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

It is common notational practice in signal processing to use normalized radian frequency

Thus, our sampled complex sinusoid becomes

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.^5.4

The spectrum of infinitely long discrete-time signals is given by the Discrete Time Fourier Transform (DTFT) (discussed in §2.1):

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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.