Normalized frequency (signal processing)

Frequency

In digital signal processing (DSP), normalized frequency (f') is a quantity having dimension of frequency expressed in units of "cycles per sample". It equals f'=f/fs, where f is an ordinary frequency quantity (in "cycles per second") and fs is the sampling rate (in "samples per second").For regularly spaced sampling, the continuous time variable, t (with units of seconds), is replaced by a discrete sampling count variable, n=t/T (with units of "samples"), upon division by the sampling interval, T=1/fs (in "seconds per sample"). This practice is analogous to the concept of natural units in physics, meaning that the natural unit of time in a DSP system is "samples". The maximum frequency that can be unambiguously represented by digital data is (known as Nyquist frequency) when the samples are real numbers, and when the samples are complex numbers. The normalized values of these limits are respectively 0.5 and 1.0 cycles/sample. This has the advantage of simplicity, but (similar to natural units) there is a potential disadvantage in terms of loss of clarity and understanding, as these constants and are then omitted from mathematical expressions of physical laws. The simplicity offered by normalized units is favored in textbooks, where space is limited and where real units are incidental to the point of a theorem or its proof. But there is another advantage in the DSP realm (compared to physics), because and are not "universal physical constants". The use of normalized frequency allows us to present concepts that are universal to all sample rates in a way that is independent of sample rate. An example of such a concept is a digital filter design whose bandwidth is specified not in hertz, but as a percentage of the sample rate of the data passing through it. Formulas expressed in terms of and/or are readily converted to normalized frequency by setting those parameters to 1. The inverse operation is usually accomplished by replacing instances of the frequency parameter, with or (Wikipedia).

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Normalized frequency (signal processing)

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