Normalized frequency (signal processing)

In digital signal processing (DSP), a normalized frequency is a ratio of a variable frequency ($$f$$) and a constant frequency associated with a system (such as a sampling rate, $$f_s$$). Some software applications require normalized inputs and produce normalized outputs, which can be re-scaled to physical units when necessary. Mathematical derivations are usually done in normalized units, relevant to a wide range of applications.

Examples of normalization
A typical choice of characteristic frequency is the sampling rate ($$f_s$$) that is used to create the digital signal from a continuous one. The normalized quantity, $$f' = \tfrac{f}{f_s},$$ has the unit cycle per sample regardless of whether the original signal is a function of time or distance. For example, when $$f$$ is expressed in Hz (cycles per second), $$f_s$$ is expressed in samples per second.

Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency $$(f_s/2)$$ as the frequency reference, which changes the numeric range that represents frequencies of interest from $$\left[0, \tfrac{1}{2}\right]$$ cycle/sample to $$[0, 1]$$ half-cycle/sample. Therefore, the normalized frequency unit is important when converting normalized results into physical units.



A common practice is to sample the frequency spectrum of the sampled data at frequency intervals of $$\tfrac{f_s}{N},$$ for some arbitrary integer $$N$$ (see ). The samples (sometimes called frequency bins) are numbered consecutively, corresponding to a frequency normalization by $$\tfrac{f_s}{N}.$$  The normalized Nyquist frequency is $$\tfrac{N}{2}$$ with the unit $1⁄N$th cycle/sample.

Angular frequency, denoted by $$\omega$$ and with the unit radians per second, can be similarly normalized. When $$\omega$$ is normalized with reference to the sampling rate as $$\omega' = \tfrac{\omega}{f_s},$$ the normalized Nyquist angular frequency is π radians/sample.

The following table shows examples of normalized frequency for $$f = 1$$ kHz, $$f_s = 44100$$ samples/second (often denoted by 44.1 kHz), and 4 normalization conventions: