Normalized frequency (digital signal processing)

Normalized frequency is a unit of measurement of frequency equivalent to cycles/sample. In digital signal processing (DSP), the continuous time variable, t, with units of seconds, is replaced by the discrete integer variable, n, with units of samples. More precisely, the time variable, in seconds, has been normalized (divided) by the sampling interval, T (seconds/sample), which causes time to have convenient integer values at the moments of sampling. This practice is analogous to the concept of Natural units, meaning that the natural unit of time in a DSP system is samples.

The normalized value of a frequency variable, ${\displaystyle \textstyle f}$ (cycles/sec), is ${\displaystyle \textstyle f/f_{s},}$  where ${\displaystyle \textstyle f_{s}=1/T}$  is the sampling rate in samples/sec.  The maximum frequency that can be unambiguously represented by digital data is ${\displaystyle \textstyle f_{s}/2}$  (known as Nyquist frequency) when the samples are real numbers, and ${\displaystyle \textstyle f_{s}}$  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 ${\displaystyle \textstyle T}$ and ${\displaystyle \textstyle f_{s}}$ are then omitted from mathematical expressions of physical laws.

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