arbitrary units), is the

*phase*in radians, and 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 are real numbers. In addition to radian frequency , it is useful to define , where is the frequency in

*Hertz*(Hz).

^{6.1}

By Euler's identity, , we can write

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):

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 . For , may be called the

*analytic signal*corresponding to .

^{6.2}

It remains to find the Fourier transform of :

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
property*^{6.3}of 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 having complex amplitude , summed with another at frequency with complex amplitude .

**Next Section:**

Spectrum of Sampled Complex Sinusoid

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Optimal FIR Digital Filter Design