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Frequency divided by a characteristic frequency From Wikipedia, the free encyclopedia

In digital signal processing (DSP), a **normalized frequency** is a ratio of a variable frequency () and a constant frequency associated with a system (such as a *sampling rate*, ). 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.

A typical choice of characteristic frequency is the *sampling rate* () that is used to create the digital signal from a continuous one. The normalized quantity, has the unit *cycle per sample* regardless of whether the original signal is a function of time or distance. For example, when is expressed in Hz (*cycles per second*), is expressed in *samples per second*.^{[1]}

Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency as the frequency reference, which changes the numeric range that represents frequencies of interest from *cycle/sample* to *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 for some arbitrary integer (see § Sampling the DTFT). The samples (sometimes called frequency *bins*) are numbered consecutively, corresponding to a frequency normalization by ^{[2]}^{: p.56 eq.(16) }^{[3]} The normalized Nyquist frequency is with the unit 1/N^{th} *cycle/sample*.

Angular frequency, denoted by and with the unit *radians per second*, can be similarly normalized. When is normalized with reference to the sampling rate as the normalized Nyquist angular frequency is *π radians/sample*.

The following table shows examples of normalized frequency for *kHz*, *samples/second* (often denoted by 44.1 kHz), and 4 normalization conventions:

Quantity |
Numeric range |
Calculation |
Reverse |
---|---|---|---|

[0, 1/2] cycle/sample |
1000 / 44100 = 0.02268 | ||

[0, 1] half-cycle/sample |
1000 / 22050 = 0.04535 | ||

[0, N/2] bins |
1000 × N / 44100 = 0.02268 N | ||

[0, π] radians/sample |
1000 × 2π / 44100 = 0.14250 |

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