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Normalized frequency (signal processing) - Wikipedia

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In digital signal processing (DSP), a normalized frequency is a ratio of a variable frequency ({\displaystyle f}) and a constant frequency associated with a system (such as a sampling rate, {\displaystyle 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

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A typical choice of characteristic frequency is the sampling rate ({\displaystyle f_{s}}) that is used to create the digital signal from a continuous one. The normalized quantity, {\displaystyle 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 {\displaystyle f} is expressed in Hz (cycles per second), {\displaystyle f_{s}} is expressed in samples per second.[1]

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

Example of plotting samples of a frequency distribution in the unit "bins", which are integer values. A scale factor of 0.7812 converts a bin number into the corresponding physical unit (hertz).

A common practice is to sample the frequency spectrum of the sampled data at frequency intervals of {\displaystyle {\tfrac {f_{s}}{N}},} for some arbitrary integer {\displaystyle N} (see § Sampling the DTFT). The samples (sometimes called frequency bins) are numbered consecutively, corresponding to a frequency normalization by {\displaystyle {\tfrac {f_{s}}{N}}.}[2]: p.56 eq.(16) [3] The normalized Nyquist frequency is {\displaystyle {\tfrac {N}{2}}} with the unit ⁠1/N⁠th cycle/sample.

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

The following table shows examples of normalized frequency for {\displaystyle f=1} kHz, {\displaystyle f_{s}=44100} samples/second (often denoted by 44.1 kHz), and 4 normalization conventions:

Quantity Numeric range Calculation Reverse
{\displaystyle f'={\tfrac {f}{f_{s}}}}   [0, ⁠1/2⁠] cycle/sample 1000 / 44100 = 0.02268 {\displaystyle f=f'\cdot f_{s}}
{\displaystyle f'={\tfrac {f}{f_{s}/2}}}   [0, 1] half-cycle/sample 1000 / 22050 = 0.04535 {\displaystyle f=f'\cdot {\tfrac {f_{s}}{2}}}
{\displaystyle f'={\tfrac {f}{f_{s}/N}}}   [0, ⁠N/2⁠] bins 1000 × N / 44100 = 0.02268 N {\displaystyle f=f'\cdot {\tfrac {f_{s}}{N}}}
{\displaystyle \omega '={\tfrac {\omega }{f_{s}}}}   [0, π] radians/sample 1000 × 2π / 44100 = 0.14250 {\displaystyle \omega =\omega '\cdot f_{s}}
  1. ^ Carlson, Gordon E. (1992). Signal and Linear System Analysis. Boston, MA: ©Houghton Mifflin Co. pp. 469, 490. ISBN 8170232384.
  2. ^ Harris, Fredric J. (Jan 1978). "On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform" (PDF). Proceedings of the IEEE. 66 (1): 51–83. Bibcode:1978IEEEP..66...51H. CiteSeerX 10.1.1.649.9880. doi:10.1109/PROC.1978.10837. S2CID 426548.
  3. ^ Taboga, Marco (2021). "Discrete Fourier Transform - Frequencies", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/discrete-Fourier-transform-frequencies.