Spectrum of Sampled Complex Sinusoid
In the discrete-time case, we replace
by
where
ranges
over the integers and
is the sampling period in seconds. Thus,
for the positive-frequency component of the sinusoid of the previous
section, we obtain
![]() |
(6.8) |
It is common notational practice in signal processing to use normalized radian frequency
![]() |
(6.9) |
Thus, our sampled complex sinusoid becomes
![]() |
(6.10) |
It is not difficult to convert between normalized and unnormalized frequency. The use of a tilde (`


The spectrum of infinitely long discrete-time signals is given by the Discrete Time Fourier Transform (DTFT) (discussed in §2.1):
![]() |
(6.11) |
where now




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Spectrum of a Windowed Sinusoid
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Spectrum of a Sinusoid