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

(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.

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

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