A sinusoid is any signal of the form
![]() |
(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:
![]() |
(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
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 property6.3of delta functions:
![]() |
(6.6) |
Substituting into (5.4), the spectrum of our original sinusoid
is given by
![]() |
(6.7) |