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 (` ') will explicitly indicate normalization, but it may be left off as well, so that may denote either normalized or unnormalized frequency.6.4
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 is an impulse defined for or , and denotes normalized radian frequency. (Treatments of the DTFT invariably use normalized frequency.)
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Spectrum of a Windowed Sinusoid
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Spectrum of a Sinusoid