Normalized frequency (signal processing)
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/f", where "f" is an ordinary frequency quantity (in "cycles per second") and "f" 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/f" (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 formula_1  (known as Nyquist frequency) when the samples are real numbers, and formula_2  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 formula_3 and formula_2 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 formula_3 and formula_2 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 formula_2  and/or formula_3  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, formula_9  with formula_10  or  formula_11 Alternative normalizations. Some programs (such as MATLAB) that design filters with real-valued coefficients use the Nyquist frequency (formula_1) as the normalization constant. The resultant normalized frequency has units of "half-cycles/sample" or equivalently "cycles per 2 samples". Sometimes, the unnormalized frequency is represented in units of "radians/second" (angular frequency ), and denoted by formula_13  When formula_14 is normalized by the sample-rate ("samples/sec"), the resulting units are "radians/sample". The normalized Nyquist frequency is π "radians/sample", and the normalized sample-rate is 2π "radians/sample". The following table shows examples of normalized frequencies for a 1 kHz signal, a sample rate formula_15 = 44.1 kHz, and 3 different choices of normalized units. Also shown is the frequency region containing one cycle of the discrete-time Fourier transform, which is always a periodic function.