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1.5. Representation of Digital Frequencies

A digital frequency can be written on paper in units of Hz or in units of radians per second as

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A more common method is to express a digital frequency as a fraction where the sampling frequency has been normalized to 1, such as

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In all cases, the value of k can range from 0 to N/2. Examples of the four representations of digital frequencies are illustrated in Table 1.2 for several values of k.

Table 1.2. Four Ways to Represent a Digital Frequency

A frequency in the analog domain has infinite resolution and therefore can take on all possible values. A frequency in the digital domain can only take on specific discrete values, which are multiples of fS/N. The value of an analog frequency can always be made to match exactly the value of a digital frequency, but since the value of the digital frequency does not have infinite resolution, the opposite is not true.

The amplitude of an analog frequency can take on an infinite number of different values, whereas the amplitude of a digital frequency can only take on discrete values. The number of amplitude values that can be represented by a digital sinusoid is dependent on the bit width of the individual digital samples. For example, suppose the bit width of a bipolar sinusoidal sequence is given by B. The sinusoidal amplitude can take on values equal to 0 and ±[1,2,3,···, (2B–1 – 1)].

The sole purpose of this chapter is to introduce the mathematical representation of frequency in the digital domain. This book contains a great deal more information on the subject. For a detailed analysis and tutorial on the synthesis of digital frequencies the reader is encouraged to read Chapter 8, “Digital Frequency Synthesis.”

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